Instructional Aids in Mathematics. National Council of Teachers of Mathematics Yearbook 34

DOCUMENT RESUME I

ED 076 388 SE 015 896 AUTHOR Berger, Emil J., Ed. TITLE Instructional Aids in Mathematics. National Council of Teachers of Mathematics Yearbook 34. INSTITUTION National Council of Teachers of Mathematics, Inc., Washington, D.C. PUB DATE 73 NOTE 442p. AVAILABLE FROMNational Council of Teachers of Mathematics, 1906 Association Drive, Reston, Virginia 22091 ($12.00, $11.00 for NCTM members)

EDRS PRICE MF-$0.65 HC Not Available from EDRS: DESCRIPTORS *Activity Learning; Audiovisual Aids; Computers; Elementary School Mathematics; *Instruction; Instructional Aids; *Instructional Materials; Laboratory Procedures; Manipulative Materials; *Mathematical Enrichment; *Mathematics Education; Secondary School Mathematics; Teaching Techniques; Textbooks; Yearbooks ABSTRACT Information about available instructional aids, suggestions for selecting and evaluating these materials,and guidance in using them are provided in this yearbook.Chapters cover the use of instructional space; textbooks and otherprinted materials; programed instruction and teaching machines;calculating devices and computers; projection devices; manipulativeaids; projects, exhibits, games, puzzles, and contests;and the teacher's role. The chapters on computers, manipulativedevices, and projects include extensive bibliographies. The appendix listsnames and addresses of producers and distributors of instructionalaids. (DT) a. . ' h.4auI: - - e: tJ:t-:_ ' ::r;t ,: . S . , ., ::......

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4-.. 98 . 9L0 13 :- National Council of Teachers of Mathematics

Thirtrqourth Yearbook ION,-'1 AIDS

.3,711-iL\ \ATICS

Thirty-fourth Yearbook

National Council of Teachers of Mathematics

1973 Correspondence relating to and orders for additional copies of the Thirtv-lourill Yearbook and earlier yearbooks should be addressed to: The National Council of Teachers of Mathematics 1201 Sixteenth So eet, NW Washington. D.C. 20036 PERMISSION TO REPRODUCE THIS COPY RIGHTED MATERIAL BY MICROFICHE ONLY HAS BEEN GRANTED BY Charles R. Hucka TO ERIC AND ORGANIZATIONS OPERATING UNDER AGREEMENTS WITH THE US OFFICE OF EDUCATION FURTHER REPRODUCTION OUTSIDE HE ERIC SYSTEM REQUIRES PEW MISSION OF THE COPYRIGHT OWNER

Copyright © 1973 by The National Council of Teachers of Nlathematics, Inc. rights teserved. Library of Congicss Catalog Caul Number: 76-189323. Printedin the United States of America. The development and use of instructional aids in the teaching andlearning of mathematics have expanded tremendously in the pastquarter century. Ushering in that era of change was a 1945 report to the Board of Directors of theNational Council of Teachers of Mathematics by the Committeeon Multi-Sensory Aids. The Council published this report as its Eighteenth Yearbook under thetitle Multi-Sensory Aids in the Teaching of Mathematics. The introductionto the Eighteenth Yearbook contains a prophetic statement anticipating the expansion that is recorded in the present volume: "It is hoped that thereport will be followed in a few )....ars by one showing improvements and changes which have keptpace with the progress of such aids in the world about us." Since World War II the school mathematics curriculum has beencompletely overhauled. Increased activity in mathematical research, advances inautomation, and invention of the electronic computer had causeda revolution in mathematics, creating pressure for curriculum reform. Efforts to incorporate new materials into the mainstream of school mathematics precipitated the writing of "new mathematics" textbooks and pamphletscontaining changed content and new pedagogical strategies. Production of supporting films, filmstrips, overhead visuals, models, manipulative devices, and pertinent professional publicationssoon followed. Beginning around the mid-fifties, a speedup occurred in the trendto individualize instruction in the schools. This evolvement promptedsignificant new advances in instructional aids in all subject fields, mathematics included. The combined effects of compulsory attendance laws and thepost World War H population boom had brought about burgeoning school enrollmentsthat included a large subpopulation of students for whom the establishedpattern was inadequate. The new population of students presented a wider range of ability than formerly anda greater variety of personal and socioeconomic backgrounds. Coupled with the swelling enrollments and changing traits of schoolyouth was society's deepening concern for the individuala concern that lookedto the schools for expression. To stimulate planning to improve the quality of education for allstudents the bright, the average, and the slowJ. Lloyd Trump publishedthe pamphlet Images of the Future. Trump proposed that schools be organized around three kinds ofactivity large group instruction, individual study, and smallgroup discussion. He believed this scheme would result in improved instruction and makemore effective provision for

V individual dillerences. While Trump's proposal was directed primarilyat the secondar) level, the effects of his recommendation were also lelt at the elementary level. Tramp's recommendation served as a catalyst that spurred architectsto design buildings that included group spaces of various sizes, and encouraged flexible scheduling, team teaching, and the use of a variety of instructional aids. Skinner's and Crowder's work with teaching machines and programed instruction marked another step forward in the enlivened move to individualize instruction. These two men devised concrete ways of adapting instruction to the individual learner. The ast potential of the computer was also felt as a positive force. Computer-based teaching machines have the capability of completely individualising instruction. To date, however, this particular use of computers has been prohibitively expensive. Presently the most popular use of computers in the schools is as a tool in problem soh ing and flexible scheduling. Government support also helped extend the development and use ol instructional aids in mathematics. Enactment by Congress of the National Defense Education Act in 1958 and the Elementary and Secondary Education Act in 1965 increased the financial ability of schools to procure instructional aids, thereby placing them in the hands ola great many teachers and students who had not previously used them. Business and industry saw federal aid to education as an opportunity to expand their enterprises into the education market. Passage of the 11)58 and 11)65 laws was followed by a rapid proliferation in the design and function of instructional aids in mathematics and a buildup of the inventories of such aids. The events and trends identified above, because of their synergetic nature, will undoubtedly induce continuing expansion of both the de\ elopment of inst.' uctional aids and their use. This makes it impossible 14 today's teachers of mathematics to keep track ol new developments, make appraisals with confidence, and incorporate new materials in their teaching without reliable, organized assistance. To help teachers cope with this perplexing situation, the National Council of Teachers of Mathematics challenged a group ol authors to produce a yearbook containing information about the growing spectrum ol available instructional aids, suggestions for selecting and evaluating these materials, and guidance in using them. From their diverse backgroundsrepresenting experience in the schools, universities, business, and industry the authors who accepted the Council's challenge attempted to prcduce a yearbook that can stand the test ol time and change. That some instructional aids will be obsolete as this yearbook goes to press is inevitable; that others will be developed and perhaps also pass into obsolescence is not peculiar to mathematics instruction alone. But the reader of this yearbook will not go unarmed.

EMIL J. BERGER Editor -)! t DRIAI \1\,, VTR

Hum HownEN, Random House, New York, New York WILLIAm K. McNA, Skyline Center, Dallas, Texas ROBERT G. NUNLEY, North Texas State University, Denton, Texas LEANDER W. SMITH, IBM Corporation, Princeton, New Jersey HENRI. H. WALBEssER, University of Maryland, College Park, Maryland DAvin W. WELLS, Oakland County Schools, Pontiac, Michigan Emu. J. BERGER, Editor, Saint Paul Public Schools, Saint Paul, Minnesota Grateful acknowledgement is hereby expressed to the persons listed below and to others who contributed ideas, references, suggestions. and editorial assistance

Paul Bcrrisford, University of Minnesota Library. Minneapolis, Minnesota Aaron Bisberg, Image Publishing Corporation, White Plains, New York John H. Bolstad, Lawrence Radiation Laboratories, Livermore, California James D. Bristol, Shaker Heights Senior High School, Shaker Heights, Ohio Lee J. Brockway, Shaver and Company, Architects, Michigan City, Indiana Emalou Brumfield, Shaker Heights City School District, Shaker Heights, Ohio William W. Caudill, Architect, Houston, Texas Louis S. Cohen, Bloomington Public Schools, Bloomington, Minnesota Eleanor Duckworth, Educational Services, Inc., Watertown, Massachusetts John C. Egsgard, Saint Nfichael's College School, Toronto, Ontario Robert Ferguson, Diamond Path School, Rosemount, Minnesota W. Eugene Ferguson, Newton High School, Newtonvillc, Massachusetts Ourania Forbes, Flossmoor, Illinois Alexandra Forsythe, Gunn High School, Palo Alto, California Emil Greishaher, Minnesota Mining and Manufacturing Company, Saint Paul. Minnesota Kcn Heggen, 'Minnetonka High School, Minnetonka, Minnesota Gregory W. Johnson, Stillwater High School, Stillwater, Minnesota Randall E. Johnson, University of Minnesota, Minneapolis, Minnesota Pamela Katzman, Annandale, Minnesota Donald Thomas King, Saint Paul Public Schools, Saint Panl, Minnesota Joan Kirkpatrick, University of Alberta, Edmonton, Alberta Muriel Lange, Johnson High School, Saint Paul, Minnesota Jack Lown, Minnesota School Facilities, Minneapolis. Minnesota Lawrence Marks, Byron Junior High School, Shaker Heights, Ohio Sister Mary Peter Marthaler, Sauk Rapids, Minnesota Sol Mastbaum, Murray High School, Saint Paul, Minnesota LaRue West Miller, Fridley, Minnesota Joanne Moses, Saint Paul Public Schools, Saint Paul, Minnesota Anthony Nogales, Seattle Public Schools, Seattle, Washington Bea Ogrens, Fridley, Minnesota William Overman, Las Cruces Public Schools, Las Cruces, New Mexico G. Franklin Padgett, D.C. Public Schools, Washington, D.C. Robert Pecha, Churchill Elementary School, Cloquet, Minnesota Eugene F. Peckman, Wexford, Pennsylvania Daniel Pedoe, University of Minnesota. Minneapolis, Minnesota

VIII Charles NI. Proctor. Jr.. Montgomery Count Schools. Rockville. Matyland Diane M. Ptak. Oakland County Schools. Pontiac. Michigan Seymour Schuster. Carleton College. Northfield. Nlinnesota Pamela Schwandt, Normanchde State Junior College, Bloomington. Minnesota Judy Schwarzkopf, Saint Louis Park. Minnesota Nlatge Senninge, San Jose, California Joseph Septa. Murray High School. Saint Paul. Minnesota Nlarla Shapiro, Cambrichie, Massachusetts Carol Shingles.l larding High School, Saint Paul. Minnesota Fern Singer. Nlounds Park Junior High School, Saint Paul, Nlinnesota Henry Slotnick. National Assessment of Educational Progress, Denver, Colorado James 'Tucson, University of Minnesota Library, Minneapolis, NIinnesota John Wagner. NEchigan State University, East Lansing. Michigan Robert R. Willson. Seattle Public Schools. Seattle. Washington Clista M. Wood, Harding High School, Saint Paul. Minnesota Lateen G. Woodby, Michigan State University. East Lansing. Michigan The following students contributed ideas: Mark West Saint Paul, Alinnesota Wendy Johnson, Dassel, Minnesota Nlalco Int Ritter. Saint Paul, Minnesota Steven Saliterman, Saint Louis Park. Minnesota Students who posed for division-page pictures are: Daniel Brown, Saint Paul. Minnesota Robert Lewis, Minneapolis, Minnesota Bruce McNamara, Minneapolis, Nlinnesout Lynne Nelson, Minneapolis, Minnesota Denise Riley, Minneapolis, Nlinnesota Deserved 'noise and gratitude are extendedto the following specialists in graphic arts: Photographer Charles Hoffman. North Saint Paul. Minnesota Artists Nfarshall Kathan, 'Washington, D.C. Judy Norman, Minneapolis, Minnesota Noel Thorpe. Saint Paul. Minnesota Also acknowledged are the generosity and cooperation of producers and distributors of instructional aids who contributed pictures or gave permission to print pictures of specialind instructional aids. Finally, recognition and appreciation are extended to three generous, loyal, and dedicated people who gave expert assistance and timely advice in resolving a myriad of special probieins 'elating to the production of this yearbook: Jack E. Forbes. who served in the capacity of assistant editor but refused appointment. Carroll S. Bt own, who served as the editor's chief consultant in planning aml producing the artwork and selecting pictures. Mary Kaiser, the editor's secretary, who handled reams of correspondence flawlessly and whose meticulous attention to biblio- graphical detail contributed immeasurably to the quality of this publication. CONTENTS

Foreword

Acknowledgements

I.PERSPECTIVE, PURPOSE, AND PROFILE Emil J. Berger Punrosr Or Tins 1300K, 4 ORGANIZATION OF Tins 1300K, 4

2.INSTRUCTIONAL SPACES 7 Emil J. Berger ORIGINS AND MAJOR INFLUENCES, 9 Stir-cosTAINEn CLASSROOM ARRANGEMENT, 13 Vmur.o.sizED CROUP ARRANGEMENTS, 17 Large Group Instruction, I7 Sinai! Group Instruction, 20 Mathematics Laboraimy,24 IndependentStudy,29 Tutoring, 32 TEAM 'TEACHING, 34 ScininuLixe, 37 ConnenlionqlScheduling, 37 Modifications of the Conventional Scheduling Pattern,37 ModularFlexible Scheduling, 38 COORDINATING FORM AND FUNCTION, 42 Mod:fy;ng Conventional Buildings,42 An Individualized Eh:menu:D. School Mathematics Program Conducted in a l'aried-Sized Space Complex,42 OpenSpace Schools,46 ErILOGUE, 5I BIBLIOGRAPHY, 52

3. THE TEXTBOOK AS AN INSTRUCTIONAL AID 57 Henry H. Walbesser PROLOGUE, 59 Isittonucriox, 59 A IIIMORICAL PERSPECTIVE, 59 EarlyForms and First Purposes,59

XI Appearance Becomes a Factor, 60 Early JlIathematies Textbooks in the United States, 63 New World Editions, 61 Mastery: the Diet of the Day, (i4 Deviation From the Rules, 66 Visual Appeal of Textbooks, 67 The Historical Development of the Textbook. III Summary, 70 TwEsTn:Tn.Cr.s.rmy CoNTRnw.rioNs To THE 1.:.voLuTioN Or TII TEXTBOOK, 70 TIN TnooKs As PRODUCTIVE INS'IRUCTIONAL Ams, 72 Observations on Teacher's Manuals, 73 Development of 0 Functional Means of Describing Textbooks, 73 Search for Feasible Criteria, 73 Advantages of Behavioral Description, 74 Recommendations, 76 Active Teacher Involvement, 76 A Srr Or PRommukrs FoR DESCRIBING AND CONS'IRUMING BEHAVIORAL OBjECTIYES, 76 INsTRucrioNAL AIDS AND BEI! AVIORAL OBJEcivcs, 88 A POENTIAL Sm.rerioN PRocrouRE, 94 Compromise Procedures, 96 Epn.00ur, 103 IhnuocitArnv, 104

4.OTHER PRINTED MATERIALS 107 Hilde Homier, John N. Fujii WHATAREOPM?, 109 Itchy are OPM Being Produced? 109 How arc 01W Used? 109 NIoTivAIoN, 110 DISCOVERY, 113 EskicnmnsT, 117 CHANCE OF PACE, 123 DRILL AND PRACTICE, 125 ArrI.ICATIONS, 127 PRovESSIONALCROWT1i,130 SUMMARY, 131

5.TEACHING MACHINES AND PROGRAMED INSTRUCTION 135 Inmes E. Gilbert, 1.2ander IV. Smith Iltmucm. Sv.n-rcn, 137 DEmoriNG PROGRAMED INS1RUCTIONAL MATERIALS: AN OVERVIEW, 115 DEVELOPING PROGRAMED INS.' RUC! IONA!. MATERIALS: SOME DETAns, 145 SELECTION AND EVALUATION OF PROGRAMED INSTRUCTIONAL NIATERIAL.S, 146 ANn AnNtiNislit.vrIoN OF PRocRANt INST RUC ITON. 148 Preparing Students. 148 Student-Program Interaction. 148 Establishing Guidelines. 148 Supervision, 150 Measurement and Evaluation. 150 Timing, 150 CONCLUSION, 150 BIBLIOGRAPHY, 151

6. THE ROLE OF ELECTRONIC COMPUTERS AND CALCULATORS 153 Robert I,. Albrecht, William F. Atchison, David C. Johnson, Walter J. Koctkc. Bruce E. NIeserve, John 0. Parker, Dina Gladys S. Thomas BRIEI HISTORY, 155 THE COMPUTER IN NIATnENtAlles INsTRucrioN, 161 Genoa! Corn puler Uses, 161 The Im part of the Com puler in the Mathematics Classroom, 163 EXAMPLES OF PROBLEM SETTINGS AND PROGRAMS, I66 COMPUTING EQUIPMENT, 179 Calculators, 181 Programable Calculators, I83 Digital Trainers, 186 Small GeneralPurpose COMPliNrs, 187 TimeSharing Swims, 188 CONCLUSION, 190 SELEcrEn BIBLIOGRAPHY, Annotated. 191

7.PROJECTION DEVICES 203 Donovan R. Lichtenbmg MOTION PICTURES, 205 Using Motion Pictures Eff ectively, 206 Developments in Projectors, 208 TELEVISION, 212 ElfediVe Utilization of ITV. 214 Classroom Arrangement for ITV, 215 OVERHEAD PROJECTION, 217 Preparing Transparenci,'s, 218 Illustrations of Transparencies, 219 Screen Placement for Overhead Projection, 223

XIII Orvzur. PitoJici los. 22.1 SLIDES AND EILMs'IRIPS. 225 SOME NIA Jolt Souitcrs Or Pi:op:elms EQUIPMENT AND NIATERim..s. 22$ SEI.EcTro ItEriltr.scrs, 229

8. USING NIODELS AS INSTRUCTIONAL AIDS 233 Donovan A. Johnson. Emil J. Berger, Gerald R. Rising THE ROLE OE MMUS THINKINC, 235 THE Roil: Oilooris Is TlIr TAcniNo AN1) I.EARmso Or MA IHENLYI les, 236 Arodels Provide a Setting for Discovery of Concepts, 236 Models Can Be Used to Focus Attentionon Ideas That Are Under Discussion, 238 Models Provide a Means for Making Independent Investigations, 243 Models Can Be Used to Provide for Individual Differences. 2,16 Models Can Be Used to GC11Claie Interest in a New Topic. 219 illodels Can Be Used to Promote Enjoyment of Mathematics. 253 Models Can Be Used to Build Appreciation fm. Mathematics, 256 A Map. Function of Models is Their Positive Like! on Retention, 258 Models Can Be Used to Teach Applications, 261 SUGGESTIONS FoR USING NIODELS EFFECI WEIN, 261 THE KEY To Errrcove Usu. looms, 266 USING GUIDE SHEKES IN CARRYING OUT EXPERINIEN'Is WITH Nloorrs, 267 STuol:NT.m ADE mobris. Bun.oixo A Moon. Cou.rcrlos, 278 Wuxi. NIODELS ARE AVAILABLE, 279 WIIAT Is Tut: FuTuRE NIonr.I.s? 280 SUMMARY, 280 FoR TwrsTv NIATHEMATICAL TOPICS AND CoNcErTs, 281 BIBLIOGRAPIIY, 295

9.MANIPULATIVE DEVICES IN ELENIENel'ARY SCHOOL. MATH ENIATICS 999 Robert I.. Jackson, Gttssie Phillips !WI:Alton Os MA:sartn.,yrivr: DEVICES IN ELEMENTARY SCHOOL MATHEMATICS INSTRUCTION, 302 Somi: ClIARAGTERisTics OF Coon MANIPULATIVE Drvicrs, 303

XIV SONI GI:NLRAI, GVIDELINESFOR .F111:USE, Or Dcviccs. 303 cLAssfricAHON DEviccs, 30.1

DcmossmATtos liomthsANiDEvicrs, 30.1 IiAci: vALur Duxici,. 306 CO warp 111;.ths.. 111.ocKs. Rom. AND Discs. 309 NUNIISER BoAKDs. 313 Gum; AND Om's. 316 r.Astatrm ENT Dr.vicrs, 318 NIoinas Or (;I:omit:1Ru: Ria..vi u )Nsnirs. 322 CAmrs AND Puzars. 325 Srmim, Com rum IoNAL Distcrs, 328 CoNcLusio Ns It rcARDING RrsrAkiral, 332 EVALUATION Or NIANIPULXIIVE DEVICES. 332 OUTLOOK FOR TIII: 1ui URE. 333 SOURCI:i Or N Asir(' i.,v1wr Maci raims FOR ELLNIENTARY SCI1001. 11::\ IATICS, 335 LO BIBLIOGRAPHY. 339

10. NI ATI IEMATICS PROJECTS, EXHIBITS AND FAIRS, GAMES, PUZZLES, AND CONTESTS 317 Viggo P. Hansen. Samuel L. Greitier. Emil J.'erger. William K. McNabb THE NATuar, Or A NIATHENIATics PRojrcr, 349 PRITARAlioN Or :\ PRojEcT, 349 Tin: ROLE Or PROJ EC Is IN THE 1.EARNING Or NIATIIENIATICS, 352 Exinurfs AND EuRs, 353 ExAmmrs Or STumsT PRo.jccrs, 354 Computational Devices and Computers. 355 Estimating the Value of r, 357 Making Measuring Instruments, 358 Applications of Mathematics in Science. 359 Making Alodels of Three-Dimensional Figures, 362 Geometry and /frt. 361 Ch'ing Geometrical Interlffelations Ia illgelffaic Expressions, 364 Graphical Representation of Complex Roots of an Equation, 365 Ceometrical Constructions and Curve Drawing, 366 zln Optical Method of Showing Conic Sections, 368 Reflection Properties of the Parabola and the Ellipse, 368 Exponential Curves, 369 Fundamental Principle of Counting. 370 Prababi/i/r, 370 Trisecting an Angle. 373

XV Non-Euclidean Geometry, 373 Map Projections,374 Sundials,374 Illaking Designs,375 A Historical Project.378 Theorem Demonstration Models,378 MAT111:NIATicm, GAMES AND PUZZLES. 379 MATHENIATICS CONTESTS, 382 SUMMARY, 383 REFERENCES FOR SIXTY-SEVEN PROJECT TOPICS, 384 No DucEus AND DISTRIBUTORS OF MATHEMATICAL GAM ES AND Pumrs, 399 A St:AA:cam BIBLIOGRAPHY OF GAMES AND PUZZLES, 399

11. A SYSTEMS APPROACH TO MATHEMATICS INSTRUCTION 401 Jack E. Forbes, James F. Gray IVIIAT Is A SYSTEMS APPROACH To INSTRUCTION? 403 WhatIs a System?403 Systems in Education,405 ANALYSIS Or SYSTEM GOMPONEN1S, 408 Human Componentsof the System,408 Printed Medium Components,409 Audio and Visual Components,10 Models, Manipulative Devices, and Gaines as Components,412 1116-.hines as Components,412 TEACHER-MANAGED INSTRUCTIONAL SYSTENIS, 413 The Teacher-Managed Instructional System in the Large,414 ClassroomSystems Thinking in the Small,415 The Teacher-Manager and the Motivational Components of the System,420 A lUsiond,420 COMPUTER-MANAGED INSTRUCTIONAL SYSTEMS, 422 Evaluation Management, 422 Management of Drill,423 AComplete Computer-Managed System,424 Plans for Making CMI Available,425 A SYSTEMS ANALYSIS OF AN ITEM OF INSTRUCTION, 426 Assumed Entering Behaviors,427 Expected Terminal Behaviors,428 The Design of a System,429 Continuing Toward a "Best Design,433 BIBLIOCRA PI IY, '134

APPENDIX PRODUCERS AND DISTRIBUTORS OF INSTRUCTIONAL. Alps 436

XVI 1.PERSPECTIVE, PURPOSE, AND PROFILE THE DIHEDRAL KALEIDOSCOPE, is an instructional aid thatis remarkable for its simplicity and for the sophiqication of the mathematical ideait depicts. It consists of two plane mirrors joined to loan a dihedral angle of 1800 n for n = 1, 2. 3,.. .,and physically exhibits a dihedral group of order 2n. The mirrors in the picture are inclined at an angle of 600, exhibiting a group of order 6. The key between the mirrors yields 6 visible images. including the key itself. 1

CHAPTER 1

PERSPECTIVE, PURPOSE, AND PROFILE

by EMIL J. BERGER Saint Paui Public Schools, Saint Paul, Minnesota

The opening chapter serves as an introduction to the Thirty- fourth Yearbook. The significance of the title "Instructional Aids in Mathematics" is illuminated with a capsulation of two affinitive ideaslearning in mathematics and instructional functions related to instruction in mathematics. The working definition Of instructional aids in mathematics that was used to give direction and organization to this yearbook accents specifically the "things" that can be used to promote learning. The chapter elaborates on the four major objectives of this book and presents-a summary of the chapter-to-chapter themes. 3

1. PERSPECTIVE. PURPOSE, AND PROFILE

Learning among human beings manifests itself 6. Communicating the structure of the sub- as changes in the behavior of inch\ ideals. ject(i.e..relatingconcepts.definitions, "Who/ Iknow something new about the assumptions, and theorems) world, I'm different. When I knowsome- 7.immediately reinforcing learning thing new about myself,the worldis S.Providing opportunities for practice different."1 9.Presenting variations of previously learned Some changes in behavior considered indica- ideas tive of learning in mathematics arean increased 10.Presenting problems in application facility in recognizing and using mathematical 11.Nlonitoring the student's progress terminology, performance indicative ofa growing comprehension of concepts, and an expanded 12.Evaluating terminal behavior in accord- versatility in applying mathematical principles. ance with accepted criteria. Other changes in behavior regarded as learning "Iso some degree, the teacher can perform all in mathematics include a stepped-up proficiency these functions through oral communication. At in solving problems, an increased resourcefulness appropriate times the teacher's performance can in analyzing relationships, andan advance in be supported and the learner's experiences en- the decd of creative ideas generated. larged with the aid of textbooks, teaching ma- The educational processischaracterized by chines. projection devices, computers, models. an instructional activity that is designed to meet manipulative devices,television, and suitably the requirements of getting students to learn. designed instructional spaces. What instruction meanscan be clarified by ex- With this much introduction, the working amining some of its functions. Among the func- definition used to give direction and organizations thatpertainto mathematics instruction tionto the production of this yearbookis ale the following: presented: 1.Motivating the student to begin learning 9.Presenting a sequence of stimuli that is An instructional aid in mathematics is structured to keep the student engaged in viewed as any object, piece of equipment, learning model, device, instrument, publication, 3.Giving the student directions to dosome- picture, chart, or any facility that serves thing, look at something. or startsome the purpose of enhancing or effecting the activity learning of mathematics. 4.Describing or portraying the kind of performance expected of the student when the learning period ends This definition refers specifically to the set of "things- that can be used to promote the learn- 5. Guiding the student in discovery learning by giving appropriate prompts ing of mathematics. It refers to everything in the learner's environment that is external to the learner, exclusive of the teacher. The teacher is I. Attributed to Helen Linden in VW: A Guidebook the human side of the learner's resources and is for Volunteers in Participation, Office of Teacheraide and responsible for helping the learner regulate his Volunteer Services Publication no. 352 (St. Paul, Minn.: St. Paul Public Schools, 1970). environment. 4 cum,TER ON!

PURPOSE OF THIS BOOK ma nship. as wellas suggestionsfor teaching classical concepts with devices that are brand The production of this volume was inspired by new. the premise that the thoughtful use of instruc- The fourth objective has been met by incor- tional aids by teachers of mathematics will serve porating illustrations of original teacher- and to in prove mathematics education now and in studeitt-made devices that invite imitation. The the future. Specifically, this yearbook was planned authors of the various chapters are aware that to meet the following objectives: creativity in teaching really conies to flower in I. To give teachers reliable organized infor- the area of instructional aids. mation regarding the range of possibilities of acquisition and utilization of instructional aids in mathematics ORGAN IZATION OF THIS BOOK To provide teachers with a basis for evalu- In keeping with the broad point of view re- ating the quality and utility of instruc- garding the nature of instructional aidsthat tional aids forms the basis of this yearbook, Chapter 2 directs 3. To provide teachers with suggestions for attention to the setting in which insult, tion in making effective use of instructional aids mathematics takes place in today's schools. The chapter seeks to indicate how different kinds of -I. To stimulate teachers to create instruc- instructional spaces can ellect or enhance the tional aids. learning of mathematics. Itis proper 10 think of To meet the first objective, this book has been space as an i:ls;ructional aid because the kind of organized according to the classification scheme spaces that are available dictate to some extent suggested by the chapter titles that follow. This the kinds of instructional functions that arc pos- scheme is intended to help the reader internalize sible. In the organizational strateg) of Chapter a feeling of organization for the variety of static, 2. different instructional arrangements are first graphic, mechanical,electronic, and environ- described. and then suitable spaces for each are mental phenomena considered to be instructional identified. aids in mathematics. The examples included in Chapter 3 concentrates on the most revered of the different chapters should serve to inform the all instructional aids, the textbook. The main reader regarding the many kinds of instruc- concern of the chapter is with the problem of tional aids that can be acquired and their mani- selection.l'Ite scheme proposedfor selecting fold uses. textbooks involves decision-making that is based The second objective deals with the problem on behavioral objectives. of evaluation. The examples of instructional aids In the not-too-distant past the mathematics presented in the different chapters embody ac- textbook for a patic:cia subject or grade tended ceptable standards of quality and utility. By to be the curriculum, but this is not as true as it giving thoughtful consideration to these exam- used to be. Today there are available a large ples. the reader should gain a sense of discrimi- number of monographs on special topics, charts, nation in selecting instructional aids suited to workbooks, drill and practice kits, pamphlets, his purposes. pictures,newsletters,periodicals. mathematics The third objective is related to the second. library books, and books that are referred to as The examples of instructional aids included in supplementaryreferences.Allthesearedis- the various chapters were chosen to illustrate cussedin Chapter under thetitle "Other particularinstructionalfunctionsandtech- Printed ,Materials." niques of nse. Thete are suggestions for teaching Chapter 5 contains a brief account of the his- modern cone pts with age-old symbols of crafts- tory and development of programed instruction PH:sm.-FRT. PuRPosF., \NI) PRonI.E. 5

and lc:Jolting machines. (illidelines lot selecting dilution of matei ials that ha% e the sainc am ;butes and e% altiating plogiamed histtuction mate' ials as inset uctional aids. .a chapter dealing %%id] these ale sketched. Also included is a frank dim ussion aspects of minx)l mathematics is included in this of the uses and abuses of plogramed instruction )earbook. materials. The general availability and wide variety of Chaplettiis cemented with oalculating de instructional aids discussed it Chapters 2 through vice, and «milliliters. Included are:thistorical 10 have enlarged the teacher's role in selecting Iesuntc highlighting major elopments in «,111- and utilizing instructional aids. The haschap inner science and descriptions of calculating terthat is. Chapter II gives an in-depth dis- devices and «nnputels that can be used as in- cussion of the expanded role of the teao her as structional aids in matheinatios. A major pact of manager of an instructional system containing the chapter is devoted to a discussion Of current man) components that «mnibute to student Ina( tices and pi obable luttne uses of comptitris learning. The toms instinct/mod sv.tcm,system and «nnputational de% ices in the schools. approach to instruction. systonA thinking, and ( :halter 7 gives an overview of the various so on, all describe an orderly process for making kinds of projection devices that have proved their decisions concerning Whatisto he taught, to effectiveness over the years. as well as those in- %%limn. and using which materials. The chapter novationsthatteachers of mathematics have iuc lodes .111 explanation of the s)stems apinotao It begun thing only lecent1). Specific instructional to) instruction and .111 anal)sis of v.11 ions instruc- aids treated include films, filmstrips, slides, oiler tional aids as components of :111 overall system. head and opaque projectors. and television'. 'Choc is a detailed description of the teacher's (:halter S deals with the use of models. The rode as systems manager as well as comments on ( hapter develops theory associated with the kinds ssible contributions of computers to the dis- of instructional aids to which much of the (:coun- charge of management functions. Finally. the cil', Eighteenth Veatbook is devoted, gives a de- chapter contains :111extensive example of the s( Option of the major types of models. presents application of systems thinking, to the detemi- ant illustrated discussion of various uses of models. nation of objective, and the selection of mate- identifies plomising practices. and contains an rialsofinstructionforapopular topicin extensive listing of models that are available mathematics. from commercial sources. Since Chapter II presents :1 planfor the (:hapter 9 focuses on the subject of manipula- orderly selection of instructional aids,itis as tive devices. The treatment presented concerns close as one can come to a summary of the the kinds of devices that are theful in helping diverse contents of preceding chapters. In a elementary school students learn about the na- sense itis both a summary chapter. which pro. ture of number. «HIM 11114. symeins of numeration, sides a stub( titre for the information acquired properties of operations With 11:11111)erS, fralii0115, in the first reading; of previous chapters. and an and spatial relationships. Considerable space is introductory chapter forase«)11(1.in.depth, devoted to a (liscussion' of the use of manipula- reading of these chapters. tive devices as concrete referents in promotin discovery of concepts and improving computa- To quickly locale topics that are of tional skills. immediate interest or concern, the Chapter 10is devoted to projects, exhibits. reader is referred to the table of con- fails, games, puzzles, and contests. Usually these tents on pages xixvi. This table in- aspects of school mathematics are thought of as cludes both the primary and secon- individual or group activities. However. since dary subheads for each chapter. these activities normally involve the use or pro- vg

STUI)VCAltILEL showing sound cquip ment in use '6461114 NI:kT1.1 ENIIICS INTEIL EST CENTER. Creek Valley Elementary School, Edina. Minnesota

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CHAPTER. 2

INSTRUCTIONAL SPACES

by EMIt. J. BERGER Saint Paul Public Schools. Saint Paul, Minnesota

Chapter 2 treats the kinds of instructional arrangements currently being used for mathematics instruction and the types ofspaces needed to accommodate these arrangements. The chapter begins with a look at the origins and major sources of influence that helped shape the various instructional arrangements and then describes in detail each of six different arrangements: the self-contained classroom, large group instruction, small group instruction, the mathematics laboratory, independent study, and tutorial. The open space school is viewed as a school building design concept that encourages flexible grouping patterns and individualization of instruction. q

2. INsTitticrioNm.sPAGEs

Instinctional spaces serve as hist' t(tional aids ORIGINS AND MAJOR INFLUENCES I)) accommodating- other components ()I institu- ("mil quite re( ent h. the most pervasive influence tion. that is. students. teachers. time. textbooks. in the design and use ()I insti fictional spaces has teaching ma( lanes. computers. proje( tors. models. been the (entuieold belief that the war 10 :111(1 manipttlatit materials. teachisfor one teacher to get together with The purpose ()I this chapter is to indicate how twentive or SO suulenIS. The dilleieut kinds 01instructional spa«'S (an en- ratio of twent) live to one (an be traced back hance and effectthe learning of mathematics. atleastas far as the fourth centuryinthe If al times the discussion seems to have a more lebrew Bobvfonian Tabuttd: genetal iefereme than just to mathematics. itis because instinctional spaces have notusually umbel of pupils to be assigned to each teacher is twenty -disc.IIthere are been designed specifically to accommodate the fifty.We appoho Iwo teachers.I Baba teaching and learning or mathematics. Rails 21:1.21 1)1 Six (Micron arrangements of instructional The oneroom schoolhouse of the colonial and components ale recognized: selfcontained earl) national periods or Amerk a was based on room. laige gimp, surall group, laborator). this credo. These schools were built to accom- dependent stitch. and tutorial. These arrange- modate from lilteen to twelny.live students rang- ments are not intituall exclusive; and it is II:Infra ing in age from six to sixteen. and they were that they overlap. for the last livelarge group, largely ungraded. The teacher dealt with each small group. laboratory. independent stud). and student individually. The usual procedure was tutorialemeiged through sel)aration. accentua- for sin:lents to come up to tile teacher's desk. tion. and expansion of instructional functions one by one. to (eche whatever they had been originally performed ill the self«,ntained class. assigned :o memorize. The exchange between loom. This differentiation of instructional func- teacher and student was largely a mocha ical tions and prcwision for spa eCs to house them one. involving almost no technique on the 0'S came about through American society's ccuttittu. of the teaclm. I his time was taken up by heat .,; ally enlaiging demands for more and better edu- lessons. assigning, new asks. setting copies. mak- cation of its youth. ing quill pens. and dictating stmts. The historkal Icsunlc that follows recounts The opening of the Quincy School in Boston some of the major influences that helped slope in IS17 inaugurated a new kind of school orga- the different instructional arrangements. `lite nization and a new kind of school building, both next section treats the self«mtained classroom. of which were to become standard in America and the ensuing sectiontreats the other five until the lastfew decades. This was thefirst instructional arrangements. Then there are ttvo school in America in which pupils were divided sections that deal respectively with team teach- according to age into groups that would stay ing and scheduling. The final section contains together for a year's work. Each grade met with a illustrations of efforts to individualize instruc- single teacher in a separate loom and was given tion by coordinating form and function.In- assignments deemed appropriate to that age. To ( hided is an extensit e discussion of open space accommodate this new scheme of organization schools. the oneroom schoolhouse was multiplied under one 1001 and .1:1( Led up egg/ ral la,hiou. The l'AerNilling was suppo.ed 10 happen inIllis Boston ..1/nunun hi 18111 olk.ed illis dem lipiion sun/1.11d his!' 01the lualtling: qudelit udiatioli,, eS. This school house. being the Iasi el et 11,1 ilidepemlem ,tuck ohtially (14)ing .1z...holm/ush in 1110,1 ol the 111(1(1('11 :Indiakiiig:1101toilet lingles1%.Ikc.11se 01W 1111prOVellIellIS. is four modes high. leat her and one gr011p ()Isl pel 101 tiled 311 and comailis Ielve school room., each of whit 11 a«-ommodaics 51; scholars. :111(1 1 Ile%e 11110:1011N in one 100111.1 his .irraligrmeni a hall furnished with whit IIsSlII 10 Ile 1,110Wil Ille ( 00111. seat 700 pupil., Ii .1150 ha, el quail re( un- rill 11 allii()%1 19111. s(11001 building, woe (011. lionroom..lls gealcst impiovemems %o wed mole or less 011 I!ue plan of the Quincy «insistillhaving :tseparale room [M- e:tellteacher. and a separate desk for St 11001. will' 1e( 1:111g111:11'( lasfo0111boNs's eachNCI101:11[37. p. 72] ranged ou eiilier side of wrridors. The archilecl

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Courtesy ofEducational FacilitiesLaboratories

rualti: 2.1. Quincy St' hnoi I N's.n:ucriONA SPACES '1'1

William Caudill describes the complete acceptance of this classroom box: 4 41 At the turn of the century nearly every start in the United States passed laws which specified what this box should look like. Laws stated the size, shape, fenestra- 40 tion, and even the orientation. It wasn't long before the classroom box and its 40 0 25-to-one pupil-teacher ratio became a 'S:\CREI) CON to educatOlb and architects all over this nation and the West- 4 IS ern World. [16, p. 8] Atter 1Vorld \ValII the construction of school buildingswasinfluencedbytheresidential rambler and picture-window fadthat accompanied the dash to the suburbs. The result was sprawling NC1100111ollhes with laige classrooms FIGUR!.9.9 and laige areas of window space. But in spite of this subtle influence for openness, these sprawl. Ur/ands:red /rum Sitclk am! tht Earn-mingPt arcs. by peimb , and ,Nintt ing schools functioned like the eggcrate schools. The idea ofthe self-contained classroom con- almost immediate') to several kinds of changes tinued to dictate insit tictional functions. Elemen- in the use and design of spaces for mathematics tary students remained in the same box with the instruct ion. same teacher for a year. and secondary students Inthe high school one change took the direc- changed boxes and Lemke' N everhour. The box tion of specializing existing standard-sized class- maintainedit'legal status, rigidity, and hide- rooms as "mathematics classrooms- I)) moving in stuctibilit), and the halo on the Sacred Cow special Itimishings and bringing in a variety of remained untarnished. insutictional aids for mathematics. Such spe- Old) during the oast two deader have edu- cialization was billowed b) recommissioning of cators and al cllitects become notic eabl) sensiti. e hfila 1 1unused spaces;INad jlinf tspaces(e.g.. to the idea that school buildings and facilities mathematics libraries anci mathematics labora- should be designed to accommodate and encour- tot its). In new construction some of the special age desirable lust! tictional functions. This has lornishings were built in as fixtures, and adjunct led to intimations in the design of new build- spaces and bt and a 1(1.Ni/cc! classiooms were put ings and facilities and modifications in the use next to each other tO loan mathematics of older ones. What has inotiated these innova- In the elementary school a modest trend de- tions? In the case of mathematics, some of the %eloped toward beinidepai t menia 1 ii.t t ion(e.g.. impetus has come 110m the drive to imprme mathematics and science) and related specializa- school mathematics that started after \Vol Id \Vat tion of Niall(1;11(1Nlied classrooms. flowerer, all II. Changes affecting education generally have these changes ,aknowledgcd the accepted status come from a steadil) glowing zeal for the goal and milstiaints or the ,el«),Imilied classroom ofindividualizinginstructioninallsubject arrangement. areas. The two influences have not been inde- The main impetus for architectural change pendent. and they are not yet history. came from a different direction. In 1959 J. Lloyd The ch.ie to improve sc 11001 mathematics that Trump opined that the challenge for (viand() was launched in 1952 and was gi% ell a boost in .\merican education had largely been met (73). Sputnik in 1957 testified ill a complete overhaul To stimulate planning to improve the quality of of the school mathematics currimitun and led education he proposed that schools be organized io -01 .11:1lor: 01,11511! !ramp ,10 5111.1, punon: 33.o11 Tort :N3111%!111: 344.11:1 (limit': oir 110!1.111I1N 11111)1AH011 pup (11101;1 110!1011 10 1101117/111:11NAH)11! Jo 115110 11011,)11 cm! ti01....0.) 5.01111113i. told patull...q: 1111:31 ;:up131.0I put! t: To Ammo.) pin: q:11 1)111: 1)0!1011"! 31(1!x3IJ '11!1)11)3113% Oil 13!13(i P `1111 .111th attialpc ppiwi IlII'.)I tiolpuiNti! :)11. 11091:zpit:`;10 10 atil ootps

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subject (or topic) at a point appropriate to his appropriate to their needs and purposes. past performance. and to have a variety of in- 2. Instructional strategies should be based on structional aids at his disposal. wellefined behavioral objectives. An early form of individualized instruction 3.Performance grouping and periodic that leans toward the view held by Nfitzel was grouping should be based on diagnostic developed by Search around 1894. Search insti- and achievement testing. tuted an individualized high school program in -1.Small gtoups should be organised to meet which the student teas placed as an individual. particular instructional purposes. in which he worked as an inditidual. progressed Students should have opportunities to study as au individual. and graduated as an individual independently. (66. p. 151). 6. Students should have access to a variety of Between 1911 and 1920 Helen Parkhurst de- instructional aids. vised an individualized self-paced laboratory-type 7.Assignments should be varied to meet in- plan(called the Dalton Plan)in which the dividual students' needs and interests. teacher and student formed a contract with the S. Every student needs dialogue with a teach- student agreeing- to undertake assignments for erthis includes tutoring. several weeks duration in his various subjects. Both views of individualization of insnuction The student was quite free to work at his own have had an impact on the use and design of rate. except that he could receive no new con- instructional spaces. Up until now the influence tract until he finished the one on hand (59). of the grouping doctrine has been paramount. The work of Skinner and Crowder with teach- This is reflected in the choice of content and the ing machines and programed instruction in the organization given this chapter. At the same mid-1950s and early 1960s also gave impetus to time. the influence stemming from the adaptive the adaptive point of view of individualization theory has not been neglected. of instruction (69). Still more recently, a technique called "indi- SELF-CONTAINED CLASSROOM vidually prescribed instruction" or IPI has been ARRANGEMENT developed in five elementary school subjects out of the work of the Learning Research and De- Unlike much of today's writing about the self- velopment Center at the University of Pittsburgh contained classroom, the treatment given here is (38: (i:3). This technique consists of prescribing one of respect. The dominant position of the for each student a program of studies tailored to self-contained classroom as an instructional ar- his learning needs. The curriculum for each rangement is recognized. While its limitations subject area is specified by a carefully sequenced are acknowledged, its possibilities for enhancing set of behavioral objectives, and students are the teaching and learning of mathematics are placed in the program at various levels in actor= emphasized. Even though a variety of innovative (lance with their performance on pretests. The instructional space designs has been incorpo- mathematics curriculum is based on more than rated in new construction in recent years, the four hunched specific objectives. Students work economics of schoolhouse financing are such that individually on a precisely ordered set of mate- the self-contained classroom arrangement will not rials built on these objectives ((i3). soon be abandoned. Instructional functions per- The two views of individualization of instuc- formed under this arrangement developed with- tion that have been described converge on asin constraints imposed by the size of the room sumptions such as the following: and by the accompanying school organization. 1. Students of different abilities. backgronnds. In the elementary school this means having or competence levels should have curricula one teacher meet with twenty-five to thirty-five 14 cilApTER,Two

students in one wont, six to seven hours per day, In an instructional space such as the one de- five days per week. Grouping is usually by age, scribed the teacher performs most of the and this is also the basis for regulating student 11111(1011S listed below: progress. Nlathematics is taught from thirty to I.Motivating students to learn new topics forty-five minutes each clay by the same teacher and concepts who teaches all subjects except perhaps the fine 9.Presenting basic content using the chalk- arts, industrial arts. and physical education. board,overheadprojector,charts,and The secondary school organisation in which demonstration models one teacher teaches one subject to one group of 3.Clarifying statements in students' textbooks twenty-five to thirty-five students in one room, -I.Stimulating discussion among students by for one fifty-five minute period, four or five times dilecting questions to the whole class or to per week is a variant of the self-contained class- individual members loom oiganization of the elementary school. F.Assigning practice work (by handing out Grouping is usually by ability level, by subject, worksheets or assigning problems Irom a or both, and promotion depends on whether or textbook) chosen to give students of dif not the student does the minimum amount and ferentabilitiesopportunitiestostudy quality of work 'Nulled for ci edit. independently The instructional space associated with the 6. Stye' visingseatwork(practice work or self-contained classroom arrangement is usually problem soli ing), gis ing students in need 11:felled to as a stanclaid-sized classroom. ofhelpbrief(sometimesmomentary) A typical unsophisticated standard-si/ed class- amounts of tutorial assistance ooin has from 700 to 1000 square feet of floor 7. Administering and collecting tests. and le- space and isI ectangular in shape, with window teaching content with which thew is wide- openings along one side wall and chalkboard spread difficulty. mounted along the front wall and perhaps also Thislistofinstructionalfunctions,is,of along the other side wall. The back wall may course, not delimiting and will vary with the have a bulletin boaid panel. Furnittne for stu- individual teacher. In addition the teacher pet- dents consists of hulk 'dual deskchairs that are forms man) hist] sectional backup functions be- arranged in lows parallel to the side walls. fore and after school and at other times when he The teacher usually has an executive desk and Iristhe room to himself. He reads,studies. a large chair mounted on a swivel. In addition, plans the curriculum, prepares presentations, thew is usually a modest sited bookcase, a stor- duplicates worksheets and laboratory guides, age cabinet, a filing cabinet, a shelf or table for writestests,plans assignments, evaluates text- models, and a closet for the teacher's belongings. books and other instructionalaids, evaluates At the elementary level the space facility de- student performance, keeps records, and holds scribed may have sonic specialised features for parent conferences. different subjects, although usually not many. What students do is for the most part gen- In the high school this room, when used for erated by what the teacher does. Students listen mathematics hist' uction, may be equipped with to and watch lecture demonstrations in which a demonstration slide ride and a graph chart. the teacher uses the thalklmaid. overhead pro, In small communities the looms used by elemen- jector, and deinonstation models and devices. tary school students and secondary school stu- Students take notes, ask and answer questions, dents di e often of the same design, located under do assignments, take tests, and gise demonstra- the same roof, and a casual observer could not tions on the chalkboard or overhead projector distinguish one from the other except for the when asked to do so. size of students' dcskchairs. Various efforts have been made to speciali7e INSTRUCTIONAL SPACES '15

the standard -sited class' oom 101 mathematic 11- has been mounted using wall 01 ceiling blackets. St I uct ion (Figure 2.3). In some instances this has II the room is lane. lower 10011 divide's have been accomplished by innodncing a variety of been placed to partition off a corner of about insnuctional aids such as models. charts. manip- 100 square feet for a mathematics library and a ulative devices. measuring and drawing insn separate space of Horn 100 to 300 squaw feel mews, supplemental) textbooks. student jour- for ,t mathematics labmator). The room dividers nals. and teaching machines: b% moving in a are equipped with bulletin boards on one side glass.encloseddisplaylasetostolevaluable and shelves On the other, or they may consist of models and instruments: by providing auopen hinged panels of chalkboard. Sometimestall display case to display students'projects and movable bookcases are used instead of lower exhibit,: and b) supplying a maga/ine rack, room dividers.Individual deskchair units ale book display stand. and book truck to dkpla replaced b) two -(hail tables in onlei to encour- and make printed materials easily accessible. In age joint exploration of mathematical topics b) addition, a demonsnation table with sink and two 01 more students and to make possible seat- drain has beet, installed(or a demonstration ing arrangements for small groups. cart moved in).as and water have been inade Specialising the standard-shed classloom 101 available, elect] ical owlets and shelves to sup- mathematics list' uction b) moving in furnittne port TV receivers have been put in, and a per- and equipment and installing appropriate fix- manent select for plojecting films and filmstrips tures en«nnaged the expansion of instructional

FIGURE. 2.3. A standard-sized classroomspecialized for mathematics instruction by use of two- chair tables, a mobile demonstration work center, bookcases, displaycases, storage cabinets, overhead projector, mud viewing screen.

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DEMONSTRATION TABLE

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Ada/Ned porn drawings by E. II. Sheldon Equipment Company 28' x 36' 16 CI IAPTEIt TWO

functions with one teacher in charge. and con- dents who need extra help. and rapid learners comitantly smed to imptoxe the qualityof should be our,agcd to sun t spec ial pole( is by mathematics education possible under one teach- perusing a file of suggestions for projec is that is er. The desire to expand instructional functions available for this purpose. in the standardsized classroom by setting aside Sometimes the teacher can follow a strategy of portions ofitfor libraty and laboratoi y spaces intelrelating total group instruction with small led to incorporation of adjunct spaces for such group instruction. For example. a teacher ma purposes in new building designs and to the present the basic part of a new unlit to the entire et Cation of such spaces through remodeling in class, give a test. and loin] small groups based on older buildings. the results. Then he can tutor the slow learners Attempts to individualise instruction inthe and let rapid learners enlarge on the basic ','ea self-contained classloom arrangement liae taken by doing problems in supplemental y textbooks. the path of making pros ision lor hulk idual dif- The total class is reassembled for the next unit. ferences by %arying levels of procedtne, by vary- As another strategy, the teacher can present ing assignments, by grouping students within the basic content to the total group. set ;ancients to classroom according to their performance, by work individually, and mentally tabulate the providing opportunities for students to attack a membership of three groupsthe most progres- topic in a variety of ways, and by conducting it sive. the slowest. and the group that is holding limited style of tutoring. its own. 'Members of the progressive group can To y levels of procedure the teacher usually Nene as tutors for members of the slowest group. works withttle entire group. For example, to As the teacher spots widespread difficulties, he introduce the topic of carrying in addition, the brings the group back together. teacher can explain Methods appropriatefor Observations of instructional procedures car- different ability levels by using concrete objects ried oninthe self-contained classroom indi- and counting, by using ten as the key. by using cate that effective indkiduali/ation of insnuction a number line, and by using the computation under this arrangementisdifficult. 'fetchers algorithm. who attempt to individualise instructionilla Strategies for varying assignments include mak- self-contained classroom end up by dealing with ing open-ended assignments that challenge the individuals, not as individuals, but as menthe's able but do not discourage the less able, specializ- of a group. One teacher cannot work individu- ing assignmentsforindividual students, and ally with each of twenty-five to thirty-five stu- making two-pat tassignments following apre- dents often enough to make the encounter worth- sentation. The fist p, tt of .t Mo-part assignment while. Neglecting students of high ability because may deal directly with the basic content of a the needs of low ability students are greater is lesson and will be identical for all students. The not a satisfactory form of individuali/ation. Ev- second part can be varied depending on stu- ery student has a right to dialogue. Small group dents' needs and abilities. assigning challenging discussion, althoughmanageable,isgenerally problems to the belie' students and noel prac- confined to 'mief exchanges between the teacher tice exercises for the less able. and a small fraction of the total group. Labora- Regrouping students into subgroups is a third tory work is possible only fota small range of way of providing for individual differences in it concepts. and learning by discovery is usually self-wntained classroom. Such regrouping ideally leset%ed fot the few students who make discov- istentative andisbased on a diagnosis of eries initially. Independent study does occur but strengths and difficulties to determine who has is limited because of the close proximity of stu- mastered a skill or concept and who has not. dents to each other and a lack of the variety, of Write-in textbooks should be available for stu- inst.uctional aids needed to encourage it. iNsTRucrioNAL spAcEs 17

VARIED - SIZED GROUP functions. Experienced teachers of mathematics ARRANGENI ENTs report success with the following: This section concentrates On the fiveinstruc- 1.Motivating a new topic by presenting historical background tional arrangementslarge giottp, small group, 9 laboratory,independent study,andtutorial. Motivating a new topic by presenting Characteristic insult( tional functions fm. eachare practical problems huolving applications 3. described. and applapriate spaces for eachare NIotivating a new topic by relating itto topics studied previously identified, When used in combination, thesear- rangements oiler the possibility of providing in- 4.Introducing it new topic by presenting test dividualized instruction for every student. This items similar to those that gill be used to does not mean providing one teacher for every evaluate achievement student. It means c twomizing a program for ev- 5.Developing, proofs of key themems ery student, thatis. giving each student an op- 6.Demonstrating techniquesforproblem solving portunity to participate in the mixture ofar- rangements considered optimal for hint. Some 7.Presenting demonstrations involving large schools include the standard-sized classroomas a dynamic devices and mathematical models medium-sized group space inthis medley of 8. Giving presentations requiring theuse of arrangements. In such instances there are six scarce or expensive equipment (showing it different varied-sized group arrangements. sine curve with an oscilloscope) Suggestions presented in this section are in- 9.Presenting a film or filmstrip and a sum- tended to be provocative but realistic. No im- marizing commentary pCraliNTS are set down. Instead, the exposition M. Presentinginformationnotavailable inclines toward Options that the teacher can use through other sourcesfor example, notes to meet particular objectives. using the instruc- from COIIVC116011S, trip", field trips, or rela- tional spaces that are available to him. tively inaccessible periodicals or books 1 I.Presenting enrichment materials by detail- Large Group Instruction ing the contents of books and pamphlets in order to interest students in reading The goal of customizing instruction for every themfor example, working itproblem student has turned into a quest for ways of pro- fromMathematical Challenges (18) viding appropriate spaces and time for small 12.Presenting information onprofessional group meetings. independent study, laboratory and vocational opportunities in mathe- work, and tutoring. To save teacher time so that matics instruction can be provided for groups smaller 13.Engaging students in recreational activi- than medium-sized and for individual students, ties such as "mathematical bingo" itis necessary to balance such instruction with 14.Developing it concept, proving a theorem, large group instruction. or outlining the solution of a problem Since ithas been estimated that about one- with which there is widespread difficulty thitd of all instruction that might be given to a 15. Summarizing it unit by developing a topical inediunt-sized group is typically one-way, large outline and posing sample test questions group instruction saves teacher time by reducing 16.Administering a test duplication.Itis an efficient arrangement for 17.Returning atest and correcting errors reaching at one time all students who are study- made by students ing the same subject. Besides this rather obvious IS. Making assignments practical consideration, large group instruction 19. Familiarizing students with materials and can be employed to effect specific instructional resources available to them '18 (:11.1PTElt '1'WO

20. Pt eseming live telephone «micaeines with 1nehensiou ofthis :Annum. Teachers should great lining mathematicians using at spe make a c;treful estimate of what studentscan be vial phone hookup with students being expected to grasp in a largegroup pi emanation able to ask questions and should tesist the temptationto cover a topic 21. Moderating a panel discussion. «mpletelv. A huge group is better usedto stimu- liesicles being typically oneway. large group late interest in it topicor to explain a small por- instruction is teacher centered. The teacher lec tion of it than to give it exhaustivetreatment. Imes (talks). uses the overhead projector, writes The meeting of the largegroup should termi on the c halkboard. shows films and slides. gives nate with an openended assignment. offering demonstrations. and plays tapes and records. students a sense of direction for furtherstudy. Sometimes the teacher is a guest lecturer, or the Below are several principlesto observe in pre- lesson is presentedover television. In the latter paring- for a large group presentation, instance the "large group" can be decentralized. I.More materials should be preparedthan That is. students can receive instruction in small ate expected to be used. in case the pt esen- rooms viewing a television receiver.or in inde union takes less time than is anticipated. pendent study carrels using smallso een receivers. 2.The number of major ideas shouldbe The stodent's role in large group instruction limited. perhaps toas few as one or two. is usually restricted to viewing visuals,observing 3. A substitute presentation should be readied demonstrations. listening. and takingnotes. Oc- to be available in case of emergency. casionally the teacher may askan individual . Several illustrative examples should be col. student to pi esem it paper he has prepared.give le ted lor each idea to be developed. Prob- a solution toitproblem. develop a wool,or lemsolutions shouldbe handled bow take part in a sociodrama that is planned to several points of view, h isa rare problem motivate some facet of mathematicalinquirt. that niot be looked at or done in several lowever. such direct involvement bya student different ways. is at the discretion of the teacher who isin Following alarge groupsessionstudents change of the large group. and. in general. the should be given opportunityto reexamine new student's role is one of giving attention andbeing material in small groups andto confront the receptive rather than participating actively. subtleties. qualifications, and exemplifications of Preparing for a large group instruction session the new material in independent study. demands more creativity and ingenuity than does The optimal length of a largegroup session preparing for smaller groups, where the giveand is unknown, but in any case should lastno longer take of discussion can hold teacher and stmlents than mental alertness lasts. That thne willvary. oget t er. but forty minutes is a goodaverage length. In a huge group presentation the teacher \Vhat is thelnaximum number of students that should rely on broad effects using simple,pene- can be instructed eflectively in a large group? trating explanations:easy,oil beat examples: This depends partly on the age and maturity of drawings, pictures, and cartoons; and thought- the students. At the secondary level 150to 200 provokingquestions (perhapsintroduced as students is the largest group advisable. butat the potentialtestquestions)shown on prepared elementary level not more than 100 should be visuals and handouts. The opening and critical given instruction at one time. The actual number parts of handouts should be displayed with making up any large group will depend phi- visuals. ately on the number of students whoate ready A huge group presentation should havea for the experience and alsoon the size of the readily discernible structure, because thestudent space that is available; students will learn best will gain much of his stimulation front hiscom- if they are comfortable and uncrowded. L.- 1,%1. FicuRF. 2..Large group space used for mathematics insiruclimi in White Bear Senior High School, IVhite Bear Lake, Minnesota, This space is equipped with an operable wall that can be used to form two medium-sized group spaces by extending the operable wall to the front of the room (see inset).

The success of a huge won') fleeting depends taken b) the teacher while he gropes 14 a plug-in not only on the careful choice and arrangement Or rei»oves from the projector slides shown up- of content, but also on efficient management. side down or in the wrong order. Itis essential that meetings of large numbers of A large group instruction space should be stiflents be run smoothly. To leave the leacher arranged so that every student can sit comfort- free to concentrate on the presentationitself, ably. hear well, see well, and have a smooth assistants should take care of clerical and me- writing surface. Rows of deskchairs or two-chair chanical tasks such as taking attendance, collect- formica-topped tables fastened down on terraced ing papers. distributing handouts. moving dem- platforms with the rear rows elevated isan ar- onstrationcarts.adjustit:gprojectionscreens, rangement that meets most of these requirements. rool»-darkening. operating filmprojectors, ad- A space equipped as described is especially suited justing sound amplifiers, and the like. to instruction of large groups in mathematics All equipment and mechanical devices must (Figure Another possibility is a school audi- be in good working older and in the right place. torium with the floor sloping downward to the The intended effect of a carefully prepared slide front and theater-style seating with each seat presentation can be spoiled by awkward 'muses equipped with a retractable tablet arm writing 20 lApTER TWO

sin lace.,\large group space ilk::thisis often !emote (01111 olsfor raising nod loweling the used b leachers of various subjects. This means %iewing screen: mailing filmprojectors.tape specialized features needed for instruction playeis. and discplaers: and wining on tele- of large groups in matheinalics may be lacking. vision 1 eceivers. Large group spaces should be air-conditioned Teat hers of mathematics tistialk make !litchi! and %vinclowless. and sound amplification should use col chalkboauls. so the amount Of11.11kboald be designed so that eve student can hear. A spa( mum be adcq tune and appropriately planed. general purpose gitinasitini-auditoriumis um (hie desirable arrangement is 10 liaw at least two suitable. huge panels mounted and «minebalan«11 so Because projectors of dillerent types are used the% can be raised or loweted atfinger Irequentl; in huge group instruction. the ellec- 'ouch. Multiple panels can also be mourned to tiveness of different projectors as teaching lools be 11105ed sidewa s (Figure 2.5). Either arrange- should be a«lirately asse.s.sed. An indispensable mein increases the total amount of chalkboard tequirement for good viewing- is a large screen space available for teacher use. suspended from the ceiling. Each type of projector should be located in a handy place, and Small Group /us/rctio dime should be a sufficient number of easily Small groupinstill( tion allotds students cer- accessible electrical outlets. Nrotion picture and lain unique opportunities for learning- mathe- slide projectors Should be placed at the back of matics that ale not prcsem in other instill( ional the room to minimize d isti act ion. Since the over- arrangements. The mos' ad% antageous of these is head projector is probably the most used of :111 dial the structure of a small group encourages pro lectors,it should be positioned so as to be students to become actively involved in the learn- iminedialel;availableto the teacher without ing process through interaction with each 01her obstnicting student viewing. and with the teacher. Sound amplifiers should he part of the school's Ideall. a small group should have from seven unal audio N1Nleill. and there should be 1)105 ision to fifteen members. .\ group of Ibis size will give for receiving television programs. To be Opti- each student a chance to preseu and deleml his mal!) functional. a large group space should have ideas. ask and answer questions, telleci on aud media tiansmission center where :1 microphone criticize the statements of milers. check hunches. and overhead projector are located, along with and form conclusions. Increasing the group be-

Fic:tiRr. 9.5 Muhl' ple-petneleel chalk boarel h each panel resling on a Arparale horizonieti frock

Comte.) of P.. 11. Sheldon Equipment ContInutl INSTRUCIION 11. SPACES 21

011(1 filieen will deprive members 01 .1(leguale Small groups can be Inganiied lot` the 11111 purse 011101 11111.1,10 take pall 111 (11s( ths1011s. RedUC- 0)1 tea( !ling basi«oncept, and skills.I:or in- ing1 he group below seven will redtue the irk- stain e. in the prinial grades .1 small group in t)11 and stimulation needed 10 gelteratc dis( us- maillematics (.111 sl*%c 11111( 11 the ..1111 11111110SCas si011. .1 guru!) 01 twelve is plobably the optimal the lamiliar reading group. thatis. to ;;he each si/e. A, and Beggs suggest, "Perhaps student sulli( kill 011101 1 111111% 10 le.11 1111.111( there is good icason ttht 1111 more than twelve through direct parii(ipaiion. Such participation ate on a jury and wh there woe (nik twelve (an be en«nuagd means 01 (ompetithe apostles- (12. p. 21) based games and the the 01 (11111 alld practi(e .1,:t general 1110tedttle N111:111 group, should (aids.IIthe student is learning concepts. his 1101 IIae afixedtegistration but should be participation (an be emouraged by owaos ora (nganiie(ito meet ',macular purposes. guided discovery approach that relics On theuse means thatit a small group serws its put pose. of manipulative materials. displa charts. and membeiship inittrillkeep changing as indi- ( hall:board demonstrations. 4111:11 needs arise and .iimt..11 run given time .1 related purpose lot organiiing a small group membership in a small g10111) Will be based on is to reinforce concepts and skills with the aid of some instructional need. plogrcss le%el. (ontnion audio drilltapes. Students listen to and write interest. or assigned task. responses to prelecolded exercises and then cor- Small group meetings tnat be convened in rect each other's work.(1 Vith certain «mutter- response to request, !).students: they (an be prepaled tapes students can correct their «nultuted on a t.:dar basis in conjunction with own work.) ,.literthis. students pair oil and lute giottp insult( lion; or they (an be 01g:tidied assist each ()the' with exn ises the %veie not at :my Mlle to handle particular instructional able to complete alone. The number of students needs perceived I) the teacher when numitoling who (an work together undo this arrangement independent sttuI(e.g.. to reteach an algorithm is limited 1), the number of headsets 110 (01111111 - to:1 sinall group of students who are having dated by the equipment that is used. Usually th,c difficult y). maximum number is eight.

runntF, 2.(i. Stud/ group wingaudio drill tapes Err-

Goth's'` of Minnesota Mining and .11 Penning Company, Inc, ) CHAPTER TWO

.1 major puipose for holding small gimp ses- lent..( brainsiouning session mar be in oidei %ions iFas .110110w-up for large giotip insult« obtain suggestionslotattacking the problem. tion. The specific functions 01 such small gimp lb..insunming can lead to inventive. imunadvc sessions ate to amplify. clarify. and give addi- solutions dial 110 suulvot would ha«. thought of tional instruction on content presented it large Inhimself. lhainstorraing is also valuable to group sessions. Students can ask and answer ques- gather ideas for applications of %Lill,. concepts. tions and shale interim:unions. They can also. and techniques and to lonnnlate gcneraliiations. thimigh gioup discussion and suggestions hum Ci hi( km and ealnation h; ebbed the tether 01 the small group teacher. gain insight into their students ate out of phue hi sessions like these. own individual learning processes. I'his can help Hie pm pose is 10 generate a qnantit% 01 ideas them become more receptive to the information including all manner of variations., «unbina giveninthelarge group presentation. The (ions. and le( ombinations. leacher should point tip «incepts and «nrect in a similar way a small gimp of students errors of factin thinking from the sidelines. it with ,1 common inieiest in a topic, related 10 but possible. rather than from the chalkboard. 'By notpartof theregtilar cuiritilum nutv listening to students' questions and hearing their brought together to work on piojec ts. reactions to the large group piescutaIion. the Stillanother purpose Ior organizing small leacher can obtain feedback to pass along to the gioups is to bring together students who wish large group presenter and identify ulen who to calry OM an experiment. plan a field trip. need specific tutorial assistance it order to make prepare a mathematics exhibit. or get teach- lot pi ogress. a mathematics contest. A small group call be organized to meet before Finally. at nearly every level there are small as well as after a large group session. When a groups of slow !carnets who need to be !nought new topic is to be developed in a large group together to go over more slowly and deliberately session the small gum') may be dhided into sub- material that %.is pacsettcd eat lice in large group groups. with each subgroup asked to take careful sessions. notes on one aspect of the large group presenta- The role of die teacher who has iesponsi- tion. Each subgroup then prepares an in-depth bility lor asmall group willvarywith the report and presents itto the small groupone ability and maturity of the group and the objec- perhaps on the proofs or techniques developed tives for whicht be group is organived. The in the laige group session. one On applications teacher can be in charge: he can participate as of the mathematical principles presented. and an active member of the group; he can be an one relating the new ideas to pi eviously learned observer: he can work with half the group and material. let students in the other half direct themselves: Following a large group session, a small group or he can serve as a consultant. With younger may gather spontaneously and ask the teacher students the teacher will usually need to give to hold a ploblent solving session. Through col- form to the work undertaken by the gimp by laboration in working problems students can providing partially completed materials, select- gain the confidence needed to findsolutions. ing and making available manipulative deviu:s. Problem solving sessions lune particularly high and asking questions that stimulate thinking. motivational value just before quizzes or tests. Sometimes the teacher and a few good students The quizzes and tests can be administered orally can serve as tutors for members of a small group. tothe small group: students oftenfindthe And sometimes the teacher serves a small group competition stimulating and learn a great deal best by being absent. With no authority present horn each other. to settle differences students are forced to rely In the case of a particularly puzzling plob- on themselves in checking solutions to problems. INSTRC"IONAL

Icn !ululatinggeneralizat ions.detentliningthe ma.' need to meet CVCI') (by WeiltV caliclitof proofs. and so cm. minutes. III the high school each student should The (leclucnc

1:mlti' 2.7. Shapes of MII4 11 grow', Apa(e.% and Ara/ big ettithp,itraliotts

Reproduced front The S(.110o1 Libiaty by ',omission of the publisher. Edunitional Facilities Laboratories 24 1 1 \ 1 oT ER IWO

:leas ale teat lers' offices (when not in use).:a 3. Checking the foulness of 10m. the InclieNs table with (hails in a «Alter of the 1,Jc:tell:I or of the. and so on on a patio. an alcove in a siandaid-sized (lass- .1. Putting pilules together loom or lilnary. classroom adjunm spa(es. (or- 5.Measuring all sorts of objects I idol- benches. or small areas meant for r elaxa- Filling spaces, pouring water or sand flour tion. Students should be able to sitfacing one a cup into various containers alloth. in order to enhance interact)11 and ex- 7.Oldel ing «nttaincis a( «tiding to(apathy change of ideas. Ideally. of «nnse. small gimp S. Ordering children :scolding to height 01 spaces should be in proximity to the mathematics weight resource center so that students can have calm' 9.Using a balance made of a coat hanger access to textbooks and other printed materials, and two little pails to compare weights films.tapes. and models. A specialized small 10.Making graphs with blocks, buttons. chips. group space should be furnis:tcd with tables. a or pi( tine repetitions to slum (omparisons mobile mathematics unit or (at. and enough I I.Experimenting with objects to determine (hairs for twelve to fifteen students, the teacher. relations such as which objects in a set ale and possibly a guest or two. Thew should be huger than. smaller than, or the same size plenty of chalkboard space. either portable or as a given objet fixed. Small group spaces should measure about 19.:^sing relations to establish sets of ordered 200 to 250 square feet and should be reasonably pails soundproofed. well lighted. and ventilated for For example. the relation "will float on" student comfort. A standaid-sized classroom call determines the oideted pairs (ball, water) be re«munissioned to accommodate two or three and (block. water), but for pairs like (nail, snail groups by installing sliding or folding water) the relation is "will not float on." part it ions. Examining the properties of a relation Does it work if you say it backwards? Mathematics Laboratory This can lead to discussions of reflexive, The. mathematicslaborator)developedasa symmetric, andtransitiveprope.rtiesof separate instructional arrangement thn nigh dif- relations. ferentiation of instructional functions originally H. Combining and partitioning sets of objects performed in the self-contained classroom. and and relating these actions to addition and these specialized laboratory functions have since subtraction jnoliferated so thatitis no longer possible to 15. Tiling a surface or making floor patterns accomnunlate them all in a single space. In view using squares, rectangles. jrzniominoes, or of this situation, several types of mathematics other shapes laboratory arrangements are described, along Measuring distances with a ruler or trun- with appropriate spaces for each type. dle wheel The English Nuffield Project stresses labora- 17.Finding areas of surfaces by covering them tory learning inthe primary grades through with small blocks or tiles informal exploration (52; 53; 51). Listed below IS.Creating or discovering- symmetry by fold- are some of the specific investigative procedures ing napkins, paper. leaves. and so On that characterize the project. Engaging in these 19.Playing mathematical games. can help the young student develop an interest Allthese explorations are intended to be in and an awareness of Inathematicql concepts. unstructured, and informal and to be carried on I.Sorting and grouping objects in an atmosphere that is reasonably free of direc- 2.Pairing off the members of two sets of tion. As students advance to the intermediate objects to determine if the sets match grades laboratory learning should be encourage-3 NSTIW ClIONA I. SPACES 25

to follow this sequence:arying out imestiga- and pilules. All these matcrials should be Ieadilv tions with models and thilaniitlevices. formulat- at tessible when needed and out of the way ing podieses based on obsel %Jinn's (n measured when not. There should be sinks hill, nem by data. testing the lipoilieses b, mniplet midi- spate. floor areas where the students tan tional trials. and «nimiimit ating the lesults. Be- gailiel in small gi oups. .1 punle and game (elite,. cause these learning experiences ale often high- and an area for cutting and pasting. lighted 1) exciting dist ussions among students :\ teacher atit midwstern tiniveisit labora- and feat hers. the, should be tallied on in small 101 y school wanted high st 11001 students to lime groups. all opportunity to design and make mathematical The facilities for a mathematics exploration models. make and solve mathematical !Ann:um% shouldint lude adequate cupboard orgatiiie cone( lions of realia. prepare demon- and suchspat e to stole an abundant supplor stiationsforpresentationbeforelaigeand variously colored objects of many shapes and mediumsiiedgroups. prepal eexhibitsfor si/es. models and manipulatie de. it CS. simple mathematit s fairs. and do independent research measuring ii2511 uments. boxes. «mtainelS, projet Is. The teat her designed .1 separate facility

Fictati: 2.$. ii group of .qiulnt., at Mirror Lake Elementary Seltool. Bmward County.Florida.exploring numlu.1relation3hipAnAingmanipulative materials 26 c1 1 \ ',TERrkv(

thatinc hided .1 clansman's workbench: waist- :I.Using c ender calipels to obtain inside and high work wunteis with water. ga,. mul electri- outside measurements

caloutlets:carpenters" and machinists" hand .1. Measuring the thickness()Iwile with a tools and a cabinet to store them: and a laige micrometer and with a %VII gauge paper ( titter. The total lloor ,Dane ()I this lacilit. Measuring weight, tisinl.; spring male, and about half the si/e of a stanclardsi/ed class- balance scales room and will a« ommodate hom seven to ten Estimating weights. %ohmic,. and distances students at one time. The facility has the ap- Using all appropriate unit. peal ance of an arts and oafts shop and could be i.Nleamiling angles withuansits. sextants. converted into one by adding metal and wood and prottactois lathes.asolder-and-glue bench. and apaint (:alculating :fleas. volumes. ina«cssible di;- hood. tames. and ploduction costs based on ac- : thild kind ()1 mathematics 1.0 ,,story is tual measurements one in which students learn to use measuring 9.Enlaiging :1 drawing using a pantograph, and drawing instruments. Laboratories olthis :t 1)101)016(mM divider. Or printer's propor- kind involvehist' uctional Junctions thatare tion Slide I tile or by the method of radia- «n11111011 to both mathematics and industrial tion ts. Some of the specific skills a student can 10.Casinga pet Lentage prottactor toc.(ake learn ina measuring and drawing laboratory circle graphs ale listed below: II.Finding the areas of irregular shapes by I.Measuring distances with :I Rulatape. car- counting mitt:tics pentel's Ink. and engineer's tape 12.Cutting paper stock. lumber. and fabric to 2. Reading electric. water. and gas metels get the maximum usage t\,

FILES DRAWING TABLES

STORAGE O O

MEASUREMENT TABLES

FicantE2.9.Al easn ring and DESK drawing laboratory O

MEASUREMENT TABLES COUNTER 11 thlopir I /tam du:triage by E. II. tihrldan Equipment Gampant 24' x 32' INSTRM'IONAL. SPACES

13, Using a scale ruler to make charts. graphs, mathematics curricula in the secondary schools diagrams. floor plans. and noiuographs and the upper elementary grades have been re- stnicturedtoinclude instructionalobjectives H. Using try squares. miter squares. and slid involving the use of tell-key electric calculators. ing bevels to lay out angles. Inthis type of laboratory the students make The size of a meastning and drawing labora- flow charts showing steps in addition, subtrac- tory depends on the number of students to be tion.multiplication. and divisionalgorithms. accommodated. A standard-sized classroomis They learnto calculate with decimals using large enough to enable twenty students to learn calculators, and they check paper and pencil and to practice the skills indicated. The furnish- solutions to problems using the calculators. A ings of the room should include flour -to- ceiling calculator laboratory can be housed in a stand- storage cabinets capable of holding all necessary ard-sized classroom or equivalent space. Enough instruments and field equipment. bilge tables calculators should be available so that each stu- for making drawings. a waist-high counter along dent. or eve'two students. can have immediate one side wall for measuring and calculating, and access to a calculator. One version of a mathe- a chalkboard along the length of another wall. matics laboratory of this type is shown in Figure In recent yeas another kind of mathematics 2.10.Occasionally calculatorlaboratories are laboratory has appeared in some schools. Gen- combined with measuring and drawing labora- eral,remedial. consumer. business. and shop tories.

FIGURE2.10. Calculator laboratory

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Perhaps the most prestigious and exciting Thissectiononmathematicslabotatories mathematics laboratory is the school computer xvould not be complete without an excinsitsou center. Emphasis is placed On having- students a combination mathematics-science labotator), solveproblems and Onreinforcing concepts which some advocate folboth element:1n and leatned earlier. thing the computer. In addition se«mclaty schools.' Itis algued that the integra- the computer can be used for drill and practice: tion of mathematics with science «nitributes to simulationofsocial.economic. and political integration within the stud) of mathematics it. sstems: and games of strategy. (:omputer-facili- sell and that learning mathematics through its micad instruction is treated in detail in (liaptet telationship to natinal phenomena fulfills the 6. The school computer centerisusuallya twofold educational need of most students to separate enclosed space set aside to house One know mathematics and to know the teal world. 0)- mole lemote computer terminals and periph- It is further .unwed that since the stud) of science eral equipment. Some school computer centels depends on meastnement and thetheor)of also include desk-top computers. The si/e of the meastnement is essentially mathematical. teach- space thatis needed for a computer center de- ers of mathematics should tot confine the study pends on the number of terminals to be housed. Inmostschools a spacelargeenoughto I. The I ationale and demi iption of I he mathematics and cictitce laboratory mese:nett in this section tale adapted accommodate a small group is adequate. flout an at tide by Engene pc(isman (tw).

Etortu. 2.1 I.Mathematics-science labotaloly in the Golden ['ell) Middle School. Golden railer. Minnesota. This laboratory is planned foruse by filth- and ixth-grade students. INSTRUCTIONAL SPACES 29

01 meastn einem to space and time and the count- ing01 'none)but should extendittothe /ON 111e;istnement of mechanical, thermal, electrical. and chemicalenergy.Furthermore,formulas Item science should not simply be In ought in as facts to be juggled. Instead opportunities for dealing with quantit) in science should follow this pathobset %ation, measuring (and count - ing). manipulating. reasoning, genealiuing, and (mall) expressing a telationshipI))a formula. When this 0« Ills the mathematics laboratory and the science laboratory are one. Accolding toIIns point of view expo iences with science engender ideas; ideas ate nextet-haft/cc! and fmall) expressed as formulas. The following list suggests some of the steps students in a mathematics-science laborator) might pursue as part of this progression: I.Obtaining data 1)) counting and measuring Organi/ing data in a s)stematic wa) "14 (.orole.sy of E. II. Sheldon Equipment Company 3.Formulating hypotheses based on recorded data FIGURE 2.1" I.Testing hypothesesI))pet [mining actual ,stnan ,gfoupof(*menhir) school sludenIAc1u in laboratory tests (e.g., do the forces balance; a malhemalics-Acience 1aboratorl does the metal expand as hypothesiied;) 5.Making graphs to solve problems that can- with electrical outlets, gas and water, sink, and not be solved experimentally(e.g.,coin- disposal for refuse. position and resolution of knees and veloci- ties) Independent Study 6.Observing and making measurements of phenomena involving growth functions itis intporcuu 111:1(a student of mathematics 7.Expressing relationships among physical hale opportunities to study independentlyto phenomena by formulas. work by himself, at his own pace, inhis own .Nny standard-sired classroom is large enough way. Mathematics is not a spectator sport. To lot a mathematics-science laboratot) and so are lean it, one must play the game. lit independent spaces half this silt. A room used for this purpose stud) the student has an opportunity to seat ch should have work-top spaces, each capable of lor alternate ways of solving a problem and to ammunodating from one to five students work- check his solutions. undisturbed by the kibitiing ing on a project, adequate storage for all equip- of others. He has an opportunity to digest ideas ment and apparatus that may be used, chalk- presented to huge groups or to plusue the rami- board, a large demonstration table for experi- fications of concepts, problems, and proofs dis- ments involving large materials. bulletin board, cussed in a small group. I-le has a chance to cabinets to stole unfinished projects, a map rail investigate a topic in which he has a pat ticular to hang chat is, glass displa) cases for models, and interest or to uncle take a creative project that security cabinets for expensive instr.unents and will last over an extended period of timeIn tools. All work-top spaces should be equipped independent study lie has the opportunity to do 30 CliApTER l'\\()

the delving. imagining, mulling. intuiting. and considerable super%ision to keep them engaged guessing that he alone can do for himself. Some- in learning because of (Itch tendem% to be dis- times itwill frustrate him. \lore oftenitwill tracted easily. Slow learners should have just give him satisfaction. Occasionally it will exhilar- enough assistance to help them keep up with ate him. the other students and gain self-confidence. For The student can study independently ina older and mote mature students the teacher number of ways. He might read a textbook, should be available to give ad ice and assistance pamphlet, or student journal. He might use but should not super% ise too closely. 'lite purpose audio cassette drill tapes. chill and practice film- of independent study should be to help students so ips and film loops, drill and practice kits, and become increasingly sell-tenant and self-directed. programs presented on electronic computer ter- The kinds of spaces in which a student can minals to reinforce concepts learned earlier and undertake independent study in mathematics will to master computational skills. He can 'Rustle vat y with the way in which he intends to stud). a programed unit, either in a printed textbook using his own choice of instructional aids. He or on a learning machine, or use an hulk idnal needs quiet spaces where he can concentrate. He filmstrip viewer to learn new concepts or to needs spaces where he can view films and film- work aheadof other students.He ca»use strips and listento recordings.Ideally,these models, manipulative materials. and drawing types of spaces should be combined in a mathe- and measuring instruments to do experiments, matics lesotnce center. The student also needs to determine relationships and to substantiate to have access to the equipment and matetials in pool's. He can write a research paper, prepare a the %atious mathematics labotatories described report on a great mathematician, or make a earlier. To be useful for independent study. the mathematical model or display. laboratories shouldbe enclosed andinnear Pon most students. independent study time proximity to the resource center. should be incorporated in their individual pro- Typically. the quiet at ea of a resource center grams. Ilowex er,it should also be possible to should contain study carrels in which students arrange extra time for students who requestit can think, read, write, listen to tape recordings or who have been assigned a project requiring (using earphones),viewfilmcartridges. and it. The amount of time that a student should consultbriefly with fellow students. Some of denote to independent stud) will vary with Ins the carrels should be equipped with listening needs and interests. The elementary school stu- and viewing devices. Carpeting is a must in the dent in need of chill and practice should proba- quiet :um and itis advisable in other areas. bly not spend more than twenty minutes a day There should be shelving, book trucks and dis- at independent study. On the other hand. a play equipmentto accommodate and display bright student may findit profitable to spend books and pamphlets on supplementary and from one-third to one-half of the time he devotes enrichment topics, programed textbooks. inde- to mathematics studying by himself. Successful pendent study packages. and periodicals. There students at all ie els can be expected to spend should be materials for students needing teteac progressively more time studying on their own ing,for acceletated students working on ad- as they mature. A few students, particularly slow vanced topics and projects. andfor students learners, ma) ne%el be able to study mathematics engaged inexploring ideas (browsing). Small on their own; insisting that they do can cause collections of materials on special topics should them to develop a dislike for the subject. be on separate shelves. The use of the materials What is the role of the teacher in independent describedcan be encouraged by display'sof study? This depends upon the ability and matur- new purchases, bulletin board notices, and ex- ity of students. Young students will usually need hibits of special collections. If books and other I NSTRUCTIONAL SPACES 31

materials are in sight and available, students will be inclined to use them or at leastperuse 4 4 1 them. Besides the quiet ,urea, the mathematicsre- somi e center should have an area devoted to E=3 audio visual resources. There should be project- 015 101 111111S,filmstrips. and slides. videotape players, and disc players. Students using sound equipment should use earphonesso as not to disturb students engaged in other pursuits. This section of the resource center is notan area in which library silence should be expected.ow- ever, if functions nominal to this area are to be carried on, talking and other soundmust be [L)-1 subdued. Specialized mathematics resource centers of LEI cfil the kind described are mono common insecon- dary schools than elementary schools; however, ,tdapied ',innthawing% in The school Library: minted by permissionofthe publither, Educational Facilities there are elemental y schools that do havespaces Lab°, atm ies of this kind. In elementary schools the mathematics resource center is usually part ofone FIGURE 2.13.Three arrangements of sindy car- large resource center (sometimes calleda learn- rels and bookcases in independent studyareas ing center) that serves two or more oreven all subject areas. Quiet spaces for independent study can also be improvised outside the mathematicsresource center. These should be individual niches away from trafficlanes, such asasmallalcove, a 32 Cl i,ERTwo

screened -oll corner of a standard-sued classroom Tutoring Or adjunct space. or even an empty !moot» closet. Tutoring focuses on the intik As tutor, It deserves mention that today's concept of inde- the teacher engages in direct face-to-face dialogue pendent study spaces originated at Melbourne with a single student. Tutoring. may involve a 1957. where the first Iligh School (Florida) in lengthy one-to -one instruction session. or it can independent study space was It custodian's broom he a student's questioning look answered by a closet (S. p. 53). This space accommodated six teacher's confirming nod. Usually it is something students per da). one student during each of AN in between. periods. Where such spaces do not exist and Tutoring may grow out of the teacher's super- cannot be improvised. study carrels should be visoryroleinindependent study. Thus the ovided to ensure privacy. These can be placed teacher can diagnose difficulties in the setting in wrridors. in lobbies, in the cafeteria. or in and at the point at which they occur, lead the special rooms set aside for them (Figure 2.1.1). student out of misconceptions. redirect him. and Students using these quiet spaces should have finally spur him on to try again to work by access to as many resource materials as possible, himself. either by proximity to the resource center or Another tutorial function is to test a student through dialing access to materials located else- individually. and perhaps orally. Oral testing is where in the school building or even outside especially helpfulif the student has written a the building. Resource materials can also be "bad- test, for the teacher can adjust the ques- stored and displayed on book trucks or rolliug tions so that he can tell what the student actually resource centers. tan do. Doing this can also help to reestablish the student's self-confidence; he may fear that since most of his test answers were wrong, he knows nothing.

FIG UltEs 4 cluster of four study canels designed for um, in a cafeteria. /11 lunchtime the barrie., «in he hneeed.11i0'he

Ilemodut ed (,oneNI ithlle Sc hools /,',scission the publisher. Educational Facilities I.010» atm ies INSRIVIONAI. SPACES 33

ie:it her scattillditnecessat% to teteach 19. Having the student lead directions and the student concepts andskillsthathe has then checking 10 see that he can follow learned eat Het and lotgotten Or that he should hem have learned earlier but failed 10 «unplehend. I lelping the student to cut down the si /c Nitwit Intoning does involve such teteaching, but of a ptoblent. to get started on it tationall) tutoring should never be allowed to degenerate rather than being overwhelmed by it. into monotonous drill. Sometimes the teacher The student will learn more if the tutoring %%will need to help a student leant or telearn the sessions ale pleasant. meaningful. and efficient. steps of att algorithm by parsimonious prompt- There are several w.t)s the teachercan arrange ing, or help him analyie a statement to he tilts.If a series of sessions is IleCeSSa IT. heIllay proved by asking probing questions or givinp wish to prepare a folder containingit re«trd of hints to stimulate the production of hooche,..1» the student's progress and also some notesper- ingenious tutor will spontaneously generate the taining tohisinterests.Ile should begin by dialogue needed 10 bring the tutee to the "I seer' %%working on the student's most important 1».01)* stage. Following are some techniques that have lents. 'without wearing him out b) tryingto find been used by successful tutors: every weakness. lie should be attentive to what 1. Varying instruction so thatthe student the student sa)s and comment on his successful doesnotgettest less;breaking up the ethos. Finall). to help the student leave the action session with a good feeling. the last task should Explaining things several times instead of be something that is vas) for him. Jost once. and explaining in several differ- Tutoring is not just for the student who is COI ways: providing just the right amount having trouble with basic «mtent. It can also of repetition benefitthe advanced student who is studying :f.Pla)ing games with the student (suggested independently. Such a tutoring session can de- games: (aid. board. and spinner games; velop into a joint attempt by teacher :md student battleship; (lomoes: simulating a shop. to produce a proof with both contributing and ping tour) weighing one another's arguments. Nfore often. I. laking tip games %%ith problem cards the tutoring session for the advanced student isa 5.Playing "content the tutor" time during which he can obtain advice. lie may The tutor should make the same mistake need suggestions for sources of informationor several time., in a row, so that the student possible ways of resolving' difficulties he hasen- gets the ide:t. countet ed. lie may have become so engrossed in the Harlow aspects of a topic that he has lost .1sking the student questions about himself that intone numbers sight of important relationships that he must consider in order to progress: the teacher can 7.Having the student try to guess the an- help him broaden his approach. He may need, swers 10 problems before solving finally, nothing more than en«magement from 8. Wol king problems with the student: taking his tutor. Tutoring can provide the opportunity turns doing successive steps for guidancenecessarytokeep independent 9. Making tip problems. usinga menu or study from becoming aimless or bogged down. sales advertisement The dialogue can be a source of inspiration to 10. Writing answers to the problems that the the student as well as an aid in teaching him student makes tip to evaluate his own work. 1 I.Reading problemstothestudent and The role of tutor is not limited to certified having the student lead problems to the teachers. Others, such as college students, par- tutor ems, and community adults can serve as tutors. 34

During each period of the (la) A and B± stu- These blown -up cartels were orightall) designed dents can be atailable in the independent study lot stall menthe's at ,Nlelbottine I ligh School. a area to !Amide help. Studcnts 5111(1)ing the same school that pioneered in the idea of independent content can also tutor each other. There are stub. studies 10 indicate that tutoring of one student 1)) another student often improves the adtiete- TEAM TEACHING

mem 01 bot dr. The technique of team teachingisuseltdin amount of tutoring a student needs will carrying 0111a strategy of individttali/ingin- vary with the individual. butit should never struction by custontiiing a combination of ar- Ire allowed to become the steady instructional rangements for each student. In its simplest form diet of :my student. The teacher should aim at team teaching is a technique whereby two or making the student confident and sell- reliant. 11101e teachers use their respective competencies capable of studying by himself and solving prob- to plan instruction and to teach and evaluate lems on his own. the equivalent oftwo or more medium.sized The kindsof spaces neededfortutoring groups. are as variedas students' needs for tutoring-. The rationale underlying team teachingis Tutorinp, (an take pface at either the teacher's that making each teacher's special competencies or student's desk. at the student's work space available to as many students as possible imin a separate mathematics laboratory. or ata proves the overall quality of instruction and computer terminal. Occasionally a comer of a increases individualisation. space designed for small group meetings will Thus. each teacher on a team is encouraged do. For example. a teacher may find111:11 one to do "his thing"teach subject matter in which or two students of the group need individual he has his greatest competence and to project help. so he may spend some time with them in himself in the role (large group presenter. inde- a corner while the other members of the group pendent study supervisor, laboratory consultant. continue nearby, working singly or cooperative- small group discussion catalyst. or tutor)for ly. Usually some space more private than these which he has his greatest aptitude. works better. perhaps the teacher's office or a Itisargued that team teaching resultsin small conference room. Since many of a student's improved instruction because it increases input needs for tutoring. occur during independent in lesson development; eliminates confusion and study. a special place close 10 a cluster of study boredom that stem from repetition of presenta- carrels where the student and teacher can confer tions 10 several medium -sited groups ("I laveI isdesirable. B. FrankBrowndescribes a told you this?"); brings about spontaneous com- -stretched- carrel. which he has found useful: bining of the team members' accumulations of The bestdescription of the stretched supplementarytextbooks, programed units, carrel is a student carrel which has been lengthened.widened,deepened.and models. manipulative devices. and so on; and heightened. The desk top dimensions provides a base for focusing attention on critical measure six feet in length and two feet instructional problems. in width. This provides ample work space for the activities in which the indepen- Individuali/ation is increased because teachers dent study preceptor engages. Located on a team can make up combinations of large above the desk top space is a consider group. small group, independent study, labora- able quantity of shelf space that provides tory, and tutorial arrangements; use to advantage storage for teaching aids and books. Miff sitedfiling cabinets (two drawers)fit the specialised tutorial skills of the team; inch ease neatly under each end of the enlarged availability of team members for giving individ- desk top,affording convenientfiling ual assistance: and use total evaluation tech- space. A swivel chair completes the arrangement. [8, pp. 121-122] niques. INSTRUCTIONAL SPACES 35

single most important factor in the suc- lift%live minute period each cla%. Students cess of.1111 team teaching arrangementisthe and teachers divide their time among huge le( ognition of each te.( her's :pedal talents b% group in,tuction, small group inqruction. allteachers on the team. hoin lime to time mathematic, laboratory %cork. and indepen- teachers assume dillerem roles. dent study. For example. on one day all The success of a team teaching rangement students begin the period 1m %ic%ing an is enhanced b) the availability of assistance from eighteenminute film. Therealter. 70 stu- inolssionalandpa ra-prolessionalpersonnel. dents receive :1lecture for the remainder Needed from Bane to time is the cooperation of of the period. One group ()I12 students librarians, miniselors. and audiovisual and tele- meets with one teacher in a small Stoup vision supervisors. 1:csource personnel such as session:2:1 %Iticlnts go to a mathematics collegeprofessors, representativesfrontthe laboratory spat c. to work on projects: and Internal Ito eime Set.% ice.architects. atm:flies. die remaining 15 students go to ail inde- and consultants how the computer industry can pendent study space. make contributions through occasional 'crimes. 9 Thitt:i; PritroitmAxca, Teacher interns.studentteachers.student GRAD]: 1.Artir,Ni.vtics assistants. and teacher aides can assist with small Three teachers begin the school ear with group problem sok fig sessions and with moni- three medium -sitedeighth-grademathe- toring independent study. Clerks can be assigned matics groups that meet during the same to such tasks as typing and duplicating tests and block of time. Alter four weeks the students bibliographies. assuming citstocl) of equipment are reassigned to three different perform- and other materials. collecting assignments. eor- ance level groups. and thereafter they are rectinng tests. and eording grades. .\ graphic arts regrouped periodically as indicated by indi- technician to prepare visuals, charts, and other vidual progress. illustrative materials and a : iniian to operate 3. An 1-Ick: TEANITENTit-CotAnr. machines will help round out a team teaching Three teachers of tenth-grade geometry effort. team together to show films and give dem- Composition of teaching teams varies widely onstations involving specialequipment aniolig schools. The bases of cooperation usually such as a logic trainer and large-volume involve decisions pertaining to the total number tillablemodel~. Study guides and postview- of students and teachersto be involved. the ingtests aie produced as a cooperative division of responsibility, the amount of stu- effort by team members. dent-contact time needed, and the nature of the I.INEitolsocrt.ixAitv TEAstNIAThcm,vrics- curriC1111k111(i.e.. main track. accelerated, or gen- Physics eral). Onemathematicsteacherand one Various patterns exist.In the high school, physics teacher team togethertoteach teaching teams can be organised for all students advancedalgebraandphysics totwo registered for it given subject. all students in one mediumsized groups. Both groups meet grade studying the same subject.allstudents during the same block of time. The pur- within a certain ability or achievement range, or pose of the arrangement is to make pos- all students interested in a special topic. The sible the use of time-shared computer ter- examples that follow ale illustrative. minals au' to work together in small group I.SINGLE Stut.icar GRADE, problem solving sessions. ALGEBRA 5. CoNTEYr SvEctAus TEANISCCONDARY I:our teachers team together to teach SCI1001.PttAcTicAt. Antis elementary algebra to 120 students in a One mathematics teacher, one industrial 36 CliApTER 'TAM

arhteacher,and one home economics the school's big loom (similat to the pods teacher team together to teach a semester ussedin (onneciion with open spa( e subject in practical arts to three medium- s( hook later in this (1181)1(1. siiedgroups. The mathematicsteacher For mat hemats the lieielOgeneous medi. teaches the basic content on Measurement tuttsiied home base group, for each grade to all students in large group sessions and woe initiallyregroupedlunnogeneously special interest content to each medium. into four ability subgroups. but in practice siied group on 'quest. students shiftedfrom 0u(' subgroupto another for spec ilulessons and then re- Cosluxr Sr 1:m(1,1A. 1.1:ANIST.I.EMCNTARY joined their own mathematics subgroup. SCII001, The superior students for each grade 'This is a descriptor of :t team teachinti level worked by themselves much of the strategy structured by Joan Kirkpatrick. time. either individually or together. After 1Vestbrook Elementary School, Edmonton. each unit the subgroups in each grade were Alberta. tested and re-giouped. Groups fluctuated. Eight teachers were given complete re- with a small hard core at each end of the sponsibilityfor planning.teaching. and performance range. It was decided to give evaluating the total instructional program the top group enrichment instead of per- of eight medium-si/ed graded elementary mitting lice ac(eleration so as to keep all school gtoups. There were three fourth -. groups on the same unit at all times. This three filth-, and two sixth-grade heterogene. decision was made in order to make it pos- ous groups. Each of these groups was iden. sible (or students to move in or out of tilted with one home base teacher. any performance group at any time. Each teacherprovidedlargegroup The mathematics program at each grade instriu 'ion for a subset()Ithe students level was planned and taught by units. from the eight medium -sized groups in Doing thislacilitated movement 01stu- physical education. music. art, Or science dents limn one performame group to an- for two hours each week. In addition the other. Each twit had a pretest so that stu- teachers for each grade served on a grade- dents could be grouped in accordance with level team for the core subjects, reading their performance. The introduction to a andlanguagearts,socialstudies,and unit was often presented to a total grade mat hemat ics. level group, that is, a large group consist- Each team employed a scheme of speciali- ing of either two or three medium-sized zation in which one teacher served as the groups. mathematics specialist, one as the reading The description that follows tells how a and language arts specialist, and one as fiftlpgrade unit on fractions was handled. the social studies specialist. That is. each Teachers on the filthgrade team planned teacher was the specialist for at least one the units. pretested the students, and put core subject. The specialist took the lead in them into three performance groups. planning units andlessons,conducting sequence ()Ilessons in the unit was then team meetings, and being the whip for his timetabled. Next the plan was discussed subject or subjects. However, each teacher with the sixth-grade team, and any sixth- on the team taught mathematics as well as grade student needing work on fractions all other core subjects. was seat to work with the fifthgrade stu- Mathematics forallthree grades was dents whenever the necessai y skills or topics taught during the same block of time in were being presented. iNsTitt.cnoNAL sPAcEs 37

SCI I EIWI,INC. Ahnlihcalions of the Conventional Scheduling Pattern To ensure Ilia! each student will hale the ad- .11 hough the «mentional scheduling pattern vantage of le«.king hist! in lion in the %arious is extienel14;41. slight change will give instructional atiagements Ines( tibed lor him. schools a more adaptable schedule without rt. his program must be systematically planned. placing the basic pattern. For example, if three This involves scheduling. or more medium -shed groups are scheduled for Scheduling is a scheme ()I' bringing together the same or related subjects during the same the larious «,mponents of instill( lion: students. peliod. teachers (an then legion!) the students teachers. time. spaces. textbooks. models. com- into three performance levels. This arangemeni pute's, and so on. Ideally. it should be focused works well. pro% ided that the performance level on the educational needs of each student, on groups are continually re-formedas students placing him with teachers who are right for him progres!,. in the right spaces for the right periods of time A second alteration in the conventional sched- and with the right frequenc. 1 :dice. how- uling pattern will give each inedium;/ed group eer. there arc many compromises because of one "long- period each week (11. p. 91). This spatial «msnaints. scalcities ()I teacho time. and (.311 be accomplished by making an even exchange a lack of adequate materials. Thus scheduling of periods on Mondays and 'Tuesdays or any often focuses on subjects. in the belief that the other pair of(lays. For example, a class that needs or each student can be met by allowing normally meets during firstperiod could be him his own selection of subjects from among scheduled to meet during 1)0111 first and second ihose in the curri('ulum. periods on NI011(11)s and the class that meets during- second period could meet during both 'flute1)pesofschedulingpatternshave first and second periods on Tuesdays. The same evolved: scheduling based on daily,uniform even exchange ruler', would be followed by length of periods (referred 10 as conventional Illird and fottril: period classes, and so oil. 1)11r- scheduling), modifications of the conventional ing the remainder of the week all !ancients would scheduling pattern. and modular flexible sched- steel during iheir regular periods. uling. Sequence I'm:Ilionis another %.ariant of the onvenlional schedule (1.1. 1).9.1). This involves Convenl lona! Scheduling having a class that regularly meets during the first period meet ditriug the first period as usual The conventional schedule has been in common on Mondays, during the second period on Tues- use for some time. In the high school it usually (ays, during the third period on 'Wednesdays, consists of a daily minim! of six fifiy-five minute and so on, All by itself I his scheme seems to have periods. with the same sequence of classes re little purpose except to equalize distractions and peated every day. Students enrolled for a par- inierruplions associated whil different periods licular subject meet in the same group, usually and different days of the week. a medium -shed gimp of iwenty-five 10 thirty- five. A more fruitful modificalion of the conen- during the same period every day. Nlay elemen- tional schedule combines %ariation of period tary schools follow a similar daily uniform period- lengili with ordinary sequence rchalion. This length paiteni, excel)! that periods arc shorter scheme permits class groups to be scheduled in because the elementary school must divide Bunt shoo, medium, and long periods. While Ibis among nine subjects. Also, there may be varia- pattern is somewhat rigid because of its uniformi- tions in period length from grade to grade in i)it does make possible scheduling a variety of such subjecis as reading and maillematics. insiruclional arrangements. 81. I1.1.(IVI ()Al

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Nchool subjett plotted (mlietloletb .1N part of the thetchs C1111.1111 C.N 111111% 11111.111/J11011 ()Iins!' tit total program. lion. The% t that flexible Nehetioling pro- Needles., to sax. thisisa :11( 01111)Ii( met! mote, glemet i nen (:.1 in 11111(1)(11(1C111 N111111. gi,e proteiltil (lone I)% thc ()I a «Hu- it:Mlt:1s inure bole lot tutoring. a lidtill iiiiatel% puter conNiderabll feduLes the tedium and in leads to Icolg attiiat ion of ()mem into hetittente. addition enhances the prospect of deNigoing and of perfolotance

FiGutt 2.16 Sal eduling Mt ucture fm 192 students in geometry. The stt in lute wnsists 01 one latg gtoup se( lion of 192 students meeting fotty minutes pet week in one Aession: eight labotalml sections of 21 students with each section meeting jot one sixtv-minnle session pet week: and sixteen small groups of 12 students each meeting fm siAty minuhr prliod., 'wire areadv. A 5/nden/ can be scheduled in geometry lot eleven twent)-minute modules pet week: foil) minutes in large gtou(t insttlion, m.x/y minutes fot laboratot) wor h and tiro siAty-minule small von!, ins!, lir- lion sessions.

I 1 I 1 1 I I I I I 1 1 1 1 1 1

I I I I I I I 1 i 1 I e I I e . I I i 1 I . I 1 1 i I I 1 1 I 1 I I 1 I I I I 1 e I I I I I I I I 1 u I I I I I I I e 1 I e i , 1 I I I I t I I I I I I t I I 1 t t I 1 I I 1 s I I I I

LARGE GROUP (1)

1 I I 1 1 1 1 1 I 1 1 1 1 1

I I I I 1 1 t I 1 I. 1 I I I I I I I I I I I I I I 1 I I , 1 I 1 I

LAB. (1) LAB. (2) LAB. (3) LAB. (4) LAB. (5) LAB. (6) LAB. (7) LAB. (8)

------SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) ------

SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) 40 (:11APTEI: I'Wo

M M M M M Time 0 Monday 0 Tuesday 0 Wednesday 0 Thursday 0 Friday 0 0 0 D D

8:30.8:40 H Home Room H Home Room H Home Room H Home Room H Home Room

8:40.9:00 1 1 1 1 1 Meet with Advisor Music Work on Music Holp First Grade (MG) my Sdencs (MG) with Reeding 9:00.9:20 2 2 2 2 Expiaiment Listen to ()S) Resource Person 9:20.9:40 3 3 3 Listen to 3 3 (LG) Mat heat les Language Spelling Tapes (LG) ISG) View Filmstrips 9:40.10:00 4 4 Physical 4 (IS) 4 4 (IS) Education 10:00.10:20 5 5 (Free Play) Work on my 5 Guidance and Language Mathematics Puppet Story for Counseling (SG) Reading (SG) 10:20.10:40 6 6 (SG) 6 Work on my 6 (IS) 6 Social Studies 10:40.11:00 7 Physical 7 (IS) 7 Physical 7 Work on my 7 Do my Education Education Social Studies Mathematics :Music Appreciarif: 11:00-11:20 8 (LG) 8 9 (LG) 8 (IS) 8 (IS) (LG)

11:20.11:40 9 9 9Meet with my 9 9 Science Science Social Studios Social Studies Social Studies (SG) (SG) Project Committee (SG) (SG) 11:40.12:0010 10 10 (SG) 10 10

12:00.12:20 11 11 11 11 11 Lunch , Lunch Lunch Lunch Lunch 12:20.12:40 12 12 12 12 12

12:40.1:00 13 13 13 13 13 Library Study 7--- Reading Mathematics Reading Language (IS) (SG) (LG) (SG) (SG) 1:00.1:20 14 14 14 14 14

1:20.1:40 15 15 15 15 Browse in the Reading L__. Mathematics Language Do my Reading 15 Resource Center (SG) (SG) (SG) (IS) 1:40.2:00 16 16 16 16 16 (IS)

Listen to Music 2:00.2:20 17 17 Do my 17 17 17 Do my Reading Records (IS) Science Mathematics Reading (IS) (SG) (SG) 2:20.2:40 18 18 ., (IS) 18 18 18

Artrt Listen to Spelling 240.3:00 19 19Work on my 19 Social Studies Mathematics with (MG) Tapes (IS) ArtA (SG) Teacher Help (Art Lab.) 3:00.3:20 20 20 (IS) 20 20Meet with Advisor20

FiGuRr. 2.17 An example of a sixth-glade student's schedule. The white areas represent structured time and the shaded areas represent independent shifty time that the student has scheduled for himself. iNsTRuctioNAI. sPAcEs 41

M M M M Time 0 Monday 0 Tuesday 0 Wednesday 0 Thursday 0 Friday D D D D D

8:00.8:20 1 1 1 1 English English Biology English (SG) (SG) (Biology Lab.) (SG) Geometry 8:20.8:40 2 (Math Lab.) 8:40-9:00 Spanish (Language Lab.) Geometry Driver's Education Geometry (LG) (LG) (LG) 9:00.9:20 4 4 Study Driver's Manual Physical (IS) 9:20.9:40 5 Education 5 5 Biology Driver's Education (MG) Biology Driver's Education (LG) (SG) (LG) ----'(SG) 9:40.10:00 6 6

10:00.10:20 7 Biology Biology Spanish Biology Physical (SG) (SG) (LG) (SG) 10:20.10:40 8 8 Education (MG) Spanish 10:40.11:00 9 Spanish Spanish (Language Lab.) (Language Lab.) Sophomore (Language Lab.) Seminar 11:00.11:20 10 10 10 10 10 Spanish Spanish Spanish Spanish (SG) ISGI (SG) (SG) 11:20-11:40 11 11 11 11 Work Out in 11 Weight Room 11:40.12:00 12 Lunch 12 Lunch 12 12 (IS) 12 Lunch Study Geometry Physical 12:00.12:20 13 13 13 Education 13 13 (IS) Study Driver's --_, (MG) Lunch Manual (IS) Work on Biology 12:20.12:40 14 14 14 14 14 Experiments (Biology Lab.) 12:40.1:00 15 World Studies 15 Plan Wrestling 15 Lunch 15 15 (LG) Strategy Physical 16 (IS) 1:00.1:20 16 16 16 Education 16 Study Geometry (MG) Study my English (IS) (IS) 1:20.1:40 17 17 World Studies 17 17 17 Project Work on World 1:40.2:00 18 18 (ISI 18 18 18 Studies (IS) Geometry Geometry 2:00-2:20 19 19 19 19 19 (SG) (SG) English World Studies World Studies . (LG) 2:20.2:40 20 20 20 20 (SG) (SG) English English 2:40-3:00 21 21 21 21 21 (IS) (IS)

FiGinui 2.18 An example of a high .sehool student's schedule-nu: whiteareas re Pre- sent structured lime. The shaded areasrepresent independent study lime that the student has scheduled for himself. (M.\ PTElt T\VO

COORDINATING FORNI AND ing chairs; 01 «)111 sc. the kinds of instriu tional FUNCTION functions that are possible under such an a- tangement xill be limited.Thereis also the This final section brings together dare exem- possibility of knocking out a nonbearing parti- plary ideas 14 individuali/ing instruction. using tion wall to make one huge group space out of dillerent kinds of plant lacilities. The filst exam- two mandard-sited classrooms. Removing corridor ple is concerned with the possibility of modifying walls near the end of a wing or using the cafe- a conventional building to make it more lune- teria or school a tidimrittni :ne other possibilities. tional for housing individualiied progia ms. The Acoustical problems can be reduced by carpeting second example contains an actual account of the floors. how an individuali/cd mathematics iwogram is cmlied on in a %aried-sited space complex incor- In an inclividuali/cd plogunn. textbooks and 1)01 ated into the structural design of an elemen- other instructional aids should be collected in tary school. The third example is really a short one plate in order to be readil) accessible. This discourse on the latest innovation in mhoolhous- means that at least one stanclaid-sized classroom ing design, the open space school. Emphasis is needs to be used as a mathematics resource given to the special tapabilit) of this design con- center (ol perhaps mo or duce such looms with ceptlot accommodating an indkiduali/ed plo- adjac ent inlet iotxalls lento% cc!). Tittle should gram in mathematics. be areas in the resource center for housing and for using pertinent inatheinatits textbooks. films. Modifying Conventional Buildings filmstrips, records of facts. drill tapes. manipulative materials. models. and construction mate- hidividualindprograms are most easily orga- rials for projects and exhibits. 'died in schools that have sped:di/cc! spaces for large group instm lion. small group hist' action, independent stuck. laboratory %vork. and tutor- API Individualized Elementary School ing. However. such piograms can also be orga- Mathematics Program Conducted in a ni/ed in «mventional buildings that have only Varied-Sized Space Complex standard-si/ed classrooms. Let is see how a conventional building made up of standaid-sized Anintik i(1ualizedelemental)schoolmathe- class' ooms can be I ecommiv.ioned to au ommo- matics plogiam dc' eloped uncle' the dilution date an individualized program. of Vere DeVatilt is planned to enable each One standatd-si/ed classioom. (.u1 best:palmed student to select the mathematics he should study into two 01 three small group spaces, each ca- and to establish his (mil pace and sequence lor pable of holding from seen to twelve students. lc:dining the content he 11,ts selected. The major Room dividers can be installed to reduce noise conceptsofthismathematicsprogramare and (lisp:Idiot]. and individual desk( hairs in two- mganized into eight strands, with each strand ( hail tables airanged inform:Ill) for the separate organized into units. Insnuctional materials in- groups. A second loom can be made usable for clude booklets listing possible choices of text- independent stud) In arm Aging deskdiairs so books for each unit, a library containing multi- that they face away from each other or toward ple copies 01 seen selected textbook series, tests outer walls. A third room can be outfitted as a with parallel forms for each unit (available at mathematics labolatory as shown in Figure 2.19. two reading levels). and answer sheets and kegs Providing space for large group instruction lor students. Record forms include student fold- presents more of a problem. A standadSiCCI ers, test performance records, assignment certifi- classroom can accommodate a reasonably large cation records, and student-teacher conference group if students' deskchairs are replaced by fold- guides. NsTRuun oNAsp. \ cEs 43

1:1 Gm!: 2.19 A slandarAsized classroom furnished awl equipped as a mathematics laboralmy

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*4' 44 (:1 lApTER Two

The school in which this program was im- work islands are art :urged for laboratory work. plemented is the Giese School in Racine, Wis- The following quotation flout "Description of «msin. The structural de.;,91 of the school in- thelnd ividua i/ed 1:itheinat icsCurriculum cludestwoinSitietiOnal NI, econfigurations.i0115. Plojec t'' by DeVault, Btu hanan. and Nelson (22) each of which is an example of .tdied-sited delticIN what ;$1 tRiculs and(cachets do when they space complex (Figure 2.20). Each of these com- are at work in the program. plexes consists of a hexagonal shaped learning About 10 students and ateacher ate center surrounded by ten doorless pentagonal seated in an informal artangement in the bilgehexagonallearningcentersur- shaped standard - sitedclassrooms designedfor rounded by individual classrooms. Some use by small and medium -sited groups. Partition students are reading independent!)in walk between duce pairs of adjacent classrooms study ctrrels or at desks, while others are can be folded back to make large group spaces. in sm;ailgroups of two or more at tables. The auxiliary aide is at work near the The learning center. shown enlarged in Figure files, records, and materials of the pro- 2.21. may be used by large. medium- stied. and gram. A few students are gathered about small groups, and has carel, available for inde- the movable library carts. looking up suggested unit assignments in various text pendent study. Some of the areas behind the seriesand selectingtext assignments in

FIGURE2.20. Floe» plan of the Giese Elementaq School, Racine, Wisconsin. Du: distinctive design concept of this school is embodied in Iwo varied-sized space complexes. / Standard Size Classrooms >:/-/

LEARNING CENTER 7Library Teachers' Workroom

Office Workroom) Area

J LEARNING / CENTER

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(Music I NSTRUCTIONA I. SPACES 45

FIGURE 2.2 I. ALearning Center in the Geise Elementary School, Racine, Wi,semisin

Student Laboratory (Countertops)

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which the approach to the topic and the a regulat text set ies for teletence and as leading level ale appropriate 1(w their a soul ce 01 assignments. In another I, tont individual study. seen children and a teacher ale in the One student has sought the teacher's midst of a small group seminar on mt. assistance with sonic material On addi- 'iteration sstems. The students have all tion. Alter questioning the student brief- tecentk finished a unit on 1111«1111111011 ly and reviewing his lec °id folder. the Illiniera ion NySICIIIN and are working to. teacher suggests that he lay aside his pre- get her on devising their own new numer- sent work tempos rill and choose lurt her ation systems. On another char one might assignments on 1 egrOlpillig tells and ones. see a problem solving seminar comprised Another studentis working a page of of a teacher and five students who ha«. problems which emphasizes an tinder. individually been working on the con- standing of the distributive property. As ceptsinthe measurement unitcalled he finishes that page, he finds the teach "I low Is Sparc Measmed?" and who now errs manual. checks his answer. tecouls have been wifironted with anniqtte the assignment in his folder. looks at his ploblem for group solution. strandbooklet marked Multiplication- Division, selects another textbook f'0111 the cart and returns to his desk. He opens Open Space Schools the book and is once again «mfronted Effortsto obtain with some material on dist ributivity. but flexible space arrangements this time a dilletent approach is taken to needed to individualize instruction ledto the explain the concept. open space concept in schoolhousing design. So The aide has itist provided a progress far this concept has been used mainly in the test and answer sheet for one student. and is presently helping another locate a construction of elenientaiy schools; however, it page of answers lor an assignment in a has also been used in the construction of a num- teachers' manual. while yet another stu- ber of junior high schools and even a kw senior dent waits momentarily to have a com- high schools. pleted assignment certified. Seated at a listening center, a student who lacks cer In most schools built accoiding to this contain reading skills is listening to a unit cept. the teaching and learning areas are sepa- test previously tape-recorded by the aide. rated into huge open spaces called pods. Usually, Another student is puzzling to himself the floor area of a pod is equivalent to that of over a decision to move to the next concept in a unit. He checks his record folder from three to ten standard-sized classrooms. Var- and sees that he has done only two ious shapes have been tried for pods: rectangular, assignments related to this concept; he hexagonal, circular, caracole (spiral), square, or also notes that when he checked his work he had several errors on each assignment. a combination of these, such as the tri-pod, which He makes an independent decision to consists of three rectangular spaces extending work further on the same concept but to out from a common center area. There may be use a text at an easier reading level. one pod within a schoolor several, as in the Still another student is about to complete aunittest. After the items are Diamond Path Elementary School in Rosemount, checked by the aide, who will indicate Minnesota. which has three pods. The largest of from the annotated answer sheet the areas these covers approximately 21,000 square feet of in which he needs more work, he will discuss the results with his teacher, who floor space and is capable of accommodating 225 may then recommend additional or back- students. Theoretically the size of it pod should ground assignments; or, the teacher may be determined by the number of teachers who suggest that the student move on to another unit but that he attend the next are cooperating as a team in teaching a particular small group seminar on the critical con- grade, performance level, interest area, or sub- cept. ject. In practice, however, its size is usually deter- In one of the rooms adjoining the mined by the anticipated enrollment for one or learning center a group of 20 children is at work with a teacher on one of the two grades or subjects. computational skillsthis group is using Some open space schools are designed to be 3

INSTRUCTIONAL SPACES 47

ccunpletel) open, nen the huger ones. For CN. A second advantage of open space schools is ample. the second floor of the Niatzke Elemen- that they wake possible a bigger. better. and tay Schoolin( piess, Te Nati,covers %MOO more:tried inventor) ol instructional aids than scpiarefeet,is capable of accommodating GOO schools of other design. and at no substantial students. and has no interior walls. increase in cost, With icsoince materials located There is. howe)er. occasionally some hedging at the center of a pod. more of the expensive on the open space concept. Itis not uncommon equipment is ieacdily available for use by largeor to see accordion or folding' partitions, and in small groups. thus lessening the need for dupli- some schools movable coat racks and bookcases cation. This leaves money available to providea are used as sight. dividers. Sonic designs provide more complete and elaborate resource center. In for adding pal tit ions in the future if they should addition, teachers and students can make fre- be desired. quent and immediate use of audiovisual equip- mem as needs arise, for the machines are always Such spaces as administrative offices, teachers' available and do not have to be obtained in offices. teacheis «minions. team planning centers, advance from it separate department. The materials preparation spaces. and storage spaces easy access to whatever audiovisual materials a teacher for expensive equipment ate enclosed by per- or student %cants should be helpful in customiz- manent partition walls. ing instruction to meet individual needs. An open space pod can be organized and fur- The principal advantage offered by openspace nished in various ways. Usually, it includes an schoolsisthat they make performance level open. easily accessible resource center where stu- grouping. more feasible than in schools of other dents and teachers call obtain textbooks and design. Open space encourages flexible giouping other instructional aids as needed. Around its patterns on an unconstrained and continuing perimeter and in its interior are located teaching basis. so that the uneven growth ofa student in stations. where small and mediumsized groups various subjects can be readily accommodated at can gather. These stations are sometimes fur- any time. For instance, a student who is reading nished and carpeted in different lutes to enable at the fifthgrade level but doing mathematics at students to locate the different stations without the thirdgrade level can easily join anappro- difficulty. The furniture in open space schools priate group for each subject (or start one of his is usually movable and is arranged in islands to own). If lie should suddenly begin making prog- avoid cot rigor ellects. Trapezoidal and semicir- ress in mathematics more rapidly than the mem- cular tables used by students are arranged to bers of his present group, then he can move formvarious configurations. Chalkboards are immediately to a group more suited to his rate usually attached to the walls and movable chalk- and level of learning. board panels supplement these. Remaining wall An open space school maximizes the benefits areas may be covered with pegboard and some attainable in a small group problem solving ses- surface on which tacks and staples can be used. sion because students working on the same prob- Open space schools have a number of ad- lems are able to recognize each other's where- vantages me eggcrate schools and even over abouts and get together. Such clustering can occur schools with variedsized group spaces. First of with a minimum of commotion, and teachers are all, the) arc more inviting to newer instructional able to observe and direct students' movements strategies. Almost any desired combination of without giving all their attention to them. large group, small group, laboratory, indepen- A further benefit of the open space school is dent study, and tutorial arrangements can be thatit encourages greater interaction among accommodated without concern for constraints teachers, among students, and between students imposed by existing walls. and teachers. A teacher is impelled to walk over Nok 4,' ,...... *.- ., , ,,,,..-..-.. -^",.....-.^:...... --...... ,7.,77 ,- '..'. .".. --.Q64110111111101111iset-4:, ..;.1.4,:: - - ...` , A ...... t.,,,,,,,....%-4:23`'.::-. ,- '''-,''- -.- :*,-4 -,.54fo., - ,.. - r - , -. .,-,:,,-...... ,,,,- 7, , ...,:,..{...."1 ,j,,,,...., +.1.;`,, 'N..%P / , , , ..--.4, . '':''.i'4.'14.'.."ri

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and talk to another teacher if the two are near it may not be possible to change it without the eat It other. The easy exchange of ideas and in- consent of all members of the team. structional aids, as well as the benefits of observ- These problems are sometimes met by incor- ing the teaching of nearby colleagues, offers a porating adjunct spaces into the open space de- spontaneous brand ofin-servicetrainingfor sign or using operable walls to enclose spaces for teat hers. And the ease of coordination among spec ial work. Actual') these increase the flexibility teachersinopen space naturally encourages of the open space school, making scheduling team teaching. easier. Adjunct spaces include student confer- The open space school brings with it certain clu'e alcoves, specialised spaces for noisy labora- problems as well as benefits, the chief of these tory etplipment (for example. calculators. coin- being noise and distraction. Since overpopula- puter teletype terminals, typewriters, and power tion can increase noise beyond apermissible tools for making models and displays), space for level, an open space school cannot be used to unfinished or extended projects, a closed -oil' room hold more students than an equivalent combina- for film viewing. and separate spaces for music tion of spaces in a conventional school.Itis and gymnasium. Some of the most recently con- inevitable that the level of background noise is structed open space schools have eliminated the slightly higher than in the self-contained class- need for a separate closed-off space for film view- room because movement, which is not usually ing by making it possible to Clint the lighting in allowed in conventional buildings,is accepted each teaching station separately and equipping as a normal part of the instructional program. studehts with plug-in headsets. Another innova- Teachers and students seem to expect this addition that permits broadcast of lessons, quizzes. tional noise and adjust to it rather easily. Itis musical programs, or tapes to various teaching necessary. however. for teachers to adjust the stations. without distracting those in adjacent volumes of their voices so they do not carry over areas, is a wireless listening system. This can be into adjacent teaching stations, and high pitched achieved by encircling eachteaching station voices can cause problems. Unwanted sounds are with a wire loop strung near the ceiling of buried partially absorbed by installing carpeting over in the floor. Using a wireless headset, anyone the entice open space. Carpeting eliminates the within the loop can pick up the broadcast trans- noise that normally results from books being mitted to that station. dropped, chairs being pushed or knocked over, In Matzke Elemental.), School, the open space the scraping of feet, and the moving of furniture. school mentioned earlier, curricula and instruc- Italso encourages students to sit on the floor tion are nongraded.2 A team of teachers is as- and assemble in small groups. Such congregating signed a set of students for a nine-month period, enables the teacher to lower his voice, thus re- and the teachers arc jointly responsible for plan- ducing VO(al interference with other groups. ning and implementing each student's instruction- A related problem in open space schools is that al program. The entire design of both building of scheduling. Care must be taken to avoid sched- and program is planned to provide an inch- uling in proximity, at the same time, groups vidualiied program for each student. of students studying subjects that require con- During the school day the students within a centration, such as mathematics or creative writ- team arc redeployed to form small homogeneous ing, and groups of students engaged in singing performance groups in each of the subjects, or viewing motion pictures with amplified sound. mathematics. reading. and language arts, with And scheduling, brings with it not only the problem of avoiding distraction but also a certain 2. This desuiption of the mathematics programin lack of flexibility. Once a schedule has been Matzke Elemental y School k based on infolmation re- agreed on by the members of a teaching team. ceived nom Kay Killongh, the principal. WORK I.C4174SC WORK

Medium-sized 5.1

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ruantr. 2.23.The second floor of the Matzke Elementary School in Cypress, Texas. The area of this space is 56.000 squat e feel and will accommodate Iwo leaching teams, each composed of at least nine leachers and 20 students. The diagram displays the deployment of one slid, team for science instmetion in two medium sized groups and fo, mathematics instruction in seven small groups. A large group of 60 students is sent to the lower floor or outdoors for physical education. A learn of the same size with the same number of students can be deployed on the left side of the open simce.

huge group instruction in art. music, and physi- independent study in the open learning center. al education. The mathematics program is built The pi ogress of each student is reported to par- mound a continuum of skills defined in terms of ents, with ratings on individual mathematics behavioral objecthes with a pretest and posttest skills rather than leiter grades. for individual diagnostic information. Students ale allowed to progress at their own rates through EPILOGUE the sequential program. The) are afforded instructional time with a diagnostic teacher and The guest for ways of coordinating form and einfoi cement time in individually prescribed function continues ... 52 (-.11 \1' I'FR

ItIBLIOGItAPIIV I. \Ben. Flank IL. ed. Tir Revoininm nt S( /,,u! 16. Wi111.111111'. Shell, and thr Film thug .11athe tus:.1 Challenge fai..111» ttt ttt inotois Thou cc. Imestiganon no. 5. I Immo!). 1( x. and TeatherA (.1 Repent of Reg 1 MienPawn dilh Rowlett 5 Scott, 1963. (:o»feientes in .11athe tut). Washington. D.C.: 17.(:atidill. tIliain c!al. "What \Volk+ and Colima ofTeachers ofMathematic 1Vhat Fails in St haul Decigit." A alum'A Schools (;thoili 1967): 55-116. 2.ID Ittic k. Lament e Deugning the Ihithe. IS.(.11mosh, latIcii, ed. .11rithematit al Chrillcirges: titsCias(!m911, Washmgt(tn. D.C.: National Selected Noble:us horn the ".l letheutettrs Sin Conti( ii of 'I- cachets ,Nlathemaths. 1957. dent Ioninal.- 1Vashitigton, D.C.: National(.(nitt- 3.I3eggs. David W.. III. ed. Team Teaching,. Bold cil of Tembels ()1latlieniati(s. 1965. New Venture Sok.. 'tioomingwn and London: 19.Ctibberley. Elluood P. .4 Brief 11:%tory of Edit Indiana Puss, 1961. tat .1 lliarny of the 1'u:rho. and Piogiesc Bcg,4,. Dm nil Edward G, Rulke. cds. aml Oigainzatirm of Education. Roston. !lough. A"(mg,aded Sr hook in dram:. Bold New \'ennne ton Co.. 1922, Set Ie.+. Bloomington and London: Indiana Doi. 20. \tigustine.(31:11.1esII. Multiple.11. ',only of

tensity Ptest. 1967. Ter:thing Mothe tits i» the Elementary 5.1 1. 5. Boyer, EmilJ.. and Donos an A. Johnson, .1 New Volk: Ilaiper C Row. 1968. Crinde to the l'e and I'1,)( faellle»I of Teathing 01. nengn on pain., inn /cc. .1 1 /utelo urrinntan put. Id'. fat .11t.thematus. Washington. D.C.: Nation. name to: Paige:tipIttess. Educational Facili. a1 Conn( it of Teacher, of Mathematics. 1959, tics Laborat(!rics. 11.d. 6.Blake. Ilowald E.. and \ nu W. NlePherson. "lit- 22.DeVatilt M. Vete\mte E. Buchan:tn.:1nd Eatl ilividnali/ed1 nsii in lion-Where Are We?" Edo. I.Nelson. "Dsctipiim; of the Indisiduali/ed .\Intlientalics tanonal Tr( huolo. '.«.mber 1969): 63-66. l'toje(1(I\ICP)."2(1 ed.ItineograpIted. Nladison: l'ilivelAty of \\Is. 7.Iltogden. J. I). flevemping Communication Shills cousin School of Education. Mathematics (am toNwpCominitted Leainds. West Nyack. rictilum Research Center. l'atker Publishing Co.. 1976. 23.Doll. Ronald (:.. ed. Indipiduii/hing lint: net S.Blown. Bartley Frank. lidmaii ttttby dpf \Vashington mow Xele .1pfoombs to Independent Study. \ssociation for Supe!vision and (:!nriculutti 1)eveloinent. National Ethic a West Nyack. NY.: Parker Publishing Co.. 1968. tion .\ssoattion. 1961. 9. The Nongtoded High Se! I.Lusk. 2.1. Ellsworth. Ralph F.. and llobait \Vagener. wood Clills. N.J.: Plenticlfall, 1963. The &lug I.:1,:ary: Facilities for 1i:dependent 10.Brow( her, S. ni,(/(nv of the Thob/ems of Study to the Soo:filmy ti,hool New Volk Edu /;dual . 2d ed. New Volk: McCraw.1 fill Book citional Facilities 1,:thommic, 19r3. Co.. 1966 25.Elaine.I. Sutherland. and John W. N1cLood. II.Butner, Jetome S. The Pau ess of Education. Ilitildings and Facilities for the Altathntalwal New Vtn Random House. 1960. Sara

30.1 [canon d. ic load. l'estabz:.i. Ilya 11 ght and 11. Niko hell. Donald "I lousing (:04)peiatie(.141o 11, Belem iiii e Today. 1.4)11(1011:Nit:ohm:1i Co.. ing Poignant..." Ad/ 1 Elementnit S41 I Pim. 1967. lipid -14 (January 1965o:11-52.

31.IIuIg.Ud, Elliot R.. «I. Theon) 15. Nlit/el.11:114)141 E. Flue Impending Instouction lint! net . Sixfl-third Yearbook of the National Itelohnioni." Phi Deltau/,/,a,1 51 (April 197(1): Smiefl for the Souk 1)1 Education. In.1. faica .131-10. go: l'itheisifl 4)1 Chicago Ness. 1961. 6.NIm.pliy. J11(11111. .1144h/1c Schools. New York: liolto. 32.!midi( ato. BI.odour. ".\11 Appraisal on Teach. (ational Facilities Labolatruies. 1965. iIItNlailliuAlooloilar Scheduling." .11iiiiiroda 7. NImpliy. Judith. andI:obi:itStutter.S t I I C 1 of Toulon of .1111 theinatits .Veirsletti Scheduling by C puler: The Stoiy of G.-1S1'. 23 (Nlaich.\prii 1970). New Yolk: Educational Facilities Labor:tunics. 33.Individualized : The Frio ,iii 1961. School: Ednia fin the 0% Minneapolis: 511n- IS.National Conin i ission on Resources for Youth. m:sot:I S( !tool Facilities Council. 1970. For the Tutor: Tutor's Crab Bag. supervisor's iiiii nrallmic 144)7" :10 laench, 'Washington. manual. New Yolk: Govern iii cm. 1970. 1).C.: Depaitment of Classroom Teachels. Na. 19. You're the Tutor. supervisor's manual. tional Education .ssociation. 19(16. New Yolk: U.S. Colernment. 1970. 35. 1:411111. Slieolv4)(441I). 11w 1:41./v /.4144 g Center. 1) Youth naming Youth, supervisor's New Yolk:Edt:tationialFacilities I.:dun:dories. manual. New folk: U.S. Coveinment, PJ68. 1970. 51. Th Xmignided School. Reprint of articles in 36. Thte High tich00/clleaicited:rlu- the Nonember 1967 and January 1968 issues of dlr1rc..114I'hei. . / X4n444. New Yook: Eduta- ational Elementary School Principal. Washing- clonal Facilities Laboratories. 19(17. ton.I).C.:1)C11:11 Intent of Elementary School 37.King, Edwaid Salient Dates In Aim:ic-an Edit- Plincipals. National Education Association. 19(18. ration: 105-106/. New Yook: Harper 3 Row. 52.Nuffield Foundation. Beginnings. Nuffield Alathe- 1966. matics P1 Iject. New Yolk: John Wiley Sons. 38.Lionhall. C.NI.. and John 0.Bolvin. "Pro. 1963. grained Instruction in the Schools: : \n Applica- 53. I1)o. and 1Understand. Nuffield tion ofPlograming PrinciplesillIndividually Nlaiheinatics Pnije4 t. New York: John Wiley fi Presciibe(1Instioiction." Piog .41 Instinction. Sons. 1967. Sixty-Nixth l'embook of the Nation- ,I Society for 5.1. 3144thematics Begins. Nuffield lathe- Stud) of Education. pt. 2. edited by Phil C. inatics Pooject. New York: John 1iley Sons. Lange. Chi( ago:Univeisilv of Chicago Press. 1968. 1967. Pp. 217-5.1. 55. Open Schools and Their Use: Thoughts on 39.Lobensiine. Tutining 'flub land Tips. School and (:1411ge (Ity. New York: National Commission on Itesoonces Ind.: Shaver 4'z Co.. 1969. for Youth. 1970. 56. "Open SpaceElementarySchools. Mimeo- 40.Loeffler. NI:lig:net Howard. The l'refuiredI n- graphed. Stanford: Stanton! University School of IntoiiiiiiL Oklahoma City:NIargaret Howard Education School Planning Laboratory. 1968. Loeffler and Tiusices of Casady School. tinder a 57. Open Spare Schools: Report of the C - grant form Educational Facilities Laboratolies. mission on Open Spare Schools. Washington. 1968. 1).C.: American Association of School Admin- 1. Lnskey. John Callum. "A Study of Flexible istrators. National Education Association.1971. Stheduling uith Surveyed opinions of NIathe- 58.l'aienGhild Educational Centeis: ;1 Facility for maii(seaching. Nlaster's thesis,Graduate RoilyChildhood Miication.1ges Infancy to St pool of the Univelsity of Nlinnesora. 1970. Seven Years. Project sponsored by Educational 42. Manlove. 1)cniald (:., and David W. Beggs. III. FacilitiesLaboratories. Tempe: Ari/ona State Flexible Scheduling (1sing the lndillex Model. I(niversity College of Education. 1970. Bold New Yenttne Series. Blocnningum aml Lon. 59.Parkhurst. Ilelen. Education in the pallor! Plan. don: Indiana Univetsity Piess. 1965. New York: E. P. 1)ution. 1922. 13. Mathematics Progium Grades 7-12, 1971)- 60.Pet knian. Eugene F. "NlailocinaticsScieme I.abo. 71. Publication no. 37.1. St. Paul. Minnesota: St. rafory." W'exford. Pa.: The Au. Paul Public Schools. 1971. thor. 54 (a1-\I i I.II vco

Iii.Petty(' . (..tvitor./ndividnah:in.;/ad, 4 71."tet..tt(.. kitt. et al. Limb lair I...111 thounghMutt stlfts.l. lembiel't u4t futt ming, Net% i .1011.111.41 tit 1 ie%.CI 'WHOM*, 01611:(:.iuucllail NIt.(rat%11ill lInnk Cm.. 1915. .1111111A [milk% PianneiN. 1969. Proohbfo;it Ss ill ant Sc!I: (.hultwille,Iluttt lamiemnairfon us the Sthuull ;lump:I 11Luststt high Schoul. Chttl:%tstlie. Teat Ietc11. Net. 111:1,. Eslusatism.d Fattlkie% 1..1hn ste%%er. nit vt.it%ail1'emit:m(7c. Similludsicil Lunde'. 1969. Itcginnal (.enter lair EtItuatinnal aide. Lab. %:;. hum!). J.I lind. /h arm-.iN1%1 S1111114lila/111111g 1.ahuratot. ti.d. IrrnH.:,h 111the\If Ifni Srhtlf II,CIlllll Ntattlott. ltnbtt G.. and JnIttt O. 1Takin. Imlf 1111 the 1!.!.1/el I llll '111:11 Studt 111 the t "tilii.uinu ail 1111:SI:111IIIthe Sel lllll 1.11%1(1111111..1111 hi 1'11,1 I ihrillnchurf . Phil:001)1113: ItC,.111 I,lair I:ctivr St hnuk. ,,.al. Ike.N.161111.11 \ lall 111 SeD111111.1: hind Nati llll lidur ninn \n( 1.1111111. 19%9. hi. The St1Ihtuswe ut the GUI. l'ittu:IReims! 71. "hulepeudent Stuck tennis:l'Iten- IitItuta 1 I artlitir1.0 Nv Relatinu in the Centi-al gullettn ui the Via k: radlitie. Lalmvatut h.,. 19611. Val IIitutat lllllr1l S,r hni urr/ hin- IsMO/D ;e1///1/11./ Votk: lithatatitusal t/pal% ;ill4 1; 191;1.9; 15-51. Fat Unit:. 1.altotatuticN. 19(i$. 7:0. -.Net lllll fat% Edtuatintt Ttnitoitte.c: rutti Seat(h. ltestim XV. "Individualte.ulting: Impel:likes fur Inaintnentent. /:////ent,ct( the PtilIn Eflutatimuti Review(rebruall Nat ;1 ,,IAos /fawn uI Sertnulat St/u1.11nin 1S9 I I:1:1-71I. Irina% 3II 1\pia

'id/Et-dun/inn sof .11ellufusstIFIttristalIltzh Wt. 1" p. J.I lind. and Dnisey 11a1 Gurgle S's hunk 1:mult sttia 1i:els:lila fur SwIthcril hr Beurt Silumb: but us tut Chanty. Chi..agf mniminit fir Cullege and Sthttl... 19711. Rand NhN.Illy t Cc1,. 1961. 77. 1' it.I. 1.Intd. and 1.11i. 1. Karasik. Ftstan tut 1;s.shldint math 711tielleA. N1u,kegull. Midi.: Slid. the huh:ish& I I ruslsatipIeIstsushtlitr. dim I/pill In CA... 19611 Natilmal . Sec 10. Nkiitlivi. 1:. -TBathing ,Nlachities."tient-f 1$ midair Sc I': int ipals. Nat innal Edut mint, *v.. (1958): 969-77. sum iatiurl,11116. 7i1.Stanthusl% fur tirhoul Mediu l'rug ;;;;; (:Ititau 7$. "hump. I. 1.1tod..nul Deltuas 1:. Ae, lllll 1 am! aNItingtini. .111critati I.ihrary WO. HD V S,I 1 Cm lit !dun, Ists.ps upessient: lhrlrunulc i;ninn and Sall1/11:11 1ithltati1111\.%\l/l 1:16111.19119. surd Proferslmf.%. 1;111111M: it 1:7 Ilatnn. 191i$. 3.THE TEXTBOOK AS AN INSTRUCTIONAL AID TIIE TEXTI;0()K achieves. on rare occasions. both function and beauty. I'lle first English tianslation of Euclid includes such a volume. Volume 2 of this translation «iniains magnificent woodcuts depicting solid geometricfigures. This volume also contains a collection of pop.tips that rivals the charts of iii;o1; contemporary ptib Ushers. Certain figures arc made of paper and so pasted in the book that they ma; be opened up to make actual three.diniensional models as shown in the illustration. Few textbooks combine beauty. novelt. and information as well as this volume of the first English translation of Euclid. Itis a model to be emulated.

VoiaNtr. 2 of The Element% of Gime/Fie of the mos: anncienl philosopher Eurlid Megnro 1(1'111111111y (now Ins° Irunslulcd hap Mr English lining. by H. Billingsly. London. 1570. One Wp is currently in the American Uniyelsity Library. 1Vashing. ton. D.C.

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4.C!'--t"':+nr 57

CHAPTER 3

THE TEXTBOOK AS AN INSTRUCTIONAL AID

by HENRY H. WALBESSER University of Maryland, College Park, Maryland

Not the violent conflict between parts of the truth, but the quiet suppression of half of it, is the formidable evil; there is always hope when people are forced to listen to both sides. It is when they attend only one thaterrors harden into prejudices, and truth itself ceases to have the effect of truth by being exaggerated into falsehood. JOHN STUART MILL

The textbook has the capability to presentor suppress; to help develop objective or bigoted individuals; to maintain a free society or to hasten its demise. This chapter addresses itself to the problem of textbook selection. 59

3.T11 TEXTBOOK ASAN INSTRUCTIONAL AID

PROLOGUE -That is. the meaning- is clear until one is called upon for an explicit description. "El:et yschool- is consideted part of a coutempo ray. forward-looking (butnottakentoex. nettles) school system locatedilla prosperous A HISTORICAL, PERSPECTIVE suburb of a not- too -large menopolit.di \Vital are the potential sources of evidence that The exterior of each school plant display., that might yield information helpful to the inter- usual blend of school atchitectutelarge pretation of the :extbook as an instructional %vindos periodically punctuated by clean. new !id% One means of acquiring information about brick walls enclosing- a most unimaginative set the textbook is to examine its historical devel- of tectatigular shaped classrooms. opment. As one walks. pauses. listens, and looks within this schoolwhat of the nutthematics instruc- Early Forms and First Purposes tion he observes? The exhilarating sounds of students learning mingle with .exhibitionsof One 01 the fits( textbooks. it not the first text- [cachetstelling. The most singularly sulking book in a modern sense. was the elementarl element of commonness, howevel. is the deploy - instructional plogram specified by the Council ment of textbooks in each of the classes and oflain/ in A.o. $13 (5). The Council's decree the obvions instructional dependence upon the requited that children be taught the Fidem textbook. In short. this is a school not so ver Calholican el Oration cm Dominican. This was much unlike every school. the first requited program of elementary instruction in recorded Western Etnopean history.It isinteresting thatthis early "textbook" de- ibed the extent of the training for the learner Why does the textbook hold the instructional as well as the content of instruction. Viewed in system in such .1 powerful grasp? Mast the text- this training context, thisfirst textbook came book be such a dominant force in shaping the remarkably closeto constituting 0 behaviond mathematic's curriculum? Are dim alternative descriptionof the tasksthelearner was to models for the instructional systemin which accomplisha most modern view. the function of a textbook is :dieted? Through the next few centuries,the text- Just what ate textbooks? What a ridiculous books consisted mostly of books devoted to basic question! Everyone knows what textbooks are litemcy training.Forthisreasonthesetexts Or do we? If a learner reads a book in 0 class. ate often ieferred to as the ABC books. In this room. isthe book a textbook?If 0 bookis developmental period.the expressed purposes chosen from a lanai y .shell with the purpose of and actual uses of the ABC books continued to browsing thtough it. is such a book 0 textbook? retain a direct relationship to learner training. If study material is selected from a newspaper, The fundamental literacy textbooks were soon is the newspaper a textbook? If a learner uses sequenced with readings of a more religious a self-111StniCti011ai program from a computer- character. such as the Credo. the Pater Noster, assistedinstructional system.isthe computer and somewhat laterthe Ave Maria. By the program a textbook? Yet. with this acknowledged fourteenth comm. thellenedictusandthe ambiguity, most individuals have some idea of Gratias had been added to this literacy-religious what textbooks are Jimd how they are used. training sequence (6). 60 CI \P 111REE

Among thefirsttype-printed books was an elementary textbook of the Roman Catholic. Church1). This %.olitme contained the alphabet, the Credo, the Pater Noster. the Ave Maria, and a lew other prayers. The earlyReformation period found the elementary textbooks mostly manuals of church services. There was, however. an occasional insert of what might be termed secular material principallyinthe form ofclassicalsayings. Hence, the textbooks had slowly undergone a transfoimation over these five centuries until they had now become descriptions of information to be mastered.

Appearance Becomes a Factor The early sixteenth century !nought a significant contextual shift in the textbook. Marens Schulte published a basic literacy text in 1532 which presented two radical changes in format (20). Filst, the ABCs were organized in vertical arrangement on the page. Second, there appeared a print and a line of leading adjacent to each letter of the alphabet. The prints il- As runs the Glass lustrated an object for which the letter could stand. The linof reading was also related to FIGURE 3.1 Man's life doth pass. the particular letter of the alphabet. Several

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ey,9ey,v e4.9eNv., e4,9 eNyiv THE TENTI1001: AS . \N INSTRUCTIONAL. \I1) 61

variations of this basic format were soon forth- clowninthemost plain and easy coming. Perhaps the most significant was the :Manner. presentation of rhymed couplets for each letter, 111. Decimals,inwhich,amongother Things. are considered the Extraction as shown in Figure 3.1, rather than simple lines of Roots;Interest, both Simple and of reading. The advent of this change signals Compound;Annuities;Rebate,and an awareness of the relationship between the Equation of Payments. textbook format and the learner's use of it. IV. A large Collection of Questions with their Answers, serving to exercise the The first popular arithmetic in English ap- foregoing Rules; together with a few peared abOut tenyears after Schulte'stext others, both pleasant and diverting. Recorde's arithmetic. The Grounde of Aries V. Duodecimals. commonly called Cross but itis a collection of rules to be mastered. Multiplication; whefein that Sort of Arithmetic is thoroughly considered, and The mathematics texts, unlike the literacy voi- rendered very plain and easy; together times, did not reflect au awareness of the rela- with the Nlethod of proving all the fore- tionship between the format of the textbook going Operations at once by Division of several Denominations, without reduc- and the learner's use of it. The organization of ing them to the lowest Term mentioned. the arithmetic text was a "rules to be mastered" conceptualization. One of the earliest editions of elementary FIGURE 3.2 mathematicsactuallypublishedincolonial Chap. V. Di cnYi SS America was Arithmetic, or That Necessary Art Madr Most Easy by James Hodder (18). This Ilhall not,(hope) need to trouble my / /f, or Learnv; foiheco Isforkini of /his Sum, or d- volume first appealed inthe New World in ny othel haying' nold6v7Juppee)fiffictently 1719.whereitwas printedby J.Franklin. &caul of Divilioni &twill /cove et &Me:en- brother of Benjamin Franklin. Hodder's text- fure of Me. experienced fojudgyvhetliers book contained an interesting and often chal- Manner of divicime: be not plain, lined, &to he lenging collection of arithmetic algorithms. One Govehl eah fewer fipres Mon any which Is example of such an algorithmis the "plain, commonly icnght:As for Exomple qiyeatelh. lineal, and wimight with few figures" division (8 a lgori thin (Figure 3.2). 07 (5 The arithmetic written by Thomas Dilworth 98153 (0 081 no (3 enjoyed a great popularity in the colonies among. 9810418X (0 alloftheearlyforeignimports. Thefirst 98/.682000 (8 American edition of the Schoolmaster's Assistant 987054909'881 (6 (12)was published in 1773. The contents of 493827'1'008M (4 246914578637654,5 (2 this volume are quite repiesentative of the l'254567895876,5A3,2X (124999999 material treated by most of the contemporary 9876543211111 TX11 987654321 works. This Compendium of Arithmetic, Both 987654322=2,ZZ 124999999 Practical and Theoretical included five parts, 010503'338 M 2499999982 58fo4aggA 3749999974 which (in 1)ilworth's words) discussed: 58165550 4999999966 1. Arithmetic in Whole Numbers, wherein 0816756 5249999958 all the common Rules, having each of 5810 7499999940 them a sufficient Number of Questions, 588 9749999933 with their Answers, are methodically 9 9999999920 and briefly handled. 11249999915 II. VulgarFractions, wherein several 8 Things, not commonly met with, are Proof 123456789987654321 theredistinctlytreatedof, and laid E 2 CHAP. 62 (..11,\PTH: THREE

John Ward's Young Mathematician's C 'tidy significancealsobecausethe image oftotal (II)was the olttme in common use at Vale master' oftextbook items was now tempered and I lan.trd as a mathematics text used ,tt these bythe introduction of aids to the 'camel: to universitieslor much ofthe eighteenth cen be sure. the shillIton1111\11 II( 6011 b Mind(

tur). The edition commonly used at the univer- t.1.5 NI11,111. NeCr(IldeS4 11161 .11TC:11.111( Cdid sity levelincludedsectionsonarithmetic, forecast the changes that wer about to occur in algebra.the elements ofEuclidean geometry. the textbook. conic sections. the arithmetic of infinities (re- Among the most popular texts published in peating decimals). and logarithms. the New World and used by the eastern sea Literacy volumes following the letter-rhyme board colonies was the New England Primer

format (letter accompanied by illustration and (21). The IllNI edition of the Primer appeared 'limed couplet) were broughttothe New toward the end of the seventeenth century. in Wodd bythePuritans. The appearanceof 1673. The hornbook and the sample' were con- illusnations ,and a more appealing isttal format sidered by some educational histot jam not to be is a significant development beyond the mete "real" textbooks because cif their physical format. physicalchange.'Theirappearan«assume. Nevertheless, the 'minima and the sampler did enjoy videspread popularity atthis time and were competition for the New England Primer. FIGuRE3.3 The hornbook was shaped like a paddle. The example depicted in Eigtne 3.3 is typical of the hornbook's appearalne. The book's construction was usually of wood Aabcdefghijklmnop qrstuvwxyz & aeiou ofheay cardboard. On the paddle was pasted ABCDEFCHIJKLIINOPQR apiece of paper containing thetextto be STUVWXYZ aciou aeiou mastered. The text was then covered b a thin. ab eb ib obubba be bi bo bu transparent ,lice of horn attached to the wooden acec Ic oc uc ea ce ci co cu ad ed id od ud dade dido du base. The student would practice his imitations In the Name of the Father and of the Son, and of the Holy Ghost. Amen. of this lesson until he had mastered it. Some UR Father, which art in Heaven, hornbooks included a simple counting- frame. hallowed be thy Name; 744Z-.'t thy usually attached to the back of a paddle as in 0Kingdom come, thy Will be done Figure 3.4. on Earth, as it is in Heaven. Give Vottlig girls !,tudying at home or in a dame icus this .Day. our daily Bread; A and schoolatypeoffinishingschoolenrolling forgive us our Trespasses, as wefro forgive them that trespass against us: young girls at this timewould construct sam- j1111 And lead us not into Temptation, but plers on which they embroidered texts. Among deliver us from Evil. Ann. the materials embroidered on the sample's were rhymed couplets. verse of one form or another. the alphabet. and usually a few names lot natural numbers. The learning emphasis one can infer from both the hornbook and the sampler one of learning viewed as mastery of materialsand mastery accomplished by repeated duplication of a given pattern. All inall,itwas not, a particularlyexcitingemit oilmen( for the learner. HIE TENTR001: AS . \N I RUC rioN \I.\11) 63

tho' perhaps the ;mist. or Ivo' kman, has but lit t le (nay, scat c e any 1 knowledge of geometry. Then as to the ackantages that arise from both these noble sciences, when duly joined together. to assist each other. and then applied to practice. (ac«nding as occasion requires) they will 'caddy be granted by all who consider the vast adNantages that accrue to mankind how the business of pavigation only. As also from that of surveyhigiu(livid. ing lands betwixt parties and party. Besides the great pleasure and use there is from time-keepers, as dials, watches, clocks, and so on. All of these and a great many more usefularts.(too many to be enumerated here) wholly depend upon the:doles:lid sciences. The firstarithmetic textbookwritten and published by a North American was Arithmetic-h. Taiga,- and Decimal, by Isaac Greenwood (17). This volume.ivhich appeared in1729.provided our interestingorganizationalformat which helped to distinguish it hom the popular foreigntextssuchasDilworth'sor Fishers Greenood's met hod wasto givetherule, inesent examples illustrating the rule, provide Early Mathematics Textbooks exampleslot-the student, and linally suggest itformal pool. Each student example had a in the United States blank space bewail) it for the student's practice The description ofthe mathematics instruc- ol the rule. In some ways this text foieshadows tional materials of the colonial and early na- the self-instructionalformatofthiscentury. tional periodsislittle (Mkt ent from that of The technique ol rule-example-practiceisstill the imports to the eastern seaboard colonies, ivithus. the hugest proportion of these being British. The contentof Greenwood's volume was Among the most popular of these arithmetic similar to that covered b'most of the arith- works were the editions of James Hodder. Ed- metic textbooks olthis period. The material wad Cocker, John Ward, George Fisher. and included work on base-ten numeration of natu- Thomas Dilworth. ral numbers, as well as the naming of positive Wad's introductory statements in his volume rational numbers by fractional numerals and (II)areof someinterestto contemporary by decimal numerals; and it presented algorithms mathematics education. considering his as owed dealing with the operations ol addition, sub- purposes of mathematics instruction: traction, multiplication, and division firstper- As to the usefulness of arithmetic, itis formed on elements from the system ol natural well known that no business, commerce. numbers and later on elements from the system trade, or employment whatsoever, even from the merchant to the shopkeeper and so on of positive rationalnumbers. The monetary can be managed and carried on without the system ol pounds, shillings, and pence also con- assistance of numbers. sumed a fair measure of Greenwood's presenta- As to the usefulness of geometry, it is as certain. that no curious art, or mechanical tion,with many of the problems setinthe work, can either be invented, unproved. or contexts of trade, partnerships. or the transfer performed withoutits assisting principles: of merchandise. 64 (,I I \ P 1-El 1111:1,1.

Asequencingofarithmeticfollowedby Materialpresented Math enla is:15( Olk% ions algebrawasestablishedquiteearlyin the of rules to be mastered. One mule per page was colonies. In fact. the first treatment of algebra about the average for Pike's volume. and this to appear iu the eastern seaboard colonies was m !wine appeals to ha\ e been quite thmoughl) combined Withallarilhillelk riihnteth aof 'name' ed b) the succeeding authors of mathemat- CjfH-KonAl by Pieter Venema (38). Ehe Nex\ i(stextbooks during thenineteenthcentury wood edition of this flume first esenied and most of the earl) twentieth Lemur). More in1730 and\vas printedinDutch. Venetia unfortunate than this organi/ation of mathe firmlyestablishedalinkbetweenarithmetic statics "by the inks- was the adherence to the and algebra I)) commenting that he had under- master) of these rules as the chat actet istic goal taken to make a "clear and distinct eshering of mathematics instruction.(This form ofin. book" and tothis tleatment had "added the snuctional torttne is. of course. no longer prac- elements of algebra." Algebra was now instituted ticed.) The description of the rules was often as a part of the mathematics curriculum. The so ambiguous or so involved as to be be)ond purpose of algebra was to be the key to un- any reasonable assumption concerning the ca- locking allthe obscure propositions of arith- pabilities of the leamets involved. One supposes metic. thesecomplicatedformatsandambiguous The first sepatate algebra textbook printed descriptions were notintentionally«infusing. inEnglishinthe United States came some although their lack of clarity and the length eight) -four years laterit was not until181.1, of sonic of the rules strains the credibility of in the publication of An Inlroduclion to Algebra such a supposition. by Jeremiah Day (10). that there was a New Most learners who were exposedto "Old \Vol Id publication of algebra in English, mitten Pike" did notacquit e computationalfacility by an American. beyond that of being able to pei foil]] the op. era Lions of addition. subtract ion. mid t iplat ion. New World Editions and division on natural numbers. Some (nn- Towardthe endofthe eighteenthcentury pctenc) was usually acquit ed with the identifi- elementary mathematics volumes by New World cation and naming. ff a few positive rational inhabitantsbegantoreplacetheimported numbers expressedi:;the form of fractional mathematics textbooks. Among these early edi- numerals(commonly calledvulgarfractions tions which gained widesinead acceptance was inPik('s time). A student was consideredre- A New and Complele Syslem of AYilluneth by markable if he managed to interpret and dem- Nicholas Pike(26). The firstedition of this onstrate the ability to use the "Rule of Three." material was published in1785. whichinPike's version went something like "Old Pike."asitaffectionately came to be this"The Ride of Three teaches, by having known.issignificantinthe development of three numbers given.tofindafourth,that mathematics textbooks in the United States for shall have the same proportion to the third, as several reasons.First, the breadth of material the second to the first." gatheted together intoa single program and described in Pike's arithmetic served as a model Mastery: the Diet of the Day to be emulated by most of the textbooks for : \s best they can be te(onstructed. the desclip. the next one hundred years in the United States. Lionsofactivities,theinstructionstothe Second,Pike'seffort servedtoaffectmathe- students, and the exercises to be practiced by matics instruction for an even more extended the students contained within the volumes of period of time in our national history than did Pike and his contempoiaties strongly suggest the model of topics to be included. The textual that the instructional emphasis was placed upon 1.111ITN 11;001. \ INS FIWCTIONM. .\ ID 65 the mastet% 01tules. The actual instructional and Us. outs. 'I heir mode is stilted to the 1)10(c:dine often assumed the foam of presenting genius ol the gmernment lotII semis 10 be the learner with a statement ol the rule to be the polic) 01 1)ants to keep 'Itchaccounts in a, hit hate and pet plcxing a 11W1110(1as masteicd. Iollciwed by one or mote illttsmitive possible: that the smaller number ol their examples. and finalhelentless repetitionof subjects may be able to estimate theirnor- the piomItite b,the learner until he could mous impositions and exactions.litilRe publican moue) ought to be simple and successful!) denionsuate H11111101' of the rule. adapted to the meanest capacity. Long«imptuationalproblemswere also chata(letisti«if the earl% and mid-nineteenth- !toot's emphasis on the decimalmonetary (entur) arithmetic, ofratite origin. Often a s)stent tem:lined the exception to the monetary long series or oompotations Ivoold he carried presentationsdevelopedinmostelementary out by the (ithe (lass ol students with ft equent mathematics %oltiiiies lor a number ofyears alter checks on one another's work. This mass chill the beginning ofthenineteenth century. A did offer one altet native to the inch, ideal ch ill traditional presentation persisted inmost %01- lira( (ice and for that reason (mild be considered miles. withthe emphasis being given tothe of some instructional met .\ II in all. however. ti eminent of imbelioably complicated business the hist' tional pi cxedin es w( le de\oted. with transa( lions imohing English coins. It ish coins. almost monotonous cm husk cites,. to the master) pistoles. doubloons. and moidotes. as well .is of given rules. manner of possible (if ()hen improbable) coin. Congress(listingitislieditselfin 1786b) binations. Figure 3.5 showsa problem of little establishing alederal munch') system based currency complication but suggestive of the busi- onthe decimal ssteinof numeration. This ness transaction problem. decisionhadlittleimmediate effectonthe DaboitsSchoohna.sier'.$ /1.,.isiani (9). by presentation 00 discussion of the monetary sys- Nathan Daboll. lollowed inthe tradition of tem in the elementary mathematics textbooks. Pike's and Root's volumes and provedto be 1 lownet. this decision ploved to have asig- anion;.o the most popular elementar) mathematics nifi(ant impact on the content of the nineteenth- volumes of the first half of the nineteenthcen- (Intuit') elememar) mathematics textbooks. tury. The extensive use 01the Schoolmaster's .\mong thefirstevidences thatthiscongres- 1.$si.$tant can be attributedto a number of sional deds;on had an impact on the pesenta- separate but significant factors. The limitation lion and discussion of the monetary system in of material to a collection of items whichwas the elemental) mathematics textbooks was the 'note realistit.alh and pro( Mall) suited to Ilk-- app :!aatu c of Errastus Root's volume Au In- actual school circumstances was one. The smaller troduction to Arithmetic (3,1)in 1796. sizeofthe volume neededtopresent such Au lion to Arilloncti( was among the materialundoubte(I!)was among the more firstelemental y ,plumes to give a prominent prominent of these factors.:kiloliter factor of positiontothe decimal monetat )systemto- some impoctatue was the volume's emphasis on gether with a strong nationalisticplea for its the new decimal currency. The value of this adoption and use. This particular plea can be particular factor was reduced somewhat by the assessed in remarks such as the following. taken occasional intrusions of problems that reverted oni the preface. to mentioning pounds and shillings. It is expected that before many years shall Itisworth noting- that most mathematics elapse. this method of reckoning will be- volumes of this period included problems re- come general throughout the United States. lated to gambling situations as well as some Let us, 1 bel.); of you. fellow citizens, no longer meanly follow the British intricate mode that had something to do with the manufacture of reckoning. f.et them have their own way and distribution ol alcoholic beverages. 66 (.11 11'1 ER 1111:14

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betweenpi oblemsforthelearnertoinsert Deviation From the Rules solutions. The turn of the century brought with it several One of the most significant breakthroughs in innovations of note. The first of these innma. the dot:1(4)111cm of .inetcenthcentur) versions tions was Daniel Adams's text The Shola's Of elementary mathematics textbooksinthe Arithmeth(I). Adams auemind to ploduce United Stales own red in 1821 with the publica- a collection of clear and simple presentations tion of Fast Lessons iu Arithmeth (8) by \\' :u'- within practical mientations. In addition to the ten and his Sequel to the Ehst Lessons simplificationandclarificationattemptedby (7). which appeared ayear later.Unlike the Adams, he included aworkbook-like feature preceding el emen Ea Ty tuathematicr.volumes. within his plesemation 1),leaving blank spaces which were in the tradition of rule, presentation. 1 11 10011.1,N1.1 SN1 NO11.1,:)1?1.1 \ 1 (11\ Z.9

por !ruin alp (I11,1 !!)ti! )0!))1.1(1 ((1 r ;;n()%';11% tisit! !mop S)H:Mal,: 10 .)II) .,11,10(1111.111[0 -tiopp.)(1.)t 1191010.)p 11(411. p) )iii W15('111 10 111111L)1111411 s 110111110101 .110111 111'111 r 1).)11M1111

10111.011 01111 11111(11(r) 01 'S',11' (id 1% I 1)111' .1111 1)01(11110 0.1(1 01 )11(% j///b,),, pa111050id .)!)!A it.)!5i1(1 5.)010!ta(IN.) 09001m .).()(1:0(1 ..1.%% trithatit01.) wows 01.01 t.)11 111,%% m.-)11 01 01 09)(1\0 r Ira! toil cyst.' art sIM '111.101.1)(1 091)0(!. no toolpriii )HI 5 Hum!! .00p

3c311.1. 515111 .0t.)m 1)011131 01 0111 31(1iss0(1 ;10!1() satolii()% atam r 11.(101) owl mo' 0.11 itonipro 10 I: T.)11111)10 .1111 1111111.1rd paidnimir 1.01p-((ropritt.)tpritt .010 ippi)% 1)1'11 1'%%s ))1:1(1 .0111 .1011.11.01 111 itoprill!s nli !el tivoi saptilon) n .0%11s3 I0 r pool! tol I1 no!ssassi: r rim tar(1)/9 JO 1'1!!),:tiro

011,1. 11:/pirnio no! jo 5.0.10(11();) 1.10,» 9).)1133 .3011110% n 15010 301 pp: 'rip alp 111 -11!

0m0ii11ii! Jo otp 11 11:121,101)oil tiop)So)(1 11:1109)0 3)11)iii a(0A w)ls titt!itt()) ptit. 1)0910(150 (1 .!//011.r.0,1 s.itto(1100 Los(10 9101p115 3011 11tH tooti aim ilopripoitt0lit 01 latimal 1.»to» )ori j() 3111 .10(11111111 1))100)10) ilop)o11911 9.)% ()it 111 stiraitt 11i.slaipm o111 :110111(101)%v 10 .noptitio0 sr 1100% alp 1)010101)1. 50.)501 10 ail! 0)10.09(1(1r JO 0(0'011 OM1 tir in/ppm!! u) 111011101010%.)1) 10)110) ,a1) 50111110% 5301) ittas0.1(13,1 00(10 9101111.0 103(1(10 awl!) .(.11:.10(Iiii0)t0)» lott -Na stilt pp:15 1191! 1131()!:)11 011.1..9srti(Itito !() 111110110 II! ol(Ititt :) so.uas(p) otp 0)1:101(1 JO 5111 as:up s!!!!/0111111 0111 .111!tlit!n.-)(1 ti! r 15,111 01111110A 1100K SP, sr 1)1111) SUOMI 0) 0I0I(III .1.)tioli!)as Jo 91.91.1(1(1E01 Jo alp 091103 .1511 spi asilas 011111'0 Iirillip0(» 1(I sill:lc:1 01 5I11 dpis 0111 '011!1(11)91) Jatt.toi pm: 40 r .0110 36 (101.11'S JO :s1).)!(10 lo alp ism] 10(1001 aport Io m41)11,1191! 'Hp% 9 nopitiptio) (Ill 01 sap all si - ippim sxmor:!!) 0111 11ope10.1 JO 101111 alp loos0.101 'cia(i 01 sox' sr 13)!Ipilit! 1100111 ipiti 0109:011: pot: 1110,1J '1111 S111.10.1 3111 i:01)! JO moot phuNif to sqoogixdi Inn: 0111, 'motion 50110%!1.)1:4111: p) -iptroltinp por )ug! I 11:)15.(:1(1 11opt:5.105(10) p1111 0111)1:1 -N.) .(.1111110)tploolottpt 9p)0(110011.)s 11;0% alp

0)11.)!1.101 0111 soloo(.ri 10 111(110:) 11'0(1111' JO 3111 11:!l0it1 poplin)) lippp, '510011 111(110:) 11 )111)01101 1)3 1111111'11 's13(11111111 1011 asoip soilitip)% 9..)% tooplas ,)%a r -r.tapIsty» (1 s111:1111;))1:1 JO calm 01 0(1 '1)0.1..)(ittiattia.t lug Hop 10 mil sioqiiir Jo>. ppm) alp cqatiap)rAllr %(1 r JO JO sapas coops:oil) 1o11:10.1 01 :op sppti.) now:nos:m(1 0(1 pairoli Sr ntio.tis itt!0:1 Jo 1. 5111 alp 11.)1su1(l -1001titiol!00 all pails!1011110) ((i 5.1;)1191(111(1 main()) 101 iir1t10mala lions': coops:nil) tptis mon ,(tiriti s(luint') siprio slooqiNal 111 mil noptip 09:11 lio 110 Alio( itinji ;you!! Aoll (twin 110 0111 imr tr.; ilittooloop 'F)pititio) Ali()% ma! .111:10 1110(1 i.totpon(» moll mu. aottr.tra(1(11: 10 1:)p0pai..1 s.ttos.tattiN 9010:11 :mug 1. ut urr .:110, mud `1 bind dzo bow r 11.111(11(0 paptillo! Ja(pittitt JO aidut!s 10;11 51.11:111 e athtuti.) it! 0t«0) 11i0 A0 .1) 0111 luinnirimIti10) sitioploid 01 xi 01101) -,(11rinatit sc0ti0.%1 ;() 'siNo) 1,.311 of ail sr. nitoittr oil) a.)11point! poiloitt au, Fiji 10 s.tios.,,,iir,1 satunio.). 1)0.n:0(1(11: 111 01 0111 sThritiotputti - 01;S1 )% )paitillipt Aritiottiolo III) TO, In pamotp)J ((1 oats Il:uOn11)I)e 1111110.)!1 JO 00.1 p0111,1 salris at1.1, os 111111 .1111!i'015 3331011 'sattinp-m S(1 :1;Jji 00tH' 9.031 o.to.)% Time

Jo 9111 rap! lo 111 1patitippr 9 ai(luiptlio)ot oi(1 ;maim sattinio, all: 0111)11111 11'11: (01 'plat oili - atioiir it:wittri atprio paioilms snit!Arip mm11.10.011 0111 :col lard smniti 11111001.1.01-) 0111_ 15.11) 30111i0A 10 S1111 1.t1:(1-00ti11 sopos 5,0111(11(0 tp!.)% sup.ton 1)adsai 0l 11:00!)-)0,1191! Dania' ir °opinion )(1104:)1: 01 0.1111)0.1(1 or -mom tiop10otit11 pm: 7400(11.No1 110111:/1111'11.10 pawir 0,"11 s.1411103(I -100(1 sPLL Si 0110 Jo pip 'sm.! 0111 '111(J Jo tit:Timm wag 53103 ()I 3111 1311.11:31 ' pairpi;p:) si.t()JJ° 01 0i!1r1!(Ir uo stiopuirstill! 111 M1.111(1100 5)1111110.5 Jo 1011 5110(1501 %11:(11,!1111 sr r ptiop)tinj iird pip irtiop)11.1191! aiinoplp 68 CI I \I' l'El: REF

in a mathematics text. prelate to Part supports the claim that Emerson purposely Com- piledthe duce volumes stli Snell a view to illustrations: -The method emplirtedloril- lustating the subject. it will be seen. is original and peculia.- The separation of an arithmetic into separate 11:11'15 represents a significant in- 110V:1110n int l'otittCed by Emerson. This particular innovation proved to be a most lasting olgaiii/a- tionalformat. as the inniunotts inathematiis series of today will clear!: attest. :\IthotighEmerson's volumes rept esentthe firstto contain a substantial number of illustrations purposely related to instruction. In:my of the earliertextbooks didcontaina few faunal diagrams. and some volumes such as Thompson 's The, Anwrilan Tutor's ( ;ride (36) even made use of an occasional print to point tip the situation in which a problem was set. One such example from the Thompon volume is shown in Figtne 3.6. with a problem setin FIGURE 3.6 velse and accompanied by an illustration. In the midst of a meadow. A second interesting, example of a preEiner- Well stored with grass. .1.011 use of illustrations for the purpose of aug- I've taken just ow acres mentinginstruction isgiven by Barnard's To tether my ass: volume ATreatiseon Ailthiantic(2). The quotation and accompanying illustration shown Then how long must the cord be. in Figure 3.7 are typical of Ilarnard's use of That feeding ali round. drawings. lie mayn't gran less or more than Such sequences of questions were terminated two acres of ground? by a question more directly associated with an arithmetic skill.In this case. the sequencecon- cluded with thequestion *°3times 3is how man). The elementary mathematics textbooksof FIGURE 3.7 Joseph 1:ty achieved the distinction of being the most popularof the elementary mathematics "Here are three boats, and each boat contains textbooks published (luting thc nineteenth cen- three men. How many men in all?" tury in the United States (using as an index nit rook s INs I Itrt.flONAL 69

thenumber()I wpiesprinted).IZa.'shist allenlined 1)11)111)1111111S 11011N% 111 the United voloms. The Hilly Mitten( (29) and Table %tales. and Rill N ..Irnharin Ira Children (12). "Ile 1)01)111:11 11 1 (11 1Z:11 textbooks as well as pealed in and served .1..111 tine the .duce_.: of the unineums levisions (011(111( led to his Erletlit (281.firstpublished during the latter half 01 the nineteenth (entlity in1837. These whunes here im(npotated 11110 and at the beginning of the twentieth centnry Nelic, that 1)101 ed 11) be the 1110%11)01)111;11' of is rellected by the estimate of 120 million copies Ray's W0110.:1 11 quirrlir. Part FirAl was in11). sold over the lifetime of these volumes. In fart. lispedin18-12:Arithmetic. Pail Serand ap. the average annual sale of Itay's volumes be- pealed a%aCllater. and :billiard-lit. Parr Third tween111111) and 191:1is estimated at 250.000 completed the series 127). copies. The appatem !mind:wily 01 the reti...cl Ra.'s texts lesembled C:ollnirifs in 111.111110 volumes most be balanced with the ubservat' adhered 10 a S.11111)11111 and (Ill et:111eNS Of 1)1C- that although these 11N:tem revisions did( 01 M:111.111011. kJ% (11(1 not alle1111)1 11) (-01111`11( 1dif- 1 et 1errors. they did not significantly affect the fid( dellIC111:11A111.1111C111:11 1( al situations and. wittent of the volumes. The title and vocalm- in tact. introduced nothing moil: ply/ling than latychanges welt: often measmes plompled word problems. The impact Of C:011)111.11 is easily more l)sales expediency than anything else. identiliable in the continents Of Ray on intended This nuni.ationischaracteristicstillinevi-

1)1111)(1%es. N111 11.1% ploviding .1 "thorough Ionise dence today. The field tryout was conducted by of nanital arithmetic by induction and analysis" those who pinch:1sed the volumes. and inthe provision for "the inductive and The appeatane of novel forms of organiza- analyticmethods of instriktion." The use Of iimi rot the elemental A mathematics curriculum "mental" and "intellectual" in the titles Of the during the latter pall of the nineteenth velum vr. selemental. mathematics %ohmic, also was meager. John French prepared a series of iellectstheColin:111influence. The titleof volumes that typify the better textbooks of this Arithmetic.PartSectoul.forexample.was period. Thr prin.!LosonsirrNumbrn, changed to Mental ;Indian6( in the 1853 edi- was the primary-level volume of this series:it tion and Ixlr llrrlrurl Arithirteliillthe 1857 required the child to pet form mostly at an oral edition. The modernity synchome so character- level. istic of the midtwentieth-century volumes also Much more interesting than the contents of manifested itself in the last editions Of Rays these volumes were the comments of French texts with the appearance of Alodern characterizing the tole of rides. "If principles Arithmetic(30)and Alodernl'ratical are understood. rules are useless" is one com- Arithmetic 01) ill1903. ment that leflets his view. as does this remark One of the significant contributions of Ray's inhis lourth book: "If the understanding- is .plumes is not a consequence of the content or thoroughly reached. the memory 1611 take care its mode of presentation. but rather its program of itself." This represents a remarkable shift of revision. Frequent and well-planned revisions from the "rules to be mastered" view of the of these elementary mathematics textbooks were earlier centuries. IVItether such remarks actually, conducted. The revision mecItanism provided affected instruction is another matter. a means of reflecting chages in the emphasis The "Oswego Object Lesson" movement in- of topics as well as correcting errors discovered troduced one of the most interesting innova. inthefieldtryout. The publishers of Rays lions ofthisperiod.Milton Goff's workis texts did not initiate this technique. but they typical of the form taken by this innovation. broughtrevisionof elementary mathematics The idea xas to use objects and pictures as the books to a level of sophistication not previously source of instruction. 70 ( 1111'1 II:1111:1 I

The HiAtorical Developmentof 101'1'1(11Nhet itleNple.ld I he 1littr.it) the Textbook. in Summary 111V I .0 Ili I tints cotttltinrtlIN WI %ClNitIll o1 tht. "krt" i. the obtiott. The putemiethcentuv textbook; in elemen- tunnelofthelea( het 'seditionttnrnnortrn t:et!' mailienuoics were ingatti/ed to be used ill miclrurntinthc ent tit telement:et tmathematic OttlerC011e statement. exampleofrtile prac tice. demonstration of mastery sequence.:few tem. What is du_ seabilits of the content nl the went to the extreme of presenting the rules like a elemental%mathematic ttsIttilic.tiptoIhr catechismstated questions followed by answers twentieth (omit% in the United States?Flu. to be committed to memory. such :is: inclies td Smith And Eaton Punt Q. How many kinds of addition are Mere? And Fot, he(( 1:n stittimat mud in "rabic ste:, 11. Twosimple and minimum!. gent mime ch.tmaticItilts ttlempit.tsis. Perhaps the (mist dramatic of tl;e'r Cnlbntn introduced -111dItCliVC- Ntrategy for iii led in in:unction which appealed to the examination rite.itch of111011e1:11% NtNielle-AliViC Welr de of situations and exercises as a source of in- lille on in.0 uc lion in !Ude' .I1 formation to be used in constructing the rules. nrouct .111(1 in lot 'iglu ex( hattge. This change is or at least to be suggestive of the reasonableness interpretable in the totitc-i of the isolation that evolted cluing Of the rules. Mental or oral arithmetic became itt our national hi.rott a part of the early exposure to mathematics for and the glowing lantiliat itswith the (let hied the child. By the end of the nineteenth century. ttionetars ssment. The suliclilic ationIllthe ob most texts wets a blend of Co Ibutil's induction it% tit Iof (lest:loping( otttpcient tin the i.r.r. not man( et,l and the older rule-mastery. .nirhnrctic oper:uiorr.onIII. the latish it t Cal iOttaIN Texts were made more attractive by the in- title:40% :Intl Well :IN demonstrating skill in the malcipulatil to troduction of frequent illustrations; andfor of decimal mutter:tic and cases ofpet( entag, some volumes. such as those mloptin4 the Os- is also ()list.' sable in the po)pott it ins u:tiled wego Object Lesson strategy. the ihustrations lit the itt% esti:tat ions. 1'h emphasis on "prat tic ai were made :t functioted part of instruction. adult- application of these skills is apt 'Asc..% - The teacher was often given advice as to how able from these data. other than in the obvious best to use a textbook by statements from the decline in emphasis on the colonial - periodappli- author. usually primed in the preface. The trttiou. such as late. duel. alld C1011: p:1111eINIttp: majority Oftexts contained some comments 0011 Italer. The ImpactK 01I he 111etttntttlItt directed at instructional use. Toward the end socialIII HiltonIlte of the nineteenth century it was becoming more teat. affil .onus decade. ow. and mote common practice for volumes to include special sections at the end of the book such as "Helps to Teachers" or "Suggestions WENTI RV to Teachers. Elementary Arithmetic by French, ( :ONTI& I 11 UTIONS T() for example. contained a "Manual of Methods EV01.1 'TION OF T1-IFTEx-rnooK and Suggestions"in the back which leferrvd The MC:titled!c onto) brought nemcitilotis to the treatment or explanation of specific prob. teases in the school pupttlaticm. It it It the «mi- lents in the text. nion sc haul growing in the (lire( lion ()I including Because of the inadequate preparation of fourteenyearsof formalitt,tritt tion."Nis teachers.the practice of publishing -key-- eloth has been accompanied by lads such its willows including the solutions to the most the al le11114 to just ii' a pla« for mathematics difficult pr,blems as well as the answers to the inthe common.c hoof runic 1 !II'11 \ 111()01, \\ (Ns (1 ION-1. MI) 71

1.A1:1.1'. 3.1

Pi'l'Cl'itiatre i,i TOW Afire%' Drvoiril InraTionsElrinestinn-.'tfji1/irina1itt 1/ii'. in Publiintinm print.to 1917

I.Rill) 1s21-51 1S77-1917 ( Ittm s(Smith Eatitti) (Smith Eaton)(1:mclicr)

.klgebra 0.7 Alliiiaticm 1.79 1.55 0.97 (.71 Alls%ter. .... 9.91 Arithmetic-. Geueral 0:29 0.30

.krithinetic Svinlx is 0.39 0.31 101-.155 garter 0.96 0 49 0.01 1>ookkeeping 0.81 9.39 Decimal...... (11.-(153 5.66 5.42 Deneuninate NumberN 16.00 9.02 -1.17 Duodecimals .. 1.23 1 . 8). il 0..7 - - Fartors ;14.C.F. and G.C.N1.). . FederalNillel: 51..(.7.;.; "Mrs 2:11 1.01 ti Foreign Exchange 1.55 1.59 0.95 FractionN 5.5!) 11.75 11.91 11.30 Fundamental Principles 9.30 17.65 16.62 20.11 General Reviews 111E11011C:6(MS... 3.57 Meams einem. of Ccom. Shapes . 1.65 3.50 5.18 7.97 NI et Fir Ststent ,. 1.35 MiNrellaneous I(CMS 4.86 1.62 1.07

Numeration 1.50 21:52171 13.3126 3.67

Partial Payments 0.71 3.7611 0.30.3!) 9.11 Partnerships . .... 1.21 0.15 Percentage ...... 12.30 13.,11 1.1.36 Perm. and Combinations..... 0.41 50.5313) 0.08

Position. 1.21 50..85,39 rowels and Roots 3.38 .1.12 2.53 Practice -1.13 1 59 0.46 l'rt sgremiolis .. 9.01 3.51 1.30 0.42 Proportion and Ratio... . 7.70 6.70 4.40 9.69 Tare. Tret. ;:nd Cloff.. 1.47 0.58 0.09 Weights and Measures ,,:?...5011) 2.80 4.11 1.45 Written Problems 4.26 3.68 0.89 (.11 xy i-FR mu].

11 to the -catclitialIIiuciples of echtiatiott.- the pie-witted the positiontlt.ttattempted to link %kw of mathematics withinthe purpose of !mildewsinelemental%mathematicstothe sot ialutility. and the view of mathematics as physical environment. the Inotuotiou ofdisc iplinay formalism and The mic etnicilt teuiuiusaw die intiocht% igor. textbooks. at the same time. under- lion ol a sit ikingl% new mode bn the «cation went let Imologic al ielittements and ionoduced ofmallielnalit textbook.then'..II-financed setetalinnovationsthe teacher's editions be- massive materials development ptojects. Trams cameanexpec led pail of anelementaty of111.11lientatitians and mathematics editcatots mathematics series. color was generously used began to c011.1601.11e 111 1(1-0(111(11(111 of Iliadic- dilongliont the volumes. print was made Litter. matics volumes fits!lotthe second:11v N(11001 tollsonlable volnitns ten students %tete mat Luted. 1Ct ancl stibsequentl% fol the elemental% school and impel back %imes were made available. let els. The spit it of Itange the content to !take The late nineteenth and early twentieth cen- it111(xlcm- swepttheland. Schoolsysiems tuies aIso Immght several important changes Bitting hint country participatedinfield in the textbook. Man's view of himself began tryouts of "expellant:mar' maielials. to midelgo a;attic alchange. wolf,of If%clue sixth decade of the twentieth cetinirx. James ( !lb. John Dewey (I l). C. Stan- die conntion-school educational institution had leyIIall. Edward Tholliclike (37). and Ivan expelienced afantastic growth.It now was a Pavlov (2:,)is of significance. PI essules big business with the choice and implment:1 10 alter the textbooks font the experimental 'ion of instrutional programs asubstantial portion of ps.vc-hologists were complemented by the business this institution's annual business. tonnittnits's clesite to eliminate the study of The elementary mathematics textbook withan what was consideted --obsolete anti tinnetessatv accompanying teacher's guide has now firmly established itself as the principal instinctional topics." Fult her Itilies 111(:"4"c 11:111AT's device. his1171/01IIS(01111):11C(1 withthe 11111TV :111(1 Smith 1::11011 :IS voila! payments. painierships. permutations. and combin- TEXTBOOKS AS PROM*C:TIVE ations were given less space than fractions. the INSTRUCTIONAL AIDS mot iesystem. and bookkeeping. .Appeadices %vete enlaiged and °lien at (wired material slated common operational definition for a mathematics curriculum is one that lIseS a descrip to be lento...ea how the next edition of the text. tionofcontentitems ;,%1:111%- times the topics that lomid their way into bout one or mine the appendices :verse old familiar coment that mathematieal specialties. 'Hie implementation of such aCIIITIC11111111 N:11(1 often hillnened the selection of a hook. The Is 10 occur by the selection and use of partienko textbooks. shook! number of rules to be inemoi/cd was reduced. anyon seriously citaileoge this witeeptielb.a. Lessons attempted to ntilire play in the designs riot. he need only teexamine any one of the of 11s1111(6011:11 activities for votinger chilchen. popular curriculum descriptions called "scope Tlw introduction of actual measuring ins:111- alIll MAIMICV Ortt:11 ilwintiCli as a part 'items for field work began to be commonplace. of a «mime« ial textbook set ies in mathematics Bookkeeping- 'elatedto personal accounts as or he could read either the syllabi by which state well as business was introduced with the in- departments Of education and school systems c of topics such as family budgeting. postal describe a mathematics curriculum or the "cre- money orders. order forms. bills. and receipts. ative thinkiug objectives- of a mathematics series In1895 all 1)ewey published 11:e as they often appea innadvertising copy de- l'Ah"f°gr of Number (22). In this text, IlleY scribing the series. Statements of this sort appear 11,X \N INst ioN \II) 73

expetimental textbook, .1, well a,ill cont The history 01 mailientains textbook develop. menial textbooks. Each of these olc,ciplion, mem coupled %s hit obsct %adult, on the (-sots in eraangedhone the.I( !nalconsumer ih oh:tram:1' of the mail ematio, ipline aigues learner.111(1.1111101 11111.11C1).(huu,t, 111%1C.1(1ut ,nongk againstthe ii,efultie,s of deseloping 10( 11%,011 111C 41.1111111.11C pte,tlll.Iliolt 01111.11 11C. textbook selection pione(ines based011111C .11)-

111:1 14S. 11C:11:111( Cul1/1111 4111:11«011C111or the ['Cramer

Such It«)11( Cl/111.111/al4111 01111C 111:1111111:114.-A of specialvow:think:try or the ongan""lion ool ( Ill 1 11111111i,(IC.11h.ilthaii,14(101s1101beoaii,e material act orcling to .1 lecommended mode of of what i, ,aid about the Irntsuit 110 ongaiii/alion pre,emation. drumlin %law of math(111.114 %. 1)1111).1 .1110of1V11.11k001 N.141 of the maillematios ipline and ol the inral tn. .11)(M1 111(11-.11 11L-1.111(1 %%11.11i, .11)%e111.1i sit.ticezies emplosed in the oomilminio a low the iliranto tot. lion of Ike dim ivrine render,. the :11'11,1n:11nm III moll criteria inefficient. observations on Teacher's Alanuals The teachers etihioll has come10 be the source Search for Feasible Criteria of the pieseription, for the snalegies (,1 instruction.IIthose who ob,eive clasmoom leachers 1;111W11:11. 111C11. C01114111CN :111%111)1CNCI aid in ;idiom are objecike about then obralsations. xviial element of des« iption (an be ein. Icw cachets inielinei the manual other than plo.ed which is independent of .iiichanges in set sing a. .111 (nal messenger. In the extreme. content on in,11110 Howl prooedines and still pro, this sit of pieraliptions i..111 unalciable source .111 objet Ike means of comparison? Quin: a of The for die leacher and by in:pealed siick ouestion. indeed'. assessment and Sc. xpo,m.s escutitalls Iwoonies such aN01111 Cfur lotion of maillemados textbooks is .1 peisistent the learner.In !hi, insidious fashion children problem fur so hoot sIsiems. The mathematics .iii iraii,lonmed flout lea(((ds into mimics. The textbooks available ioda are (pine versatile. textbook41/.1101an in:Amnion:II system(1C. There ate texts fur almost anlearner audience 10'11141C%the«)111C111111011111 01llelulC0111C111 one (an imagin, .11 the elementary or se«mclary VI L.:A:m:1600 for the leainer and. of course. for school level. lu tact. maity of the latge publish. the elementary madiemasics teacher. Therefore. inglionises have numeiotts mathematics text- working within:Invinstructional sysieni. the books competingfur11se among the same development of a set or pro(rdorrs for the se- audience. lection 01 a textbook becomes a responsibility Suppose. justfor the moment. that these that must be a,,igned the highest of priority condition', did not hold. liklead. int:1,611w that rating,. all mathematics textbooks intended for a given age gum!) or teamensi)0,,emed these four char-

Development of a Functional acieristios:(1)ecnitained al one-toone corre- Means of Describing Textbooks spondence between the content and the order of Ilow doe:. 011C(1(i 001) a11111( 14)11.11 (ICS( l'111)14)11 pre-niation. (2) were written in the same stle.

01 /.11.111.111:114 textbook which ( an be applied (3) used the 'MUM (ma Im la ry to describe mathe- 41 III(' ( .111N1111( 14)11 01 .111 C1rel ivc procedure for matical tasks. and (I) provided equivalent space the selection ofuntillematics textbooks: The for similar topics. 'Thatis. imagine condition, lesson to be learned from the hisior of matlic. for which assessment and selection would be mad(textbook de% CIOIMiCiiiis dear: the ton- simpleit mol(((l be based upon reliably observ- lent and the instructional pnwedutes have tin able criteria such as the cost of the volume, the dergone continual change. and there is every onsittion of the binding. the attractiveness reason to believe they will continue to change. of presentation, the weight of the paper. the 74 (11 \T I 1.RI iim.1

size of the print. and so on. Sucha set of condi- 4. What is the evidence that the stated be- tions. fortimatel). does not prevailfortunately. havioral prerequisites assist the learner in because content and organizational differences the acquisition of the behavior described produce competition. and competitionis one by the objective? principal catalyst of innovation. Thefourquestionsarepurposelylearlier- Why bother with such a contrived situation? oriented and attempt to focuson the question of Because of the clue provided by the character- assessing success in terms of reliably observable istics of the items which did remain unaffected outcomes. and do provide possible categories to beem- ployedinthe judgment. The categories are .4dvantagesofBehavioral Description characteristics whose description. yieldstor- liably observable criteria.Is it possible to em- Consider the consequences of accepting there- ploy an element of description that would lid(' sponsibility of pro idling a detailedresponse to to icliztlthobservable criteria for the descrip- each of these questions. What athantages will tions or textbooks. even though alltexts are accrue from accepting such a responsibility? In different by varying degrees? Clearly content is other wouls. why undertake sucha task? not a candidate for such an element, nor are any The advantages of having availablea be- of the non - learner - related categories, butreliably havioraldescriptionforeachcollectionof observable performancedoes come through loud mathematics materials ale especial]) thamatic and clear as a likely candidate. for those charged with theiesponsibility of Suppose we agree to describe each textbook designing or selecting :1 mathematics curriculum in terms of reliabl) observable performances the fora school system. Such descriptions would learner is expected to acquire from the instruc- provide the school personnel with an objective tional lessons. Further. suppose we alsoagree means of comparing various instructional pro- to provide descriptions of the situations that grams in mathematics. The school system will can be structured so as to maximize the learner's be able to select a mathematics curriculum by probability of acquiring these specified (and. means of a well-ordered sequence of events: in the author's opinion. desired) mathematical (I) a collection of human capabilities thatwere behaviors. What does this mean one would need desired as outcomes of an instructionalprogram to provide in a textbook description, and how in mathematics for that school,system would be would it be accomplished? identified by the local school personnel.(2) a Four quzstions may be employed asa means systematic examination would bz made of the of characterizing the requirements: behavioral descriptions and the supporting evi- I. What (that can be reliably observed) will dence of actual learner accomplishments pub- the learner be able to do after he has lished by each of the mathematics curricula completed work with a particular instruc- being considered as candidates for adoption. (3) tional sequence that he was unable to do a sorting of the mathematics curricula that best before he was exposed to satisfy the behavioral requirements specified by 9. What is the evidence that the learners the school system in taskI would be conducted. exposed to the particular instruction actu- and (4)the selection of a particular instruc- ally acquire the stated behavior? tional sequence would be made on the basis of 3. Given the statement of an objective in experimentally verified responses to the exami- terms of a reliable observable behavior. nation. task 2 of the sequence, includedas Part what capabilities must the learner already of the package of instructional materials. Figure possess which will optimiie his probability 3.8 traces this series of events diagramatically. of acquiring the desired behavior? Behavioral descriptions of the intended out- IDENTIFYING THE PARTICULAR INSTRUCTIONAL SEQUENCE WHICH PROVIDES THE MOST COMPREHENSIVE COLLECTION OF BEHAVIORAL MAPPINGS OF LEARNING SEQUENCES

TASK 4

DISTINGUISHING BETWEEN THOSE COLLECTIONS OF INSTRUCTIONAL MATERIAL WHICH BEST SATISFY THE BEHAVIORAL REQUIREMENTS CONSTRUCTED IN TASK ONE AND THOSE WHICH DO NOT SATISFY THESE BEHAVIORAL REQUIREMENTS

TASK 3

IDENTIFYING THOSE AVAILABLE INSTRUCTIONAL PROGRAMS WHICH CONTAIN BEHAVIORAL DESCRIPTIONS AND SUPPORTING EVIDENCE OF LEARNER ACCOMPLISHMENTS.

TASK 2 IDENTIFYING AND CONSTRUCTING STATEMENTS OF THE DESIRED MATHEMATICAL CAPABILITIES FCR LEARNERS IN TERMS OF RELIABLY OBSERVABLE BEHAVIOR.

TASK 1

FiGultr.3.8.Tasks in iIu seleclitmofan instructional sequence

conies of an instructional pro ism in mathe- mine gill be obseiled whether the oltinies are matics arc as valuable to the cla.ssloom instructor intended for the elementary or the secondary as they are to those Who must select its N(11001 level. Textbook publishers simply do not programs. IVith this information the instructor provide such information about their products explicitly hammed as to the desiied outcomes and do not appeal to be int.:tested in providing oleachinstructional componentwithinthe such information.Itis an interesting exercise entiie mathematics curriculum. for which he is in futility to write to the publishing houses in Its;isihle. the United States and request. for a particular So now you are saying. "OK. ()F., I'm con- textbook series,the evidence supporting the vinced.you've sold me On the possibility of claims. The ambiguous nature of 'understand- using reliably observable behavior as die fun- ing,- "Inatheinatic:ailiteracy." and "appreciat- damental unit for describing- instructional Ina- ing" are seldom better illustrated than in the terials such as the 'mullein-ides textbook" (so responses one receives as a result of such tegnests. pr_tend if you'le not. convinced or go on to the NVIlat can be done to alter the situation? One next chapter inthe vearbook)but now you possibility, of course. is to refuse to purchase ask. "What can I do about it?" If one attempts any textbooks that do not provide a behavioral task 2 in Figure 3.8 using the mathematics text- description and supporting evidence of actual books commercially or experimentally available. !carnet ammiplisliments and to encomage the one would probably find that he has identified funding agencies to refuse funding to "experi- die empty set of mathematics texts. A like out- mental" mathematics development projects that 76 cl I \l'ER11

(Ii)1101 prmids.imtil.tdemipt ious and e% idem e. der dated of the put( hase of elemental% and seo- . diastiostep. )es! But atie( essai% one Jr the olidaimathematios textbooks on roller lion. to! (1,,iled (kluges ale to be I eali/ed. ht realit% the insttmrlion.tl maul ials until the audio! oi 1)1 otessional«mmitinit%of mathematics edu- pithlishel in o% ides a demi iption of pm lormano e (atm. could he asking for very little from the objectives and evideno e or niulli.hntenl oI 1»oultio c15 UI insti it( 11011:11 motet ials. If one %%1 hes the staled perfoinian«s for the mate) ial.Ihe insti noio11.1 I mate' ials .111(1 c:\pect, WW1:, 10 use e hien( c should be obtained lioni a sample of theseIllateli:11S,he stud% should he able to the intended population of uses. Roththe N.1111- identil v lea' nos ofhis material b) plcanddiepopulationshouldhe cleat I% the perloimances they exhibit. All the commu- tlesci ibed. nityor1)1.01e1S1011:11 users 1V0111(1 be I ailing for isthat the author shale this information.Ir. Aclive T('achrr Involvemen1 howec r,.111 author does not supply this infor- The authors anddistributor, or instructional mation. we are being asked to use instructional materials are not. hownel. the only ones 111 need materials for ,which men the author does1101 of change. Consider the problems inherent in an 'Amide a means ol distinguishing a star( esslul attempt to no wmplish task I in Figine 3.8. The learner from an unsuccessItil one. Furthermore, .111051 pertinent question would appear to be %%idiom, evidence of learner accomplishment we. low does one describe mathematical capabili- the consumers ol textbooks. :ore being asked to ties in terms of reliably observable behaviors? Win base materials whose uort Itis unestablished. Assuming that active learner participation is desirable. the following sequence was developed Recommendations with the objective that after completing this sequence )ou, the tender. will be able toi this juncture,inthe discussionitseems .t l.distinguish between abehavioral and a appropriate to pause and make three reommen- nonbehavioral objective: dation; based* upon the observations described 2.describe one definition. of a behavioral ob. tip to this point. jective: ItlicommENDATioN I. The National (:otincil of 3.describe one procedure lor constructing a Teachers ol Nlatheinatics should construct. en- working set of action verbs: dr.s.e. and publish a set of technical standards -1.construct abehavioral objective hone for Hi.heinatios ruiriculum evaluation. given nolibehavioral objective. 121:commEsDATI)N 2. 'Ile National Council of 'l'eac'hes of Nlathematics should be chalged with theesponsibility of reviewing all instructional A SFr OF PROCEDURES FOR material intended for elemental-% and secondary DESCRIBING AND CONSTRUCTING school instructional programs in mathematics. ll EH AVIO R A I.OBJECTIVES' Each leview published bv the NCTNI should just what purpose do statements of objectives include astatement of whether the material serve? Do you use the statements of objectives meets these criteria:(I)performance descrip- found in textbooks or courses of stud% to plan tions of objectives.(2) data on the acquisition %our instructional program? of the behaviors described as objectives. and (3) Yeas or no", a statement of %vliere the performance list and data are available. The criteria established by I. insmutional material iN adapted firms "SeNsions- thesetoftechnical standards recommended I and 2-of colonic Iof the MarylandE1l:711011m y Afalhe- above should he used in constructing the review. mai irs lizteivire Program(MEN!!!') of the University of Matheinatits l'iojert inpaied muler a RECOMMENDATION3.A moratorium should be giant from the U.S. Mite of Eli, ation. 1.111. I \ I 13001.\ \\ INS I 11*(.110NAI. \II) 77

Be honest nou. no one is going to collect your ics. Among these competem ics ale the abilit !espouses. Suppose nut plan an ins!' uctional ses- idetitil ambiguous dem iptionsofobjectives skin -Iron' a tem-IR:Cs comment:It% that contains and the ability to charge ambiguous descrip- the usual statements of objectives. Now suppose tions of objectives to instructiolialh functional all the statements of objective.; in your book wet e descriptions of objectives. eliminated. for example, by covering them with For the temainder of this section ou will be tape. How wouldout instructional planning be asked toI espond at vat ions intervals by vv I iting -7 obset vabl) Aimed? \Vold(' your planning be on a response sheet.II ou ale to acquire the (fillet en' competency expected hom this portion of the Yes or 110 exercise. ott must respond when asked to do so in the 1)11Ni:int.Itis important that you be an There ale few teachers bout the inexperienced active participant. to the experietued who would say that the usual The objectives of an instiuctional materials broad statements ofcurt iculum objectives (as should be stated in a clear. unambiguous man- they are usually constructed) actually contribute ner. Lel tainly 'hoe are few who would refuse to to their planning for. or execution of. instruc- acknowledge this as an important characteristic. tion. Why is this? Is it simply that objectives can applicable to all statements of curriculum objec- serve no useful instructional purpose? Must cur- tives. Statements of objectives for mathematics riculum objectives Ienttin an instructional win- programs usually do not satisfy this rquirement dow dressing% or can they be made to assume a of specificity and charity. more functional role? Consider the illustration in Figure 3.9 with the You ate about to pal ti( ipate illall instruc- statement of a familiar objectiveone common tional program. The piogiant is intended to aid to expel imental and «mullet cial mathematics the leader in the acquisition of certain competen- curricula available today.

The learner will build

an understanding of the

system of whole numbers. 78 cItArrim

What characteristics would you ascribe to an statics ploglains which you ale almost (main instructional program in mathematics ifitis to recognise. attempting- to achieve this objective: Is the state- What Jul% hies would be ne(cssal% to achieve ment of the objective phrased in such a manner this objective: Do you suppose ()dietmathe- that several other mathematics teachers, working matics cdthatots %%otild identil) the same «n11- independently. would arrive at the same inter- ponents as ne( cssalto ac hieving t his obje( ti% e? pretation of the meaning of this objective: Yes or no: Yes or no? Have you made a wtitten response? Fine: Is Have youmade your selection? Go ahead. the vatic() of possible interpt etations for these write down a choice! two objectives surprising: Not at all. when one The illustrationin Figure 3.10 identifies a «msiders their lack of spedli( it).in lat. that statedobjective of numerous modern mathe- which is truly remarkable is the skill testbool, authors have demonstrated itr «mstructing ambiguous statements of objectives! Perhaps the most Flotati:, `3.10 startling observation. however. is not the widespread use of these statements, but lather that most teachers so complacently accept these statements! As teachers. we acknowledge. or at least tacitly accept. these statements as reasonable descriptions of our goals and use them as the basis for justifying the selection of certain instructional materials or the performance of particular instructional acts. Each of these decisions is madeor action initiated even though thereisthis dis- agreement as to the meaning of these objectives. Consider the statement in Figure 3.1 1 of an objective for an instructional program in mathematics. What specific instructional activities in mathematics would you design to achieve this objective? 1)o you suppose other mathematics instruc- tors would reach a similar decision as to the meaning of this objective: Come on now. make a choice. Yes or no% Certainly this third objective is unlike the first two in thatitnames itparticular portion of mathematics. namely arithmetic skills. However. narrowing the content from all of matheniatics to arithmetic skills is not a solution to the inter- ptetation dilemma. A large number of varied interpretations can still be made. Such specifica- The learner will acquire tion is useful but is not sufficient. The three 1weviotts illustrations suggest that an appreciation of the the description of an objective needs to be specific if there is to be any hope of attaining uni- STRUCTURE of mathematics. form interpretation. The need for each mathe- 79

manic, objective to be interpretable is esecialk impoitant for those clagt.d with the construction and or implementation of an institutional program's objectives. However, implementors of Mal lielllals CM ricula are not alone in their acceptance of ambiguous objectives. Consider this statement (Figtire 3.12) ofall objective of contemporary mathemat ics textbook authors. Now suppose are one of four «munittee inembes (hanged with the task of independently observing students who have been exposed to in sn uc tional materials designed to aid the learners in acquiring this objective. Further, let's suppose that. based on these observations, you me to make a decision as to whether each student )ou observed had or had not attained the objective. Does the description of the objective in the pi e- viousillustrationidentifythe specificperfor- mances which you wouldlookforinyour observations? Yes or no?. Just what perforances one would be expected to observe in learners who had acquired a fami- The objective is to strengthen liarity with these properties is not contained in the statement of the previous objective. his arithmetic skills by relating The description of an objective must identify the observable behavior that a karma who has them to basic principles. successfully achieved the objective is expected to

FIGURE.3.12

The learner will acquire a familiarity with the properties of a field of numbers. soot.,JAVI"VV/At111/16%.,, ( 80 II\r ru-p.riII EE

The purpose is to help the learner haw lc (juiced. Read the objectke suited in Fig tit e3.13 with the purpose 01identikiitg the gain an appreciation of the structure obser% able behaiols ;1 MtIllt.111 shotlid he Ale to CXIIII)ilithe has ac hie; ed thec Ompetenc de. of positive fractions. scribed b) the objective. Are diet e observable behariots idelli Hied in

3 i he "i nielnelli or Ili, ObjeCliVe?: \ IC 111CIC11)e

11:16)11 (ICS(l'Ibedin:I 1%:11 111:11 1V0111(1 enable VOII 10 eI he sttcces,Iut h0111 the «-ssful ones? Yes or no? Sincc ott decided es. the description does

identik the desited observable behaviot S.:111(1 1011 Call. Or ((misc. name them. Oh? l'on say sot, decided no? (slued for you! The statement does not contain all) such performance specification, and therefore the appropt iateI esponse is no. The statement of an objective should desc ribe desited learner helm% iors. In order to be able to interpt CI an 011jeC111C Ihese bell:U*101'S should be clearly described. intent of :m objccii.e, is reliable led hr (Iescril)1 ions Of observable behavior. Consider this next objective cue 3.11) in tlw context of how effectively the statement communicates the desired behavior of the objecike. Does this objective describe the Iwhavior to be acquired by the learner? Yes or tio%

Fictnti,. 3.14

The purpose of this material is to describe long division. 'Fillri:x I gook \s, Rt nio\ "ID 81

Does the statement in the previous illustration Present a detailea analysis of finaing describe all envitonment that lewd' es the pies- en«. ()I a heal iet: the square root of a number. Yes or no: I !me mu node a written lesponse: Don't lead on until vote .\s tots ale likeh to haw ((nu:tided alleadx. die description of the inm-iotis objective does not 1(1(161% the behaviors the leafier is to acquire. Nor. pe(ttliarl enough. is a learner even titles. .alt. %hue the insti tido' might des( I ibe without all' learne:s being Inc:wilt. Read the des(ription()Ithe objective con- tainedillFigure 3.15 and (le( ide whether the !canter is ne(essary to the achievement of this obje( live. \VItat did ton (le( ide about the necessit% or a learner in the NellieVellIelll Of Otis objective: IS Ile Ile( ess:11'. or is he not ne(essary:

.\ learner is ne(essalto the acquisition of the behaviors described in the p,eviotts You don't :twee: .\ detailed ilvsis might be FIGURI:. 3.15 given (ten though I.() learneris present. Objectivemust be constructed so as to be specific descriptions of what a learner is to (10 or * builds an understanding say. 0111. by fulfilling this descriptive require- * appreciating ment of learner performat.(e can objectives be. mine functionalfortheinnovator. planner. * developing a feeling for developer. teacher. and learner. Objectives must be constructed so as not to allow for the excht. * pointing to sion of a learner under :env interpretation. * having an awareness of Ambiguity is ()hen cloaked in the garment of prestigious phrases. .\ few of the most «minion * conveys the concept of these phrases ale shown in Figure 3.16 tog,ether with .tlinget -one phase that does not belong be( arse it conves spec Hi( ally a desired behavior. Identify the phrase that does describe an ob. servable pet lorntan«. Whi(li one is it:

Select one. Don't hesitate. write it down. Now! Did ott select "builds an understanding'? It's not. you are on the right track. Perhaps you picked out the "appreciating" phrase, or the "feeling- phrase. or the "awareness" phrase. or the FIGultr. 3.16 8") \ II IR El

°cunte%s" phtast (iood. -Pointing to-is the al/Hopei:1W choice :Ind should have betbn identified within!' (fan ultt. Suppose tin ee.dinnsiontd °hie( Is such as [host. in Figtoe trete planed in limn ititilt ol ton. ItititityIN kit:, 44,11.11411441410 1144 :111\11-14iltr 11-11erit4ivrovrIre,rrx Let's suppose you have been asked to identify **Of the cube. Vould identifying- be interpeted 111170.- II 4 ititititir.. ,C11 ***kilt t.44% dellIall(111% some sort of an observable actionon your part? Observable or vague?

1 t tecluites an ohservalwle at lion. of cum se. Von might (Ally out the identiling In pointing- to an object, 01 h plat ing out lingers1111 :111°hien. or by actuall picking III) an object. ht Figule two objective. de:sti ibcd. IZead them tatelt:Ilt and select the (testi ipt ion of the objective which ivonld have the les,am-

1)41 ton select statement A or 11? 1:u:ritt.: 3.17 Recluse "full% undetstand- has many po:sible meanings. the mote specific objective would seem to hr statement I;. A B The learner will comprehend The learner will be able and fully understand the to identify the closure procedures used in the property in finding the division of fractions. quotient of two fractions.

I:uuTRE 3.1S 111 1. I IA rnook \,".\N INs 11*(.1'10,N \I,\It) 83

x.imitte the Mush atitai in Figut e 3.19. hi( It Should .111 obi Able pel 101 111.1111e 4)1.1( 11011 lir 0111.1:11. um des( iptiom 01 obje( t 'hrs. \lach ()I identifiedinthe des(I iption()Iabelta% io1.1l these ()hie( Ike. is the mote spec ili«lesci iption object i%

tleNhed leartie perfoi 111:111( e: Yes or no? - _ A or - 1:.eit[lie 1110SL(a.11:11(onsidetation of the Isuppose that%()11 decided that neither of net es..IlIetplilenlents of .1 pet lormance destrip these was .1tuella viol al objective. No: Oh! Yoll 1i011 suggests that each des( iptioll must (0111.1111 Illad lioi(c di.tt mil% statement .\ was 'w- NOIlle .116011 WOld Of phrase. \Vhat class of lords

ild\ iota1 des( I iption of th.tt onl; statement B was most ()ken dusciibes .1(11011 111 1:11051-11o1111.. behavioral descriitt ion. No? Good! Did oo volts. at Or what? (10 is both .\ and B ate helm viol al des° iptions ()I desired student perfoi mance: If so. yon have 01 course. most often a( tion is communicated .1«itthed the behavior or being able to identik b verbs. But which action verbs shall we use? behavioral objecties. All 01 them or just a few? Ate dune some par- tiula tactics that are mote effective?

The learner will be able to identify names for ten. A The learner will be BEHAVIORAL able to demonstrate )R NON-BEHAVIORAL 7 7 7 7 7 7 a procedure for finding the sum of two integers using the number line. B

I:mt no: 3.111 84 (AivrEit II [REF

The following material describes one tech- What ale words that have similar meanings in nique 111;11 has been used with satisfactory results. English called? The tactic is to define each action verb in midi a --- tray dial a class of performances is defined. The Think back. you are certain to have encoun- best way to discuss the technique is to have you tered the term before. Since words that have Firth ipate in it. So let's have vou do just that! similar meanings are called synonyms, action You will need the following mann ials for sour verbs that elicit similar behavior are calledbe- part., ipation. havioral synonyms. Rectangles: red. 5 cm b. 8 cm; white, S cm by Are the three verbs identify, pick up, and II) cm; bite. 5 cm by II) cut selectbehavioral synonyms? Triangles: red. equilateral: tvhile. obtuse: blue. Yes or no? isosceles Of course they arc! Spherical piece of bald candy Why use all three of these action verbs? If behavioral objectives are to be specific and describe One necessary component in the description of observable behavior. would it seem sensible or a behavioral objective is an action verb. But now not sensible to use one verb in place of all three? just wait;tminute! How many possible action yobs are ilime in Englisha few or a gm eat inany? The sensible thing is to agree upon one action verb and use it as the name for a class of per- You «mid decide to use any of a large variety formances. Which one of the three verbs shall we of action verbs. The variety itself. however, may agree to use? Since it does not seem to make contribute as much or more to maintaining the much difference, let's agree to use identify. ambiguity than to facilitating clarity. Identify all of the triangular regions in the The problem would now appear to be one of materials and place them off to the side. Do you I educing the number of possible action verbs have them all? Arrange the triangular regions usedin the description of objectives without from the one with the least area to the one with reducing the xariety of learner performances the greatest area. Now order the triangular re- being Gilled for by the objectives. gions from the one with the longest base to the Spread out the materials in from of you. You one with the shortest base. gill be asked to make several performances. Carry After identifying the two sets of objects, you out each task as best you can. performed two tasks. The instruction for each in- I.Pick up ablue rectangular region. (Go volved an action verb. One of the action verbs ahead. don't be bashful, pick itup. Put wasarrange.Name the other action verb. that object back with the materials.)

2. Now select a blue rectangular region. (Have Were the performances you exhibited alike or you made a selection? Good! Put that ob. different? ject back with the materials.) If you are reading this before you have writ- 3.Identify a blue rectangular region. ten a response to the last question, you are not Notice that three different action verbs were playing the game. Go back and try to write a used in initialing the three performances you response to the question. The conclusion that made. One of the action verbs wasselect.What seems justifiedisthat these two action verbs were the other two action verbs? arcbehavioral synonyms(call for a similar performance). Let's agree to useorderwhenever Did you write pick up andidentify?NVonder- such a behavior is called for in the description ful! of a behavioral objective. THE l'ENTI;001: AS AN INSTIWC:110NAL All) 85

What do )011 ( all an object shaped as shown naming? That',«illection of acceptable ispon- below? ses. Now you're really catching on Look at the tape/oids in Figures 3.20 and 3.21. Show how %011 would decide whether the following statement is true or false: Segment m is longer than segment n. .)emonsti ate how you would decide whether this What is the name of a theedimensional ob- statement is true or false. ject shaped as shown below? m

'Fell the number of triangles pictured here. FictmE 3.20 AAA n e the performances required by these three tasks similar or different? FiGuiti:, 3.21

Similar, of C011INC. XVIlat action verb would you use to describe these behaviors? One of the action verbs used to elicit these two performances wasshowand the other ws Any number of different action verbs are pos- Demonstrate and showare sible candidates. A few of these behavioral syno because they elicit the same type nyms are(ell, specify, call for,andname.Let's of performance. Let's agiee to usedemonstrateas agree to usename. the action verb for this class of performance. The agreements about action verbs made up Consider the sphericalobject among your to now mean that when you describe a behav- materials. Suppose that someone in another city ioral objective and the performance is had an object just like yours, together with a I."choosing the rectangles," you would write number of similar objects. Your task is to iden- the rectangles'; tify and name a sufficient number of character- 2."classifying the objects from the heaviest istics of your object so that the other person will to lightest," you would say " be able to identify the object you are talking the objects from heaviest to lightest"; about. Start naming! 3."telling the colors in this painting," you would write " the colors in If you have identified and named only color and this painting." shape, your description is not adequate because If you're reading this before you have respond- it fits most of the objects the second person has ed to the previous three tasks, go back and res. in front of him. Add a few more characteristics. pond. Did you writeidentifying, ordering, and If you added mass, volume, diameter, and thick- 86 ( \N-FR nki,y

The learner will understand place value.

FIGURE 3.29

ness you would be witch closer to a satisfactory The preceding actkities should suggest the descript ion. procedure that might be followed in the mo- What shall are call such behavior? What action st] uct ion of a working set of action verbs. verb should we use? Could we use iden/ijr? Now that we lune agreed on the meaning lor a few action words in the construction of behavi- Could we Ilse order? oral objectives.let's turn our attention to the Since identify tequires the individual to select ploblem of constructing a behavioral objective. an objecttilat has been named for him. this Read the objective in Figure 3.22. action verb does not seem satisfactory. In the Is the objective described in the illustration same way. order does not seem an appropriate above behavioral? choice. The distinctive characteristic of this new Yes or no? behavioral class is that the learner identifies and No. of course it is not. With the use of one 01 names the characteristics or properties. More our agreed-upon action verbs newt ire the nonbe- than one characteristic is usually included, and havioral objective described in the illustration there must be a sufficient number of these charac- above and make it a behavioral objective. teristics so that a second individual will be able to identify what is being discussed. How then shall we name this class of behav- If you have not completed rewriting the ob- iors? Suggest a possibility. jective. ch»tot leadthispart. Go back and do it now. When you have completed the task of rewriting the objective. lead it over to see wheth- Many choices could have been made. The par- er you have: (I) used one of the action verbs. ticular action verb that scents most appropiate (2) described the situation in which the !carnet is describe. The describe behavior involves the should exhibit this particulat bank», and (3) individual's identifying a sufficient 'number of indicated the nature of the pioduct the lesser characteristics of an object or action so that a is to produce. Learner products stay be quite second pet son would be able to identify it with- varied: a sentence, a word, a drawing. a series out having it pointed out. of check marks. and so forth. 1111. IT\ I In)01,\ \NI \11 RUC FION \I.\11) 87

Figure :I.23 mittains lea 01 the possible de I. \amine the obje( e des( ibed in Figuie st iptions or 1 >ell.ii01.11(thic(IicsIviiith ,0111,1 1:evviitc this obje(Iie so thatitis !Rita\ iot.11. liac been «)11,11II(led from the lionbeii:nioral obje(Ii've. Noti« thatitis not .1elillieull task to \\licit \on Ih.t\e «ntipleull the task of re\viit

nanslate this imiti(111:11 objective into a !wkly. ing the obje(iic and makingit .1 bchaviotal iota! obje( Mc. The translation is awntwlished objc.(iie. te.td these next statement;; by stating tvlutt one would look lot in terms of I.!)id "re One 01I he .I( 6011 %el bsle learner perfmmatue rather than in terms of the have agieed upon?

nonperlorinam description that the Word Yes onto? _ _ _ ( on% e% s. 2.Is the situation in whi(h the learner is to exhibit this perlormame specified? Yes or no?.__ 3. Is the nature 01 the output that the leaner is to pro\ ide e lead. spy( Hied. as 'well as ame rest' i( lions On that part i( ular output: Yes orno?.-- The learner will identify the units, tens, and hundreds place, given a numeral. The learner will acquire a familiarity The learner will identify thewith the commutative property. position of the 512's place for base eight numerals.

FIGURE 3.23 FIGURE3.24 88 cif pTER rfiREE

If you wete not able to respond yes to each of INSTRUCTIONAL AIDS AND the three questions, go back and correct what- BEHAVIORAL OBJECTIVES ever difficulties you have identified. Does the description of the objective in the illustration Instruction and instructional aids are intimatc1) identify the specific performances you would related to one another. In particular, instruction- look for in your observations? al activities include the selection of instructional N'es or no? aids, the design of the instructional sequence, The preceding material on behavioral objec- the identification of prerequisite behaviors. and tiNe,, SCITCS as an introduction to beim\ ioral de- the deso iption of those observable perk' 'minces scription. The instructionisincomplete. The that signal suc,:ess and those that signal failure. reader should have acquired sufficient compe- The textbook is an excellent example of an in, tence so that it is possible to pursue the relation- structional aid that exhibits these influences on ship of behavioral description and instructional instruction. The magnitude of the textbook's in- aids just a bit farther. One way to characterize fluence on mathematics instruction in the United the potential effect of behavioral objectives, evi- Statesisclear from educational history. The dence, and learning hierarchies on textbooks is textbook, more than any other single instruc- depicted in Figute 3.25. tional aid, affects the content of instruction, the

I BEHAVIORAL DESCRIPTION MENTAL ARITHMETIC r I I

INDUCTIVE FORMALISM AND RIGOR RULE MASTERY411)

DEDUCTIVE DISCOVERY AND INDUCTIVE

SOCIAL UTILITY

FIGURE 3.25, The shifting emphasis of the mathematics textbook THE TEN 11;001. \N INS fltit'(.110NA1,\II) 89

sequence of instructional activities. the level of possible purpose 101 the des« ipt ion of expected expectation tor the learner. and often even the outcomes that do not identik observableper- insunctional environment. To observe that the formances is to display an ignotance of desired textbook current l\ occupies a position of singu- effect without the penalty of public ridicule. lar importance in mathematics instruction is to Tlte conditions that allow the teacher to iden- bolder on under statement. This should not be tify %viten he has succeeded and when be has and is not intended to be an endorsement of this failed are hindamental to the instructionalact. state 01 allairs, itis merely an obsmation of the Basicto this identification is the description of curlew reality. what behaviors (observable performances)ale The ptesent activity directed at the study of expected to be acquired by the learner. If the teacher behaviors may ultimately bring about description of objectives for instructional activi- significant changes in the way teachers teach, but ties in behavioral language is desirable, and it this has not yet occurred. The manelous tech- would seem that such is the case. then the teacher nolog,y of the twentieth century may completely must he gi% en assistance in acquiring the coin remold our everyday means of instruction as it petencies necessary for the construction of be. has our means of communication. but this has ha\iwal objectives and learningsequences made not happened yet. The data on success or fail. up of behavioral objectives. A previous section ore are notyetin the literature. And these of this chapter briefly describes one collection of realities leave us with the present, where the proceduresfor constructing behavioralobjec- textbook remains the dominant instructional tives (23: 39). There is a growing literatme that aid (and on occasion. the instructor as well). deals with the acquisition of these competencies The textbook communicates instructional pur- associated with the construction of behavioral poses. Both the teacher and the learner are objectives. The literature on learning sequence expected to receive and act on the purposes construction is small and less well known. but communicated by the textbook.Instructional some representative work is provided by Gagne purposes are expectations. Some would agree (16) and Walbesser (40). that certain of these expectations (the "really The assumption now is that the reader is will- important- ones) cannot be translated into ob- ing to accept (at least for the remaidet of this servable performances. The existence of such chapter) these axioms:(1)instructional goals learning expectations is questionable. Because can be described in terms of tenably observable such desired outcomes cannot be translated into behaviors. (2) each instructional act has at least observable action, it is not possible to Iccog,nize one behavioral objective associated with it,(3) who has achieved the objective and who has not. a minimal set of behavioral objectives can be It would appear that such statements of pur- constructedfor any instructionalactivity.(l) pose fill lines of print in a text but have no sub- mathematical learning is subject to the accumu- stance in the context of learner outcomes and, lative acquisition of behavior, and(5)behavi- therehwe. serve no instructional purpose. oral objectives can be sequenced into descrip The ploponents of such objectives (those that tions of accumulative learning sequences called cannotbetranslatedintoobservableperfor- learning hierarchies. mances) espouse a most peculiar position. Some- This set of assumptions affects five activities thing 01 other is a purpose of instruction, but associated with instruction: (1) the selection of the author of the instructional activity cannot an instructional aid, (2) the organization of in- tell the teacher how to recognize when the ob- struction. (3) the recognition of success or fail- jective has been achieved. But don't be too criti- tit e in instruction, (A) the revision of instruction, cal of such stupidity: perhaps there is a place and (5) the design of instructional sequence. for vague, ambiguous statements of goals. A By way of illustrating the relation between ci R EE

101.1\101A Obi and ilhIl1hli011.11 aids, Ille selection procedure fails to deal with the Ties. teNIIMOk is (hosen as the lust tuitional aid undel lion of !carnet acquisitions. then the woith of discussion. 'Flue selection of a textbook is (IC:Ill the !no( echne is negligible. \%1111 in Mall) published guides. Stu It guides Ma) -Him ate (Cl lain conditions external to any be lists of steps 10 be taken. 01 the ma) be de- list of procedure which often affect the selection tailcd illustration, of the procedure being ap- of an instructional aid. Frequent!). these «mdi- pliedol something between these two. These lions ate the principal factors influencing the guides come in 111.111)(.tried foimats. State de- selection of pallittliar textbook. How often pal 'mews 01 education. educational book divi- halve you heald any one (» more of the icasons sions of publishing houses. local school sstems, illustrated iiiFigures 3.26-3.32 being advanced andprolessionalorgani/ationssuch as the in support ()I' one choice over another? NC: 01 are included among the sources issuing textbook selection proceentes. The procedtues lecommended vary to sonic extent, halt 1101 as much as might be expected considering the variety of sotnces of authorship. Some te«,mmended procedmes lequire the user to employ a list of objective criteria: some mix objectivecriteriawith subjective value judgments. The deficiencies of these selection practices ate well known and need not be enumerated here.I,e1itsuffice to observe that the criteria commonly suggested are items such as these: FICORE 3.26.b..ndemementby And:wily time cost considerations. general attlactivellesS. "IVell, this text is endorsed byso.an el-so and judgment of the relevance of examples, level of soand-so, who are well-known figuresin VO(almlarN. graded variety of exercises from easy mall:en:en ic.s. So the text certainlybe to difficult. variety of examples. authority (per- effective." son.project,or schoolsystem)endorsement, publication (late, durability, need'ed training. and lequired equipment. Although such alist contains many characteristics relevant to the selection of a textbook appropriate to a par- ticularcollectionofinstructionaltasks,the chalaclerislics aleincomplete. The character- 000 istics identified are adjuncts of an effective selec- Textbook tion procedure, but they ignore the fundamental Selection question of kat ning outcomes. An effective selec- Committee tion procedule must contain some criterion for dealing with the question of what the learner will derive from the instruction. The reviewer THE TRUTH must be provided a procedtue that results in answers to these two questions: (I)When will tin' learner be observably able to do after instruction with these materials that he could not do before instruction?(2)What is the evidence for any claims of learner acquisitions?If the 91

icimr. 3.27. The Bandwagon Effect "Look at the list of school systems that have already adopted thistext.it surely must be good."

FIGURE 3.28. "l'he Advantages of High I. Q. (Identifi«ction Quotient)"I like Ibis seiies because these authors have always written good material." (.11.p mit ismtEl.

"What was good enough for your father is good enough for you!"

Fic.nzi3.29. Ohl Frietul.s Arc Br.q "Irr .shill((/ adopt this textbook because Ilta impwlattl, familia). itlea. ale ,slill treated as ! hey aheays have been.-

"The decor is lovely."

Ficour. 3.30. "1 /u'iIhtstrallons are magnilhenl and Ahoubl )eally help motivate and retain interest.- rfil n\ rr,00k\s,vas t-Gi 10\ \I\In 93

"Tell you what I'm gonnado----'

ne.tutp..3:31,, Isn't Relief Wonderful; "The sales representative's preAenhv lionzrosderailed.wallorganized, thoughtful. and authoritative. lie said that the sequence of exer(i.w.s was (are. fully thought out and the vo«thuhur level is aimed right at our ,students,'"

"Here is the proof that the sequence works!"

FIGuRr. 3.32. " l'hy, this material is by someone who worked on the writing team that produced XEI if. /1 follow.% their outline but doe.cu'l have the typographical errors; it has more exercises, and the vocabulary is pitched at a more applopriate level," 94 LI I XI' I I !MEI'.

of cotmse. not an t)I these Conditions pie. theobje( likesofthe selectedtextbooks ate tailat each tvxd)ook adoption. and nom: ina) a( hie% ed.I he e% idem e i,te,ented N110111(1 .11ie.111. be piesent at Nome. The chances are. though. CIearibe the IRA ( COM* 01 0011:4%111% eN110,431 to that some one or mote of these are lecogni/able. the material who a( quit e each beli.kioi. .\ de- Itis deplotable that the selection 01 such .1 Cl hi- s(liption of the population styph big the data (.11 hist' me 6011.11 aid might be influenced bit such should also be pm% ided. If cnidcine is piesented outside c t het id. The (Illesi 1011 110W be(0111CS one and the letel of acquisition is ac(eptable for the of %%lietlietitis possible to olgaidie .1 poicednre population assz.ssed. then the text should be te- of textbool, selection %%hick is objet conceit- mined for !miller consideiation. If no evidence 1.1 It's on ins« action. ICI( 11st's 011 expected IC.11.11C1 of !camela«piisitionispiesented.thetext 0111 10111CS. Mid is III-MC(1111C 111.11 d 101a1 school should be discaided. If the lexel of acquisitions NSielll can Carly 0111. reported is considered too low, then the text should be tentoved Iron immecliate considera- A POTENTIAL tion but not completely discaided. Au a« eptahle levelof auptisition will SELECTION PROCEDURE vary among systems. Perhaps a reasonable (6'c:tutu can be that 90 Suppose atextbook selection committee has percent of the exposed learners acquire 90 per- been given the change of choosing arithmetic cent of the belt.kiots desci ibed In the objectives. texts for wades K-6. What might be the se- For those texts that have not been eliminated quence or p1 ocedul ch if itis agleed thatinsttui- a content check should now be made. Each text- tional objectives are to be stated in performance book should be examined in terms of the integ- language? thy of the content. More than 2 percent error For a while. at least. the first step in the selec- (2 emits per 100 items checked) would elimi- tion procedure is to see to it that each member nate the text from further consideration. of the selection «numittee a«ptires the compe- The learning sequentes of cat iof the texts tencies necessat )to construct beim% im al objec- remaining altet the «nitwit examination should tk es and leaming sequences based on the stated be examined. togethet with the supporting exi- objectives. Oncethesecompetencies ateac- deuce of learning- dependent v. Texts withmit quired. the second step can be takenthe mini- such information vould be eliminated from the mal peiformance expectations for each of the list of possible choices. IZemaining texts would seven years of instruction can be constructed. be ranked for acceptability on the basis of how The descriptions of these performance expecta- many of the desired objectives are part of the tions for each year a e statements of what a learn- piograin and the efficiency of the learning seer is expected to be able to do after the year of quence in assisting the leanters to acquire the instruction that he could not do before instruc- statedbehaviors.Figure 1.33 summari/es the tion. These statements should, of course. be suggested procedure. written. This reduces the likelihood of misinter- Because few (if any) currently available mil- pretation as well as facilitating memory. tnercial or experimental mathematics textbooks The third step is to examine the behavioral contain statements of objectives written in be- objectives of each textbook considered a candi- havioral terms, the suggested pocedute would date lotadoption. Those textbooks whose ob- probably eliminate all of the possible textbooks jectives match at least a simple majority of the at the step that calls for behavioral objectives. objectives stated by the committee would be le- This %vould result in the outcome that no text- tained for further consideration, and those that books would be purchased for a short time. "A do not meet this criterion would be discarded. short time" is predicted because the profit mo- The next step is to examine the evidence that tivewill supply the necessary motivation to ARE DESIRED LEARNING OUTCOMES DESCRIBED IN BEHAVIORAL LANGUAGE?

O YES O NO

DOES THE TEXTBOOK PUBLISHER PRESENT WRITTEN BEHAVIORAL OBJECTIVES?

O YiS O NO

DOES THE LIST OF BEHAVIORAL OBJECTIVES INCLUDE AT LEAST 51% OF THOSE LISTED SY THE LOCAL SCHOOL SYSTEM?

1 O YES O NO 4r

IS EVIDENCE PROVIDED OF ACCOMPLISHMENT OF THE STATED OBJECTIVES. AND DO THE OBJECTIVES MEET THE SPECIFIED MINIMAL LEVEL?

O YES O NO

IS THERE MINIMAL CONTENT ERROR?

O YES 0 NO 4

DOES THE TEXTBOOK PUBLISHER PRESENT A SPECIFIED LEARNING SEQUENCE AVID EVIDENCE OF ITS VALIDITY?

0 YES 0 NO

CONSTRUCT A RANKING FOR THOSE TEXTBOOKS REMAINING. HIGHEST RANK IS GIVEN TO THE TEXTBOOK WITH THE GREATEST NUMBER OF OBJECTIVES IN COMMON WITH THE SYSTEM'S OBJECTIVES AND THE HIGHEST EFFICIENCY OF ACQUISITION.

1 ELIMINATE SELECT THE TEXTBOOK WHOSE RANK IS THE HIGHEST. THESE TEXTBOOKS FROM FURTHER CONSIDERATION.

FiGtutr :;.33. Flow chart of procedure for the .%ele(lion of a textbook. based on behavioral objectives and learning hicrarchic%. Input is all textbooksunder consideration. 96 \ VIER Ii 11: EV.,

cause such information to be provided. This working won') discatded the «meutional pro( cdtnes of textbttok selet tion and de( ided to Coin /promise Procedures include behavhnal objectives among their criteria to be applied in the selection of textbooks. should lou Wish to delay the advent of mathe- Members of the committee tecognized the lack ma tit s textbooks containing behavioral objec- of stated behavioral objectives and accompany- tives and supplying evidence of the objectives ing evidence of acquisition on the part of the accomplishment by a substantial portion of the available textbooks. but they decided some selec- exposed le:mien, you may have the selection t ion n"eded to be made. The) also acknowledged committee try to second-guess the authors. The that there was neither the manpower nor time committee might try to describe the objectives available to write objectives for all the available for each potential selection. The limitations of texts, but they wanted to try the behavioral pro- this compromise procedure are obvious: (1) the cedure. So it was decided that the behavioral guesses the committee makes may or may not be objectives and learning sequences would be con- the objectives the author intended. (2)no evi- structed for a few selected content themes they dme of leaner accomplishment of the objec- consideredcriticaltothe elements n mathe- tives is available. and (3)the time involved in matics curriculum of their communit). Samples constructing the behavioral descriptions is be- of the products produced by the committee and yond a reasonable expectation for such a com- used in the comparison of three volumes are mittee. However, if there is a pressing. desire to included next. The illustrative material deals change textbooks and no willingness to wait out with the description of behavioral objectives for the assumption of responsibility for objectives meastnemem. Observe the dillerencesinthe and evidence by the authors of textbooks. com- number and quantity of objectives in the three mercial publishing houses. and the development textbook series, as well as the differences in se- projects, then an attempt at the local level may quential mganiiation. be the best recourse available. If the local school system undertakes this task TEXTBOOK A the principal difficulty is the time required to produce behavioral descriptions for each avail- GRADE 1 able textbook. The service proposed by the I.Demonstrating measurement of length using a nonstandard unit (does not introduce inch, PennsylvaniaRetrievalof information in foot, etc.. and advises against these; no line Mathematics Education project is one example segment: inequality signs used. but poorly of how thistask might be accomplished for related to concept of mde) elementary mathematics, grades K-6, and the GRADE 2 results shared among school systems. The United 1. Naming and identifying line segments 9. States Office of Education's project ES 70 sug- Naming and identifying standard units of ineastu einem gests another way it -vhich behavioral descrip- 3.Demonstrating cutler on the number line tion may be made available. using inequality signs (number line shown One of the school systems in the state of Mary- by rider) 1.Naming the length. width, and height of an land, when faced with this problem, adopted object stillanother strategy worth describing. The 5. Naming closed figures Frederick County school system appointed a com- GRADE 3 mittee to select clementar) mathematics texts lor I.DemonstratingmeasurementwithI inear oracles K-6. The committee invited the state measuring devices 2. Naming and identifying inch, foot, and yard supervisor of mathematics. Thomas Rowan, to 3.Identifying circles, squares,triangles, and assist them in the selection, and he accepted. parallelograms 1.1. 1 :1 N.1.1 s\ aiv L6

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Iec't to pick. and miles to rods 5. Demonstiating measinement ofheight by 12. ldentiling and describing- square inch two males on sline tide, 13.Idemiling square loot 6. Identiling and naming an inch 11.Describing square loot 7.Idemil) ing and naming a centimeter 15.Appl'ingj the for finding :ileain square S.Demonstiating the measurement of length units (loot. aid. mile) using an inch and a centimeter (onsame 16. Idemil ing :nu! ruler) 17. \ppfiing talc lot findingal ea of rectangle. 9.Demonstrating me:ism mem of length that nsiog at l es does not collie out "just right," hence the IS. Appling rule lor finding alCa in square identifyingorzippoximalion units to solution of sun y problems GRADE 2 19. Identifying and naming a cube 1. Identilying circles and polygons 20.identifying and naming cubic inch, cubic 2. Constrating ruler and pasting on cald- loot. :wild cubic ard boa rd. naming units on black scale inches 21.Dem nnstatlug volume ofrectangular and on red scale centimeters 99. 3.Demonstrating measure of height by two Stating rule for finding volume of recningu- scales on the same ruler lar prisms "I.identifying and naming an inch 93. Applying rule for finding volume ofrectan- 5.Identifying and naming a centimeter gular prisms with given dimensions 6.Demonstrating the measurement of length 21. Applying rule for converting all dimensions by using the inch and by using the centi- of closed figure to like units meter (both scalds on same tiler) 25. Constructing rectangles with sides of given 7.Demonstrating the measurement of length lengths that does not come out lust right," hence 26.Giusti ucting squall e given length of side the identifying of approximation 27.Constructing a tria»gle with sides congruent 8.Constructing lines of given length to given segments 9.Distinguishing between inches and centi- 28.Idelltik lug units of measine in' metricsys- meters for a given measurement u'in 10. Naming the unit along with themeasure- 29. Oidering 'mils of ineastne in metricsystem ment 30. Applying 1 the for changing larg:: to small 11.Demonstrating the measurement of a given and small to large measures in metricsystem length by using both scales 31. Appl)ing rule for finding area of a rectangle 12.Demonstrating the measurement of liquids in metric units and of time 32.Identifing and naming relationshipsbe- Wee!' English units of linear measure and GRADE 3 metric units of linear measure I.Naming and identifying a foot 33. Appling mules for converting English units 2.Naming and identifying a yard of linear measure to metric units of linear 3. Naming the length of an object measure 4. Naming and identifying the area ofa sur- 31. Consti ncting a circle. given the radius face by counting square units used tocover the surface 35.ldentifyi»g and naming base and altitude of a triangle 5. Naming and identifying the volume of solids 36. Applying rule lor finding area of triangle by counting number ofpiecesof cake (blocks) . 37. Applying rules lor finding area ofa rectangle 6.Demonstrating informally the and area of a triangle to find areas ofpor- three types tions of rectangles and triangles of measurement: linear (wires), area 38.ldentifii% and naming an octagon (squares), and volume (blocks) 7. 39. Applying rule for findingarea of a circle Demonstrating measuring of area bycover- ing a '10.Applying ride of scale drawing using various surfacewithcongruentshapes ol);ects (squares) 8.Distinguishing types of measurement using TEXTBOOK C examples from environment GRADE 1 9.Naming, and identifying unit of lengthas a 1. Identifying squares. circles, and triangles segment 2. Naming unit of measure-nonstandard unit 10. Naming and identifying unit ofarea as a 3.Demonstrating measurement of height using sq oa re blocks 11. Naming and identifying unit of volumeas 1. Constructing ruler with red and black scales a cube (red for centhneter, black for inch) 12. Naming the length of an object 100 GI I \r I'EIri 'REF,

13. Naming and identil)ing the area of a sur- -I I.Constructing geomen isfigni es containing a face by counting units used to cover the giN en number of squat e units surface 19. bemonstrating and naming an estimate of 11. Naming and identifying the volume of a the number of square units in very in egular solid by counting the units used to fill the figures solid 13. Naming and describing a cubic inch 15. Demonstrating lengthby countingsteps I. Naming and describing a cubic centimeter (segments) across classroom 15. Demonstrating and naming an estimate of 16. Distinguishing between small, medium, and the number of blocks in an array huge steps .16.Demonstrating and naming the volume of 17. Naming the step as a segment block arrays by counting the blocks IS.Demonstrating that the smaller the unit, the -17. Nami», amid identifying the volume of a greater the number for a given length variety of geometric solids by counting the 19.Constructing individual units of measure number of cubes (piece of string) GRAD': 1 90.Naming and identifying- a mile 1. Applying rule changing feet to inches, feet 91. Demonstrating an inch as approximately the to yards, and yards to inches length of a section of the forefinger 9.Naming and identifying area 92.Demonstrating a foot as approximately the 3. Naming and identifying volume length of the forearm . Naming and identifying terms used for mea- 23.Demonstrating the mile as approximately suring length (segment), area (square). and the length of a train with 120 boxcars volume (cube) Demonsn sting the yard as approximately the 5.Identifying and naming types of meastue- distance ber.veen the wrists when arms are mem with nonstandard units outstretched forming a straight line 6. Naming line segments 25. Naming an appropriate unitof measure 7.Demonstrating linear measurement using the given that which is to be measured and a inch as a standard unit and accuracy to the choice of units (inch, foot, yard, or »mile) halfinch 26.Demonstrating that a centimeter is shorter 8. Naming and identifying lengths of objects than an inch 9. Naming and identifying half-inch on a ruler 97.Naming the length of pictured objects using and with objects both units, inches and centimeters (all an- 10.Demonstrating that a centimeter is shorter swers ate whole numbers) than an inch 11. 28.Constructing centimeter ruler at least13 Intelpreting distance On a map centimeters long 12. Interpreting scale drawing, using a map on 29. Naming the measurement of length to the which a centimeter represents a mile nearest inch 13. Naming and identifying a square inch 30. Naming andidentifyinghalf-inch asa 11. Naming and identifying a cubic centimeter "pinch" 15. Naming and identifying the area of various 31. Naming the -length of pictured objects in objects "pinches" 16. Naming and identifying area by counting 32. Naminginch on a ruler units of area 17. Naming area of a given figure using a grid 33. Namingcentimeter on a centimeter ruler 18.Demonstrating theperimeter(giventhe 3.1. Naming the measurement of length to the term). using a standard ruler and using a nearest L:inch centimeter ruler 35. Naming the number of subunits contained 19.Estimating- the differences in perimeter of in a huger unit (I in. = 2.51 cm, Ift. = 12 various objects in., I yd. = 3 ft.,1 yd. = 36 in.) 20.Identifying a triangle 36. Naming and describing a square inch 21.Constructing a squat e 37. Naming and describing a square centimeter 99.Naming a length using a centimeter ruler 38.Naming and identifying the area of a variety 23.Applying the idea of using :t ruler to find of geometric figures by counting the number time perimeter of units (squares) Distinguishing between volume and surface 39.Naming and identifying the area of a variety area of geometric figures drawn on "squared" 95. Naming a square centimeter paper by counting squares 26. Applying a rule for changing a mile to yards .10. Demonstrating that two triangles (isosceles 27. Naming the number of smaller units in a right) make it square larger unit 1'1 IF, 1 It001.\'\>.INS' Rl'CTION \ 1.\II) '101

25.Distinguishing the approptiate unit of mea- 26.C:onstucting space figure cube) Irons plane sin e-inch, loot, yard, mile fignie and again finding sur1ace area by 99. Identiling liquid measures counting same :Alum es as in item 25 GRADE 5 27. Constructing rectangular solids and demon- 1. Naming and identifying unit of length as a so a t ing situlaw in ea segment 28. Constructing a cylinder and demonstrating 2. Naming width of index finger as a segment surface area 3. (.oast tic ling t vith width of index finger 29.Demonstrating sm face arca x% ith a arieo of as the-Segment space (iglu es, given their dimensions .1. Demonstrating measmement of length by 30. Naming and identifying formall) time (1). (miming number of segments rate (r), and distance (d) 5. Constructing halfsegmcnts on ruler (where 31.interpreting the formulas = / x 1. segment is width of index finger) r =d- 1, and / = d 6.Distinguishing which of two measurements 39, Describing the use of the alume distanc e for- on the hand is the greater mulas 7.Demonstrating measurements with results to 33, Naming and identil)ingvolume by count- nearest half-segment ing cubic units 8. Constructing a "N hoe- segment and dividing 31. Naming and identil) ing xolume of it into eight eqnal parts. each called a "toe" hat solid by multiplying the three dimensions 9.Demonstrating measurement with "shoe" 35. Interpreting the formula = 1X w Xh tiler 36. Applying the rule to find the volume of 10.Demonstrating use of accepted units such as rectangular solids centimeter, inch, foot, and yaul 37.Applying the rule for changing cubic inches 1 i.Demonstrating measurement with standard to gallons units 38. Applying the ride for changing cubic cen- 12. Stating a rule for finding the perimeter of timeters to liters a polygon 39.Demonstrating measurement of an angle by 13. Applying the rule for finding the perimeter counting angle units of polygons the measures of whose sides are 10. Constructing and counting measure of right whole numbers angle using unit selected H. Determining the meastiles of the sides of a Naming and describing standard angular polygon by using a rider and then applying units: radian and degree the rule for finding the perimeter 19. Describing the protractor as a device to inea- 15. Determining the measures of the sides of a sie angles polygon by rolling ixdvgon along the ruler 13. Constructing a protractor and demonstrat- and thus finding perimeter ing its use 16. Naming and identifying circle, ciicumfer- ,1,1, Naming the angle unit on the pi otractor as ene and diameter a degree 17. Demonsmiting that both perimeter and cir- 5. Applying the rule for changing one linear cumference mean the distance arounda measinement to another circle 46. Demonstrating the conversionof smaller 18. Naming and identifying area of rectangle units into Lager units (as a result of addi- by counting squares tion) or larger units to smaller (as needed for 19. Naming and identifying area of rectangle subtraction) by multiplying two dimensions -17. Demonstrating the reading of a ruler with 20.Interpreting; the formula for the :ilea of a various degrees of accuracy:s inch,; inch, rectangle, /1 = 1 x 'I,: inch, centimeter, 1111 centimeter 21. Naming and describing parallel lines and 18. Demonstrating relative lengths of two seg- parallelogram ments 22. Demonsuating the formation of a rectangle 19.Demonstrating measurement of sides of poly- from a parallelogram by cutting triangular gons to nearest;inch and finding the pe- section lion> one side and fittingit along rimeter other side 50.Naining and describing metric units: meter, 23.interpreting the formula for the area of a decimeter, and millimeter parallelogram. 4 =b x h 51.Describing use of decimals in measurement Naming and identifying surface areafor 52. Applying rule for changing one metric unit space figures to another 25. Demonstrating surfaceareaby counting 53. Naming and describing the comparison of squares in a plane figure two sets of objects as a ratio 1. ZO \II') d :11.1 I X111

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51. Guist' tt( ting squares on three side, of light des(tiptions 10justsunk a purpose. I'he ad. triangle :111(1 finding area of each ""cage II" ing to the nse "1 ele( don 1)1(' 52.Naming the 11)potentise and legs 01right triangle (Mute hosed on behavioral descriptions ale the Stating andapplyingthePythagol can dia"ge 1"11 a subje( Ike base to objet ke theolem 011C. the a1:tilabilh Or C1 ideme that an institu- 51.Interpt (ming word pntblents involving mea- tional aid accomplishes its stated purposes. and NUIVIllent 53. Naming andidentil)ingunitoflength. the (onsirtu lion of instill( limed sequences from meter: unit of weight. grant: and unit of onsiderations of what is to be learned and the (op:idly. liter condition, tinder whichit istobe learned. 513. Naming and describing metlic units Under stuh a procedure the burden of wool 57.Intel pleting %void ploblems involving metric and English measures that an instructional aid does accomplishits 5S. Naming and identifying cirennifelence staled purpose %void(' test with the distributor. 59.(:onstt titling circle 3 centimeters in diam- Two recommendations concerning the selec- eter. cutting it out. ond measuring circtini- lereme by rolling it along a drawn line 30 tion of a textbook follow. (einimetels long: repeated for circles with REcomNicNit.vrioN 1. The procedure employed diameters of -1. 5. 13, 7. 8. and 9 centimeters in the selection of a textbook should include (I) (;O. Naming spec ial fa( tor in each ( ire tun let ence by dividing circumference by diameter the description of the desired instructional out- (H. 1)escribing resulting factor as mole than .; comes in terms of behavioral objectives.(2) a and less than 3.3 and naming it r «)1111):111SOil()Ithe stated behavioral objectives V. Stating and applying the 'tile that circinn- in available textbooks with those stated by the leren«. r, >, diameter selection committee. in connection tvith the pro 63.lientonst rat ing procedure for est limiting :ilea ofa ircle by counting ;lumber of square cen- cedure outlined in Figure 3.33. (3) examination timeters (this will be an estimate!) of the textbook publisher's evidence that the be.. 13.1. Naming and identil)ing radius of a circle l'aviors des( ribed objecti \ es Were .1( ed and 1)emonstrating area of circleas close to h) whom the were acquired. and. finally. (4) 3 X r2 fill.Stating and applying rule Al = an examination 01 the leanthtg sequence and the evidence 10 support the hypothesis thatitis a The technique emplo)ed by theFrederick sequence. Count) textbook selection committee did result ItcommENI),ATIoN 2. The selection of every in501111' ObjeCIIVC description of the various instructional aid should include at least the first textbooks under consideration. Sequence defi- duce steps as suggested in the previous recom- ciencies in terms of missing prerequisites were mendation for textbook selection. identified for the topics examined. Content errors %vele also identified for these topics. The limitations of this procedure are alp:ti- EPILOGUE ent. The restricted number of topics examined The advantages of behavioral descripl;anS and may produce a distorted description of a text. feat ning hientichies extend beyond the selection I fowever. for the topics examined the applica- ()Itextbooks to include assistance in the design tion of the plocedure does yield an objective. of instruction as well as providing guidance in detailed description. the supervision of instrtution. A bold proposal, Clearl) the procedures adopted in the selection indeed, that as teachers we spec if) the expected of hist' tutional aids can be substantial!) altered outcomes of our teaching. Imagine proposing I)) the use of behavioral descriptions of expected that the meastne of out competency rests with ()townies. The piocedme de..ailed earlier it this Midence ()I obsenable learner acquisitions! chapter is one means of emplming behavioral The challenge is made. The next step is yours. '104 ( .1 1 \I' I LIZ1111:

1 1 1 11 . 1 0 G :k1 ) 1 I1' I.\dams, Daniel. The Si holm', Alithmetri. ;ink: ( I.11(.11111'ml. 1915 tow 1(41111 helms.. 1808. 2.2- \Id ell.m, 1.11111, 2,hat mod. I.\. P.I 1 tiothe ..Ittllonetti. Ilatt ..1)(1 Hui 1)(,*(1, Tile I'''. 14n114..1, o! I . ott».: D. l.. \Vomit till k atimi% to 31 eth 1830, oil% in lrarbin,,I r 1th New k: 1). Apple 3 Ruttt,1 latold 1..11suot " \11\11.11,i, 01 1.uk , co \mctir,un Arithtnelii1 extbtroks 11110101 1510." 23. Dissenation. 1IIitet,in 01 Pittsbutgh. 1958, AI,I,:,,:14'scri,.,1t"h,\''11:0.1. Gathithi 21. hifyriltit, 1911 ed. SA. "I he C.oc 31111vIand larmnta, Alathematn, /mil;n e CIti,m id the Count il of Tient." Alum. (.'!"r14c Palk: °1 _ SA. -Count iI of 1:1111/." Land \Lithematits I'roje(I. 1968. .S.%. "Peter (..misitts." 25,P.olo. hart P. Conditioned nellexe.s.London. 7. Collmtu, Wallet!. ;1i:1bn:et:I. /long a senor! to (1siold rtin et 11, %. 1927 the 11s%t 1.3-%%mi% in .1,:thmetn. Ito,ton Out: 26.Pike. I Neue and cimiplete Sv%ternul ming, k !lila:nil. 1822. Intl: methNewhtttvitott. Nlass.: John Nht all. 8, ,Iv huh:net:I im the Plan oI 1788. with Some Inip,mvemenh. tailed I 11:1 Itay. pm:1th. .4titlitnetti, 1.1,%I. Pent So (mil, /(///1 ill 11 h Met 111,11\10w I .111(1 15111 Thnil, Ciminnaii: W. It.Smith k Co.. Haul. 1821. 1812, 1813. !I.Daboll Nathan. flabitll's Schoithninter'. 28. . 'Arita .4,ithmetai, W. It. hint. New Lon(fon. Conn.: Samuel Cueen, 1800. Smith k Co 1837. 111. Day. let entiall. butodui nun ti,Igebta. Neu 29. . The / title .11t1hinetti. Cintirmati: \\". 11 :n en: I I.:we k Deforest. 181.1. It. Smith k Co., 1831. II,I)eve. lulu:. nem"' m'Y an't :to, ,11inle,ii giementaly .11ithmeta. New Voik: Co.. 1916. Yolk:\ffleritatt Book Co.. 1903, 12 I) ii 4041 th. Thomas. Si hoillmii,te,',.1%%1%tant.I'llila 31. 1 ode, n14,m to al . it 1, met . New Joseph (a 1110.14111k. 1773. Volk: Atm:titan Book Co., 1903. 13.IInC114111. Fie Icnitk.. \'u,Il:.Ime,ur:::.I,:Ilunrl:r, :o2. - Table.% and Il,rles in Witlimetu fin Pint 1, Purl 2, and Mitt i, Roston: eak. k Palmer Childien. \1'.It. Smith k Co., 183.1. 1830. 1831I. 33.I' n de, Robot, The Grounde of /1, tes. London: 1.1. Fient h.joint. The Filsi 1 e%%on% It. \l'olle. 151:1. New Yolk: I ht per lhotheis, 1870. 31.Root. lot astus. 1ntioduition to .1,i1hmetii. 15. Emilter, (arilF,\ "Ati\ nabsis of Sch.( ted N m i h:1796. Elententaty Wallin:titTexts Published in the 35. Smith. I lent y Lester. and1(.1till Eaton. Bulle- rnited States of America during the Period 1877- tin in the Si hoot al Piling:lion (of Indiana Uni 1917," Disset union. 1"ttiversity of \t:t yland. 1961, etsit.1 18. 111.4. 6 (November 1912). 16. Gagne. Robot The Conilitionc of I.eiunnig. '16. rhomp,ou. Ceinge. The arne,loin 11,C., ( Rev. ed. New York: Holt. \Vinston. 8 I loslind Co., 1808, 1970. 37, hoindike, Eilwald L. ../ttintit/ Ittlelligettie: 17. Gteenwood. ..1,:thmeti(1: l'ulgo, and 1)o penmen/0/ SludieA. New Volk: 1:tentillati Co,. mal: with the .1 al:on thin eof to a mislay 01 1911. Can., ,n Tlade, and Commene. Itostim: ,I lam 38.Venetna. Pieter. .1ollutteluit of Cyffer.A'int.q. New (ink. 1729. Yolk: J. Peter Zenger, 1730. I lhal(ler. fames, .111llinteni, illThai X el la1101V 39.\ 'albemer,1 lem yII.Constituting Ilehavioral .I)( Made .1 1 0%t1:054.Boston: S.Phillip,. N. ()Nei live,. Rev. ed. College Palk:Atat).1:111d 11(mi, Ituttolph. It. Elliot, k N. Negua, 1719. Exchattge. 1970. 19.,James, of 1'ch"1"gV. .10, fiitn/itattint Allude/ and Its ilppina. New Volk: licitly llolt k Co.. 1890, tam. Soignil \\*ashington. 1).C.: Commis 20.Littlefield, Ceinge Emery. Ern/ 110013 arm :ion on &lent e Edna:taint 01 the Auto ion Ass°. School Rooks of New F.hgland. New Volk: Itus t iation for the Advani anent of Sciente. SAAS sell k Russell. 1965. lisiellaneitus Publication 68-1, 1968. I.Li% ermot e. Ceinge. The Origin. 111310,y, and 11.\\raid. John. Young .11atheniati(iines Guide. 1.on Character o1 the -New !England Primer." New don: Samuel Fuller. 171)7. 4.OTHER PRINTED MATERIALS ;:,

fp 107

CHAPTER 4

OTHER PRINTED MATERIALS

by H Lot.: H ownEN Random House, New York, New York JOUN N. Fuji' Merritt College, Oakland, California

A variety of printed materials thatcan be used to supplement. a mathematics program is examined in this chapter. Reasons why such materials should playan important role in a well-rounded mathematics curriculum and specificways in which they can be used effectivelyarc also discussed. Fifty detailed examples illustrate exactly howa teacher can use a wide range of Other Printed Materials. Each example specifiesan OPM, a function, a technique. and the pule level recommended for the introduction of these materials.

A FILM THAT SPANS ANY CLOSED FRAMEassumes a contour with minimum area. This property can be illustrated withsoap film. When air drying plastic is used, the resulting film ispermanent. The models shown in the picture were formed by dipping wire framesin a substance called Formafilm. 109

4. OTHER. NIATEIZIALS

Have you ever wished you couldvary the con- thingthereistolearnabout mathematics. tent and approach of the mathematics program As a result. much of the materialthat is not you are teaching without abandoning its basic required by various curriculum guidesis often mganiation? Have you ever wishedyou could published only in OPNI form. divide your time so that you might help students Somematerials.particulalythosewhose who need extra practice and at the same time primary purpose is recreation, application,or renew the motivation of students whose inter- individualised drill and practice.are not adapt- est is waning? Perhaps OPM (Other Print, d able to the standard textbook format.These can Materials) can help you'realize your wishes. be presented most effectivelyas OPM. In summary, OPNI ate producedto help keep WHAT ARE OPM? the curriculum up to date andto give it balance. How OPNI can be used to accomplish thesegoals In this chapter. OPM ate consideredto be in- is the most importantcmicern of this chapter. stuctional aids inmathematics consisting of pamphlets, library books, reference books,re- How are OPM used? souice books, paperback books. skill kits. independent units, curriculum guides, charts, pictures, To learn how OPM arc actually used in todafs "free and inexpensive.' materials,newspapers. schools, the Yearbook Editorial Committeesur- periodicals. student journals. bibliographies. and veyed nearly a thousand mathematics supervisors so on. Standard and programed textbooks. pre- across the country. Here are illustrations of what pared to be used for regular class workas basic the committee found. instructional aids, are not considered to be OPNI. In one junior high school,a teacher uses the local newspaper in general mathematics classes. Iifiry Are OPM Being Produced? Clot-cry ads are coirpared in _preparinga shop. ping list. Advertised prices of various kinds of Curriculum development in school mathematics cats are compared with the total cost when pur- is in a state of continuing change. Since it takes chase is made on an installment plan. time for textbooks to be written and published, In another school, a teacher capitaliseson the OPNI, which can usually be producedmore enthusiasm of airplane bulls. Attention is tut ned speedily, are being used to obviate the time lag. from tossing paper airplanes out of classroom Furthermore. until the experimental aspects windows to reading The Great International ofcurriculumdevelopmentarethoroughly Paper Airplane Book (OPNI Example 3, de- tested,itis financially impractical to include scribed late* From this book. sunk:tits whoare them in standard textbooks. However,many of interested in airplanes learn about the relation- these experimental concepts and approachescan ship between the cent, r of gravity and thecenter be published in OPM form and usedin con- of the lateral area of paper airplanes. Mean- junction with a standard textbook until they while, students who are not interested in the arc popularly accepted and integrated into the mechanics of airplanes arc challenged by the curriculum. problem of creating fabric designs that illustrate Another reason for the ptoduction of OPM the renowned four-color problem, moirepat- is that authors of textbooks frequently follow terns. or one of the network problems from curriculum guides produced by state and local topology. These concepts can be investigated in school systems. and these do not includeevery- one of several OPM. 110 (:11 \ PTEI:

In another junior high school. teams of four (and man( mole than ate described in the es students each competed for two months in all amples) can be«imbincdxvitheachfunc- investment project. Each team began with all tion. .1Ithough many OPM ale multifum tional. imaginary 5500. which could be hucsted in ally none was classified under more than one func- wav the team members :tweed on. Sales and new tion in circlet to litchi& as peat a lepresentation pm chases were also agreed on by the team. After of OPM as possible in the fift examples given. the initial olganiiation in class. all investigation of news media and professional literature that MOTIVATION ledto agreements concerning the imaginary transactions and all record keeping were per- Motivation is a very personal thing: what moti- formed outside of class. except for a weekly up- vates a particular student may be quite apart dating of progress that was kept on a chart in from the comse content. Within the course con- the classroom. lute:lest ran so high that several tent. however. motivation is generally more pre- of the teams continued the project for the school dictable. For the purpose of the examples. thei year and even longer. fore. motivation will be considered the presenta- These and other examples that could be cited tion of content intended to promote student in- ale perhaps somewhat unusual. Others that were volement in learning. reported in the sill vey were more closely 'elated EXAMPLE 1 to the curriculum wider study. of shorter dura- OPN1: Soap Bubbles, :id ed., by Charles V. Boys tion. or less demanding of the students. Most (New York: Dover Publications, 1959) examples repotted can be classified under one or GRADES: 7-12 mole of the following seven functions: TECHNIQUE:Class demonstration (presented 1. Motivation 5. Drill and practice by two menthe's of the class) 2. Discovery 6. Application Refer the students to the OPM and ask them to 3. Enrichment 7. Professional growth experiment with the following problems in mind: 1. Change of pace 1. Why are bubbles that are blown from a pipe The techniques for using OPN1 ran the gamut or flicked from a small metal loop always from occasional inclusion of interesting asides spherical in shape? by the teacher to independent student research. 2. How do the areas of soap films compare when The following sections contain examples that the film is bounded by a loop that lies in a illustrate how various OP,N1 can be used to serve plane and when is bounded by a loop that each of the seven functions listed above. To does not lie in a plane? Why? make the ex:imples specific enough to use as work- 3. Can you predict the shape of the soap film able models, each is based on a particular OPN1 that will be formed if the wire frame is made and a particular technique. Where the OPM is into various geometric forms such as a three- used directly by the students, as in the case of dimensional figure-eight loop or .1 Spiral- workbooks and drill kits. directions for use arc shaped theedimensional loop? not detailed. They are given in the teacher's ma- 'I.Can you predict the shape of the soap film terials that accompany the OPM. For other types when the wire frame forms a cube? A tri- of OPN1. the examples are explicit. angular prism? A tetrahedron? These examples are by no means exhaustive; Incidentally, although the OPM suggests for- they constitute a small sample of the available mulas for soap solutions, the solution included OPM and the many ways that OPM can be used. iusoapbubble kits that can be purchased in any Since the function of an OPM is a basic consid- variety store or toy store works very well. Per- cration in its selection, the examples are classified manent models (an be made by using Formafilm. according to function. Any one of the techniques (See the picture at the opening of this chapter.) OTHEI: 1'121 X l'El) NIATEItIALS 111

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EXANIPLE 2 3.Each contestant must demonsuate the fliglit OPAI: Malhemali(al Snapshots, 3d Ametican chatacteristics ol his .tit plane it .1 designated ed..te%. and enl..by Hugo Steinhaus (New time and place. YOl*k: Oxford University Press.1969) 4. The winner will be judged on: GRADES: 9-12 a) The completeness and accuracy of the en- TECHNIQUE: Introducing the next lesson try specifications sheet A lew days before a topic is to be considcted by b) The workmanship and neatness shout iu the class. ask one member of the class to investi- the airplane gate a related "snapshot- and to report on it the c) The flight characteristics of the airplane day the topic is introduced. demonstrated at the designated time and . \s anintroduction to the study of meastne. place. ha\ e .t student report on the following theorem: Alew sample impel airplanes should be shown Ice, me The am ea 01 polygon w how and demonstratedtothe students.For each points ol a lattice is equal to one less than sample.has ea prepatedentry specifications the stun of the number of interior lattice sheet. In the earlier grades, it may be necessary points and hall the number of lattice points On the bottler. for you to show the students how to find the information On the specifications sheet for each This them em is discussed and illustrated on sample plane. Older students should be referred page96ofthe OPM. As more conventional to the OPM itself or any other sources they may methods of finding areas of various polygons are wish to use. studied. their results cant be checked against the Set the contest date about a week born the results found by using this theorem. time that the rules are given, and reserve part of EXAMPLE 3 a class period in the interim for clarifying the OnI: The Great International Paper Airplane t ales and answering questions. You may wish to Book, b J. Nlander, Dippel. and II. Gossage encourage lecognition olthe participants in a (New York: Simon 5 Schuster.1967) newspaper story (school or local) or at a school GRADES:6-9 assembly. An exhibit of the airplanes inthe 'EECI INIQUE: Contest school display case would also be an appropriate An active involvement inlearning can be form of tecognition. gained I)) proposing a papenairplane contest to EXAMPLE 4 be held at some future date. The stage for the Sphereland, by Dionys Bulger. trans. Cot.- contest may be set by preparing a contest plan nelie J.Rheinboldt (New York: Thomas Y. and drawing up a set of rules. A sample of such Crowell Co.. Apollo Editions.1965) a setorrules follows: GRADES:7-10 I.Each contestant must make his own paper TECHNIQUE: Question of the week airplane. Select one of the books from a library or book- 2. An entry specifications sheet must accom- cart collection. and devise a question concerning pany each airplane, describing as accurately a concept thatis discussedin an interesting as possible: manner in the book. Post the question and the a) The :ilea of the lift name of the book on the bulletin boatd. At the b) The lateral area end of the week. discuss the answer briefly in c) The center of the lift class. d) The center of lateral area The following are a few sample questions con- c) The center of gravity cerning information contained in Sphere/and: () The weight (determined with the aid of a I. Why isitthat anyone who continues to good balance scale). travel due west will eventually approach his OTHERPRIN-Ii 113

starting pointfrom the east? (P. 61.) The differences in ages and mathematical ma- Is the stun of the measures of the angles of a turitof club members limit the kinds of proj- tr iangle always 1800? (Pp. 115-1.11.) ects in which the e»tile club can participate. 3. What istheshot testpathbetweentwo The explicit directions for making and using an points: (Pp. 163-70.) orientalabacus,asetof Napiecs bones.a EXAMPLE 5 "stepped-wheel" calculatm a digital compute'. OPN : Geometric EA Cr( i.,e.s in Paper Folding, b) amid an analog computer provide a challenge T. Sundara Row (New York: Dover Publica- for every member. Club members, can demon- tions. 1966) strate the resulta working exhibitin mathe- GRADES: 7-12 matics classes. TECI 1 N IQU E:Reviewing properties of polygons by paper folding DISCOV FRY Distribute several sheets of irregularly shaped If discovery is viewed as student investigation of paper to each student in the class, and challenge concepts and 1 'oblems that are new to him. then the students to fold the paper so that the folds every student can have the satislaction of dis- Form a square. In Dicier to form such a squat. covery in mathematics whether he is led by a the students will need to Iecall the definition and programed approach orlefttohis own de- various properties of a square. Depending on vices. The techniques in the examples that fol- their age and ability, most students should be low may be varied to satisfy individual students able to fonn a square, a rectangle. a rhombus. an or individual classes. isosceles triangle, and perhaps a regular hexagon EXAMPLE 8 and a regular octagon in the same manner. opNI:Explorationsin illathe»ultic.s, by Robert EXAMPLE 6 B. Davis (Reading. :\ lass.: Addison-Wesley Pub- OPM: Flatland, by Edwin A. Abbott (New York: fishing. Co., 1966) Barnes C Noble. 1963) This discussion guide can be used individually GRADES 9-12 by students. or ita» be used by teachers as a TECHNIQUE: Making a movie source of novel ways in which students can ex- If your school has a photography club(or. plot c a variety of mathematical concepts. The better yet. if it oilers a course in photography),a technique discussed below isbased on pages joint project of the photography and mathe- 77-83. matics groups could prove exciting to both. The GRADES: visual nature of its contents and the first-person TECI INIQUE: Guessing functions stylein whichitis written make Flatland a Ask a volunteer who has made up a "rule" that natural subject for such a project. Nlaking such describes the relationship between two variables a moviewriting the script and then directing to step up to the chalkboard. As members of and producing ittakes much planning. cooper- the class suggest values For one of the variables, ation, and hard work: but it can be an tmfor- the volunteer lists them and uses his rule to find gettable, involving experience. the corresponding values of the other variable, which lie also lists. The object of the game is to EXAMPLE 7 guess the rule. To avoid having one or two stu- OPM: I Can Learn about Calculators and Com- dents always guess the rule before the majority puters. by Raymond G. Kenyon (New York: of the class has discovered it, any student who Harper k Row. 1961) thinks he has discovered the rule should write it GRADES: 7-12 on a slip of paper and show it to the volunteer. TECHNIQUE: Nfathematics club project If the rule was guessed correctly, the guesser is L PTER FOUR

le«)gni/ed as having dis«mered it, but the rest time is to look for patients, Seatching lotpat. 01 the students can «unlink: the game until terns (.111 1C:1(1 to (11S(01e1.1, o1 impottant new most (or all)()I' them know the rule. mathematical ideas. Some mathematicians leel EXAMPLE 9 that the study of mathematics is a search lor patterns. OPNI: Shape and Size. a Nuffield Nlathematics GRADES: 8-1I oject publication (New York: John Wiley g: TECHNIQUE: Class aril% it,' Sons, 1968) The activity outlined below is based on pages GRADES: 3-7 16 and 17 ()I the Onl. It is particularly effective TECHNIQUE: Working with models when studying multiplication of polynomials. Divide the class into groups of two or three I.Multiply; (1)(2)(3)(1) = students each and plovide each group With Add I: 2 I ± I = 25. paper fasteners and strips of stilt paper punched near each end. Direct them to make frames Note that 25 is a perfect square. Thus: (1)(2)(3)(0 = 5= I. ofduet:, four.five. and six sides by joining Repeat stepIfor (2)(3)(1)(5). the strips kith the fastenets. If a frame is not for (3)1)(5)(6), rigid. additional strips(struts)should be fas- for (I)(5)(6)(7). tened until the frame becomes rigid. One student in each group should record the information shown in the table.

Least Number 3.Generalise for (n)(n+1)(n + 2)(n -1-3). of Struts Number of Number of Needed for Triangles When (n)(n I)(n + 2)(n + Sides in Flame Rigid Flame Frame Is Rigid = ((n2+ 3n + I) [(0l2 + 3n -I- I)

3 0 I)(x + I) 9 v!: I, where x = u= -I- 3n -1-

5 9 3 If step 3 is omitted, the described activity is 6 3 appropriate as an investigation or as motivation for drill in the lower grades. The number patterns suggested by the first EXAMPLE II two columns and by the first and third columns OPNI:LaboratoryManual enable the studentsto make predictions for forElementary Mathematics, by W. Fit/ger:11d,D.Bellamy, frames with any number of sides. These predic- P. Boonstra. J. Jones, and W. Oosse (Boston: tions tan be tested by making more frames or by Prindle, Weber k Schmidt, 1969) sketching polygons. Older students may be asked to express these patterns for a frame with n This OPM was designed to be used in college courses for elementary teachers or prospective sides; that is, to represent the number of struts teachers. However, the activities can be adapted by n 3 and the number of triangles by n 2. for elementary and high school classes. Further experimentation leads students to dis- GRADES: 5-12 cover that the stun of the meastues of the interior TECHNIQUE: Laboratory lesson angles of a polygon of n sides is (n 2) x 180°. Have each student, on graph paper, outline EXAMPLE 10 square legions with one, two, three, four, five, OPM: Number Patterns, byWilliam H. Glenn six, and sevenunits on a side. The smallest and Donovan A. Johnson(Manchester, Mo.: figure contains one square region. The next Webster Publishing Co., 1960) figurecontainsfivesquareregions,four of Oneof the things a mathematiciandoes allthe which have an area of one square unit each 0 E1 al& mu:\ I El) NI \ FERE\ I 115

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and one of which has an :tea of four square leftto right, respectively. A fourth grid may units. Alm Ae«trdig the number of square have equallspaced vertical grid lines. while the regions contained in each of three or four of intersecting grid lines all meet in a point. the smallest ligui es, the students should in edict After the students have worked sufficiently the number of square regions contained in the with graph paper consisting of square grids.

next larger figure. The prediction can be checked inoject one of the other grids onto the chalk- by counting, the square regions. The results mall hand and ask a student to map a diagonal line be generalized as shown in the following table, segment from a square grid onto the projected with the first column representing the number nonsquare gt id. of units on it side and the second column repre- EXA \IPLE 13 senting the number of component square regions. OPM: Geometry, Experiences in Mathematical Numbcr Discovery, Unit -1(Washington. D.C.: National Number of Regions of Units Council of Teachers of Mathematics, 1966) This is one of a series of ten self-contained 9 S = + 12 booklets designed for use by ninth-grade general

1-1 = 2'-'-1- mathematics students. However. many of the

30= + 37, -1-II: concepts are appropriate for use with younger students. GRADES: 5-10 TEC1INIQUE: Class discussion Each section of the OPM includes a class-discus- rt n: (n + (n 2): ... 12 sion feature that is composed of sequential clues. ticnis based On some activity the students per- EXAMPLE 12 form. The questions lead the students to gen- OPM: Experiments in Mathematics, Stage 3, by eralizetheresults of theactivity and draw J. F. F. Pearcy and K. Lewis (Boston: Houghton concl ttsions. Nlifflin Co., 1966, 1967) EXAMPLE 14 This OPNI is one of a series of three manuals OPM: Mathematical Discovery, vol. 1, by George designed to be used by students in a laboratory Polya (New York: John Wiley K: Sons, 1962) situation. The experimengs in each book vary in This OPNI. written for the teacher, can be subject matter and difficulty. Many of them can used as a source of problems and ways of investi- be integrated into the curriculum and adapted gating problems that teach students the methods for more traditional classroom use. of discovery. GRADES: 5-12 GRADES: 8-12 TECHNIQUE: Designing your own graph paper TECHNIQUE: Special unit Prepare a few examples of nonrect angular co- On the first day, analyze the following problem ordinate grids on, acetate sheets. For example, in class: one such grid mai have equally spaced horizontal grid lines intersected at angles of 60° by equally 1. A farmer has hens and rabbits. These ani- spaced grid lines. Another such grid may have mals have 50 heads and 1.10 feet. How many equally spaced vertical grid lines, while the dis- hens and how many rabbits are there? tance between the horizontal grid lines doubles Identify the unknown, the given data, and the from line to line. For a third grid, space both condition that relates the unknown and the the horizontal and the vertical grid lines at dis- given data. Then assign several problems such as tances that double from bottom to top and from the following for the students to analyze: (mum PRIN I ED \I VEERLU,s 777

2. One pipe can fill a 600gallon tank in12 EN R WI M ENT minutes: anodier pipe can fill it in 20 minutes: and a thild pipe 'can fill the tank in 30 Enrichment is usually considered to be an) ex- minutes. How long will it take all three pipes tension of the curriculumeither horimmal or to fill the empty tank? vertical. As such, it is probably the most general 3. A dealer has two kinds of nuts: one kind of the functions of OPM. Many teachers like to encourage nonstructured enrichment .opportuni. costs 91) tents it pound. the other costs 60 cents a pound. He wishes to make 50 pounds of a ties by giving students time to browse in the mixtme that will cost 72 cents a pound. How mathematics section of the school library or by many pounds of each kind should he use? giving them time to select one of the books from the classroom collection. Some teachers, on the .1. If the area of a right triangle is 37 square other hand, like to make definite assignments to inches and its perimeter is 30 inches. what is meet specific instructional objectives. the length of its hypotenuse? Where a classroom collection is not practical On the second day, use the groping method to because of change.of-room procedures and the solve problem I. Have the students make a few need for many classes to share the collection, stabs at the answer and tabulate the results. book trucks that can be moved easily from class 1Iens Rabbits Feet to class have proved very successful (Figure 4.3). 50 0 100 A schedule can be planned to coincide with 0 50 200 25 25 150 class use, and students tan be assigned the re- From the table itis clear that there must be sponsibility of delivering the book truck to the more hens than rabbits. Continue the table on scheduled class. this premise. EXAMPLE 15 liens Rabbits Feet OPA1: Measuring Splems end Their History, 26 24 118 28 99 by the engineering staff of Ford Motor Co. 30 20 10 (Dearborn, Mich.: The Company, 1966) Thus, there are 30 hens and 20 rabbits. GRADES: 7-12 Assign problems 2. 3, and 4 to be solved by TECHNIQUE: Group activity as introduction the groping method. to a unit on measure On the third day, use the algebraic method. Although the OPM can be used beneficially at Begin by stating the problem as shown: any one of the grade levels suggested above, the in Algebraic following suggested activity is most suitable for In English Language grades 7-9. There are a certain number Assign each class member to one of five com- of hens mittees one committee for each unit inthe and itcertain number booklet. Each committee should plan how best of rabbits to report on the unit for which it is responsible. The liens plus the rabbits Maps, tables, drawings, and other visual displays have 50 heads x y = 50 should be prepai col to illustrate an oral report and 1.10 feet 2x + 4y = 110 to the class. Then solve the system of equations. Alter the initial planning period, the work Assign problems 2, 3, and 4 to be solved by should be done outside of class. The oral reports the algebraic method. can be given one each day for it week during the On the fourth day, compare the problems for study of the unit on measure, and the displays similarities and differences. can be used as bulletin-board exhibits. 111111111

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EXAMPLE 16 can be expressed with 7 as the numerator OPM: Ahnof Mathesnatio, by EricI. Bell 9. The set of integers; that is, (New York: Simon Schuster,1937;paper, ( .... -3. -2. -1, 0. 2. 3.. . 1965) GRADES: 9-12 (e 0- when n is othi. n 4-4 ; when nis TECH N 1QU E: Supplemental r eating even, n. -,-)rr This OPNi can be read for enjoyment atany time. Itis the kind of book that should be in- 10. The set of powers of 2; that is, cluded in a collection of OPNI that students may

( , read inclass after completing an assignment. I.:, I, 2. , 8.. ) Student 'shose interest is aroused by the IBM (e.g., when n is odd, n chart,OM Example 25)will enjoy reading 2 "1-1)12; when n is about the lives of mathematicians represented even, n .4 On the chart. Point out the need to arrange the membersof Reference should be made to the OPM in the sets in some order so that there isa first conjunction with the discussion of concepts that member, a second member, a third member, and are studied inclass.For example, when the so forth. The students should see that if the Cartesian plane is introduced or when the equa- members of a set can be arranged in suchan tion of a circle is discussed, suggest that the stu- order, then they can be put ina one-toonc dents read Chapter 3. "Gentleman, Soldier, and corr spondence with the natural numbers. NIathematichm." When prime numbers are Then ask the students to investigate whether studied,recenintenclChapter1,"Prince of or not the set of positive rational numberscan Mathematicians." When complex numbers are be put in a one-toone correspondence withthe considered in class. recommend Chapter 19, "An set of natural numbers. After they havespent Irish Tragedy." a sufficient length of time on the investigation, EXAMPLE 17 suggest that the interested students read Chapter 3 of the OPAI and OPM: The Gentle Artof Mathematics, by report to the class the next Daniel Pedoe (New York; The Macmillan Co., time the class convenes. 1963) EXAMPLE 18 GRADES: 8-12 OPM: What Is Mathematics? by Richard Cour- TE,CIINIQUE: Adding depth to classwork ant and Herbert E. Robbins (New York: Ox- During the class discussion of one-to-onecor- ford University Press. 1941) respondence. lead the class to describe howa GRADE: 12 one-toone correspondence can be established be- TECHNIQUE: Question of the week tween the natural numbers and various sets of The following question, together with thesug- numbers, such as the ones listed below. gestion that the OPM may be helpfulas a refer- 1. The negative integers ence, may be posted on a bulletin board. 2. The square numbers Can you show that e", +1 = 0? 3. The positive square roots of the natural To find the sum, the students should review the numbers infinite-series definition of cr, as givenon page 4. The even natural numbers 449 of the OPM, and should verify that 5. The odd natural numbers elr= cos x sin X. 6. The nautilil numbers divisible by 3 Page 478 may provide the hint necessary for this 7. The set of positive rational numbers that verification. Replacing x by 17 gives can be expressed with 2 as the denominator e:71 = cos 17+ i sin Tr 8. The set of positive rational numbers that = - 1 + O. 120 (:11 ouR

EXAMPLE 19 Patterns and Puzzles in Mathematics. by Sylvia OPNI: Elementary Mathematics: Emit Innen!. Horne (grades 5-7) by Lola J. May (New Volk: Harcourt Brace From Fingers to Computers, by Margaret F. Wil- lovanovich, 1966) It:tiling (grades 6.8) These materials were prepared to be used by Probability: The Scien«: of Chance. by Margaret students. They ate self-explanatory and were de- F. Ink/ming (grades 7 and 8) signed to extend the average cm riculum for each Mathematics: Man's Key to Progress, by Richard designated grade. A. Denhohn (book A. grades 6-8; book B, GRADES: 3-6 grades 7 and 8) TECI1NIQU F.:individualactivityorclass Learn to Po ld-Fold to Learn, by Janet S. Abbott project (grades 3 mull) Individual activity is suggested (for better stu- Mirror Magic, by Janet S. Abbott (grades 3-5) dents while less able students are concentrating Paper and Pencil Geometry, by Susan Roper on drill) when class sets are available and for a (grades 4-6) class project when class sets are not available. Making and Using Graphs and Nomographs, by Such a project for a fourth-grade class can be Richard A. Denhohn (grades 5 and 6) centered around "calendar squares," which are GRADES: 7 and 8 for the technique that follows discussed on pages 12-15 of the OPAL. TECHNIQUE: Studying in small groups com- If actual calendar pages are available, supply posed of students who do not require help in :t one for each student and direct each student to current unit outline any 3 by 3 array of numerals on his page When less aide students require extra help in with a square. Then tell the students to find fundamental skills or on the current unit. stu- the stun of the numbers in the center row, the dents who do not need such help can study Fro:n center column, and each diagonal of the square. Fingers to Computers (fourth book of this OPM Although the sums will vary for different stu- series) in small groups. The membership of each dents. each student should find the same answer group should remain constant during the study for each sum. Furthermore, the center number of each of the five parts of the book. should equal one-third of this sum. The study group should function indepen- Similar results can be found for any 5 by 5 dently with almost no help from the teacher. calendar square. Interested students should be During the study of Part 1, "Finger Computa- encouraged to investigate 2 by 2 and 4 by 4 cal- tion." the students should study the descriptions endar squares. and diagrams and then master the computations EXAMPLE 20 by performing theexercises.Discussion and OPAL: The Franklin Mathematical Series (Pasa- group criticism should be part of the learning dena, Calif.: Franklin Publications, 1968) process. This series consists of the following eleven AlthoughPart2,"The Abacus," canbe books. The first seven are case-bound, and the studied front the diagrams in the OPM, a com- remainder are paperbound. The series was de- mercial abacus-or one the students in the group signed specificallyto extend and enrich the make-would be even more satisfactory. mathematics curriculum. Each book is activity- In Part 3, "Napier's Rods," and Part 4, "The oriented and directs the students to construct, Slide Rule," the students are directed to build experiment, analyze, and generalize. instruments and use them in computing. Learning about Measurement, by Sylvia Home Part 5, "The Electronic Computer," may be (grades 3 and 4) separated into several smaller units to allow the Mathematics around the Clock, by Margaret F. membership of the group to change during the Willerding (grades 5-7) time it is under study. OTHER PRINTED 121

EXAMPLE 21 Ne(111C11«' converges to the value Continued Fractions, by C. 1). Olds, New Mathematical Library. bk. 9 (New York: L. \V. x = 9 Singer Co., 1963) Although concepts discussed in this OPM can found by using the quadratic loinuila. be presented to students as soon as they can add EXAMPLE 22 fractions.the OPNI is not intended for inde- OPNI: Mathematics in the Making(Boston: pendent reading by students below the ninth or Houghton Co.. 19(17, 1968) tenth grades. This OPM consists of the rollowing booklets. GRADES: 9-12 The booklets were designed .0 be ..ised by the TECHNIQUE: Extension of Glasswork students.If classroom sets are not available. 'When studying the solution of quadratic equa- many of the activities described in the booklets tions by factoring, some equations are found that can be adapted for use by individual students. cannot be factored in the usual manner. Ordi- 1. Pattern.tern. Area and Perimeter, by Stuart E. narily, this situation is followed by the immediate Bell development of the quadratic formula. You may 2. Binary and Other Numeration Svdems. by wish to introduce an intermediate method., as Stuart E. Bell described On pages 5-7 of the 01.1. The equation 3. Looking at Solids, by Stuart E. Bell 3x 1 =0 .1. Rotation and Angles, by Stuart E. Bell mar be expressed as 5.Curves, by Stuart E. Bell x = 3 + Scale Drawing and Surveying, by Stuart E. Bell providing x 0. 7.Transpamatimis and Symmany, by Stuart Since x = J3 thisvalue may be used to re- x E. Bell place x on the right side of the equation, giving S. Networks, by Sunni E. Bell 9.rill.sorts of Numbers, by St win E. Bell = 3 + 10.Sets and Relations, by Stuart E. Bell +1 x 11.Graphs, by G. Long and K. A. Hides Continuing to make the replacement gives 12.Statistics, by Irene Campbell GRADES: 3S x = 3 + 1 TECHNIQUE: Laboratory lesson on drawing 3+ curves 31 Adapt the directions Outlined in Curves (bk. 3+ 1 5). pages 22-25. 31 and 32, for classroom use. 3+ 1 x EXAMPLE 23 OPM: Theory of Equations, by Cyrus C. Mac- Note that this continued fraction gives successive Dull'ec (New York: John Wiley & Sons, 1954) approximations for x: (Actually, any textbook on theory of equations could be used.)

1 GRADES: 11 and 12 3, :1 + -.3+ , 3 + 1 ,.... TECHNIQUE: Independent research 3+71 3+ 1 3 After st tidying methods for solving polynomial 3 + equations. ask the students to suggest ways to 3 solve the cubic equation Evaluating these fractions suggests that the + 3x? 15x 127 = 0. 122 C11 NPTI'l:

\\lien the familiar methods of factorisation and tips. the resulting charts tan be easily lead from trial by synthetic division fail, suggest substitut- a distance and can be nride part of a permanem ing y -1 for x. The result gives collection.The AncientIlelnewnumerals. y'- - 110 = 0. shown on page IS of the OPM. c :ald be used to make a companion piece for the chart described Then let y = z +to obtain above. z" - 110z3 + 216 = u. Such charts. made by members of the class. Since the resulting polynomial equationisin will be mole than classroom de( orations and are quadratic Colin,it can be solved by using the likely to be studied by students before and after quadratic formula to give class. = 3 VI and z, = Nlan other suitable materials from the publi( ro (elate :1 and z, to the original equation, domain can be (opied in this manner 'without 6 violatingcopyrightagreements. Arithmetic substitute the values of zi and z into y = Charts Handbook andCharts for the New togive by Dumas,C. I:. Howard, and J. E. Dumas. Y = 3CCI + provide collections of charts prepared to he re- 3V1 -r - Iis a solution of the original produced by nieans of an opaque projector.Mak- cubic equat ion. ing and Using Charts (rev. ed.).by A. Liechti Ask the students to verify this solution and and Chappell, is another source of informa- then to generalize the solution procedure for tion on charts. These three books are published by Fearon Publishers. Palo Alto. California. ax3 + cx d =0. Interested students may wish to compare their EXANI PI.E 25 lesults with those in the OPM. OMI:Men of Modern Malhentaltes, a lli. imi, EXANIPLE 24 Chart of Malltentalies front I000-I900(Armonk, OPM:N'umberA Old andNew, by Irving and N.YInternational Business Machines Corp.) Ruth Adler (New York: John Day Co., 1960) 'Ilk 150 by 2'l inch chart illustrates the devel- GRADES: 3-6 opment of mathematics on a timetable back- TECHNIQUE: Student preparation of a class- ground of important historical eventsinall room chart fields. This activity is particularly suitable for days GRADES: .1-12 when absence is very high because of bad weath- TECHNIQUE: Wall.chart reference er. a religious holiday. or high incidence of ill- This chart can be a valuable addition to any in- ness. Any new work introduced on such days structional space and particularly to small group would need to he retaught later; but students spa( es. One of the easiest ways to use the chart who are present should be able to participate in is to make frequent reference to it as one of its a meaningful learning experience. entries is mentioned during a class discussion. An opaque projector can be used to project the Students should be encouraged to consult it fie- numerals of the ancient Greek numeration sys- quently and to do research on the lives and work tem. as they are given on page 11 of the OPM, of mathematicians listed on the chart. Onto a sheet of tagboard. By adjusting the dis- The amount and level of research inspired by tance of the projector from the tagboard, the the chart will vary with the grade in which the chart can be enlarged to cover the available chart is used, but the feeling that mathematicsis space. Individual students can be assigned to a vital part of history and that it is created by trace parts of the projection. If the students use Ilesh-andblood men and women can be fostered grease pencils or felt-tipped pens with broad at all grade levels. o-nwit ritiNTED .NIATER1.11.ti 123

CHANGE OF PACE This OPNI is a compilation of HO selected problems from theMathematics Student Journal. 'Mere comes a time in every class when a change together with their solutions. The problems are of pace is needed. A change of pace can take mganized into geometric, algebraic. inferential. many forms such as a quickie pluzle at the end and miscellaneous categories. of a class period. recreational material during GRADES: 7-12 the last class period below vacation, a competi- TECHNIQUE: Class opener tion between class teams. or a special unit. Write one of the short. simple problems plc- EXAMPLE 26 seined in the OPNI on the chalkboard so that OP? I:Coordinated Cross-Number Puzzles, by students who arrive before the class bell rings William II. Crouch (Cincinnati: McCormick- can begin thinking about the problem. Use the Nlathers Publishing Co., 1969) first five minutes of the class period to discuss This OPNIisan excellent source of many the problem anditssolution. The students car.fully selected cross-number punles. Each stu- should then be ready to settle down to the lesson dent in the class should be provided with a for the day. copy. GRADES: 3 and EXAMPLE 29 TECHNIQUE: End-of-class-period activity OPNI: IVsBuilt Our Own Computers, ed. Albelt Have students start a pnitle about ten minutes B. Bolt (New Yolk: Cambridge University Press. before the end of a class period. This will pro- 1966) vide time to give them a good start and to gen- GRADES: 9-12 erate interest in finishing the puzile outside of TECHNIQUE: Using logical diversions class. Either (luring or after the study of truth tables. EXAMPLE 27 ask the students how they can use the methods OPNI:Mathematical Models,2ded.,by H. of logical reasoning to solve the murder de- Martyn Cundy and A. P. Rol lett (New York: scribed on pages 16 and 17 of the OPNI. Inciden- Oxford University Press, 1961) tally, the complete solution is given in the OPAL. GRADES: -1-10 In the next class period, challenge the students TECHNIQUE: High-absence-day activity to a game of tick-tack-toe under the following Discuss tessellationssometimes calledtiling rules: patternswith the ancients and let them form 1. The teacher always plays first. their own. There are various types of tessellations. 2. The teacher always opens with 0 inthe Regular tessellations consist of patterns of con- lower left-hand square. gruent legular polygons, all of the same kind. During the third or fourth game. begin to Semiregular tessellations contain two or more question the reasoning behind the various moves. kinds of congruent regular polygons arranged so Lead the students to express the reasons for the that the same combination of polygons meet at moves as implications. "With the students work- each vertex in the pattern. Other tiling patterns ing in pairs, have them analyze the various pos- are formed by congruent nonregular polygons sible plays in a game and express each move in and by congruent curved regions. Examples of logical form. In order that results can be com- each type are illustrated and described in detail pared, number the squares from left to right in on pages 59-68 of the OPAL. each row, starting with the top row. EXANIPLE 28 Examples of various logical conditions used in OPN1:Mathematical Challenges(Washington, game are shown in Chapter 8 of the OPAL, D.C.: National Council of Teachers of Mathe- and a computer-wiring diagram for the game is matics, 19(35) given on page 71. Mathematics 124 YES, in the Modern World MATH. 1111.4`:. The Mathematics Student Journal CAN , EN BE Cdkulator .Pah141. Activities gLf.1441 MM=

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EXANIPLE 30 students to read it independently. OPN1: Game., for Learning 1)) GRADES: S -12 Donovan A. Johnson (Portland. Maine: .1. Wes- TECHNIQUE: Going oIl on a tangent ton Walch. 1960) After discussing the definition of a polygon. de- opm «titian], descriptions and &lit ec bons fine a complete gtaph ,is one in which each pair lor se,enty games that involve arithmetic. alge- of ertices is connected by an edge. Then ask the bra. and geometry. Most of them are recom- students to determine the gieatest number of mended for use by mathematics classes. Ilom- vet rites for which a complete graph can be di awn mi.. one chapter contains party games suitable kith its edges intersecting at some point that is lor mathematics club meetings or class parties. not a ertex. The answer to this question is illus GRADES: 5-9 orated on page 7 of the OPNI. TECHNIQUE: Ouii panels for a chapter re The OPM contains many other subjects suit- view period ablefortatigentialconsideration,including Two panels compete in identil ying a subject. bonus puizles at the back of the book. The panels take turns asking questions to which the moderato' (teacher) can answer either yes DRILL AND PRACTICE c» no. Eke no answeis is the limit per panel. Whetheiitis in spite of the curt ent emphasis Rules may ,:try to a«ommodate special chc urn- on understanding basic principles or because of st ances. it. the fact romans that the need for drill is an ever-present one. How to vary the routine na- EXAMPLE 31 On': The Scientific American Book of Mathe- ture of drill and how to individualize it for the matical Puzzle., and Diversions. by Martin Gard- maximum benefit of students arc, likewise, ever- ner (New York: Simon k Schuster. 196,1) 'resent concerns. GRADES: 6-12 Cross-number puMes.variationsof magic TECHNIQUE: Prevacation activity squares. number sentence flash cards. computa- Provide the students with strips of paper (con- tional skills kits. and practice problem kits can all be used as variations of drill. struction paper works well)i inches wide and approximately 32 inches long. If ',cher aide EXAMPLE 33 is available for such tasks. each strip should be OPM: 1, 2. 3: 4 Book to See, by William Wond- prefoldetl into 19 adjacent. congruent triangular riska(New York: Random House. Pantheon regions, as described on page l of the OPM. Books, 1959) After making the hexahexaflexagons in class. GRADES: K and 1 the students can form designs by coloring the TECHNIQUE: Tell a story laces. as described on pages S and 9. These de- lime the childien build a story :wound the ob- signs can form the basis of an amusing diversion jects that are the characters of the book. Number for younger students as well as the impetus for and numeral recognition(from 1 to 10) and setious research lot older students. order become an integral part of the story. After the book has been used several times. consider EXA.\IPLE 32 starting at the back and working forward. or OPAL Puzzles and Graphs, by John N. Fujii working from start to finish and then back again (Washington, D.C.: National Council of Teach- for a given story. ers of Mathematics, 1966) EXAMPLE 3,1 Although this OPNI could be used as a com- OPM: Individualized illathematu.,: Drill and plete change of pace unit. time does not Medici' Kits, by Patrick Suppcs and Max Jer- narily permit such (fix ersimis. However. using man (New York: L. W. Singer Co., 1969) ideas Irons it occasionally in class may inlluen«. Each of lour kits contains 500 different plac- 126 CI IA PTER FOUR

licelessons, 20 different pretests, 20 different a level. After the fifth lesson, the student is di- posttests, 4 sets of posttest answer cards, 30 stu- rected to the posttest. He records this score on dent-profile cards, and a teacher's guide. By vary- his profile card, from which the teacher can ing the order of the blocks of lessons, each kit evaluate the student's progress at any time. can be adapted tofit almost any elementary ma thema t ics curriculum. EXAMPLE 35 GRADES: 3-8 OPM: Alike Unalike Strip Books, set 1, by P. B. TECHNIQUE: Independent student activity Radnore (London: Philip & Tacey, 1964) ;dis- On the basis of his initial pretest score, the stu- tributed by Math Media Div., H and M Asso- dent is directed to a specific lesson card. He then ciates, Danbury, Conn. works independentlyadministering, correcting, A series of four books, each of which stresses and scoring his own lessons. His score on each recognition of shape and size. The strips in any lesson determines whether he will continue on one of three horizontal sections into which the the same level of difficulty or move up or down pages are cut may be flipped independently until

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N. 707 Cross Number Puzzles o-rilER PRINTED NIATERIAI_s 127 the shapes and sizes of all three exposed figures Each student can proceed through the vot k- are matched. book at his own speed. Because of the 'mune of GRADES: K-5 crossnumber puzzles, every puzzle contains a TECH N IQUE: Small discussion groups built-in check on the student's answers. That is. In grads K-2, a teacher or teacher aide should if all the answers that are to be recorded hori- work with each won!). \\Then necessary, the adult zontally ;tic entered first, the blanks in the puzzle can ask questions that lead the childi en to tewg- columns are automatically filled and should 1w nize similarities and differences inthe shapes the correct answers to the "down" operations. If and sizes of the various figures. they are not, the student is alerted to the inac- In grades 3-5, groups of students should work curacy of one of the "across" operations. independently. One member of each group may be designated to sketch each group of- matched APPLICATIONS figures. Not all students need to see frequent applica- EXAMPLE 36 tions of what they study, but all students get a OPAI: !Mars the Number? (Pasadena, Calif.: senseof accomplishment from applying the Franklin Teaching- Aids) mathematics they learntoout-of-school daily In Box I, each of 50 cards (11/, by 9 inches) contains an addition or subtraction combina- life situations. tioninequation form. The answer and the EXAMPLE 38 related subtraction or addition combination is OPM: FIFC Alone), Alanagement Program (Chi- given on the reverse side. cago: Money Management Institute, Household GRADES: 3-6 Finance Corp., 1968) TECHNIQUE: Contest GRADES: 6-12 Separate the class into teams A and B, and dis- TECHNIQUE: Special unit play the first caul. If the first person on team A This program is a self-contained unit available gives the correct response, show the reverse side in both English and Spanish which includes and discuss the :inswer and the related combina- twelve booklets, with a teacher's guide for each, tion. If the correct response is not given, call on and five film strips. It can be used in place of. a volunteer hom team B. If he gives the correct or in conjunction with, a chapter on related answer, discuss the reverse side of the card. If he content in the text. does not give the correct answer, insert the card EXAMPLE 39 at random in the unused pile of cads. Record a 0 PM : The World Almanac and Book of Facts. point lor the team whose menthe' supplies the ed. LumanH.Long(GardenCity,N.Y.: correct answer. Doubleday k Co.) Reveal card 2 for the first person on team B The statistics concerning heights of moun- and repeat the procedure. tains. areas of states. average temperatures, and soforth. make excellent informationfor bar EXAMPLE 37 graphs. circle graphs. histograms. and Inoken- OPAL Cross-Number Puzzles(Oaklawn. Ideal School Supply Co.) line graphs. This OPM c onsists of cards that list operations GRADES: 4-8 to be performed and individual workbooks in TECHNIQUE: Class reference which the students record their answers in the EXAMPLE 40 form of cross-number puzzles. OPM: Catalog of Sears Roebuck k Co.. Niont- GRADES: 4-7 gomery Wald, or any otter r large mail-order TECHNIQUE: Individual drill house 128 CI IVIER Foul;

GRADES: 5-8 Assign groups of stitdents to prepare informa- TECIINIQUE: Use of activity sheet tion necessary for filling out the forms One If a classroom supply of fliers is available, stu- group can be responsible for information con- dents can work independently. When only a few cerning wages. including all the deductions. An- copies of a catalog must be used by the entire other can be assigned to home expenses, such as class, students may work in small groups. interest on a mortgage if the family is buying a Since a catalog is an item that is not ordinarily home. Other gimps can be iesponsible for med- kept after its expiration date, it would be easy ical expenses, contributions, and so forth. When to accumulate a classroom supply. This would all the information is assembled, it can be dupli- make it possible to compute the percent of in- cated for each member of the class. (lease (or decrease) in the price of comparable Work on the forms can be done individually, items from year to year and, in any one catalog, as a class activity with the help of an opaque to compare shipping charges for different items. projector, or by small groups. A sample activity sheet is given below. EXAMPLE 42 Name Date OPM: Map of your state, available from various BUYING A Ruiz_ IGERAToR sources GRADES: 3-6 The refrigerator in your home needs to be re- TECHNIQUE: Special projectplanning an au- placed. It must contain a freezer unit, and it tomobile trip mist fit into a space 30 inches wide and 68 inches This project can be as extensive as you wish to high. It may be either white or coppertone. make it. After the itinerary is decided on in class, I.Select a refrigerator that meets these require- students may write for hotel. motel, and other ments and record its price. information on which to base expenses. All ex- 2. What is the shipping weight of the refrigera- penses, such as cost of meals, rooms. tolls, car tor? expenses, car repairs(ifitis decided by the 3. What is the shipping point of the refriger- class that those will be necessary), and admis- ator% sion to museums and theaters should be care- 4. How far do youlivefrom this shipping fully considered in conjunction with a piedeter- point: mined budget. 5.Use thefreightinformation table inthe EXAMPLE 43 catalog to find the shipping charge. OPM:Belier Homes and Gardens Cook Book 6. What isthetotalcostofyourpur- (Des Moines: Meredith Publishing Co.) chase) GRADES: 4-8 TECHNIQUE: Special projectplanning a class Questions con«.rning sales tax and installment picnic menu, complete with recipes buying may also be included for the appropriate Lead the students to include some canned foods, grades. such as baked beans; some foods measured by EXAMPLI.. 4I weight. such as hot dogs and marshmallows; and OPNI: Individual income-tax forms (Washing- sonic foods to be prepared fromrecipes, such as ton, D.C.: Internal Revenue Service) potato salad and cookies. GRADES: 9 and IO Use the tables of weights and measures and TECHNIQUE: Special unit the table of can sins on the inside back cover of In preparation for filling in the tax forms. de- the OPM in planning the quantities needed. cide with the class on vital questions concerning lime the students select recipes and then adjust an imagined family, including the number of the quantities of the ingredients as necessary to persons, kind of housing, and so forth. feed the entire class. OTHER PRINTED MATERIALS 129

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FIGURE 'L6 OPill that have applications in out-of-school daily life situations 130 (:II \ PTElt FOL'It

PROFESSIONAL G RowTH TEC:I IN IQUE: Reading for fun In these essays Lenten) examines such prob- Too hequently a teacher's professional growth lems as outmoded teaching methods that squelch isthought of exclusively interms of course the excitement of mathematics. the education of work. This. of course,is not the case. Profes- thewell-roundedman, andthe-vanishing sional growth should take plate with the prepa teacher." ration of every lesson. Each time the mathematics to he taught is reconsidered. new insights. EXAMPLE 46 new methods of approach, and new applications OPM: State curriculum guide for mathematics can be found. But professional growth should TECHNIQUE: Informal study group also be more extensive. Meetings of professional Most state and local curriculum guides for oiganintions, curriculum study groups,text- mathematics reflect the concerted efforts of lead- book 0:1111a11011 committees, workshops, semi- ers in mathematics education to adapt current nars, and many other similar activities should trends to local needs. The guides ordinarily con- foster professional growth. Of course, individual tain an outline of the content for each course or study should not be neglected. A few hours of grade, courseobjectives,and suggestions on reading and study each week can lead to a more approach. satisfying and better teaching career. Unfortunately, many teachers are not as well acquainted with such guides as they should be. EXAMPLE 44 Academicsubject-matter supervisors or depart- OPM: Yearbooks of the National Council of ment heads might encourage the formation of Teachers of Mathematics (Washington. D.C.: informal groups to examine the local guide and The Council) discuss ways in which its aims can be met in The Learning of Mathematics: Its Theory and individual classes, Practice,21st Yearbook. 1953. The Growth of Mathematical Ideas, Grades K-I2, EXAMPLE 47 24th Yearbook, 1959. OPM:An Introduction to the History of Mathe- matics, Evaluation in Mathematics,26th Yearbook. 1961. 3d ed., rev., by Howard Eves (New York: Topics in Mathematics for Elementary School Holt. Rinehart C Winston, 1969) TECHN IQUE: Lesson preparation Teachers,29th Yearbook. 1964, Historical Topics for the Mathematics Class- Historical references concerning subject matter being studied make lessons more interesting and room,31st Yearbook. 1969. give students a greater understanding of the TECHNIQUE: Reference library overall development of mathematics. For exam- Every mathematics teacher's personal reference ple, when studying the sum and product rela- libraryshouldcontainrelevantvolumesof tionships of the solutions of a quadratic equa- NCTM yearbooks. The representative volumes tion, consider demonstrating Euclid's geometric listed above could be used as the nucleus of solution of quadratic equations as illustrated such a library. Periodic refresher browsing, as and explained on pages 73-74 of the OPM (pp. well as a search for answers to specific questions 67-69 of the 1964 edition). that arise in the classroom, results in the kind Such a demonstration will review geometric- of professional growth that contributes to satis- construction techniques and foster a greater un- faction in teaching. derstanding of the relationship between the solu- EXAMPLE 45 tions of a quadratic equation. It will also make OPM:Random Essays on Mathematics, Educa- the students appreciate the convenience of alge- tion and Computers,by John G. Kemeny braic notation and help them to see algebra and (Englewood Cliffs, N.J.: Prentice-Hall, 1964) geometry as branches of the tree of mathematics. O TIER Ilk I NTED \I ATER] \ I s 131

EXAMPLE 18 TECHNIQUE,: In -seree course OP,M; Atathernatics: The Man ,ltade Cu/verse, As the content of the traditional high school by Sherman K. Stein (San Francisco:\\'.H. geometry course continues to be critici/ed and Freeman k Co.. 1963) altered. itis important for high school teachers TECHNIQUE: Reading for extended hori/ons to lie familiar with geometries other than Euclid. Teachers of mathematics in upper elementar Can geometry. These nta soon be part of the through senior high school will find this OPM a high school (our.se. In this OPN1 the author ex- source of many fresh mathematical idea, that amines many geometries: pre.Euclidean. (an be Nliai ed with their classes. Furthermore. Can. metric. vector,transformational. non- the interrelations and unusual applications of Euclidean. and projective. number theory. topology, settheory. geometry, algebra. and analysis that arc presented in the OPN1 will provide many hours of pure enjoy. SUNIMARY ment for anyone who is interestedin natter In recent yeats the revolution in mathematics unities. education has resulted in many changes in the EXAMPLE 19 curriculum.Despitetherapidity with which laior Topus in modern mathematics: these changes o«111. it still takes three or more Set and Group Theory. by Donald E. Mansfield years for a textbook to be written alni published. and Maxim lint( kheimer (Nev York: Harcourt. Howe er. OPM provide a means of keeping the !Ware World. 1965) curriculum up-todate and balanced. TECIINIQUE: independent study There are a number of bibliographies that can This text was writ ten expressly kr ma themal les be used as source lists of OPM. The publications teachers in England \im ate caught up in cur- listof the National Council of Teachers of riculum refonn and need to judge the mathe- ,Nlathematics is an obvious source. The NCTM matical content of the many new courses being also publishes The High School Mathematics suggested for use. The authors never forget their Library, by William I.. Schaaf, in which about wadersthey anticipate the kinds of questions 800 entries are classified by subject and alum- that a teacher would raise when faced with new tated (this is now in a 1970 edition) and ilfathe- ideas, and they include approaches and exercises truni(s LibraryE/emernary and JuniorHigh that he can use in his teaching. School, by Clarence Ethel Haldgrove and Her- The content of the OPM includes work with bert F. Miller, now in a 1968 edition. Another sets. equivalence relations. mappings. cardinal veryusefulcompilation isBibliographyof numbers, groups. isomorphisms. homonurr- Mathemaths for Secondary School Libraries, by phisms. geometry as a study of properties invari- Robert A. Rosenbaum and Louise J. Rosen- ant under gimps of transformations, and "high- baum. published in 1964 by Wesleyan University, er" structures such as rings and vector spaces. Middletown. Connecticut. Mathematical Book- EXAMPLE 50 list for high School Libraries is published and On I: A New Look atGeometry, by Irving kept current by Mu Alpha Theta, the National Adler (New York: John Day Co., 1966) High School and Junior College Mathematics Its clarity. style. and content make this OPM Club, University of Oklahoma, Norman. It lists quite versatileit serves as a challenging study 10 periodicals and 170 books that constitute a for the amateur mathematician. as a reference for well-rounded mathe natical library on a limited the high school mathematics teacher,ts supple- budget. The annotatian for each entry contains ment:11y reading for the very able high school the price and the publisher's address. The Teach- student, and as a text for the high school teacher er's Library: How to Organize It and What to and future teacher. Include, whichis a jointpublication of the 132 CI IA PTER FOCI:

Autcnic.ut Association of School Librarians and timetable that shows the distances between sta- the National Commission on Teacher Education tions. and many other similar hems. and Professional Stand:n(1s. contains a list of 50 Many teachers have 101111d (aid files, organized books and 5 journals considered most essential by subject and cross - referenced for function. to in a library for teachers of mathematics.Itis be of great %Ate in using OPM effectively. Files available front the NEA Publication-Sales Sec- of laboratory lessons and activity sheets ate also tion,1201 Sixteenth Street NW. Washington. very useful. With011t such records. the work in- D.C. volved in preparing for each use of an OPM is It would be impossible to compile a list of all lost; and. understandably. enthusiasm for their the available OPN1. Useful items whose primary use wanes. purpose is not related to mathematics instruc- No chapter or article can tell you how you tion might be overlooked in preparing- such a should use OPNI; it can only make suggestions. list with its unlimited possibilities. Such a list The when. where, and how of using OPNI must would include Continental Can Company ads of necessity vary with the individual school, that feature puzzle-type problems, the financial teacher,andclass.However. the betterac- section of the New York Times, a finance-com- quainted you arc with the available OPM, the patty informationflierthat shows atable of more you will u,e them and the richer your loans andpayments, an AMTRAK railway classes will be. 5.TEACHING MACHINES AND PROGRAMED INSTRUCTION PRETEST FOR FIRST UNIT i NO ANALYZE PRETEST RESULTS

ASSIGN POSTTEST NO PREREQUISITE PREREQUISITE OBJECTIVES OBJECTIVES I ASSIGN ANALYZE YES ,OBJECTIVES POSTTEST FOR UNIT RESULTS

NO ASSIGN POSTTEST FOR UNIT

ANALYZE NO CALL POSTTEST TEACHER RESULTS

YES

ASSIGN POSTTEST NO r REMEDIAL REMEDIAL OBJECTIVES I WORK YES ,:l1 ANALYZE PRETEST FOR YES POSTTEST NEXT UNIT RESULTS

NO I I AlMIff I A BASIC PROGRAMED INSTRUCTION PROCEDURE 135

CHAPTER 5 _

TEACHING MACHINES Er AND PROGRAMED INSTRUCTION by JAMES E. GILBERT Purdue University, Fort Wayne, Indiana LEANDER W. Small IBM Corporation Princeton, New Jersey

..,

Chapter 5 reviews the history and development of programed instruction and teaching machines, and it discusses some uses and abuses, of these instructional aids in the mathematics classroom. The production of PI and criteria for its selection are sketched so that teachers may better evaluate the materials they encounter. 137

5. TEAcm ING MACHINES AND PROGRAMED INSTRUCTION

A sequence of items, each requiring a response programed materials for mathematics and to hom a student which he can then «nnpae with provide further sources ofinlormation about a given answer or which leads hint to another programed instruction and the devices dial have item in such a way as to increase the efficiency been developed to present materials 10 students. and ellecliveness of learnilig,isbasic to pro- In 1958 B. F. Skinner wrote, "There are more gramed instruction. The need to control such people in the world than even before, and a far sequences ledto the development of devices greater part of them want an education- (2.1). called teaching machines. Meanwhile, the prepa- The decade that followed the publication of ration of learning sequences has made a signifi- Skinner's article witnessed a revolution in edn- cant contributionto twentieth - century educa- c:.zional technology with many of its roots in tion by focusing the attention of authors of texts psychological laboratories not unlike that which 011 the goals of instruction and the specification Skinner operated al Harvard. or those behavior, a student should exhibit 10 A CAVEAT: Bear in mind that we are still in demonstrate mastery. By structuring, or pro- the Model 7. stage of this instructional medium; graming, learning- experiences. a student may be for, while teaching machines have evolved from led from whatever skills and abilities he pos. scroll-fed viewers to complex computerized units, sesses to whatever new skills and abilities he and programs have passed from sentences with should possess as a result of his interaction with properly selected words omittedtointricate branching patterns involving both student-con- theinstructional module. Programed instruc- structedresponseitemsandmultiple-choice tion, then, is a method of learning acid teaching items, itis still not clear what direction this geared to individual capabilities and using new evolution will take. and dilletent methods which involve new concepts of preparing instlional materials, new H ISTORICAL SKETCH methods of presenting these materials to students. new aspects of control of the learning The Socratic method, recorded by Platoin situation, and new techniques for the determi- his dialogue Meno, is often cited as a forerunner nation of subject matter mastery. Programed in- of the conversational method on which pro- structionisbased, in part, on three concepts gramed instruction is based. Such history is in- agreed upon by psychologists and educators: teresting but of little real value to us, for if the 1.Students best learn at a rate that best suits responses made by the student are indicators their capabilities. that behavior is being shaped from more or less 2. Students best learn if they are responding random responses toward something se fight as to and interacting with the subject matter "terminal," the Menn offers a poor example. actively and appropriately. In 1926 Sidney Pressey, of Ohio State niver- 3.Students best learn when they receive im- sity, developed a device by which his students mediate confirmation or correction for each might be tested and shown the results imme- response they make. diately (10). This device is pictured in Figure Flow programed instruction got started, how 5.1. Pressey's work remained relatively unknown it developed, what technological advances have and of little consequence in educational circles been made, and how these forces combine to en- until the principle of immediate confirmation hattre mathematics instruction is the concern of was exploited in the military trainers used dur- this chapter. A further goal of this chapter is to ing World War II. For such devices to work, it suggest some criteria for the selection and use of was necessary to have the questions prepared in 138 (31.1,TER

anutItiplchoite format. Thischaracteristic spurred:t teneed interest in statistital ptoba. bilities and stochastic processes. Nfeanwhile B. F. Skinner and his colleagues. after years of behavioral teseatch on infialut- 111.111S(111A'CO111,.. lunar. en .1. suggested that stn. dents ,,,01,1,1 it.; if the, wtre rttr :uded for the «notruction of a collect response to a guesiion. Br Nowt:ming the questions (hence the term programing)and gradually withdrawing the clues. behavior could be shaped. An exanyle from an eal) tonstructed-response inogutinis shown in Figure 5.2. In the example we Nee the retptired repetition of a desired response. The claim made for such repetition is often based on the assumption that Cunt rr.t rrl&op 'at .11`elr It Ar teottftr if 90 portent of the students construct the cor- Ficuto: 5.1 rect response 90 percent of the time. leatning SidnyvP.ss'stesting arhinc

norm:.2. Examplr from an at* rmislrucled-rcsponsr program

No. I. The numbers NC use to count oith are called 'counting numbers' or 'natural bets'. 0. I. 2.1. etc. are natural 'lumbers. is not a natural number.Is ", a natural number? 1 (1 es or no?)

natural 2. 0. 1..tud 2 are natural numbers. 1. 4. 71. and 10.001 are also r numbers.

naturol 3. D is a C771 number.

1. Which of the follnuing is not a natural number) T.- 0. 12. 627. 2 63. %. 5 -1

8011 5. ghich of the follouinu is a natural number) ELI 10.2. 1'.. $001. 'I'

Yes. 6, ^ is a natural number: 5 is a natural number. If )mu add the tuo natural numbers '2 and 5. is the stun a natural number/ Ell rs or no?)

Yes. 7. Vick any tun natural numbers: add them. Is the resulta natural number ? (les or no?)

No. 8. Is ', a natural number? I:-.771 rs or no?./

No. 9 Man) )ears ago men used only natural numbers. Hut the). found eventually that natural numbers were not admit:ate for all their needs, such as etpressing a part of a whole. Vor example. if a baker dititled a loaf of bread into tun equal parts andgave one part to each of his tun friends, did each friend get one thole loaf of bread' j (rrs or no?)

Prom Modem NI;Itlicmatits. No g::1111:11 Textbook. BookI by Lends Eigen. Jerome I). haplan. awlMaltEmerson. Copyright C)by Science [tort:n-1s Itsiocintec, Inc.Reprintedhpernsicsim:of the Intblicher. I'li \CIIINC: NIACIIINES ANI) PROGRAI1) INsiRircnoN 139

has taken ',late. A plot iat wthisting of .t set of ,- .1tott.self 'len ia I.single-response. student -constructed items was olten inept cal on a scroll for piesentationiua simple viewing device.in cyltiell each item (called :t firmly) would appear inan apertut e and the correct response would remain concealed until the student had com- pletedhisresponse and advanced the scroll (Figni e 5.3). The ulent would then compare hi, response with the awaver shown. which.it was hoped. would reinforce his decision. Such devices and the programs written for them precluded extensive skipping. reviewing. or branching and hence were termed linear. They were also referred to as Shimierimt. after B. F. Skinner. who initiated this format. .,-,zmv.-axTR37,srazzyteemsateecormaimummx=ii

1:16tmr. 5.3

.11/A. .11.,1.V III icaching mahinc Cour tesy of Teaching Machines Cm par at ion

slann-rOr.P.

FiGrRE 31.,1STleaching machine

Couric.syof Mast Development Company '140

1:1c.m. 5.5.Cyclo-tracher learning aidlinear program, constructed-response device. This ma- chbre is mannalll operated. Students lean at their own pace, and handwritten responses me recorded On a .seka, ale answer sheet. The programs. on circular paper sheets. are reusable.

The multiple hoice lomat iequited b) Pres- whorthId bookprogiain or as the Gunerblian sey.:, early device was used inthe1950, for program. alter Not man (.1 (nyder. who suggested troubleshooters and military trainers. Although this format. the student might use 0111) a small portion of In the early history of plograined insti union. the text, his incorrect choices afforded the authm minielons studies compaling the relatke Inuits of the program the opportunity to correct bits of linear and blanching plograins were con- of misinformation and to provide for a wide cluded (-I, pp.1 16-1 7). The controver.$) raged, range of student ability. An example from an and the results tended to show 110 significant earl) branching program is shown in Figure 5.6. differences (II: 23). That is. when the two pm- In this example the bit of text is followed by gimping methods were «mpaied, it was seen a multiple-c !mice item. a list of altediatiNes, and that the clegi cc of stotlen t mastery tended to be a set of directions for finding the next bit of independent of the program used. information. The format suggests. and was ks positiveplogramed-instructionresearch of tenincorporatedinto.amechanical device findings mounted. in the earl)1 Mins. it Was no utilizing a microfilmjewel undo ple«Kied me. sot 1)1 isc that both I'ICSNI and SN1SG mulct look didoit.11 «mool in order to facilitate the locat- programing efforts.Both groups. pioneersin ing of the .111)101m iate institutional media (Fig- mathematics curriculums ICN IS1011, used subjett- ure 5.7). The text sersion. with page-turning rmttier spec ialists. teat gars. and ps)chologists in performed b)the student. was known as the cooperative %%Thing efforts. The results fallored a 1

TEACHING N1A(.:111NES AND PROGRAMED INSTRUCION 14'1

28 Urom page $3)

a YOUR ANSWER: = 1. a

You are correct. You should notice the exception of 0 from this statement.The 0 a expressionsand are not defined.That is, they are never used. 0 0 The reason is that all kinds of foolishness can be proven if we assign any value to the result of a division by zero. In algebraic expressions, when two letters or a number and a letter are written side by side, they are meant to be multiplied. Thus, 2a means 2 times a, ab would mean a times b, and so on.For clarity, the numbers or letters to be multiplied are sometimes placed inside parentheses. Thus (n)(1) means n times I. Now, here are a number of very simple statements about numbers. FIGoiti:. U.O.Example from an All but one of these statements are true. Which one is not true? early hrnnehing program

n n O. page 32

(n)(I) =n.page 37

= n.page 40 1

n + 0 n.page 42

(n)(0) = 0.page 45

n + I =n.page 46

LCH Repiodined, by pel liom N. :1. C/ metier and G. A C. Malan. Adsentincs in Algebra. 1960 by Doubleday Co.

FtGolty. 5.7,A niolulor Mark III

O O

Gain Lesy of Salgenlll'eleh Scientific Company 7-5. The Additive Inverse

We have pointed out that 0is the identity element for addition. This is simply another way of expressing the addition property of 0: the sum of

any number and0 is equal to the number. Suppose we have two numbers whose sum is 0. These two numbers are

related in a special way.

0? 1 What number when added to 3 yields the sum -3

2 What number when added to -4yields the sum 0? 4

In general, if x and yare real numbers and if x + y = 0, we say thaty is an additive inverse of

x and that x is an additive inverse of y.

is an inverse of -2. additive 3 Since (-2) + 2 = 0, 2 additive 1 4 (4) is an of inverse 0 5 If y is an additive inverse of x, then x+y 5 6 An additive inverse of (4) is

7 An additive inverse of0 is 0

If t is an additive inverse of s, is it also true

8 that s is an additive inverse of t ? yes (yes, no) Given two real numbers, each is the additive inverse of 0 9 the other if their sum is

10 Consider the following pairs of real numbers. Which pairs are pairs of additive inverses?

-7 and 171

4 4 and =

0 and 0

4 + (-3) and (-4) + 3

(-5) and -(-5) (A) all but one (B) all (C) all but two

Since for each pair we can verify that the sum is 0, [Ei] is the

correct choice. If you answered incorrectly, convince yourself

that the sum of each pair is 0.

252

Refooducrd.brbow:swot,. pow .Sr /roof Mothrwaths Study (donb. Plogiamcd Fn.( Comm in I)1.1 (I(e xi.ed form I I). (i) 1966 br :;MtiG.

FIGURE 5.8. Example front a hybrid program I'l \CHING NIACIINES AND PROGRANIED INSTRUCMON '143 mixture Of linear ,tact Inane hung items. called clearl) leads to the introduction of the computer. hybrid oreclecihprograms (27). A typical page The wmputet has opened up the potential lor bon] the SNISO hybrid prognmi is reproduced teseatch on !canting and teaching and has. itsell. in Figtne 5.S (2(i. p. 252). become a sophisticated tea, A machine. Stmed Note that the textis followed by constructed whit materials of the sort we hase seen (conven- response items and then a multiple-choice item. tional1»ogratnedinstruction),the computer Variations in this pattern prevail throughout the ma) operate in a tutorial mode. capable ol «m»- text, making it difficult to present the program paring.Neril)ing. suppl) ingclues, branching. in a sc roll-like device. However, efforts has c been and tesiewing, )et maintaining a record of stu- made to execute and present hybrid programs dent actions at all times. The Imo] ial progiant. using such devices. One example is shown in insofar as it simulates the actions a teacher can Figure 5.9. perform. might he exttemely difficult to plepate NIcanhile, William Una!, then a psychologist were it not fen author languages that requite at the IBM Reseatch Center. suggested that "au- minimal knowledge ol computers ott the pint of tomated instruction may be looked upon as a the subject-matter specialist. problem in simulation, in which the goal is to Ile potential of the computer as an instruc- most completely imitate the mmersational in- tional aid is oohing bout an industrial re:watch tet action between the student and his human emironment to a school-unkersit) research en- tutor(29). To imitate or simulate the complex vironment where federal monies arc available pa-Jerns of which the human mind is capable tounderwrite dcclopmentcosts. Suppes at

norm.: 5.9. Honor teachingmachinehy- brid program multiple-choice response device. 111111111111111111 This baile)y-powered machine features push buttons to advance programs from which the student learns. The device hasitlimited Ii lmi»ching capability. In the branching sec- 100, A lions of each Mogram a different button is .( Tot. 4,1k listed for each possible answer. Pressing the Holm. tot owed button aulmnatically advances the scroll past the explanations given in response to the incorrect answers.

Como toy of Honor Produces Company '144 ClIAP I El: VIVI.

Stanford. Mimi atPennsylvaniaState, and materials of instruction (I). Lambe at State University of New York tepre- As earl)as19(i I.researchps)( hologistsat sent a few of the universit) researchers in the SNstem Doelopment Corporation reported the etneiging field of «mipmer-assisted instruction. construction ol "(:(»tiptiter-based Labotatot ) Experimentation at the Brentwood School in for Automated School Systems.- called Cl..\SS Palo Alto, California, has shOwn the computer (8). \\line dune appeal to be man) achantages capable ol controlling a segnence of drill and in such a total s)stent, the Ividespread a« eptanc e pot( lice exercises: expel imentation in the Board and use 01 such systems has been dela)ed of Cooperative Education of Westchestet Coun- factors of cost. technical cleAelopment. and the ty. New York. has demonstrated the computer lack of instructional mate' ials imitated in this as a simulator for use in games with elementary Way. and secondary schoolstudents(:30):and CN A new era emerges; a new level of sojthistica- perh11111 In sacral N Clr England RI el tam- lionlotplogiatned insouction and teaching tory schools has 001 ed the feasibilityof the machines appears; anditnew applicationof computer as itptoblem-solving- tool. The re- technology in the field ol education (ltos the scald] goes on(I.1). and the impact of pro- school situation by way oltesearch. The de- gramed-insouction techniquesisfelt permeat- Nelopments in this field are vapid and exciting. ing the development ofitnew generation of holding much promise lor the futureof

FIGURE 5.10,IBM 1500 computer-assisted instruction terminal \CHING :\IACIIINS \ND \\IUD I NS 1.1:t"c HON '145

struction. Tui dun informationabout these Onc e die dec ision is made to utilise prcrgi.uncd developments is given in Chapter 6. "Electronic. institu don a, a «»»ponclit of the instinctimial Computers and (:alculators." s)steni(the (hitch or low th alteinatice above). the next questionis "Are programed instruc- DEVELOPING PIOGI:.\NI tion materials available that will meet the in- INSTIWCTIONAI. ,NIATEIZIALS: structional lequirement?" Ilibliogtaphic sources AN MI.:I:VIEW of available piogiams include those of Fenton and Gilbert (9), Iliac) and Newton (15). and Generalrelei elites onprogram development Schramm (23). We have included in this chap- abound. Those who seek to develop l»ogramed ter excerpts from a lew of the mathematics 10 o. materials WillI111(1 much general information giamed materials available. iuIliethowr (2).Brethower etal.(3).Fry The question that then arises is "On what bases (10). Glaser (13). 1,tunsdaine and Glaser (17). should a program be selected?" We shall ad- I.)saught and Williams (IS). and Taber etal. chess this question mot e in a later section (28).In this section the characteristics of Iwo- of this chapter: but it should be home in mind grained learning will be sketched: the subsequent thatif programed materials do not exist,the sectionwillprovide more specificdetails of preparation of such material, necessitates a thor- piograming based On guidelines used by the ough empiricalvalidation. as detailedinthe SMSG writing efforts in mathematics (27). following section. Stich validation isor should (:racialtothe developmentofprogramed be. an integral part of the development. of pro- materials is the specification of a set of instruc- gramed instruction materials,Itis a tinte.con- tional requitement,. goals. needs. or problems. stinting and difficult task, but one without which Mager has written extensively on the identifica- programed instruction may fail. tion of objectives (19). and Briggs et al. have addressed the question or setting objectives as a vital function in the design of an instructional DEvriorim; pRocRANIED system(1). SNISG (27) includes explicit refer- INsTrulc:TioN NIATERIALS: ences to mathematics goal setting in the Manual SON1E DETAILS for Programerswhich was used in producing the Based 011the experience of constructing pro. SNISGPropamed First Course in Algebra(26). gemmed instructionmaterialsi»mathematics, After the instructional iequirementis recog, 0 the authors propose the following steps: ni-ied and spec Hied in terms or student behaviours. one must ask the question "Is programed hi- SIT.). 1. Specify the objectives of the pogram in st" uction an .appropriate medium?" At least foul terms of beim% iors that ate to be performed b) alternatives are available at this point: the student. The author of a program must be 1. Another medium is more appropriate. able to answer the question "What should the 9. A noninstructicmal system is "better." student be able to do as a result of completing 3. Piograined instruction is part of the opti- the program?" The answer must be explicit mal solution. for example, "lie must be able to factor expres- '1. Programed instruction is. per se. the "best" sions like ,Ix= 6,1." Gagne has suggested that choice. such objectives 'quire the author to go one Suchalternativesthen demand attentionto step further and state those behaviors prere- economic. management. administrative, and cur- quisite to the specified goal: to wit. "Ile must ricular constraints, as well as the technological be able to factor expressions likeI.r 12 and considerations concomita»t with the use of pro. X2 ,1" and "He must illustrate the meaning of -1ained instruction. factor" (12). As tedious as this seems. such a 146 CILAI,TER Fl

16N:cull% ma) enable the author to addles' his Does the contentsuggesta pedagogic audience mote apploptiatel. strateg? Sit r 2. Prepare a set of (Iiterion items, that 3. Who ale the students for whom the pm- test attainment of the specified goals. gram is written? Si Et, 3. In terms ()I steps I and 2 abme. prepare I.flaw ptelequishe skills been a«plited? an initial instructional sequence. At this point 5.Isthe question clear? Too long? Too the author must acconn»odate both cognitive short? style and desired end result. That is. the author 6.Is the answer to a constructed response must be :dm to the was in which a student unambiguous? learns (from the manipulation of objects, from 7.Does the 'espouse require a change in be- visual images. or ft 01» verbal or symbolic state- has icui. ments) and the terminal behavior of the student. 8. Ate multipleq hoi«clesponsestealistic? II the studentis 10discriminate between integer Disfra( ling? Ohs huts? squares and nonsquat es. the items ma be ()I' 9.Is provision made for re% iew? Indexing? multiple-choice response tpe: ifthe student is Ilelp? to constrict the squares of gken integers. the 10. What will a student be able to do after items Ina be()Iconstructed-responselpe. completing the program? Usually.itis some combination ofitems that Rea( ling to the abme questions will1101 make will work best. a good author of a bad one. The qttstions STep I. `Fr% out the instructional sequence on a represent owl) a lew of those that occur: but small number of students. with an observer to heare not 10 be ejsii. dismissed, Whileques- note student difficulties. Ask students to react tions I and 10 appear to ask the same thing,

to the selection of wording. the length of the itis worth »ming the dilfetem e. Question I ad- sequence, and the amount (more or less)of dressesthe progiam goals while question 10 material needed to communicate the notion. At addresses student attaim»ent. all costs. let the student be frank! There is110 substitute lor writing. revising. STrt. 5. Revise the sequence. Take into account testing. and remitiag. A 111°10(1;1 knowledge the student reactions, the effect of verbal change. of content and methods of teachingare pre- and the mathematics being taught. While most requisites which "those expelt in programing research indicates little (fillet enc e between oert but not in mathematics" are inclined to forget. (written) and covert (mental) responses. overt response has an initial advantage in the fact SELECTION AND FA7ALuATIoN that it can be assessed (II. p. 123). OF PROGRAMED STEp 6. Try out the revision on more students INSTRUCTION MATERIALS who are of the audience to be addressed by the Most teachers will find neither the time nor program. This trial will result in further sug- the energy to develop programed instruction gestedrevisions andwilltendtorefinethe materialsotherthanshortinstructional se- program. quences on discrete topics. In this section, there- The process suggested here is not easy. It re- fore, some guidelineslortheselectionand quires intellectual honesty and strains the pa- maluation of existing plogramed instinction tience of an author, The efforts can be justified materials will be sketched. Teachers of mathe- if the result is student achievement. Sonic ques- matics who contemplate using programed mate- tions that may help authors avoid frustration rials should read Kahn (16) and view the NEA follow: slide/tape presentation (22). The article by Kalin I.Are the goals clearly defined in behavioral relates specifically to selection of mathematics terms? materials. TEACHING1,\C.IIINES \ND PROGRAMED INS! kmrioN 147

The first step in selecting mathematics mate- should \yolk Onough the materialas a student ) ials,pro:Jo:lined or nonprogramed.isthe de- before selecting itlor use. Selecting a text. the termination 01 an instill( tional need. This should teaching ()I\\ hid) relies Ilea\ ih on the teachei. inc Inde «msidelation of the «intenttobe ma tequite less time for es aluation than select. taught: tl Itequenc01 the ollerings: the num- ing a set ofIWOgraitted instruction matelials. ber of students to he served: the requirements where much of the teac het's control is assumed 01 the students. !acuity. and administration: and by the plogram. the availability ()I materials. Sele( t ion of programed instructional materials. Let us consider the situation where a ntimber then, must take the following into account: of available programs do exist and (an poten. I.Student needs: to satist course lequire- batty meet the instructional tequiret»ents pre- ments (a course or part of a course being vioush defined. The task then becomes one Of taught): to piechule «mrse deficiency (pTe selecting one or morelc.es Amnion. Data semi\ e medicine): to impro\ e course per should be «illected about each pn)gram 81)(1 its lormance (raise a passing but low level of administiation. Judgments must then be made achievement): to augment it course of in- about the level of each program, the time re- struction (supplement or «nplement a quired to «nnplete it, the mode of presentation, «mise being taught): to pro\ ide new infor- the manpower required to administer the pro. mation (establishItcore level of under- grant. and its economicleasibilih,. Sonic data standing for a «mrse to be taught) may be obtained from the authors or producers 9.InAnnoor needs: to complement instruc- of the various programs. but some must be tion(replace aportion of conventional gatheled empirically. Itis the empirically gath- instruction): to assist instruction (provide eted data that will. in the end. determine the instruction in addition to conventional in. local value 01 each pro,f4ram. Sonic questions to solution): to augment instruction (provide be resolved include the following: material that is not or cannot be provided I.Are the authors known to be expert in the by conventional instruction) content and pedagogy? 3,Administrative needs: to extend offerings On what population was the program vali- beyond local capabilities;to provide in- dated? structional control (uniformity of exposure 3.Is the local population similar to that used whet e wide teacher variation has resulted in the validation? in student, receiving unequal teaching): to I.Will the program satisfy lo( al needs? meet curriculum demands in areas where F. ;kw the authors' goals consistent with those loads may be exceeded by student needs. of the local school? After selecting a program that meets the con- G. Are pretests and posttests as ailable to assess tent demands and the student, Imult%. and ad- student achievement? ministrative needs. the evaluation begins. The While it appears trite to ask if the content is programed materials should be compared with applopriate,itis worth considerable effortto other media and presentation modes: films. mass determine the tesidts of using a program on it lectures. special classes, and so forth. small nt»ber of students before its widespread Variables within a program which bear at- use is contemplated. .\ very impoant step in tention include: the vele\ ame of responses: the program evaluation and selection is a trial run variation of pedagogical patterns: the use of of a few students before extensive use is con- simulation: the use of induction: the appropri- sidered. ateness of criterion items: the allowance for re- The teacher of mathematics who contemplates view and practice: and the overall consistency using a set of 1)1.0g-tamed instructionotterials of st)Ie in the program. Variables that are more 148 ( \PI I I:Fi

difficult to assess in( Jude: student attitude over self should be deal.IIihr 111(11itilitislicv,to an extended pctiod of lime: ictention ol mate- students. .1 samplc poi tion ol the plogiain may- lial:teachei attitude and bias; administiative be extra( td as.1 stalk% Ite«nd keeping. l- problems (liov%to grade work done. how to sponse techniques. and methods of individual schedule ( lasses. etc): and hidden «isis ()Icle., mud) should be ( oscred ill detail early in the vices :till materials. «Hirst.% The goal hoe is to be stile the student As mole clIolt is expended in the es alnatiou is pit:limed to begin leaining mathematics with- and selection ()I plogiams. less %sill be spew in out being disti.uud the medium. he should designing .1(Iiiiiiiistiatic ,(IltiiioiiIllical lime 11.e the light attitude. 1)10(1111,in s. and male instill( Holm!!mildews. Flu.die nin-IIthe Hal, to in( lease his piobabilit of sins ess. student fails. the hist' uctional s)stem has failed- %sill impale continuous esalitation of plogiams St wlent-Ilrogrant Interaction and possible leclehilition of goals ()Iinstill( lion, IIthestudentdoesnotintelat twiththe Inbrief.the Ni(l),inselecting plograined plogram. nothing ma\ be leagued. Although this materials ale these: statement sounds ohs ions. its mull is often ig- I.DOM(' uInIIINII'llid011:11 noied by mans who woi k with plogiained mate 2.Idntils possible solutions. Progiams (I() not (mile packaged with con- \Voris through the plograin (like a stuclent). tagious human excitement.l'he instructor and I Is :Cie W0'41,1111 on less the educational ens honment must pioide this. 5.liased result,. 51/W1'1111e or select an al- Student-0°120:m intelaction can be enhanced ternalise. in mans was so The lea( Ile, who supelsises p1 o granted study may icinfoue the inters( (ion dur- usvAND AmilxisTRATioN ing studentinter views; other institutors may vizoGRANIED INSTRV(:11ON show an interest ill student plogiess: and patents The sank piogram ma\ bode ill to one teacher may support self study and :1111(1e111 11:0 111 11):111.011, and goodto another. Smith (25). May (21). CA 110N : l'rogram perfmman« re«ads (Of and Schiamin (23, die uses and abuses of this time spent, frames emit meted, errors, quiz scores. medium in Ieseatch and practice. Se\ eral sug- test results) a:.0 important by-products but must not become whips by which students are held to gestions follow which ma) seise :is a guide to a program. On one hand, they Call give guidance, teacheis ()I mathematics who decide to tryplc) (i measure of progress. (i sense of achievement: On giaincd-insti in [ioninatelialsin(hell c lasses. au. other hand, they (an be( male ends in them- The suggestions ale al billaril) classified 1111() six scloes. If the obje( live is learning, it should be areas: pieparing students. studentprogram in remembered that many paths will lead to it. A multiple-ehoice path may provide enrichment or teract ion,establishing guidelines. supervision. Hari [nation among its alternatives. 'rims Ihe measurement and esalualion, and time. slimtest route ( fewest responses) may not neces- sarily be best for each student. Preparing Students A perceived need On the pail of students in- Establishing Guidelines vokedillthe instruction must be present it . \n instructional systemis an mg:mi./mien of the bilge' population. Es en though a ical need materials of instiuction, e(luipment. and far exists, if itis not per( Cis ed by the students, sub- together with a set of plot cdtires that gmern sequent activityto'elle\ ethe need 111.1) be them. to satisfy specifiable student goals. Itis severely limited. The program must become a 11111)01(am for both students and leacheis to be chicle by which a need is satisfied. The goals well hammed about those plocedin es that are of the tuck'. the plogram. and the student him- imposedbyplogiainedhist! action 'FE CHING MACHINES 1NI) l'ItoGRANI El) INS Eln'CTION 149

Knowledge of:t pogrom's objectives. materials. 1:amiluations of using the ptogkun ma% methods. and ( haracterisths inoy be communi- in(iude 101105511p studies, retention stud- cated through printed materials. although many ies.I equisiie pretesting and posttest ing. other methods exist.slit Itas ;poly meetings. and obsel%ations that ma) remr as are- seminars. hulk idna 1«muscling. a udiotope or sult of using the program. %ideotape sessions. and small glotip (lis( tissions. S. A statement t)1referen«s to other cur- A student's guide mav facilitate the learning of ricular materials the student. A teacher's guide Ina lacilitate the 9. A time plan to guide assignment andse- adminiswation of the pogrom and alert the in- quencing in the program structor to possible student dilli( fillies in either 10. An indh allot) of the goals to hemet by content or use of the program. the students ill each unit A useful teacher's guide for a program should I I.Test and problem-set answers include at least the following eklen Some other hell),141items that might be in- I. A brief nomcchni( al history ()I the devel- cluded in th tea( her's guide are supplememary opment of the pogi:nit. including a ra- problems and tests. work sheets. glossaries. dia- tional) or philosophy underlying the crea- °rams. and work and flow chat ts. tion ()I the plogiont In addition to the teacher's guide. a student's specili( anon of objectives guide may be useful. It should include features These should appear inhelm% iora I that will make the program enhance learning. alms to fa( ilium. observation of student The following Ipes 01information could be perfointan«.s. suggests one op. included: 08(11 to Inc-poring such objectives (19). I. A stt«inct statement as to why this pro. :3. A statement of pi et ecitt isites required as grant and this method are being. used entry behmior (absohnek essential) Shale some of the tea( hers thinking with Stull knowledge and skill requirements the student. should be specific enough to enable the A description Of what programed instruc- diagnosis of student deficiencies. tion is and how it should be given to stu- 1. A statement of attitude. aptitude. and in- dents with dilletent backgrounds terest loeIs that may be essential to com- Some students will hose been exposed pletion of the program progionted material before: others will 5. A statement of plogrant validity be seeing it for the first time. Noted should be such data as a descrip lion of the fieldtest population. achieve- Ast:1elell)f(jectivesexpected and sainPle of ment-test smies. and statistical data to These should enable the student to see indicate central tendency, disposion, and what he is about to encounter. signifuan«.. Testreliability shouldbe 1. A sample sequence on how touse the considered as well as conditions under program which field testing was carried out. This will result in fewer questions later. (1. A statement of pogrom effectiveness The student should see what records are to l-low good is the program? Questions of be kept and how they relate to his learning. cost. tine. chore. and manpower efficiency 5. A table of contents or index for the student should be addiessed. The degree of out- to facilitate review side supplementation needed should be 6. An enumeration of supplementary mate. ment lotted. rials to augment the program. 7. A statement of how the program was used Pros iding materials of this type to th-2 student and shotild be used (vital,? may lessen subsequent orientation and adminis 150 cliu,TR Fivk

ratite time. The cautions thatlollow should Minimum test ing should hi( hide .t prerequisite be kept in mind: or diagnostic test, periodic unit tests. and post. I.Do not discourage the student from his tests. Some quizzes mat he sell -cored by students: task; learning is his icsp.msibility. others should be group- or two her-scored.In lie honest with the student by indicating addition. a delayed retention test would be of difficult spots and potential problem areas. va lue. lie are of molesting. The learning ()I 3. Reduce anxiety by pleparing the student mathematicsis'medicated on active student for selfstudy. learning without the burden of meremphasis I.Keep busywork to a minimum. on evaluation. 5. Do not promote the program as :t "quick and easy- way to learn. It may not be! Timing I low long should a student work on a prop,ram: Supervision !low long 5110111(1 he work on each unit?IIa It has beat the experience ()I many educational student sets his own Nitechtle. the teaher must institutions that the use ()I' 1)10;41a:tied materials provide guidelines to prevent him from spend- must be supervised. On the whole. students us ing too much time On any one unit and to pre- ing program without supervision' either rush venthis rushing through the more difficult through it or are poorly motivated and do not units. Pacing may be set by suggesting a speed finishthe plogram.In either casethey may ()I from lilt) to seventy11%e frames an hour The react negatively to the sponsors of the program. teacher must judge whetbet this seems to need The teachcr must be ready to diagnose deficien- reduction or increase. cies.plcscribe academictherapy,. and arrange Another aspect ol timing concerns the opti- administration to facilitate learning. The a:ach- mum length lot each session. A rule of thumb e' must maintain the student-1)1°gram interac- is one hour pc' sessionlatigue can easilyset tion if the program is to work. inif no time limitis set.Itis better for the Supervision is more than monitoring progress. student to do some work each day of the week It consists of ( ailing pet iodic rev iews of material, rather than to do a whole week's work in one testing.schedulingsmall group meetings or session. seminars. and branching a student in and out of a program as needed. It aiso involves num ing CONCLUSION both the less able and the more able to prevent It remains the obligation of the teacher. together difficulties and to capitalize on the learning ac- with content :01(1 learning. specialists. to be alert quired in the program. Periodic reassessment of to new and promisin; tends that emerge as goals is essential. 1)eadlines should be set for teaching- machines :n1(1programed instruction completion of wilts: awl peer pressures should assume a proper pole as instructional aids in not be discounted. mathematics. Much research remains to be (lone (1.1). and many instructional problems rentlin Measurement and Evaluation to be solved. Only by care in selection. evalua- Few single elements are able to cope with the tion. and use of this medium can its misuse be total instructional processes. Diagnosis. teaching. prevented. and evaluation are parts of a cycle that takes The bibliography that follows will provide :t place in education. The testis a part of the start toward a better education on this subject. process, but only a part;it should sample be- It is not exhaustive but has as its goal the in- havior. not attempt to do the whole instruc- clusion of references for those who would use tional job. or develop programed materials. TP.ACIIING NIACIIINES, \ND PROG1tANIED INS1'RUC1 ION 151

RIBI,IOGRAPHY

I. .1iitIcrson. R. C. "Educational Psychology." ..In Ili.I:alin, Itobeit. "Some Guidelines for Selc(ting a nna/ Reniete of Psychology 18 (1967): 103-61. Piiigranicil ext in Nlatheiliaths.Mathematics 2.Iti edu mei..I).NI.Pt og muted Instruction:.1 Teachi 59 (Joinialy 19.l6): 11-23. Mannal of hog:an:mg T('chniques. Chicago: 17.Litinsdaine..\.\.. and Robert Glaser. Touhing Edwational Nlethods, 1963. Moan liCS aril Piogninined Laming: tionrce 3.Biethower, 1), NI., et al. 1'10g:tuned !.earning: Book. Washington. 1).(:.: National Education \ lhorlrruur.. \un\I bor. Nlich.: .\:in .\tbor Pub. striation.Deli:mitten(of\udirnisualInstruc- lisheis. 1965 tion. 1960. 1.Briggs. L. J.. et al. /mtcnctiona/ NItnio= 18.'As:night. J. I'.. and C. \I. 11'illiams..,1 Guide to graph no, 2. Pittsbitigh: Atm:titan Institutes for hog:amin! Instruction. New York: John 1Vilcy Iteseatch. 1967. Sons. 1963, 5. Cram. David. Explaining Tern hi ng Ala( hi nes and 19.NI:Igo. 1Z. Preparing lintzuctional Objectives. l'oogiainining. San F.andsm: Keaton Publishers. Palo .\Ito. Calif.: l'earon Publishels. 1962. I96:. 20. Nla. 1:enneili 0. Pi ()gra wed Lea in ng a nd th- 6.(:iowder. N.\.. and G. (: Nlartin. .1dventtn eznalnal Edinalion: ,1(:1;.11 Stad. San Prands. tttflgbta. Ca:.den (Ity. N.Y.: Doubleday k Co.. to: Nlathematical .\ssociation of America. (:otii. 1960. mittee on Educational 11(1:a. 1965. 7.Eigen. Lewis I)..)ctotne D. I:aplan. and Ruth 21. . "Pioglain ,\ utoniation.- Math. Emel yin. Modern Mathematics. a l'rogunnd einalics readier 59 (Nlay 1966): .I.1.1-51. Textbook. bk.I. (3ticago: tie ;en( c Reseal( h. \s. 22.National Education Association. Se/yet/an am/ sot iaies. 1961. I'seofPtograunnedInstinctionalMaterials. 8.Englund. I). E.. and I). P. Estavan. GI.iISS: The 11'ashingtion. 1).C.: 'Ile Association. Department Jutoinated Classroom. SP 5.1.1.Santa NIonica. of .1iiiliovisual (ion. 196.1. Calif.: System Development Corp.. 1961. 23.Sthramill. 11'..S. Ile.search uu Piogianied Inftriu 9.Fentim. C. and J. E. Gilbert. Ptogunmeed In. .In Junotatd Itiblinguiphy. Washington, 31Inction Guide. Nelintypoit. Nlass.:Entelek. 1).C.: Coveinem Printing Office. 1969. 1967, 21. Skinner. B. F. "Teat !ling Nlaclii,es." Science 128 10.1.1 y. E. I. Temhing Machines and l'ingiainined (2.1 00 lobo- 1958). lust: zillion. New folk: NIcCra -11111 Book Co., 25. Smith. L. W. "The Use and Abuse of Progr amed 196:1. Inst ri it tiro]." Mathematics Teacher 58 (1)e( ember II.Pund for the ,1dainenient of Education Four 1965): 705-8. Caw Studies of Progun owed lust: 10 lion. New 26. School fork: The 1:011d. 196.1. Nlatlicinatics Study Gimp. Programed First Course err Algebia: lieuised Form 11. Stan- 12.Gaglic Robert NI. The Goinlitiom of I eon tong. 2d ed. New Yolk: Holt, Rincliait k Winston. ford. (:alif.: SMSG. 1965. 1970. The SMS6 l'rogianied Learning l'toi. 13.Clas(r. Robeit. ed. Teaching Machines and I'm- eel. Report no. I. Stanford. Calif.: SNISG. 1966. giant:tiedLearning II:1)ata and 1)irections. 28. Taber. Julian L. et al. 1 .ea :zing and Progiannud 11'ashingu 1).(:.: National Education Associa. I ncti in lion. Rending. Nlass.: :\ thlison.11'esley Pub. lion. Department of Audiovisual Instruction. 1965. fishing C:o.. 1965. H. Meitner. Ralph 'I'. "Designs for Future Explora- 29.Utial. 11'. R. ()// Convelsarimnd interaction. RC thinsinPriigraincdInsti uct hm." Matheinalicc 532. Yowkniwn Ileights, International Busi Teacher 59 W(binary 1966): 110-1.1. ness Machines. Th,nuas J.WatsonResearch 15.I I it key. :\ !bet tE.. and John NI. Newton, cds. Center. 1961. (.'omfontr,dssisted Ms/nu/ion- :I Survey of the :10.Zider,R. Games for School UM:. Yorktown I iteraluw. 9th ed. Neburyport. Mass.: Entelek. I leights, Board of Coopetative Education 1968. of Westchester County. 1968. ,r '153

CHAPTER 6 THE ROLE OF ELECTRONIC COMPUTERS AND CALCULATORS by 101;KR-1- L. ALBRECIIT Nlen10 Park. California \"11.1.1Am ATCIIISON University of :Maryland, College Park, Maryland D: wit) C. jot iXsox Unkersity 01 Nlinnesota. Minneapolis. Minnesota \VA1 I-1'xJ. K orrt:r. Lexington High School. Lexington. :itlassachmetts Bittcii E. NIsEavr. University of termont. 1'mi-tin:Awn. Vermont jolts 0. PARKER Palo Alto Senior flight School. Palo Alto. California DINA GLADYs S. TitoNt As Arvada High School. Arvada. Colorado

Chapter 6 gives a brief history of electronic computers along with a description of their uses, particularlyas they rciate to mathematics instruction. Examples of where the computermay be used in mathematics instruction are given, and actual simple computer prograws are listed and explained. Comments are also made on the various types of computers available. ranging from the small desk size computer to time-sharing systems. Some discussion of calculators is also included. '155

6.HIE Roi. 01: ELEcrRoNR:ComPuTERs AND cm.cuLAToRs

Electioni«ligitalcoutpntct, embod ingthe .\unong the earl} names associated with the basicconcepts of'nuclei itcontinuo, did nut histoiol di it.tl computers me Pascal. exist thin} ogo. hie .1, the eat I% 1950, and Babbage. Pascal. in 11;12. designed %dial ap. the experts lel! !hal nu %%mild noel need mole pears to liae been the first mechanical (output- than .t lew computers. Yet the.lantial% 11)72 issue ing device. It was .1 simple adder operated me- of Bu.stu.. intomaiton icpcnied that there were lianicall and using wgged %%heels. His inspira- then 70.000 computer, installed and more on tion came from the fact that he became bored tinier. Not otl ittony continuos aailable. %rich the job of helping.; his lather with his tax but can now perform in excess or 1.11110.0011 computations. InIti7lLeibiti/ designed a ma- calculations per second. chine that ((mid multiply as well as add. It was It was olds .1 len liars ago that the «unpuler not finished. howe%er. untilI (i9I and even then as called a -glorified slide rule. The head of did not functionreliably. olleg mat helium ics depot !mem "mai ked that Ilabbage. %%Ito nas bout inI 792. built .1 simple ill%1.15t 0111 se:. in1 Ile slide Mit l% Cie %.iiiishing- "data cote engine" that actually worked but was liont the college ctilrit !dim!. so %conkd courses onlyit small model (Figute 1heti he at- on how to use the computer. This has not hap. poled. Ve ile)W lune colleges and universities filleting the cleglees ol bachelor of science. ma,. FtcuR 6.I. the Babbage 'WO:react: engine- ter or science. and doctor of philosophy in the new field called -computer science. The 11 11111- her these schools is inc leasing rapidly. Now the use of computers is moving rapidly into the high schools and junior high schools and.in some instant es.into the elementary schools. At ordittgI%. we have thisI hapier on computing devices and computers.

BRIEF I IISTOIZY

Since the pli111:111(011( UM of this Chapteris de( 'void/ tligard compliers. the discussions that Follow icier only to them The histm y of digital computations leaches taeback 11110 history to the time when quatailuation was lust conceived by man and he devised schemes for computing things. 'These schemes mulotibledlv could also be related to the beginnings of -modern :age. Ina.- as 1 Ile% Set 111) a one- 10011e (0111'111)011(1e11Ce between sticks. or fingers and itimals or whatever else early men may have been trying 10 keep track of. This led 10 ( 011111ers of %arions from comom to :Rider.. from :older :: lo (alcohoor, :111(1 Jimmy to computers. CmirIrsy of the.Boman!:iirchitre 156 SIX

tcniptcd .1!mgt. }sinking model. he csas Inla He it lean} ot Led! Imptoscnienis on thus(' designs to make it uork bet ause 01 a lack of (hafting.trul lot the use punched caul, continued.Ist 1»11 mechanical skill at the time. I leinian Hollerith ins emed a machine for sort- In 1833 Ilabbage insenied a mine he (Alec ing punched (aids (.1 pscscnt (I.1)(15101) 4)I ands an "analytical engine. Had it excr worked. it .1 (ta d is shown in Figuic 6.2). and he built the would !lax e done man) mote things than his flIM NCI. Of ILIl pundit:di:n(1 tabillating ma engine. It included the concept of a chines. These ina( !lines were rasedits woik on memory: that is. it had a section whet e ntimbeis he cenSlis Of 1890. «rald be sioted. It also included the concept ol Anothel stage in the decelopitient 01 (output- plogranting. ,\ progiant lot.t «imptiter is a set cis is :Mistimed I)} the II1M 60 I. slum ti in Fignie

1)1 instructions clitectinA the compute' }dial 10 6.3.Hiis equipment had the abilit) to exec me (10. His "plogiani- (.en in( hided the «incept of 'went}or more opeiations between the sra((es- plogiants being able to change its own instt tic- sise runlits; in ol (.11(15. The wiling ()I.1 (4)111101 lions. These ate the main seasons 11abl)age panel deemitted the sequence of opciations is thought 01 as being one of the prime con- that }vet c out I)) the machine. It had the ibutots 10 the ideas of out model ti computing capability oflooping back to icpeat a sulks of machines. 1111101-Itinately. he never completed his calculations. machines and (lied in 1871 after many years 01 The IBM 60 Was01 I °Wed it)the ( frustration. Clearly. however, he was an out- 40 anted alulatctr (CI'(:). which }sas mote ad- standing- theoretician. vattced.Its()pc]ations}vetecletetinitied by One of the major factors contributing to the reading its a (leek of cards containing coded in-

earl) decelopment 01 the computer field was the s11 boats. This leptesented .1 significant step use 01 punched catds. Their use dates back to rot watt!. since it was considerably easier to work 17'25. ecru below the Indust] ial Re} oltition. with decks of cards than eith }sited plugboaids oa automatic loom was cleceloped b} jacquard in conti 01 panels. Note het e that the basic ideas of 1801 which csas (ratione(I I)) putt( lied caids. and Babbagc. dating from 1833. }vet e being emplo}cd.

Fictint 612.The Hollerith card

it IF A-1-6>C *11 THEM 13 III 11II

1 1 0000000000010000100000100000000000000M000000000000000000000000000000000000 0 0 0 0 tis cr Is11 rain Is rani 10.01 rinnistsilis:0,11ilmustsiltutilo raturas alums! siststssstratstilcutuctiscs'mum' Imamn111313 I 1111111111111111111111111111111111111111111111111111111111111111111111111111III 2 2 2 2 2 2 2 2 2 2 2 2 22 2 2 2 22 22 2 2 212 2 2 22 2 2 2 2 2 2 2 2 2 2 2 22 2 2 22 2 2 22 2 2 22 2 2 2 2 2 2 2 2 2 22 2 22 2 2 2 2 2 22 2 22 2 33333333333313331333333333333333333333333333333333333333333333333333333333333333 4444444444444444444444444444444444444444444444444'4444444444444444444444444444444 55555555555555155511555555555555555555555555555555555555555555555555555555555555 66666616616161666666666666666666666666666666666666666666666666666666666666666666 11111117117117777777777777777777777777777777777717777777117777777777777177777777 88888888818181888188888888888888888888888888888888888888888888888888888883888888 99999199999999999999999999999999999999999999999P99999999999999999999999999999999 I Z1436 I 1 I 13111211:41OIIIIIIII:211/1/ZOSIOINZO)1112/3141010/11114:41043444$4:41443SOSIMISOISOI:ISMIOZSIIIISISIOSIMIIMIIISIIIIISIO: 157

Fugutt: 6.3.113111 604 and 521

Coin tem. HI info national 1:tiatic.' Mar blurs Cm trim:lion

Another eally computer that_ used many of was available. liabbage's ideas was the :kutomatic Sequence The Electronic.NumericalIntegiator and C:ontIolled Calculator, or Mark I. bah' In Pro- Computer (ENIAC). designed byJ.Plesper lessor Howard Aiken of Hal ald UniS'ersity and Eckert and Joint W. Manchl atthe Moot e completed in19-1-1. He Ivo' ked in conjunction School or Engineering of the Unieisit ()I Penn- with MAI and used electromechanical devices syk ania and completed in1915. was the first such as relays, counters. and cams: and reading. electronic-omputei (Figure 6.1). It was «nisid- punching, and printing- units. The sequencing ()I instructions was mainly by means of papertape input. :\ limited capacity lor program branching Fumr. 6.4. EN/AC computer

3 158 s)IN

eled .1 malt" sit ide ""."(1. I" Ii( "I." I` in e"gi- Itis mu% on displa) in the Ntnitlisonian Ithaint- !leering tee linolog). The l.\ fAC: was a specialiied Thc fits( installation lot a plicate cotpota- computer built lor the U.S. Arm) to calculate lion. also a UNIVAL I. went to the General ballistic tables lot the Oldnan«b Department. It Electric Appliance Park at I.ouis%ille. Kentmk). occupied a ,pat c of .1hoth .h1) 'cc( by 50 feet and in1931.Inthe meantime (1952) the Alome contained about 18.000 cat stunts tubes. It )%0111(1 School had «mildew(' the ElectionicDisnetc not open ate vet y long. however. berme it would vatiable Automatic Computer tDVALL bleak down. The elecnonitcomputers mentioned ;dune A number of well-known computers (wick!) Sete but the list of a whole set ics of computels followed the NIAC. This is the point in time using cat twin tithes. Others %%etc the UNIVAC where the ideas of John von Neumann and his II. the U\ 1\''.\(: 1101. 1102. and 1103; the IBM work on the logical design of computers signifi- 701. 702. 701. and 709: the Bioroughs 205 and

cantly influenced the field. Ills design work was 220: and the Bendis GISto mention old%a appatentl)influenced by the studies inpine few. These computels using cacutun tubes are logic by Turing. \rollNetIlliallil (le%eloped the all eNamples ol ,shat are tailed Jo.q-genetahon concept of .1 siorrel program capable or modil)- (01111m105. Rs 958-SO the mrcalled .sonnel-gcn- ing itself. This idea. and ()diets. helped sulk rration comptucts began to appearcomputeIN many computes design problems and permitted made 55ith nail 515101 5 instead of tubes. EX.1111pICS the development of computers with gleatly in- of these are the IBM 1620,1.101. 7090, 7091. (teased problem solving capability.. From these 7070. 707.1, and 7010: the UNIVAC 1107; the ideas came the firstelectronic stored- program Conn of 1)a ta 160 and 1601: the But roughs 5500; computer. the Electionic 1)elaycd Stotage Auto- and the Honeywell 200 and 1800. Maintenance matic Computer (EDSAC.). completed in England ptoblems with the Na ond-generation computers in 1919. lit 1051 the first commercially aailable wele consi(lerabl)less than the) wete with the electronic stored-progt am computer became as a i !- fitst-generation tube machine, since the useful able. namely, the L/NIVAC I (Universal Anto- lifeof iransisuasis considerabklonger than 'italic Computer). shown in Figuie (i.5. The first that of vacuum tubes. one of these went to the Census Bureau in 1951.

Goorloy of lfrolington Rand Univac II 1,t()Li. ELEcTRoNic com pr.i.i.:Rs AND GA Lc( LAToks '159

The I Um ihna-g, netairon compute's appeared inI!162. nse (oniptiteis emplo) the new ill le- aicil chi nil s, This, nicails that man) (Fillet ent It""Ii'ms "ill be 1)(110""d "" '11"11 the di( (.11(1 shown .11 the hullutn of I-.gilic 10: is (.11hible Of doing the %yolk of the II piffled tit( nit (.11(IS .11)Me it.I: \Mill/ICS 01 the thild-gcnetation computers are the 360s and the WM; the Burroughs 6.000 and 8,000 set les:and the SI) \series machines. here now esist imitable continuos with inte hated tin nits. sw h as the I lewleu-Packaid and the Vang Labotaunies progiamable cal( ulators. that pet lot In as well as main 01 the arliet large machines. :\s each new genetation oI computers comes along. the problems ol maintaining themare lessened. Sc hook contemplating the purchase or rental of mm1)11611;4- equipment should be percep Ike about the type of computer they are getting. Fictlizi:,(;.6. Tbe inlegraled card al Obtaining older equipment at "baigain" prices the bollom of ale piclare is capable of doing will not be :Ibargain atall in the long run. the work of the 11 1»inled circuit cards Equipment in this rapidly growing field can very shown above il. quick!) become antiquated and expensive inre- lation to the state of the art. Several Computers now athrtised as "educational computers" are 1106011ti(11111e into (0111111011Ilse. This was obsolete machines that actuall) cost more than actually a big stride forward in that it reduced new equipment! laititenatice of computers is errors. costh and has 10 be taken into site0g considera- Current progiaming methods almost es.clusk e- tion in the put( !lase of a «nninuet. It is alsoyen l) inohe the use of what ale frequently called important that the tight sir of compute] be /right)' -lrucl or problcm-otienird languages. Es- obtained. amples or these ;Ie FORTRAN (Formula Trans. Another ter) important aspect of the evolu- lation). (;0110l, (Common Business Oriented tion of the compute] field has been the inc reased Language). and ALGOL (iligoritlint Oriented ellicienc in the use of the machine. that is. in /.anguage).Ehese computer languages attempt to the wa of (oninumicating tsith the machine or. have the compute' statement of the problem 115 tousei Ile MI11111011tenth of programiv the (hoe to the normal mathematical-formula state- machine. Speaking ratitesgenerally. the early ment Of the problem as possible. as in the case first.generat ionmachines required 111:11 both of 1:()IZTIZAN. COBOL was designed to be very Ihcil inset uc [lofts and 1 kit (1.11.1 be pill in bin.ii). close to the businessman's statement of his prob. octal. or hexadecimal number form. Each opera- hem. ALGOL is a language designed by an intion. arithmetic or otherwise. required a sepa- ternational committee in an effortto devise a rate instruction in this lorm. Several stages in computer language that would be universal!) the development of programing followed this. used. It is an excellent piece of worebut it did

Methods were learned for using dec imal instruc- not achieve universal acceptance., ally there tions and data. Mnemonic letter codes for the exist many higher -lm el languages.t these ad- '160 C11 \rER SIX

%anced languages a single statementIsIINII.111 CHICAGO TRIBUNE eglikAlelit to man\ of the eaiel, mole [clump, number «ides. In the past few \eats e%en simpler languages have been doeloped for educational (and smallbusiness)use.Languages such as

BASIC. TELCO NI P, I (U 1 KTI:A N, CUPL, and APL enable the 'Iser to learn pm- graining in a matter of a few hours. These languages ate well suited lor educational use. Work in the area of programing and the various (0111- inner languages is normally (ailed Aollwaw work in contrast with work on the computer itself. which is spoken of as hardware \cork. The previous paragraphs indicate a %et y rapid de\ elopment of both the hardware (the machine and itsrelated equipment) and the software (the programs and computer languages) for the

Coto to). Ch k agoTIHume computer field. 'Him is something of a merging of these two aspects as the computef-hadwate FIGURE 6.7. "And another thing about computers. they dwi'l goof off behind my back." design goes further and further toward reducing the problems of programing or of man's communication with the machine. One may think of computers as having human characteristics (the cartoonist responsiblelotFigure 6.7 certainly does!) and may fondly dream of the day when he may just vocalise a problem to the machine and the machine will solve it. The state of the art is, in fact. quite jar born such a reality. Wm k is going on in I his wren. :nut devices have been constructed which do "understand" some spoken Nvords and can thus acti\ ate a computer for action: but this is only a small beginning. Suffice it to say that compute's ate now very widely used in government. industry, and education and that these uses will cleatly expand very rapidly. Society is at a point where if all computers were suddenly removed from the scene. the change would be crippling to our entire economy and progress.

Funnui 6.8. A compuler center

Cour Loy of Control Data Corporation 111 OF ELEC:1-liON IC CWII'Ll'EliS\ NI) C \ "16.1

IIII. C.ONIPC IN 111- NI VII l'ICS INS 1.1:UCI.ION -vor.

The ptrctiding sec lion MI111111.111/Cd the (ICVC101)- eleclicmiccht;itaic ontputing machines. The itttpac Iof these high speed «unputing and data Inctecssing mac lines On modem icty .otel its at ti( Ili(, III( It .0,111;2, al 1.1111.INCR 1.11( Is°.

IIIC I01 ut c tipalions 1)( ul,Ih must be able to %%od,clitecil% and indite( II%frith comptnets isine teasing %et%rapic11%. Thus it would be .1 dissct %ice to students it the% did not have ample °ppm tenni) to use computers eluting then plemllege of pietoc,ttion,tl education. This see Lion inc hides a minima' %ol the c intent genetal (IWS 01 computer, and a disc itssion. frith .tit \amides. ol the applications of com- Imleis in «mittIteliOn ith Ito& mathematics clasNes.

General computer !.,e%

...idjutun.tls.11C lullol statements that indeed continue's ate becoming ut integial pad ol the educational poccss. In pat tie ttlat. much attention is being gi%en to the use of .tomputet as .1 Lunn to incliichtali/e instruction.Figure 6.9 pictures a center where pepaation lot such instruction is taking place. This use ol a continue' is generally referred to as CAI (Computer-Assisted lnstruction) or (:AL Corn toy of PhilroFord Corporation (Computer-Assisted /.earning) and in a crude sense means the1» escntationof plogramed- learni»g materials by means of computers. For FR,IrRE, 6.9.AI lhe Philco-Ford plena inIVil- CAI there are usually several terminals and other hnt'(;)ove. Pa.. co))-icalum Jo) a system of auxiliatdes is es «mnected to .1 c elm al computer. «»npule)-ai(1edinsh utlimi is beingp)epthed These terminals may involve many different for the Philadelphia schools. Use of a tom- kinds of equipment. such as visual screens, type- pule)language called INFORM permilA writer keyboards. audio-instruction devices, or leachersIvIffepare lhe curriculum wilhoul even films or slides that ate synclnonized with the special knowledge of programing skills. regular terminal display of programed materials. The computer as a tutor is an infant today, bin as more is learned about how to use this development effectively it will become an invalu- able aid to the classroom teacher. CAI is considered further in Chapter 5. "Teaching Ma- chines and Progranted Instruction." 162 CIL\ l' I EI. SIN

A «unputc: ith terminal de% i( es such as those stud) population trends. economic gross thind describedfor CAI ma% also be used inthe mail\ other statistic al matte's. Language tea( hers following: have been using computers for the study of I.Peparation 01instt uctional materials linguistics. XI usic teachers lime been working for 2.Reinforcement 01 concepts by means of many years On computer-composed music. Art simulation of physical, social, economic, tea( hers have investigated the use of computers politic-al, and business system.; (tor exam- to mathematically organise certain types of alt ple, b) playing computer-based economic work. Librarians are using the computer to cata 0 political games the student makes (deci- log the books available in a geographical area, to sions based on his knowledge of the far- process book (utters, and to process interlibrary tots that influence the situation) loans, thus making aailable more books to more people in the region. 3. Acquisition and storage of student infor- Computer usage is rapidly becoming an inte- mation (responses) gral part of many courses offered in %ocational 1. Retrieval and presentation of information education programs and business education pro- stored within the system grants. At the college level a variety of programs 5.Administrativedata processing.record combining computer science with other disci- keeping. counseling, and research. plines is evoking to prepare students for profes- Computers are used in the educational field sional cat eels in the computer field, pat ticularly b} many people from many different subject in the areas of systems analysis and business data- meas. Social science teachers use the computer to processing progt (Figure 6.10).

FIGoRE 6.1 0.Business data-processing facility

111-111-

GourleyofButt oughs Corpmadon I III. ROLE 01: El.a. IONIC CONIPt'l RS \ \ I.Cr I, \I OR:, '163

Anothe of :;eat potential lot wputer loom. pen cntage of m hook thionghout the titiliiation is that of ptoblent soling. continuos «ninny that ate inNolvcd in the use of computers and computer 1»ogiaming are being used to in thelasNI 00111 IS stea(lil\111 leasing at an int- soi e1)1 Oblell1SinS( iell( 50( 'alSl lldieS..111(1 pessiNe rate. What sous 01facilities ate mail- 1 1 1 : 1 1 1 1 C 1 1 1 : 1 1 1( S. 1 1 isill1 Ilk :Ilea that equipinem able: Where does the computer aid in teaching and 'wadi:11s ale available today to enable a the present curriculum: How does the use of a school to get sun a:(1. The next section giNes it «mputer change the curt it ulum: Why has the desuiption 01 «nitputeis as a problem soking use of.t compute' in the classtoom been wel- tool in school mathematics instruction, comed so enthusiastic:ilk? These ate some of In s( 110015 ( 01111)111C1S are :1\ a liable. one the questions that ate considered in succeeding (an find almost cerone using the machine: sections of this chapter. I. The administrator(lob his bookkeeping. In the next section sample ptograms ale pre- scheduling. and student record keeping) sented.usingseveraldillerem computer lan- 2. The teacher(lor demonstration of key guages. However. before examining specific ex- ideas in the classroom) amples. itis pertinent to disc uss the reasons lor using «nuptial' 'no: paining ill s class- :1. The student(lor the application of the theories he has learned in the classroom) rooms as an integral part of the presenturric- Very often the student will center many ulum. The use of 0otminuets its teaching aids does not in:less:1i il) alter the mathematical con- ofhis extraunticular about the computer. Nlodels of «nnputers are lie- tent of topics being inesented bin often will stig- gest modification in the manner of piesentatim. queenly consult( led as plojects for 5( ienCe 1:111:1. Dating bureaus. computer -1» Ogralined A mathematics class that has access to a com- by the students. are now common on many. puter or computer terminal is affected in seveial ways. One of the often observed differences ,be- campuses. Several effective uses of computers have already tween this and other classes is the extent of student motivation. Nlans students of all ability been demonstrated on the educationalscene. Education is now at the stage of rapid expansion levels at all school grade levels become highly and development of these uses. motivated .hen permitted to useit computer. The reasons for this motivation are difficult to specify. but its results ale frequently dramatic. The Impact of the Computer in the In now Mathematics Classroom cases the machine itself is at first the prime target of student interest. However, this About 1952 the modern methods 01 teaching interest soon transfers to the magnitude of the mathematics etc:lied an upheaNal in mathematics p1oblern soh ing power that is available. The fact classrooms. Textbooks were changed (and are that many students will do interesting, challeng- still being changed); teachers slatted going back ing. and significant mathematics when allowed to the «dleges fn 'one training in modern to use the computer to pursue their own special methods; and «die,,,.5 started changing their interests has been repeatedly demonstrated. plograms lor the training of teachers of mathe- In working with low achievers, many schools matics. The reverberations ate stillfeltin a lore found a desk-top pot table computer to be inajmity of school systems asthe impact of an essential piece of equipment in the intake- modern approaches «nullities to stimulatere- nottics classroom. In using the calculator 01 com- visionsof elementary and secondaryschool puter the studentisableto concentrate on mathematics curricula. process in problem solving. for he has a device About 1960 another loolution started with that remmes the drudger) of computation. Often the use of compacts in the mathematics class- flowcharting is taught in preparation for prob- 164 CIL\ VITI: SIX

lem solving; then problems ale solved by using ing in a very short thlle. One might ask why a the flow chat and the computer. The students program should be written, since the student can study mathematics and mathematical appli- has «mstructed a flow chart. When a computer cations rather than do drill on computation. program is written, the student can have the and the interest leel is significantl) Itighei be- compute' executeull) his proglain and lack e cause the indhidual learner is now able to con- immediate confirmation as to whether 01 not his centrate on meaningful problems. algorithm is correct.Ifitis incorrect, he can 'reacheis of regular classes that have access to then study the output and 'elute his program. a computer htue discovered they can now con- During this work, the teacher can observe the centrate on the problem solving aspect of their I aiming of each student's program and can im- subject and not the alithmetic, which can often mediately see which students ale hating trouble obscure the real problem being discussed. For with ;t particular concept. For example: Which example, consider the second-year algebra prob. students had difficulty when A was negative? km of finding the solution set of the inequalit) \\Thick students could not handle the case with ax:: bx c< 0 for any values of a,b,and e. imaginary loots? Which students did not con- If students ale each asked to write a computer sider the magnitudes of the roots RI and R, ogram such that the computer will accept any when they occurred? Which students failed to A. B, and C (in computer language) and then think of any of the special cases? The answers to type out the solution set, they must approach these and similar questions permit a teacher to the problem in a general manner, since no spe- analy/ethe problems of individual students cific numbers have been given. A valid flow chart quickly and in detail. for the solution ofthis problemis given in Modern methods of teaching mathematics re- Figure 6.11. (ibe the student to discover the generalization The student who can develop such a flow chart :as tell as to do the particular problem. Com- understands the meaning of the disuiminantiu puting methods require the student to mulct- the quadratic function, the elect of the sign of stand the generalizations in order to program A on the graph of the quadratic function, the the particular !noble'''. Because of this relation- elect of dividing or multiplying an inequality ship the student findsthatprograming,isa by a negative number, and other special cases natural part and extension of the development that can occur in this pioblem. In addition, he is of his mathematical thinking. Programing forces motivated to consider a general mathematical the student to be specific and definite. relationship (he uses variabl,:s). All the relevant While a computer can be effectively used in concepts need to be considered without lefeien«! enhancing the teaching of the regular mathe-

to specificnu ical problems and thus without c ricultim, dune ale also many situations the confusion of atithmetic ell-cm. Although the where teachers and students can use a computer teacher's original assignment to the student was facility to extend the study of a particular con- to write a plogram, the flow chart lepresents the cept beyond what is traditionally done. Seventh- teacher's purpose in giving the assignment: to grade students, for example, can investigate 01 promote a complete undeistanding of the mathe- research a mathematically interesting number- matical ideas involved. In addition, the assign- theory problem such as the frequency of primes ment provides the student with the opportunity in an arbitrai y interval. to design his own algorithm. This aspect of computer-assisted problem solv- To expedite the transfer from the flow chart ing also leads us to a consideration of appro- to a computer program, itis desirable that the priate teaching methods for using a computer. student have available a simple computer lan- Teaching that makes full use of a computer's guage that allows him to mite the program cod- potential requires some changes from the tradi- OBTAIN NUMERICAL VALUES FOR A,B,C

NO YES

IS NO B2 4AC < 0 YES

NO YES

SOLUTION SET IS ALL X

SOLUTION SET IS _V ALL X SOLUTION SET IS NO X SOLUTION SET IS

IS NO YES SOLUTION SET IS B2 4AC = 0 ALL X > C/B

B+,/B2 4AC R1 = SOLUTION SET IS 2A ALL X < C/B

4AC SOLUTION SET IS 2A ALL X EXCEPT X = B/2A

SOLUTION SET IS NO X

NO YES SOLUTION SET IS R2 < X < Ri

SOLUTION SET IS ALL X < R1 AND ALL X > R2

FiGuRE6.11. 1 ow char! 1 66 (31 \1'1 1.0 si\

actual presentatioirlollowed.btchill routine 01 acti5 ities: a ((nuptial in the athealics class. matematic. instruction.Students shouldbe !Doti can be .111 ex«.11ent demonsnatio, des is e. encouraged to make conjectures and develop The teacher might write and run a program to (and !cline) their own ;ago' ithits and 'hell use form the partial sum of an infinite reties o to the computer to test their algolithns. This ()lieu plot function that is being discussed. Students means from fifteen to twenty different student Ina) ask questions such as "The appcAimation

progiams for a pat titular problem. The teacher mat be this for1 On (elms. btu what isit101' inthis situation provides "test" data for the 1,000 terms:- or "Winn happens to the graph it student and may help him in the development I change this «nistam or militate 111.0 term?" The of the program; the computer is used to check speed of a complier ',emits almost immediate the plograin. ()hen the "giving" of a final answer response to these gnestions %%id' specific lather should be delayed until the student has made a than general .11.%ers. A student questionWhat

number of trial computer runs. if... .r a r e l ygoes unanswered when the power In addition to enabling actual programing of et com puler is readily available.

EXAMPLES OF PROBLEM SETTINGS AND PROGRAMS This section contains a set of examples of pro. Although some annotation is given to the light giants written and exectted by students. Each of the statements in the programs. it is assumed program was written ;cs part of a student's ex- that the reader has some knowledge of one or perience in learning mathematics. Each example more computer languages. Es en if he does not. includes a statement of the problem and a dis- this section should still be of inkiest because it cussion of the pm poses that motiated the pro. includes a discussion of the purposes of the dif- gram. In the first eight examples, the program ferent problems and shows the computer solu- and its output (a computer printout) are listed, tions. If 'the teacher has bad some contact with and the name of the programing language and a computer language, he will find that with only the computer on which the program was run are a little effort he can read many of the programs. identified. Examples arc presented for various since languages for most computer systems have many similarities and there is extensive carryover grade levels and for programs run on different in knowledge from one programing language to computers. Almost any computer, from a small another. Text materials designed to teach com- desk-top model to the largest computer, can be puter languages are available from commercial used to solve intosting problems. In general, publishers, computer companies, and education- the easier the access to a computer and the more al institutions. An annotated bibliography in- versatile itis, the more useful it will be; but al- cluding such texts is provided at the end of the most any computer can be valuable. chapter. 1:01.1: OF ELECTRONIC: (:Omlq : \NI) cmcI-1.-voRs 167

EX A 11)1.1.:

Crude: Problem: 1)rilionstration of scietnific: notation i'urpote: IIIthe inlet-mediate glades students are usuall intioduced 10 sc ientiliclunation. The loiltnving Iwo one-slaienteni pro:pains were nsed in .1 sixilt-grade class 10 augment the (list appt oach Iu stieniilic notaiion. :\ compute' prints the small numbeis ill intege: 1(11111 Inn at sonic- !mini changes to the exponential folw for larger numbers. In the first prograin only nonnegative iniegral poweis of 10 are used. For telaiivel sin,111 ntunbets the continuer types the. number in decimal Iuiiii. but alIt).1100 the coin-

lintel changes 10 I 10'. Since die lust program is lintiied to powets 01 10a ciictint-

Nla II( e which tends In suggest lhat one need only add /est: 10 intilliplI)\ I%101, lowed b wore general numbers in the second prctgram to illtisttale the genet-al use of scientific 0011110n. Language: TF.I.COMP Computer: PDP 7 TIMESHARE

Program:TYPE I.10.100,1000,10000.100000,1000000,10000000.100000000.101)0000000

Output: 1 = 1 10 = 10 100 = 100 1000 = 1000 10000 = 1*1011 100000 = 1'10;5 1000000 = 1*1016 10000000 = 1*1017 100000000 = 1*1018 1000000000 = 1'1010

Output: TYPE I.12.123.1231.12315, 123156,1231567.1234678.123.156789.1231567891 Progm711 : I = 12 = 12 123 123 1231 = 1234 12315 = 1.2315* 1011 123156 = 1.23156*1015 1231567 = 1.23.15679016 12315678 = 1.2315678*1017 123156789 = 1.23156789*1018 1231567891 = 1.231567891*1019 (.11.\I.TER SIX

EXAMPLE 2 l;rade:

Problem: To find the intersection of two finitesets specifically. to find the intersection ol

{ 1. 5. 7, , 29} and {2. 5. S. I I. 1.1 -11; PUrpoce: Pt imarill algorithm design. a consideration of whatone does in finding the intersection of two sets

This problem servesIt)reinforce the concept of set intersection and10 Ill:111011Niran' that seztl'Cilittg" for a pattern is a desirablyway to approach the task of generating a see. Lung:mgr.: BASIC Computer: IIONFAAVE1.1. 235 TIME-SI !ARE Program: 10 FOR = I -FO 29 STEP 2 il'he vatiable 1, takes on all values of the ele- 20 1:01: K = 2 TO 11 STEP 3 ments of Net A and the variable K takes on Al 50 II: K THEN 50 values of the elements of set n.) 1t GO TO ii0 50 PRINT Iii) NEXT K 70 NEXT.( 80 END

°Input: 5

II 17 23 99

EXAMPLE 3

Grade: 9.10.I 1

Problem: To demonstrate computer printing of graphs of trigonomenit !unctions usingstand- ard library progtats Purpow. In the ninth, tenth, or eleventh grade, studitZ normally begin the study of trigono- mettle functions. In the following onstep plogratn. the graphs of several trigonometric functions ate plotted on one set of axes 10 show the effect ola on the plot of the function y = sin OA:. The plot routine comes as a standard libraryprogram for most large computets. The availability ol a quick and a fairly accurate graph of almost any function encountered in high school mathematics isa tvry useful feature of a classroom computer or terminal. Language: -rELCOM P Compuler: PDP 7 TIME-SHARE

Program: PLOT SI N(X /2). SI N(X). SIN(M) ON 1 P(I 00*X)/ 100 1:01: X= SI'1:SPI!25:SPI Otapta: (See graphs on next page.) nu: RoLE or Eux:TRoNic comm. IERSNI) (:11.CULATORS 169

-5.1.1 . * -1.01 . + * -9.89 . + i; -2.76 . + * -9.6:: . + * -9.51 + * -2.38 + * -2.9t; + * -9.15 .+ * -2.01 + * -1.88 + * -1.75 + . * -1.63 + * -1.5 + . -K- -1.58 + . * -1.25 + * -1.13 + * 1:tcutzt- fi.12. Programmama .- --! A. + . -.87 * + . [The numerals on the kit t epreNent value:. of N. -.75 .11. + The 'tilled 011.11,.1 |,u, indiCated bv the pint:. -.69 * 4- . (3Ithe graph range front -1 to -I- I.) -.5 * + . -.37 * + . .- A. + . 12 *+ . ,: .19 .+ * .95 + .37 - + * .5 ÷ .62 .75 .87

1 +* 1.13 1.95 * 1.38 1.5 * 1.63 * 1.75 * 1.88 * 2.01 * 9.13 * 9.9(1 * 2.38 2.51 * 2.63 * 2.76 * + 9.89 * + 3.01 * + 3.1,1 * 70 CHAPTER SIX

IiXA.\1I'I.E Grade: 9.II

I'voblem: To ..olvt: a ststent of tt linear equations in it wilsoowns (Gat NN.J01(1:iii tedta( lion)

i'urprisi.: the student is tequired to know .111 important algorithm and to Ilseitto soke the f/1"fible111. The output Will he the M)hlti011 Set of the :,...le111 of etilelli(111 or the tedneed form til the equations in case the NyNteun of equations has 110 101111i011 ur 11.(S .111 infinite NCI of

Lunguage:TELCOMP

Computer: PDP 7 TIM E.SI R E

Program: 1.0115 TYPE it. ""YOU 11:1171.: N EQl1ATIONS IN N UNKNOWNS. WHERE"' 1.01 DENIAND N 1.015 TYPE NOW GIVE ME TI I E COEFFICIENTS*" 1.0" DEMAND A11.J1 FOR .1= I:I:N+I FOR 1= I:1:N 1 m.)5 TypEc # tipawes the paper up) 1.01 K O. 11 = N 1.1)1 K = 1.05 To srEp 1.09 11: Ark.k.1z 11 1.01i A[K.JI = AIKJI 'DIV FOR = I FOR DIV = AfK.K1 1.07 Do pART 2 Folk= 1;I:N ;1 FOR AILKI IF >< K FOR 1 = I:1:N LOS TO STEP 1.0! IF K N (Note: means 1.09 TO STEP 1.11 IF K N 1.10 C= 1IF AILK1 Th.< FOR.1 = K ±I:I:N FOR C = =1) 1.11 S =-. S FOR 1:1:N+1 IF 1.12 T K K W =IIFC=0 1.15 TO STEP 1,01; 1.1.1 W = 1IF A[N.N1= 0 1.15 TYPE -"NO SOLUTIONS- IF 1V :z1 AND AIT.N. lin TYPE -DEPENDENT- II: = 1 AND AIT.N÷ 11 = 0 1.17 T\ PE -"REDUCED ,MATR IN CON: A INS" IF 11 1.IS TYPE All.j FOR .1 = 1:1:N÷I FOR I:I :N IF W = 1 1.19 TYPE "'SOLUTIONS ARE"" IF W 1.21) TYPE All.N+ I] FOR I = I:1:N IF W 1.21 TYPE #. # 2.1 Ali-.11= Alk..11*N1U1. 2.2 A11-11= n A11..11"<" 101 8 output: Do ',ART I you HAvE N :N N UNKNOWNS. ww,:j.tE

NOW GIVE NIE THE COEFFICIENTS (The with-t ihio! athols are typed in (luring execution time.) THE ROLL: OF ELECTRONIC CONIPLITERS AND (:.11.COLATORS 171

.\12.11 = 7 s01.1.-rioNs ,A11.31 :\12.31 2.1 I)0 PART I MI LVVI N F.()J'ATIONS IN N UNKNOWNS. 11111:11:. N

011" )IE II I I' COEI F:CiF.NTS N1.11 = 2

AIL:ij-- I =

:\j2.2J = 5

A[2.51 .1 \ f2.11 =

AP.I1 =S = I Al5.51 Af5.-11 = NO SOLUTIONS I: II) I CED \i\II. IN CONT \ INS (When tittle Is no solution. thiN program gixes

A11.11 I the roelli( lent of the reduced Immix.)

\EL21 (1

AILNI .5 I..11 = .227272727 :42.1] = 0

A[2.2] -2 I .Al2.31 = .5 A(2..11 = .8030:36561 0 .\13.21 = 0 43.5] = 0 A[3.11 = 6 -172 ( :HAVITI: SIX

EXAMPLE 5

GYadc: I I.12

Problem: To find the ./eros of a polynomial up to degree 5, using Newton's method

Purpose: The atudrnt IN to «nistruct an algorithm to sole the problem. The final plograin miliies the following techniques: The computer is given the coefficients of a polynomial function. the upper and lower bounds of a Nearch :ilea. and an increment of search. The plogrant then has the computer search for c:hanges in sign of 1(x) on each given increment between the given bounds. When a sign change is detected. the program branches to a Newton's method tontine that prints x and 1(x) for each ef eight passes throng!' the routine. Mole sophisticated programs of this type can set their own bounds. check for looping in the algorithm routine. use the zeros of the derivatives of 1(x), or employ many other techniques. However, this program was introduced in a more elemental y setting dim the more complicated input but simpler program.

Language: FORTRAN

C0711puler:IliN1 1620

Program: It) READ 105.:\. B. C. I), E. F. S. T. R (A, B. C. D. E. and F are the poly- PR INT 110, A. B. C. D. E. F. S. T. R nomial coefficients. S and T are = 0. the lower and upper bounds of = (MASBrS 4- CS DS ErS F the search area, and R is the in- V = S 4- R crement of search.) 20 IT = (a(A V 4- B)*V C)V D)*V ErV F (Steps 20 to .10 scan between the IF (FS* FV) 70..10, 30 bounds looking for sign changes.) 30 FS = FV V = V 4- R IF (T V) 10, 20, 20 0 IF (N) 30, 50, 30 50 PRINT 120. V. FV 60 CT = CT + IF (CT 5.) 30, 10, 10 70 VI V 1)0 IOU 1 = 1, 8 V1 = ((((A*VI B)*V + CVI +D)*V1 + Er VI 4- F VI = VI FV1/((((5.A*V1 1.11)V1 3.C)*VI 2.*DVI E) 100 PRINT 120, VI. IN I (Newton's algorithm, x = x, f(x,)/('(x,).) GO TO 60 (Thecomputer prints on each loop.) 105 FoRmAT (9F5.2) 110 FORMAT (11.10, 9(F7.2, 2X)) 120 FORMAT (I HO, 11-1X = 8H F(X) = .F1,1.8) END, 1'1 IE 1:01.E OF ELECTRONIC CONIPUTERS AN!) CALCULATORS '173

Output: 0.0.00 1.00 -12.00 0.00 2.00 -6.00 6.00 .10 N = -.40158730 1:(X) = .01600000 (The first line lists the coefficients of the polno- N = -.40158103 1:(X) = -.00003310mial (0,0,I,-12. 0. 2) ,the bounds of the X -.10158403 1:(X) =. -.00000010 sem eh (-6 to +6), and the increment of search -.40158103 F(X) = -.00000010(.1). The output gives the values of x and 1(x) N = -.10158403 F(X) = -.00000010each time a root is detected.) N = -.40158403 F(X) = -.00000010 N = -.401:;4105 1:(X) = -.00000010 = -.10158103 F(X) = -.00000010 N ."9""991 1:(X) =-.87500000 =.415i)5568 F(X)=-.0639891(1 N = .115505181:(X) =-.00017750 .41550519 F(X) = .00000010 N =.4/550519 F(X) =0.00000000 N .415505191(X) =0.00000000 N =.415505191(X) =0.00000000 N = .41550519(X) =0.00000000

EXAMPLE 6 Grade: 8. 9 Problem: To find the slope of a line. given the coordinate:. oftwo points on the line Purpose: In elementary algebra the student is introducedto the idea of the slope of a line. It be. comes important to know how to find the equation of a line, given a point anda slope. given two points. and so on. This simple program 'quires thestudent to use the general notation for the coordinates of two points to finda slope. The definition of the slope of a line is used. It is an easy step from this program to the generation of the equation ofa line through the two points. Or, if algorithm design is the objective, amore complete algorithm would aiso test the x and y values to make sure that the points are distinct andto learn whether the slope is undefined (infinitevertical). Language: BASIC Computer: HONEYWELL 235 TIME-SHARE Program: 10 READ XI, YI, X2, Y2 20 PRINT (Y2-YI)/(X2-Xi) 30 GO TO 10 0 DATA 0. 0, I, 2. 3. 5. 7, 7. 3, 12. -7, 72 50 END Output: .5 -6 174 Cll.\ VII: SIX

EXAMPLF

Grade: I I.I9

I'aob1e n: To lied an approximation Of the 11111111)er

l'IllpOSC: The lumber r is introduced to junior 0 senior students in older to ch.( u.s the fun( lion log, x. Man} texts mention that

e = (1+ .

rti but 01( ourse no proof is (Meted. The simple program that follows prints(1 +-) II/ 101 n I. n = 1001. n = 2001. n = :1001..\Ithouh this (1,P)onstration is Ito woof of the convergence to r, it is convincing- and simple.

1\ n The function(I + --1niapproaches r very slowly, and round -off error becomes very important. thus a 28digit mantissa was nsed in this run. (The approximation is still only accurate to about 5 places for n = 10.001.)

Noic: This same problem can also be solved effectively on a desk-top machinepar- ticularly if the machine has an av routine or key. However, the output is ohn restricted to a fixed number of digits. The program below illustrates the use of a FORTRAN program that provides flexibility in the number of digits desired.

Language: FORTRAN II

Computer: HMI 1620

Program: 'LIST PR INTER (This control card changes the mantissa length *FAN D K280 I to 28 digits.) DO Iii N = 1. 3001. 1000 E= I. (In this program (I 1/N) is multiplied by itself N times. thus avoiding the exponential function A = N using e.) P = I. + I./A b020I -1,N 20 E = E'P 10 PR INT 30, N, E ( ;ALL EX IT 311 FORMAT (I I-10. H. F32.28) (The 111(1lot spaces the printer in FOR- END TRAN II.)

Output: I 2.0000000000000000000000000000 1001 2.. -.;2875345538925213607420 2001 2.7176029086275165583155130720 3001 2.7178290707,1776058836733294,10 THE 1:01.E 01: El.1:.(2.11:0NR. AN1) CALCUL \TOItti 175

ENIAIPLE s

Crack: 12

Prob/cm: 10 approximate the value of an integral. specifically.to find the :ilea bounded by

Purpose: SNIA;\ PleMculary Fun( lions and an) one ofa number of new senior texts imroduce a scheme of appioximating the area of a region bounded b)a parabola y + bx + c, two vvi 1.10 al fines X = ni and X = 1/. and the x-axis. This methodcan he expanded to Sh»psou's rule for appioximate integration ofany continuous function. IIIthe following program a student used Simpson's ruleto sole the inoblem !risen above. Language: FORTRAN II Computer: MAI 1620 Program: = SIN(1.) + EXP(1.) DC) 20 I = I. 99. 2 P X 1 = 1. + P /100. N2 =+ (P + 1)/100. 20 = + -1.NSIN(N I) + EXP(X1)) + 2.*(SIN(X2) + EXP(X2)) (1" SIN(X2) EXP(X2))*.01/3. PRINT 30. Y (Calculates the last or right-end value and null- GALL EXIT tiplies by AX*1/3) 30 FORMAT (I HO. 1:13.3) END Output: 5.62721760 (The correct value is close to 5.627223.)

Selected Problems sev.:rai additional examples or computel-oriented problemsare now (Acted. without their programs. Each of these problems has bern programed by students.

EXAMPLE 9

(-;rade: 7 Problem: Generate a table of factorials. Purpose: To demonstrate the idea of factorials and t-) provide the student witha chance to see how rapidly factorials increase in size The limitation of the word length (number of significant digits)of it computer can become an overriding factor. The program will approximate fora large number (20! has 19 digits) and will fail for a ver) huge number. 176 ca s1N

EXAMPLE 10 Grade: II, 12 !'rob/cm: Given the coordinates of n points on the Cartesian plane. find the equation of the best linear approximation for these points. Purpose: In science (muses there is occasion to collect data, to plot points on a graph. and to attempt to extrapolate or interpolate front the data. If in studying the graph the relationship seems linear, the student then obtains the following information: I. The equation ()Ithe best linear approximation lorthe points (methodofleast squares) The equation of the line is + b. where the patameters in and be defined as follows: = XV - n

2. Some indication of how near these points are to the line Error can be indicated by the maximum distance ()I' any point Irons the line or the average of the distances of all points front the line.

EXAMPLE II Grade: 7-12 Problem:Generate an arithmetic, geometric, or some-other sequence of numbets. Purpose: Many different goals can be emphasized: 1.In the junior year. infinite series for at-cont x, sin X. Jog (x + 7, etc. are often introduced but their usefulness is seldom demonstrated. The sequence ()Ipartial sums of a series can be printed by a computer to any desired number of terms to demonstrate that the use of a series to calculate the salve of a function or number is feasible. The series shown below is typical:

7;2 1 I I = . .. 6 12 22 32 2. Geometric and arithmetic sequences and seriescan he studied with a computer. For example, before discussing underlying mathematical proofs many terms of the series +(4,.).5 +(2..1.2. r canbecalculated and their sum compared to 5 / 9 5 which is the algebraically determined sum of the infinite series. 3.In the seventh and eighth grades, students are often given part of a sequence and asked to guess the general rule governing generation of the terms of the sequence. The computer can be programed to genchte a particular team and stop, then generate another term and stop, giving students a chance to check their hypotheses without any practical limit on the number of terms generated. IHE ROLE OF ELECTRONIC: COMPUTERs \ NI) \ LCCLATORS 177

rXAN1PLE 12

Problem: Calculate the number of possible combinations n things taken r at .t time (where n and r arc positive integets) and the number of permutations ofn things taken r at a time.

Purpose: To enable the student to obtain quick answers to these calculations when workingin probability and statistics: iiCr = r!(i r)!

nPr = n! (n )! A program is easily written for any machine froma small desk-top calculator to a large time-sharing computer and will save time in doing the calculations.

EXAMPLE 13 Grade: 7-12 Problem: Find the ( nbc root of any number u. Purpose: This problem can be programed in severalways to meet various purposes: I.The teat her may want to demonstrate howPVTvaries. Newton's method willoen crate a about as fast as any algorithm. 2. The teacher may want to demonstrate the general idea of iterativeprocesses.The following formula might be used for this purpose:

\ITT X, X, + = 9 3. The teacher may wish to demonstrate theuse of Newton's method of finding the roots ofx3 n =0. In this case one of the following equations can be used:

+ r xt (x,3 ?)JU?.

EXAMPLE 14 Grade: 7, 8 Problem Test associative. comm walk e, and distibutke properties of addition and multiplication mod ni. Purpose: To reinforce the concepts of associativity, commutativit). and clistributitity and to apply an important technique in proof, the search for a counterexample The general idea of mod m arithmetic is reviewed in the writing of the grog am. '178 (.11 \PT1:1: SIX

EXAMPLE 15

Grade: 9.11

Problem: Find the roots of the general equation ax-; bx

(impose: To re% he general solution of the quadiath equatim and to encourage the student to consider special cases of the solution, such as a = lac is negative. and so on.

EXAMPLE 16 Grade:

Problem: Find the prime factors of a positive integer. Purpose: Toencourage the student to review the following hleas as he writes the program I. The meaning of"ais a divisor ofb" The student may have to devise a test to decide whether ais a divisor ofbwhen the remainder of the division is not available. 2. The meaning of"ais a prime number" The student must choose some algorithm to test for prinies.

EX:MPLE 17 Grade: 7-9 Problem Write a program to print the absolute value of any integer. Purpose: To encourage the student to use the mathemath al definition of absolute aloe when writing the program Although the sub-routine "ABS(X) is available on most machines, it should not be used here. The student should test to determine if x is positive, rem. or negative, and take appropriate action in each case in order to get the absolute value of x. Doing this will motivate the use of the definition 01 absolute %aloe and teinforce the student's comprehension of the concept.

EXAMPLE IS Grade: 9-12 Problem Mitea progiam that leads the numerator and denominator of a rational number and prints the nonrepeating digits and the repetend ol the infinite decimal expansion. Purpose. I. To lc% iew with the student the general idea that lational numbers can be repiesented by repeating infinite decimals and to "discover" some algorithm for finding the nonrepeating and repeating parts 2. To provide examples that promote the "discovery" of theorems which state that the decimal form of every rational number lepeats and that the number of digits in the repetend is a function of the denominator. 1:01.1.: Or IRONIC CW;I'l-1 ERS 1Nll (, I ORS 179

EXAMPLE 19

Grade: II

Thobieni: Calculate the area abo" the.s -axisand u;"ley the cur,e v I %. for between and b were a and b Purpose: lit man% modem eleventh -gtade texts. the logat him' function is inn Od tU ed as the area desuibd above-with-a I and b = x. Using this definition of the logarithm function the hind:internal rule oflogarithms can be demonsnated. IIa computer program is written before formal proofs arc. offered, the following, ideas can be introdtued: I. The area under a curve can be interpreted as the limit of a ,,,quen«e of ateas of rectangle:,. 2.Se% old tests can be run and. if desired. the rule -discovered- that log a log b = log ab.

Gomm-rim:EQUIPNIENT Direct access is provided by transporting. stu- Although some computer topics can be taught dents to It computer (enter. by installing a coin- pwer in the school. by bringing a small portable without actual use of a computer. the real belie- fits of a school program involving computers computer into the classroom. or by installing a terminal in the school to «mununicate directly, can be obtained onlyifteachers and students actually use computing equipment. Many de- with a time-shared coinputer system. % i( es are used to tea( h «mtpu to -I elated «nu epts. The :RR antages and disathantages of actually including «ntipmet sh»ulatot"i0).' leas ing a computer in the school must he related logic !maids. and digital trainers. Some of these to the costs of existing computers. With older, do ices are useful in tea( lung about computm obsolete computers, the cost per student in the thatis,inhiss' m tional programs or units in program is quite high. Howe% et. new equipuu .11 is being introduced that is much mote useful and which the ("11""e' itselfis the 1)1i01.111' object of instill( tion. lu general. lume% et. these devices (onsidcrabl) less expenshe than the computers have vet)limitedusefulnessinmathematics .t durinp, the past few years. Also. many instruction. Tlua elm(' this section %% ill describe of the new computers are small enough that they the types of computing equipment that are useful be !nought direct!) into the classroom when as problem solving tools and instructional aids needed. in the regular mathematics curriculum. If indirect access isused, student pi ograms t\ school that has decided to include the use are sent to a computer center by mail or by Of computers in its mathematics ctu tic ultun can (miller. In this oite of access. an important con- do this in many ways. These can be categoriied sideration is the turn-around time, that is, the as follows: elapsed time between sending the program and I.Direct m e.ss. Students and teachers operate receiving the results. The means of using a the computing equipment. This is some- computer in an individual school is determined times called "hands.on" use. by the type of problems to be programed. If 2. indirect ess. ograms ate sent to It com- students are working on hulk idnal projects and puter center for processing-. and (ne results do not need immediate results, then inditect ale returned. access by courier or mail service to a computer '1 80 CII NI' ITIZ SI \

!milli)is often adequate.II.however. iestIlts being used for actual transmission.itmat he are necessary for immediate use. as lor a particu- used by students IoI the preparation andcritic a- lar ( lass period. as turn :wound is essential and tion of tapes of (aids lot later the a time-sharing teletype or onsitc computer is program is leproduced On tapes, cards, 01 mag desirable. netic tapes at the computer center;iti, pro \ (rnuier servi(C provides an cllicient and cessed. and the lestilts are transmitted back to low -costmethod oftransmitting studentpro the student. Student programs might be sent in wants. Programs ale scat1)) messenger to a the evening. with the results a)ailable at the millittter center for execution, and the lestilts beginning of the next school day. ale ieturned in die saris way. Piograins may be ,\Iairt- schools are now using computers in a sent in handwritten or typed forin..\, an alteina- mode that is ieleited to .1, time sharing. .\ school like. equipment su(Ii as a Le) punch or ((nit elle' distii( tcan install a terminal 1))pa)ing a flat (lor mark-sensc (aids which can he converted niondil)chaige phis additional charges that punched). lot pteparation of plograntin «int- depend on how melt the computer is used. In pulerlead:dile form. (an beinstalledinthe this tea). the cost can he geared directly to the hoot. 'ruin-atomic! time ma) range a number of students participating in the c.Itita- lew howls to a few (lays, depending on the prox- tional program and the inannet in which the imity of the computer center to the school, the piograin is conducted. means oftransportinginformation. andthe Terminals may heinstalledinindividual efficiency ol the «impute'. center. Some school schools or in a central location (e.g., the admin- districts already have computers for administra- istration building). Students ma) hate direct tive data processing- that could also serve the access or indirect ac eess to the computer. or a needs of students. Another possibility is buying combination of both. For example. it a terminal computer time fro)), «nnineicial service bureaus is installed in a 500(11. students may be given or from «imputer users in the vicinity of the primary access clin-ing the school day; then at school. .\ good clioc is a college or unit-mit) night admin istrat it c !row ams can be rim 50 thaw center: in fact, several universities now provide results 1611 be :tvailablc the next morning. III computer ser) ices l0 s( 110015 'Ii their geographical man) cases the terminal itself can be used as an aleas. In sonic parts of the country, educational program-preparation deg ice.Programs data centers are being established to provide a an be prepared "oil line" (that is, with no con. variety01 computer servicestoparticipating IleCt1011 to the computer) and transmitted :ita schools. In many cases,these services include later time. This results in reduced operating processing of htudent programs. costs. If a school districtis geographically 'emote Projects are under way to evaluate the several fioni a compater center, indirect access can 'e ways of vol iding student access to computers at achiet ed b) sending and receit ing plowains 1); a reasonable per-student cost. It ma) happen that mail. The main disadvantage of this method is some blend of two or more approaches will be the longer turn-around time. the best way. The availability of low-cost communication The following paragraphs describe several equipment permits direct telephonic, transmis- types of computing equipment in current use sion of student programs to a compute' (enter in mathematics instruction and give at last one for fast turn-around processing. The school in specific example of each type. The listis far stalls a paper-tape, ptinchedcard, or niark-card from complete! It includes examples of the fol- device that is directly attached to a telephone lowing types of equipment: line for the communication of information to Calculators the compute: center. When the device is not Programable calculators THE ROLE OF ELECTRONIC CONIPUTERS AND CALCULATORS

1)igintlra s Small generalpurpos comptliers Tie-sharing

Ca/or/a/ors Conemional. eh tically driven rah:III:tuns have been alotind along time. They are used in litindiedN (,1 whool distlicts. paictilaly in in- m111(6(111:11 programs for low achievers. Figure 6.11 k a ',hum: of aprinting- calculator in common liNU. A more recent development is the eleconic calculalor. Machines. of this inn: arc far more powei rid ploblen solving tools than the conventional machines. Of course. they are also more expensive! Figme 6.1;is a picture of an demonic calculator system Ilea can be used sinillhalleously by four people Working on four different problents.

FR: uttE 6.11.Prinling calculator Courtesy of SinithCorona Marehant

FunntE 6.14.Elrelronir calculator sync»)

.S1Z.7

'101 to 1r woo'

Courtesy of Wang Laboratories (:II \ PTI-At SIN

diagram or oneofthe ke%boaldsofthe lilt' SC(1000 e 0I 1..C%s110kes 11'ang 1,31)0EalorieS NV:4CMIN ShOW0 in Figure ictinned 101 eat 11 of sescial appliLitions of the l;.15. Alt ulator. In Carp (ase. die lsidi (an htie.id In addition to the usual operations of addition, in tlw disp1:1\ following die last 1,esook. Niltuaenon, nuillipliitation, and division. 11114; Ordel NOkC .1 10 Oble10 llIC (al( itlalor also porkies direct evaluation (one one must_ first know what happens when emit kesinike) 01 X. (.\:". I n x. and (4'. kes is pressed. Then he intim design 3 -,e(illeme 01

DISPLAY +1 2 3.4 5 6 7 8 9 0""

PRODUCT LnX X2 MAR MULTIPLIER ACCUMULATION Air( 'mg( 7 8 9 CIS ARY Alt ACCUMULATION SWITCH SWITCH -.- .._. bb = 4 5 6 as ONOFF 111IM, 111011111(OMR ENTRY SWITCH MAN MAR ACCUMULATION AMR 1 ADOIR 2 SWITCH /1111116. 0 + = AIWA RICaTT ADDIR I ARAB 0 ADDIR

FicREli.15.Wang 1.aboraimi.v ealerlalor keyboard

APPLICATION KEYSTROKE SEQUENCE (Left to Right) DISPLAY (Answer)

Cl(AR 37 x 89 ALL 8 3293.000000 (NT(R a air

Cl(AR 18057.3 ALL 8 0 Ei3.141361257

Ct(AR 7892 All 8 9cm 622521.0000

Ct(AR V-6747 ALL 6 2.539685020

WAR In 22.5 AU a 3.11351530

1.36 Ct(AR 3.896193301 e Al( 6

FIGumi 6.16.Keysiroke .cequenre itoi.r.01:ELEC:ntomc comrtriT.its\ND %IAA].vroits 183 ke St1 I )ke% I() %Oh( II le !)) 01)1e111t11.0. is, design Pragramable Galculal(as a piogratit.If the plogiant (sequence of key- .X progiamIonat (intentional(al( ttlator strokes)is long.itis written down for future executed (carried out)manually. flatis.the Id el ellt e. 1)10'41.1111 to evaluate \ i operator mannally presses the instruction Le... 10 W enters data. and records the result;. STEP INSTI:UCTION Sonny' elect' on i« lett la torseau execute Tres. the "CLEAR ALL" key. 1)10141:1111o1secetal steps aatomeinally. -These 9 Input the value of a on the keyboaid. ate referred to as 1»0g,ramable calculaims. The 3 fees the "X2.. key. comp:wies that build these in:hines liequentl .1 hes, the left-hand "+" key. call them deshiop romptticts. Input the value of bon the keyboard. The plog antable Calculator hridges the gap Press the "N.'" key. between the nutnuall. operated calculator and Tress the left-hand "+" key. the generalpurpose computer. Programs that S Press the "VX" key. would take hours to execute manually can be °lime canbe satedblwriting theinstruc- tallied out in minutes or even seconds when tions in alibi eviated form. The program now executed automatically. looks like this: Two methods of programing ate it: common use. In one ipe. used bl the Wang caltadator. STEP INSTRUCTION programs are ininched into a program card. I "Fo use the program. the program cat(' is inserted 9 KW) a into a card programer and the startbutton 3 pressed. The system executes the program automatically at high speed, pausing whenever itynt a b of dataislet-mired. The caul programeris shown in Figure 6.17. 1, S Miami: 6.17 Courlew uJ Wang Lobo: atilt les Wang keyboard and card programer 184 CHAPTER SIX

SEATCARD SQUARELY TOPEDGE FLUSH WITHuls.ritmur, cErricssr.s:

.0-0047J5:C2DIOrD=00102:12020,020202020CC=ECCCCCCCCCCC!E3 U.ciaUrOzati_JIJUza134341U202020:043rECCC:LCCOCC:LICC:DECC 4..34JCII_JU::_AU:OCUUCC:LIC:12CC4]:LIC:DLIC:LiCCCUU::::2;zzz4:::zzz ZZZZZ -0-0-0-0-1100-0-0-04,M43404]U-LIC-0-C-0-124.14.]-013

4]-00-0-FJ{YJ-0-0-04104434j4343-0-0-04343404343 43204Xli3C14-JCth:21.320434MCCC20204DCIA3204.1122[2CCO 4320CUCUeJUU=04.04J4jC1:002021JCC:Dr.0211COCCUUCCCOCC43 CUU:MUUthjeL1043thj:DOCUCUCCCUtAAAJC4AjCUCCI_AJ...2.Z..Z.-ztzZtZtt:ZZZ:Z::::: ZZZZZ ;;;ZZ 0.0071143

FlotazE 6.18. Wang Laboratories program card

The programcandissimply api escored (punched)HollerithGild. showninFigure 6.18. The crudpictured can hold up to 80 program steps, each represented by a pattern of holes punched into the card. The holes in:ty be punched into the cad manually by using a paper clip, pencil, or stylus. A different type of programable calculator is shown in Figure 6.19. In this calculator instructions and data are stored internally in a magnetic- core memory. The memory consists of three "working" registers and sixteen storage registers, each capable of storing a 1-digit decimal numeral. Three of the registers arc displayed on the television-like smell. They are im of ed in the actual work of computing and data manipulation. The other sixteen registers are used to store up to 196 program steps, input data, and intermediate results. This inachiee features tinect e alua-

don (1-3 keystrokes) of these functions: 12,x, log x, e, sin x, cos x, tan x, arcsin x, arccos x, arc- tan x, as well as the hyperbolic functions and their inverses, It also has a special arithmetic unit that performs computations with complex num- Courtesy of Hewlett-Packard bers or vectors, including single-keystroke con- Ficmui 6.19 version from rectangular to polar coordinates Hewlett-Packard Model 910071 programablc calculator and the reverse. THE ROLE OF ELECTRONICCO11PUTERS AND CALCIQATORS '185

The machine can be used manually as a calculator or automatically asa stored-program computer. In using the machine mannall. Ley., ale pressed in the sequence required to solve the problem. just as with the IVang- calculator describedpreviously. To storeaprogiam. switch is set 10 "I'ltOGK \NI' mode: then the program is enteled by !messing ke.s in the required sequence. As each key is pressed a corresponding (ode is stored in the memory. The followingisa progiam to compute H. where n! = n (n I) (:;)(2)(I). (Ifele n is meted manually and then the pm- giant is executed: therefore n is ill the X register before step 00.which copiesitintothe Coto tesy of czelei (-Parka? ti register.) FIGURII 6.20. Hewlett-Packard plotter

CONTENTS OF REGISTERS STEP KEY COMMENTS X

00 Is 01 Is 02 I --..-03 (Subtract) n n(n I) (n I) = (04 I 1: X > Y (Conditional 05 0 branch to 06 address Od) 07 ROLL 4- n k Pa-1 08 X (tMultiply) n k 09 ROLL Is n k Pa (Oa GO TO (Unconditional 0 branch to 1_20c 3 address 03) Oct END 0 /' = n!

Alter entering the program, the switch is set The usefulness and efficiency of the program- to the "RUN- mode, n entered, and "CON- able calculator is in,:reased by the addition of TINUE" button pressed totellthe machine peripheral units. For example, it printer and a to run the piogram starting with step 00. The plotter call be plugged into the Hewlett-Packard computer computes the value of n! and stops machine, pioviding the 111C8115 for automatically with the value displayed on the screen. The printing or graphing results (Figure 6.20). program can be repeated as often as desired for In an educational setting, the programable different values of n. calculator has the following advantages: The program in the memory can be copied onto a magnetic card. Then, when one wishes Low costand a minimum of maintenance to use the program again, he merely drops the Portability card into a slot on the calculator and records As portableasatypewriter,itcanbe the program back into the memory. brought into the classroom when desired. 186 LH WIER SI\

Capability Program storage ItIi is suflu lent oompttting powet to solve The usual limit in plogiant storag is hunt most of the problems in the standard mathe- 100 to 200 plow:tin steps. matics curriculum. For example. these ma- Numeric onir c Dines canbeusedtosolve almost every The plograntablecal.nlatorisdesigned pioblem in the previous section: the only lim- print:nil% lor numerical oalculations, Itis not itation is that 111 sulying sinutltaiieou, Ntla designed to handle variables. 1.1011N the number ()I equations cannot exceed three. l'rogra in language Programing is generally in a )man hive Ian Teachability guage or an octal type of coding- rather than Average and even below-average students :t simple compiler language. This will present can quickly and easily learn to use a program- considerable difficulty when the inathematioal able calculator yet it provides plenty of chal- problem im ok es a large number ()I algebraio lenge to bright students. manipulations :111(1 decisions. Tra11Sferlbilily Methods learned on the plogiamable cal- l'rogra in library culator are readik transfened to more power- "I -hose small continue's may not have intim.- di. telavailable a library ()I plow:tinsa set ful equipment. of plogrants (business. statistics. record keep- There are also, of course. disadvantages. The ing. scheduling. et( .) stored in the computer programable calculator has limi:ations inthe and made :mailable to the user when needed. following- areas: The program library must be stoned exter- Data storage nally on punched (aids or magnetic Girds. On Data storage is usually limited to a few lo- larger computers Inc plogram library is avail- cations; therefore. problems that requite in- able "on line- as part of the system. Such pro- ternal storage of large amounts of data (as lol- ;II ants at e often useful in so ietto e. so( ial studies, l:lige matrix problems and most business (lata- and mathematics. processing applications)cannot be handled. In brie!, the plogiamable calculator probably provides an economical and effective way to introduce computers into the mathematioh Ctlit1111. It is generally deshable. howcner, to sup- SIGN t-ACCUMUUTOR plement its use with access to more powerful 0 M CrO 17.p. equipmen t. OVERFLOW - E CHANGE - VEVORY ADDRESS 0 EE 1=1::0 CI;0 0:L

LOGIC 20.0 eAE DISTRIBUTOR STPUCTION Digilal TrainerS O. 0 03:1 CII;E Digital trainers are small computers that are ADDER COUNT DOW' 0.-I'ROCRA/A ;MASS CONTROL I;GP I1 D0 MC MO. designed specifically to help wadi fundamentals in particular areas ofcomputer-science edtu a- T11111111000ill 0EMI., tiou, They are pat ticularly useful in teaching binary arithmetic, computer logia. computer cir- cuitry. computer operation, and tronbleshooting and repair of computers. These ale subject areas in which the computer itsellis the primary ob- CourtesyofFab, tTeh, Into, ',orate(' ject of instruction. A digital trainer in wide- FIGURE 6.2I. Bi-Tran 6 digital trainer spread use is shown in Figure 6.21. 1111.1t01.1; OF ELEc:ritoxic compt' 1-Lits \ND\ (ri, '187

The digital ttainet is not designed to be used desk) and too hem) to be easily portable: the as a ploblent soh ing tool or as a practical. -teal newer machines are sma:l. compact. and por- lifc- computing Kist' ument. I fence its usefulness table. The lollcm ing 1)601) describes mo small in the mathematics ( niTil 11111111I 1111thed. computers that meet these requitement:. II : ,:ner is .1%.ilable. lume%er. it Ina) be used Figure 6.22 pictures the Digital Equipment occasionall)to dentonstiate computer concepts Corporation PIP -S, Icomputer.Itis athin!.

to students %%110 ate using °diet ecptipment for machine integratedtgenetation chinks. mathematicI problem Nuking. expel intentation. -1 he unit slum n includes the( omputing unit. and dist mci.%, cutememory.connectionsfor inputioutput equipment. and an operator's console. Also in- Small General-Purpose Cmnpulers cluded (not shown)is a teletypewriter that is connected to the computer by a cable. A com- One way to extend and expand the use of complete at ray of peripheral devices is mailable as puters in the M IWOl is to acquire and use a small optional equipment. Software is available for general-pin pose computer, thatis,a computer' running programs writtenin BASIC, FOCAL that includes each 01 the billowing: (similar to TELC:MIP). and 1:0ITRAN. Se,,eralthousandstoragelocations,e:,r1) capableof storing one alphanumeric character The mentor) is magnetic cote or equivalent. Ficukr. 6.22.PDP-811 computer Expandability Additionalstorage can often be added. Abilityto use storage locations inthange- ablyfor data or instructions Alubly to use most of the standard peripheral drim essuch as teletypewriters, paper-tape cadet., and pun( hes. card I eaders and punches, !white',plotters,cathode-tay.tubedisplays. magnetictape units. and discstorage units "Theseshould beavailable asoptional equipment.

Softwarefor1»weessingstudentprograms written in an algorithmu language suchas ALGOL, FORTRAN. BASIC, TELCONIP, Al'!.. or 'PEACH Languages such .ts BASIC or TELCOMP ate particularly desirable. Great care must be exercised in the selection 01 this type of equipment. Some school districts ale still purchasing old, obsolete computers that actual!)cost me than modern,t hd.generation machines. The older machines are slow and are terriblyinefficientinprocessing algorithmic- language imtgrams. Furthermore, they ace generally too large (e.g., the size of a standard office Courtesy of Digital Equipment Corporation 1 88 ( .1 I \ P

Time Sharing .S\,\/etits A prayerful nettt\ pc of°minuet s)stetii has heel), (le\ elopedone that petinits man \ user, to share the simultaneous use of Lute, last s\ stein in a «Hi\ eitient ;old pa( ti(al inannet.I his t\l>c ofstein is called it time-shaiing s\ stem.

Eat. lnuser communicates 5\ ill] the «nnputer

s)steniI)) meas 01 .1terminal such as .ttele- t \ pe. an electricI) pcx\-1 kyr, ena Lc\ boatelIon input and a 1/ 1'11111112 dC\ I( c or small IT screen lot otuptu. Tra initials ate connected dill cull \to the systemIt\coaxial (able enindite( thIt\ means of ordinal \ -telephone lines.

If the connectionIS b) 1110'4111 Oi(Chi/11011C lines, a terminal (an be geogtaphie all\ remote Iro the computer system. It can be in a different city or exen in a different state. T \ pie;11 time-shah ing s\ stems ser\ ice Imin 2 to 200 terminals each. 01 these s)stems include leatures designed spec ilk all \lor educa- COUJi< NI' of h TrIc I-Parka+ d tional use. and terminals ale in use at seeonclary 1:te:»RE 6.23. Heudett-Parkard 2115/1 computer schools. colleges. and tmi\ ersities tInoughont the United States. Sra»e of these systems operate nearly twenty-lour hours a day. every day.

rigtire6.23 picturestheI Iewlett-Packard If the number of users is large, the time-sitar 2 I5A along with ateletypewriter in14s)stelnwillle(Ittirea \ er)lastc°1111)Iller and an optical (aid-reader. The optical emel- with a large (hum on- dim attNilial*storage unit. t-cadetreads i»lormation horn Hollerith cards In addition. it must ha\ e hardware devices that that have been prepared by marking preprinted many terminals to be attached to the boxes with an radinary soft pencil. Figme 6.21 diagrams a lime sharing sys- The unit on which the computer is sitting contains the power suppl) and empt) compart- TIIC I 1111e-sit a I ing s) sh2111 Opel a I Cs till d C (lt- ments for additional core storage and a disc stor- I 1 01 01 aset)p(>werIttlsetof soltwate. The age unit. E et) thing is on wheels so Can the Ittnctions normally perframed b) «n»ptitation- systemcanbe easily moved towhereitis center personnel are columned by the time- needed. sharing software. 'These administrative Ittnetions The snyt11 computers shown here have suffi- include scheduling theuse01 the computer cient apabilit)lot almost eve])insure dental (dctert»ini»g winch user gets to Inn his prob- use in se«mdar) school mathematics. They ma) lem and when), reccweling automatically(loi

be used as pi °Wein soli ing tools .111(1instruc- (halging put poses) the computer time used. and tional aids in theegnla mathet»atics curric- managing the Hitt-my of progi;inisthe setof ulum. In addition. they may be used in elective 1»ogiams(business.statistics,recor(1keeping, courses in computer science and in vocational etc.) stored in the compute' and made asailable programs involving- computers. to subscribers few their use. THE ROLE OF ELECTRONIC CO,NITI-TER`,\ ND (\ LCI'I \TORS '189

LARGE DRUM OR DISK STORAGE UNIT

INPUT OUTPUT COMPUTER DEVICES DEVICES

COMMUNICATIONS CONTROLLER

TELEPHONE TELEPHONE LINES LINES

TERMINAL TERMINAL TERMINAL 1 TERMINAL 1 2 3 N

FtGrut:. (1.21.Diagram of lime-sharing system

Note should be made that smallet sense cat ds that 11.1%e been prepared pleyiously) systems ate ctirtentl)available (lor 2 S users) through a (aid reader. As the program is ell. that require far less equipment and. of ( :ourse. ((net!.itis stored on the drum or disc storage ate intuit less expensive. Such s)stems ma) be unit. me(! ellecthel) in a single school or slimed by Now suppose that the user. liming entered his two or three schools. progialn. Iequests the computer to inn his pro- An understanding of the use of &time-sharing gram (which exists now on the (listunit). If the system can perhaps best be gained b) examining computed is not being used at the moment by the pro(Atne lot soh ing a problem on the s)s- an) other use'. odicr than lotoutine input and tem. nsiug one of the terminals. Assume that output(, the inogram 1\111 be running in a matter the user(a teacher or student) has written a of a Traction of a second. II his inoblem tequires program in an algorithmic language and now only a second or so to complete (the case with wishes to process his program on the s1SiCiii. AIN() many student exet, ices),it will be tun to «un- a`Mlnie that he is thing .1 terminal tonne( ted to !Action and then the output glinted on the ter- the system through an (natal-% dial telephone. minal. This output ma) consist of the desired le- (Seeillosnationoppositetitlepage ofthis stilts or may be one or mote Cl rot messages point- chapter.) ing out typing Or other ertors in his program. First. he gets the attention of the system by If the computerisbeing used when the run dialing. He is then able to communicate with request is made, the tequest is held in a list with the s)stem by means of the terminal. acertainpriority. The time-sharing software He enters the program by typing it on the contains scheduling portion that determines, keyboald of the terminal. !uniting a punched among all those programs requesting computet tape that has been plc:paled ine\iottsly on the time, which program is to be run next and how terminal. or submitting cards (punch or mark- much time is to be allocated. Eventually (in a 190 \ iTER SIN

sC«111(15 01 less) the useris scheduled 10 he memory computers that might he acquired It\ a next. The sof [wale s)stent causes the computer school for classroom use.I lencc. a plug' am that 10 cease Intining the piogiain then being 1%011..ed might inn for st.Acial minutes on a small di um- On teen ititis not finished), causes itto be memocompute' would he executed in a few written off °Ivo the dim unit, brings in the new seconds on the time-sharing S111e111. inogram front the disc. and starts running it. 111 M1(1111011, the cost 0111 dine-skating system Ifthe new program requires longer than a is determined lthe amount of use. )pically, second or so, the problem may in turn he inter- the use is charged old\ for the time he is -on rupted. hien off on the disc. and held in abey- line" to the computer and for the rental of the ance for a short time while other users are ser- terminal device. \iced. Later (again. in a matter or seconds) the program MU he In ought back from the disc and CONCLUSION continued from the exact place where it was interrupted. This process is repeated until the In view of the rapid (le\ elopment of comput- program is completed and allthe results have ing equipment. Wan) of the facilities described been pinned. in this chapter will probably be improved on or °litetime.sharing system isespeciallywell even replaced within:t\ cry few yeats.itnot suited to situations in which many short pro- months..\Ithongh c hinges are not quite so rapid grams are lo he processed. This. of tonrse, is Ille as Figure 6.25 depicts. the exaggeration has some kind of situation typically encountered when a basis in truth. computer isused as :01instructional toolin However, tegardless of the equipment a\ ail, teaching elmentar) and se«mdar) mathematics. able, the mathematical expel iencesconsideled It should be noted also that the type of com- in this chapter will he appropriateas long as puter used in a time-sharing system is usua11) these ale includedinthe school'scurriculum 100 to 1,000 times as fast as small drum- or disc- objectives.

ef,

110:91

"Sad case there . . brilliant computer mantook a six weeks vacation and fell too far behind in his field."

FiGultE 6.25 Courtesy of F. D. Thompson Publications I III: I201.1: 01: ELECTRON1( H.:RS \ NI) CNI.C11, roits '191

SEI.F.(: IA.11)ICI RI.IOGRAPIIV n notated

This bibliography is di% idol into font sections. Pau ,\ motains%mines oflassfoom maief ials (student or teacher) :Ind lor classroolli the of the computer .1, a ploblein solving 1001 in s. Part R lists telefetu es 1111Ille :urns Of C01111)111(1":, in echn.itioll and colilluttt Science. Part C. «mtains other inlotination. luiling articles 01 interest to teach .111(1 student,, ..10111 bil/110;4111/111( :11Illalel la 1. Pall I) lists orolessional societies.

A. CL:\SSIZOOM oils disc ussed are illustiated b lulls doiumitted \ Robot I.. 'read/ rion.ceh /1...1,5/(:. 2 pi% Igrams. c.dil.: Tinica Education 6. cla%mau. 11);,%id. Mathematics Laboratory Wort: Corp.. 1970. Papetbaik. /took. Tewksbury. Mass.: 'Wang Laboratories. (attident text) 1967. Paperbaik. \tiinnothittionfor junior high school or (Tea. her e,oil(. book) slither high ,(pool students. lhe pace is rely Flfilly.sn mathemathal topics that ian be slow %viol ofiltintied (lion, 10 1114 lease student taught using the 11'ang calculator. .fiot i a I ion. 7. Coonnince on the Undelgradttate Pli)grani in Netedencr. August 1969. 2..\11)fecht. Robert I... et 31. (:ompnier .11e/hodc (Teal her relele1110 rrr,llatheminus.Palo Aho. Cant.:Addison. Wesley Publishing Co.. 1968. This issue, WWI Ille theme "Calculus (Student text) Computers :' contain, illf01.111:111011011 the use 01 compute', in litstyea cahuiu, coniss For high .(pooljuniors or seniots or ha ill iollgs and tutivelsities in the (Juiced Scales. iolleg livslifften: designed lor Ilse 8s a stipple. mental y text in inathentatiis courses in Paid outlines ale plesented so that individuals «miptitets ate used lor demonstration or to can iontait applopliate institutions for ft/Idler salter ploblents.It may be used as a prinfaly information, 8. Computing and Nlathematics (:catiitiltana text for all ele(tke (04115e ill (01111/11leOlielllell comidex Numbers, Fro(tions. Limits, and S. -Vatinal 1;aces for I )gmithins, 1)enrer: (Iniver- 3. A !Owe. RichardV. Computer P/ogramming sity of Denver. 1970. Paperbacks. and Related Mothematic:. New York: John (Student supplefiwtits) Vile Sous. 1967. These 11/111' 0/1111/Iner.Nteildedinstruction (Teat( her maim ial or college student text) units use the BASIC language :111(1 :MIMIC the .A book on programing with it good ihaptr availability of computer facilities. They were of compuiertelated mathematics. A college. written especially for secondaty sihool seniors level text. but a number of the examples ale or iollege freshmen taking a calculus or pre. :11/1/1010iale 101' se0 Mdal y s(1,o1 the. calculus (muse. 1.Ltrutti. Ludwig. andlariatt Visich. '1/0' Uses 9. C4)111, Samuel I). Meilientar). ,\runicival of Computers in High Shoolc. S soh. Brooklyn. sis: An Algorithmic Approach, New York: Mc- °.1'.: IluutinrtonPtoject. PolytechnicInsti- Graw.11ill Book (:()..1965. tute of Brooklyn. 19(19. 1970. Paperbacks. (Teacher reference) ('Teacher t e) Imileigra(htate textthatispaiticulaly A guide 0Wel ills man' topics III mathematics, clear. FOItTltAN IV isused in plop:lining scieme, and smial sciefue which gives useful examples. pmglants and teaching strategies. 10.Corlett.Peter Norman. and J.I).Tinsley. 5.Cal/fah:tn. Brice. et a I. rip/died Nan/erica/ Meth- Practical Programming. New Yolk: Cambridge ods, New York: John \'iley Sons. 1969. University Pless. 1968. (Teac her ere' ence) (Teacher reference) An hue/mediate treatment of the theory and Thisis an introductorytextemploying applications of numerical methods. The meth- A1.001. as a language. It has both numerical 191 (.11 \P1-l.1: ,I\

.111(1 IMI11111111r11(.11 applilation. but the enr stMl111. 'sill 111111 this 1111140M 114111 10 1111111(1 1(il

pit.tsts is sittongl% Ott the ftnniet. .111.11%sk .1 him 111.11111g(11.11101.4( 1 he 1(51.e( CR !GIs \NI. Pim e«hti.0 of 1,1 Ini111111Mial (.011 .11111n..1(((.. Ilo s(1111(INIO(411 0)1111011111g 1.1(1111% (e'er:dd. (111( oh Wm and llte Com p titer. 'Fall& .111(1 (.11111411Iii((411(141r1( 11.j11%., 0111% peIlt .111(1 basset.: flotilla State rui%eisity. 1970. \41 ;41(41.1111111.; 1.1014111..411(I 111( leN1 ( I rat her It let en(.) Is i41ntlr.11ibl' shit all\ (1101011Illg1.14 1.111%. 1 rails( ript and stuninat% of.1 duce da% 16. Dion. 1Villiam S.. :111(1 Judith I. I dweuls. "hitt! Irtrn(c 141 (11,(11v. all as11.(Is 01 114118 the Sten- ;lig the Best solution%l.1(outputer.- Immo! beig and 1Valker (:121(:1SANI (Aldus test. 'I he 1;dmationol Irato Prod:son: 6. no. 2:90 107. suggestions telly( ted (luring the tonlerent C. ale leather teletentm though ille ate made site( for this '1 his ailide dem:does how the «mtputer tan ate %Amble to atnone teat !ling (al( tilos. be used in .01% ing the ploblem(s) 01 masimik 12. Dalton.I.thIntal Med:am:v.I lanoverN.11.: ing an arra. Itis an extellent example of bow Sc t ondats StIntol Dalt:nom!' Col. III( tomputer (.111 he 11.(11 10 (All111(1 111,111111AM' lie hiesru ("111"1"6"" (:culer. 1971). P:11)(1' to ,stolveinteirsting pitoblentsin mathematits. bat k. 17. Dont. 11'illiam S.. and Ile:belt J. Cir:11)(1g. (Student t(St) .11alhtma s and Willi 1(11t7'lt 1 111 ex(ellrnl explmation ol the ph)'sital Ione, PPogiontining. Neu S of k: John k Sons. imohed in orbital flight.Better students of 1967, sconnkear algebra or more advanced courses 'Stu(lent text)

tan undetstand this rewanliint matetial. Good 1 eNtellenttestfor glade 12 students or application of trigonometty and analytit geom. tollege Iteshinen width takes ammo° olthe V. tole ol the tomputer illmathematits instim I. Dantaskos, N.I.. and NI, Smyth. You and don. (SeeteviewintheNI.1.5's Tech Chester. l'a : PNIC (:011eges. 1969. lira! Monthly. Imittar%1965. it.10:1.) Pape' balk. .11.0 a good .0111«. 01 ideas for teat bets. (Student test) 1$.Engineeting (Mtt«.pts (:tit tit 'dont 1'14+1 t he purpose 01 this textisto teach about (1:1:C.I'). Man Math. Wm/t/. :1 pis. Nen Yolk: enginect Mg andlet linolog% usingthetam. NItGtallill Book Co.. 1969. study appoiat It, Seeral ploblents are nth:tined (Student lest,1.11/111:11111.11, Mid1(..1( her 111.0e b% (Himmel' mailability. and the problem solv- Bahl ing appotadt is sets useful cult without avail. 1,(:(.11' has des eloped .1 101( e W41441.1111101* able «mtputer ladlities. billed with ideas for high '1(11001 %111(1c111S. Stile( Ind 5(1 1 1011, (I(.11 .sill 'eating, inteiested inapplications. the ICA( 11111p, «41;11/111n (OM Ins mid using 1.1. Ilaitinotali (.011ege isietit Computation Center. the (omitting' in sot% big ptoblents. Seto/oh:iv&hot,/ Puilect(Inid.ear repots) 19. Betitaid 1. The lanwnagr Comptit, and Ifemonstlation and Experiment:dim:in Neu Volk: Nit Craw.' lill Book Co.. 1962. (:inaphrepTraining Use in Seionda eadier :detente matetia Satools.I lanmt. N.I I.:11:titmouth College. Soon. vet) good 111(111.I(111:11- problem settings 1968. for senior high s( 110°1..1 tee( ter would find this ( leather 'detente and soma. listing 015111 a good motility ol interesting !noblest's.I In book dell) 111.0(1 619 is quite readable and would be an appiopriate The booklets gke .1 (ICS( t 1111100 01 the I);41 addition to the .01001 libtaty. 111011111-Se(011(1.11% S(11001 (.01111/111en PlOie(1. In 20. General Elect] (:o. NfalkI I inn Sliatim; Ser- Addition 141listing many matheinaths ite Teat !ling (:nides. Isuds. Bethesda. \ Id.: hint .(as with watputer potential. times pro - General hie( it it litho manor' Set sit el)ept,. 1969. tide information tegattling the top4 °tidbit's (leather tele:clu() and other institutional materials ol the pot I.tdt of these lour papetbatk booksllgebto jet t. I.llgelno Trigonometry. and Ph;stilo tow Dodes, !Ring 1., and (;icitiel...\'uott.orni tains a number 01 piograuts for ploblents nt .1iiti/) ow with it ..lpplhations. New Ym : mathematics t.tslicthe, along %%lilt smite excl.- 11:nden Book Co.. 1961. (ises. The tit.:' tial was developed h) teachers (Student text in the Altoona Atea School District. Collegc ft c.finten and better high school math 21. Gine:the:ger. lied. and (.cotgefallray. hob I Ill ROLE 01, ELECTIWNII. (.ONI1.1 in;\NI) ct1 101; 193

lroIN Ito Lion pole 11111 . Neu Yolk.jolia originall% utitten Ioiland published Ina Oli \Vile% sz Sons. 19i6, Papeibat k. well l'udemood Clop. tor listAith its desk- (Ieacher I eso 111.11111:11! top tomputels: thus the pootv.noing

\ good stillsol !mildews 01 Willek ne tied tinsels 141 Out111111(dill l' 111111 dint( tilt% tan ht used to (imple- giamma 101. Hie imitheanatit:ilsetting- me menttheaeu,tilatw.uheut.tIitst tat aitultuu or .c.oulald to the tegtelar 7-12 tuatittiltint. loon the nun lens of an eletti;e «mse on tout 26.Ileminet \ J. Statistical Cenninitatinnt puler applitations. on a C ;;;;; \V;11111:1111. Mass,: Mak- 22.Giialtieti. Domenit N. Bath Math, tics: ..111 dell Publishing Co.. P967. Isler .11.111 ((((( h: (;reoll Mather:min s: .111 (Teat her 'idea en«-) Ho noon 11,11) ,,,,, and The WO . \ brit: \n athati«.(1 text that might be used for pal. .1nI:le Ho mu .1.eWklon y. lit ula, topic% apploptiate to atRained student% \Vain.; Laboratories. 1968. 1969. Papeibacks. in setinidaiy (Student texts and le:ulna's editions) 97.Hit key. AlbeitI'. -The I'm- of the Computer These hooks ate part of a lise-vidume seties its laflaelitatits Institution:" Tzro.l. no' College piepaied lotuse with \\',m; desktop tomptit- .11athmnIns Imo mil. Swim; 1970. pp. .11-51. rhe two hooks on basic anti general nettlie. ('Iatli- 'detente.) mat its empliasiie business applicationsof The :it tide gkes a general dist ussion of the mathmatits and tonesponcling use of salt tilat tole of the (imputer in the dassloom and indi lug w (imputing equipineut. Real boy cites . number of Nye( ificsettings whete the itultult.,(list 11,,ions and poildt.m settings ap (impanel. tan be used as a [ambient solving tool. 'Hopi bit' lot student, in glade 7 through junior 28.Iliggins. G. A. Thr Elemenhoy hoot : college. The material on Nell Illding A pprourh. If:mover. Secon- 1":11CS %linen( C)is 11:1E6( tikirl approptiate for dat St hootPublitatiotts. tutotath College «almoner a pplit Kictsil Computation Ct i.ter. 1970. 23.1 laag. James N. C ro%hro Shorrintd FOR. (Student Text) TI Plogie ;;;;;;; New Volk: Ilayden Book A (muse in element:fly him lion% in which Co.. 1969. Pape' bat k, lIst701 :I(01111011(Thulas, 311 1Illiniatepart. (Student text) Knowledge of 11.SIC and :availability of tom. One of the lengthiest itittodiutions to R)1;- puling fat ilities are assumed. This text. written TRAN plogiatning. The book is intended for liar set midat y sc himl or first.ear college NIII. setondal school students or college tinder( lass. dents. is a good simile of ideas for teathers of men who have winpleted two years of algebra. students inall setondary stltoot mathematics Many interesting applications ate illustiated in (muse,. the example problems. 29.Holden. Het bet t L. Inbodortion lu FORTIMM 21.I lamming. R hatd \V. (:1:b.uhrs mid the C II'. New Volk: millan Co., 1969. Paperback. puler Bemire' . Houghton Mifflin (Student text) Co.. 1968, Text designed to wadi 1Y)1:RAN 1V for ("Feather inatetial or material for selected bill. use as a st ientifie tool. Good for the begimier. dents) Inn not appoptiate as a referents wink. The T1111isamt expansion of an earlier mono- lads of flow chatts is No110:11111CS «01111sitig. giaph by the N:IIII title published by the CLIP,NI. 30.Hull. T.E.The .V ical Integrationof The booklet disiusses some aspects of comput- Ordinary1)illrent Is'Inalions. I tet keley. ing :Is they :11 e related to the beginning calcu Calif.; Committee on the Undtagraduate Pro. lus «Hirst!. The material is well written and easy gram in(atlieniatits. 1966. to lollow. (Teacher material or college student text) 25.Hansen. V. P. Dis«nering Nlathematits through (:ollegolevel mathematits and applications of Computeis series. 6 vols. Hartsdale. N.Y.: 01. the «mtputer. tint No ward. 1968. Paperbacks. !l.111111. T. E.. and David I). 1)ay. Cplums (Student texts) Problem Sohn:1;z. Reading. Addison. \Fes. Titles in this series are as follows: linlionul ley Publishing Co.. 1969. Numbers. (pt. I andpt.2). illgebm, Onorn- (Student text) etry.Arlin:lod:11gebtri mid Trigon lllll CUT., The initialise of the hook is to explain the and Shili.slies. The material was function of a «onputer and show how the coin- '194 (:II \I' l'Elt SIN

Inner tan be used to Nolte a %vide variety of Reinhold (:0.. 19711. hut:testing ploble; \\Idle the bookis (Student text) of tot inner m lent e. mato of \ tonapteltensie soul:el:00k frith ex.:mples the exett i.e. Ihne implications for the mathe in: hided !tom matheinatits. st lent'''. engineer nett it lassionni. ing, and business. 32.In0t;flitiona1BusinessNlat lane:(:ot 1).C . 38. Kenten. John :and Thomas B.1S/C 31allonntliiN rn See ltny Schooh. Publi- 1'onzilimming. New York: John SIMS. taii(m uo. (:"11.Iliti7.0. 1967. P:11)(1 1):1( k. (Teat her matet 1:11) matetia! or supplemental v:indent (:outme administiai and turrieuhun guide ale:lest tiltedlora (output:.madiemati's .\11 intiodut lionpr tedmicitte: of tomputer toms. The book «mtain: 'mit: on histm of pit:graining 'king the language I3.\S1(:. Mete the tomputer. How dent:. marlin language. is an applit Alms set lion that tontait's a vat lei% detked language. l'OP:I.R.\ N. and 3(1v:tined of ploblem setting:. inalltematit almirk. (number ihern. ploba- 39.Killen. Nlit bad. lnlon1m-1 bilk v. man it ec. (u.). ;I.1 lal lila! .1 1,)noach.uksbur. Masc.: 33. bilooluclitot 111 Eirginring .4n- 1ang 1.abotatorie:. 1969. idyNis. 11,1Comptili.N, Plank. N.V..1 11:\ 1 (Student text and 'ember's edition) edmitai Ptiblitation Dept. -This book is :me of a reties pi:Tate:1 lot use (Teat her matetial) with \ ang desktop «Miplitrs. It1 nild1:11/C5 :\ (ollege.level text :showing(11010:icr :11)1)1i- computer Imulamentals trialexamples taken :mime: in engineering prblenet. It tun mat hemat 11%. 3'1.Iverson. Kenneth E. Elentiamy Fitnaiont: .1ii '10.Knuth.IbmaldE. r I nItil.11gotilhms ..1 Igmill TinImen1 Chitag:r StienteRe- and Sna. .11goollonI. Reading. M(tt h AssOt imes. 1966. \ddison VeslevPublishing Co..1965. (Student text) 1969. \textbook for pletaltnIu: students at the ( Feat her 'act cut e) high school milege A1(10)":411 Ihe Pro- These air the filsttlo olnutesIII/II)III graming Ligi! tge. API.. is tilt and rotund. jet led CI %el le%. The Alt of (.0m. km %et),ligmous. the textis a good sour::: puler Ptogramming. designed to explain and 01 idea: for tealbers. illustrate most of what is known about basic 35. Johnson. David LaityI..Hatfield. Pamela computervograming techniques (exclusive of 't'. 1:aiiman. Thomas E. kitten. Dale E. 1.a lust %ohmic tontaitt: Envoi, and John W. Walther. Computer Assist- many excellent Net t lug% for illustiating alga ed 1:Itlictuati'sPlogram P)series. 6 Iithm design. col:. Glenview.Ill.:Scott.Nut:sin:tit8: Co.. 1 I.I:0tke. %Valter C juttis in /hi (:iitstroont. 1968-1970. Pap:11)3(1c. Mavnattl. Mass.: Digital Equipment Cot 1).. 1968. (Student text, lea( heriiiii mentaries) Pape' bat k. Supplement:1'y situlent booklets for cath of (I cachet resomte manual) 111e grades 7 thiough12. The material: ton- \n indexed icicle:1(C lvoik «mtaining spy. win many excellent exettises for use with the :Bit suggestions on holy and when to use the stand:n(1 ntatheutalit, 1 ut 'it tilum: students mite tomputr xithin a sthoos pteseni mathentati's po)grams in BASIC: 10 study «mcepts mritultun. and soli e ploblenk. '12.Kovath. 1.adkI).(:ontpute.Chinied Mat/u 36.Johnson. Donovan :\.. and Gerald R. ltising. maths: !Wiwi:triton lo Numeii«tl Gniriitnes fro Teaching Mollie lies. Belmont. S311 1 101:1:11.1)ay. 1961. Paper. \.:1(1,,,,.,Ilipublishing co.. 1967. ( :alit.: bat k. (eat her refet rut e) (Teat her tefelenee or supplementary stlulcut Theset am "The Role 01 Computers- inthin se«nulaty schoolmathematits method:text 'flit111)0k ill dealt with "IIIIIIIelICII I shows some intetesting setting: at the junior Mai C and I:lest:tit: a number of inteteming high school :nut senior high school levels. purl:1ms. %Vitae the material is written to be 37.Kat/an. Hat y. l'iogiamming Com- done xvitl Ia computer facility. the content p:art' Trchniques. New Volk: Van Nostrand has implitatiotk for «rummer we. THE 1:01. 01: ELECIZONIC COMP1' 1lts\ ND C \ HMS '195

3. M(Ciat ken. Daniel ft. and \William s, horn. \n1 uleti.tn Deselopmnt \Re. seatchPiojec t HighSchoolStudents," tAtiintii2.1".%111:qt11,":"11'01:1";\114(1)el 1 homas Nos ember 1970. pp. 597- ( I eat ter idet ent et 60s. The book de.igned lor undergraduate stn. \ Student (Ai:opt:ter That 1:eall \Voi Ls." dents in engilireiing or scien«.. Students should \. 1)ecember 1971). pp. 681- base analssis or calculus bac Lgiound helots SI. .111C1111)1111g111.111% 01 the 1)101)1011S..'1 lie11001. "Inioulticing Nlatris.\ Igebra %%hit Computer 1% 1«0111111016ed .1%ar1C.11 110'1 CICICMC illthe Piogramining." Montague. Jantialy .11C:1%01 0:1111:160110111111( 6011% :111d.)1% in}; 1971. pp. 65-72. equations and Nl111% of equations. ..(:arts.. (:omputerassisted." Iteleti S. Ittigliei. Nt,ii,. Um B.. ;Ind E:111 J. Scliwepl.. .1 l'elinais 1971. pp. 155-66. InImilm Sum to 31elhads I'mng the 46.NItillish.1 hairs. ,!intent l'Ing)amming: 11.. , awl:1m maisdeli publish, Lumzun*.. New .\ millan Co.. 1966. Paper bac L. imp. Co_ 196S. Par:Iliac k. (Trache: icicle:1(e or college student text) (Student text) \ (011C4C 1CX1 11) 1111101111W .\11 intindm lion to plograining for the be. N11114-111% 11) 1110%C 1110110(1% 111.11 .11C :114)14:11)1C 11) ginner which can be used by students who base

111(' 11111)1C11111:16011% 01 :11g1)1 1111111% 011 (0111 completed second-yea algebra (although some pilling mac hines. eNall11)" Would 11:1"SII)be kil)1)(:(1). (""1 45. Madre lo, 'readier. -CAnnputer.Oliented problm development in later chapters. Nlathematics- depai intent. Edited by \Walter J. 7. National (:otnicil of Teat bets of Mathematics. E%1)C1ke. hatodudionInau Language (Teacher Iefeien« and student (11.1SI(:). \Washington. 1).(:.: The Council. 1968. . \m lit les des( rating Inns tea( Bets and students (reacher «lei cot c) used the «impute) as a problemsolving tool in The pamphlet «nitains easy-tole:1d intio the mathemaths classlooni ale included in the duction to the programing language following listcil«nitiibutions to the depatt- with illustiations of computer applicationsiu mem. school matheinatits.Itis intended for teacher "Quadrat it Equat ions-(:omputer Side." use. but students in grade 9 or above could use Thomas E. Apia 1969. pp. 305-9. it to learn how to piogiain in 11.\S1C. "Generating Itancloni' Numbers Using i\lodu. 48.Pennington. R:111)11II. lubmInelm-r Compuler larAt idmiet ic."Mother ArthurIndelicato, Alclhods and Numerical "Inalysiv. New York: May 1969. pp. 355-90. Macmillan Co.. 1965. -Pi luteTriplets.- JosephIliisch.October (reacher lelerence or student text) 1969. pp. 67-71. A postintegralcalculustextforcollege: "Computers in NlatheinaticsEducation." however. [hoc ale settings with implications for Chat les J. 'Zoo. November 1969. pp. 563-67. secondary school mathematics. "A Case Study inMathematics-the Cone 49.Price. Wilson '1'., and Nferlin Miller. Elemenls Nikander J.Daniaskos. December or Mita Mor,ssing Malhemalics. New Yoi k: 1969. pp. 612-19. !loll. Rinehat k \Winston. 1967. "PatternsinAlgorithmsforDetermining (Student text) \Whethe I.aige Numbers Ale Prime." Aaron I.. For students who have «mtpletedfirst- or Buchman. ana:my 1970. pp. 30-11. second-year algebra with a vocational interest "Computer.extencled Instruction: An Exam. in the fields 1)1 data processing and computer plc." \William S. Dorn. February 1970, pp. 147- programing. The major emphasis is on prob. 55. Ion solving lather than proof. No specific pro. -1'opicsinNumericalAnalysisforHigh graining language is taught or assumed. School 'Mathematics." Stephen I). Schet. April 50.Ralston, Anthony. and 11. S. Wilt, eds. Manic. 1970. pp. 313-17. rrrulitol Methods for Digital Compolerr. 2 vols. "Polynomial Synthetic Division." Irwin (loll- New York: John Wiley k Sons, 1960, 1967. man and Larry 1:auvar. May 1970. pp. 429-31. (Teacher reference) "Predicting the Outcome of the \World Se- An advanced reference text too difficult[or ries. Richard Brown. October 1970, pp. 91-500. students. Format «insists of a mathematical dis 196 ( .11 I' l'IzR 11\

cession of .1 topic followed ht aNCI IC. Of Id:lied thehist %ohmic%l'Itesccond seeral 101)1(, 01 11:1111(111:1' 11101CA l0 11 «)111101011lse:. computational Ago' ithins and pest-titss0111C

5 1 i(11.11.11nett.Modem St hool 1111/11,1.1 1111114111.111( s %%111111 .1cstill:1111e1-01 J'Ir limo(.1 plooach: A lgel»aI.Teisslon.y. acl.nn('(I sl1111C111s. mass.: ';,,,,g 1970. l'apelback. 57.seiisk,. Ah.1%ll.Luru(1uh.jhah (Stu(Ient text) 1 ladsdale. N.Y :()holt 1:01uard. 1970. Paper- ,A compiete Iiist-)eo algebra «nnse built back. mound the %Yang calculator, This is.1 good ( I eat her I chi eli(e) idea iesotti«. fond, for teacheis using :my type The book is urine!' tor use with the ()1netti ol electIoni«alc ulator. Progiamma 101 deskaop coulmtei. Itis a «d- 52.Rosenthal. NlyionR. .Vmnero al Methods in id tion of plograins from dilleient mathematical Com Imiet I'mgonnming. Homewood. ill.: IZ.i( h- let els. 110111 algebia to talc illus. including some old I). Irwin. 1966. business and iecreational mathematics. (Teacher relelen(e) 58. Shot pe. Ncilliani P. BASIC: In I :Mod m tom to The plc.-emotion assumes :1 good high .4 hoot Com pub.). l',OvalnIng I "%lug the Lan- mathematics bac kgiound. Included in the book (?rage.New \ of k: ghee Pless. 1967. l'opelback. :ne sections on tectut en«. it:lotions. 1 elaxation (Teacher fere! etc e) andNIonte(:arlo methods. randolimiumber This textbook is on element:1i% intioduction genelatois.deleiminoitts :111(1 :111(1 matrices. to ptogramingindie time-sharing language simultaneous equations. (:.`.SIC. am! itincludes exam ides taken (tutu 55.Rule. viiii.ed i mathematics. science. and business. The book BOSt( )1I:I)i Mile*\Veber k Schmidt. 1970. Paper- is easy 14) head :Ind understand. The emphasis bac k. is not on lialdale but on getting the header (Stlident text) plogramingasquickly as possible. Contains many sample engineei ing opplica- 59. Smith. ItobeitE. The /taw% of /.1)/(T/(.4.V. dons that can be 1111(1(.1,1°0d by se«niclary sc hoot students. No pievious computer experience is .\ finite:11)4)1i.: Combo( Data Coup.. 1967. Paper. bac k. assumed. and almost all mathematical motel ial (Supplement:11y student text) is suitable for students with a solid bac kgiound An excellentsowce ofpioblems(mainly 111 Ne«)1111. tcar algebra. munbel theory)for «uttputer solution-junior Si.. Sage. Edwin R. olden, Solving trill, a (:om- high .4 hoof and senior high school. The author puler.Neburypoit, ,;\ lass.: Entelek. 1969. ismelt' cleer ill Papel back. the -"Ihe motivates die ploblems it a story setting. (eat her tcleten.e(11'student text) The book identifies ploblems for computer 60. -_ BASIC Ide. Nlimicapolis: Intona- solution through the mathematics curriculum, tional'riiiie Sharing (:01 p..1969. Paperback. grades 8-12. (Teacher teleience or student t('xt) 55. School .Nlatlietnotics Study Gimp. .11govribmc. The test includes lot t-olie lessons and fifty, Com lodallon. and Mathematics' (with student ievie pioblems attempting to lea( li the 11\SI(: supplementsforFOItTILAN and ALGOL). language and various applications. holden] set - Pasadena. (:alit.:.1. C. Vloman. 1966. Paper- ling, ale mu intetesting os those in'Thor bac k. /;acesof listed above. (Student texts with teacher (Oitimentaries) 61. Steinbach. Robert C, Progoonining LxerMel 101 For a glade12 mathematics and computing PrOblefis.()IiellierlIanguages.BeverlyI I ilk. worse. Spends ex«.ssive time Ill how 0111- GlencoePiess. 19(19. 1)istibuted by Mac- WIWI' %%111ks: however. the later sections iitiliie millan Co., New York., the computer in studying appropriate mathe- (Teach(l famine and source of problem,) matics, Good sonic e of ideas. (See also l'oisythe The book provides a series of labolatot)' exer- 'dereli(e in part11) . cises designed to "develop the student's ability all.St lett«.ResearchAssociotes. Comp/ding to communicate beely «ntipme Coucepfs in .1la(Itemalies. 2 tots, Chicago: Sri. using a problent.oriemed language.- The state. elice Iteseardi .\ssociates. 1968. went ofeNet iSei is independent Of anyspecific (Student texts and teaching guides) language. Included arc a number of business- A good history of «niiptuels is piescined in type applications. Fl I E ROLE OF ELECTRON IC CONI'TERS \ NI) ( )RS 197

62.Stenbeig. %%alienIt.. and 1:obeitJ. %Volker. ruts. Nels 1(>11:.\(.11letni(PleSS. 1969. Paper. (."(//, Wm:ICompute) Otienterl hesentation. back. 2 sot.. I .111:111.1`"r( : FloridaSlat( I .it). cx(ellIllt ("Het ti°11 of P.'Pers on .1 variety

( :R !CIS \ AI. 1911g. ol topics111 101111/111e1:1V.INlell111%111( 11011. 110111 U.N.!) general MI" eN paper. to %Pc( al( aPPlitati" in II( 1(1(..1%of t.dtulu..II(introduced using 111S1111( 11011.including dis(ussiotts of lialdware. algolitlitni«on«.pts.\ !though the otder of languages. and c«monlics. topi(s (fillets 11010 that in other filst.year calcu- 69.Bat ker. P. J.. and W. F. BeN('iidge.!taut Com-

lus [/' \'s. 1 igor (0111(111 .11e 1101 1ellIllell but pute) Studies. 1.ilinbingli; ()liner &7Boyd. 1970. (1111.111«11 jib the help of the (outptiter. The I his is a general intioduction to the tonstrtto textis independent ())a spedlit programing operation. and applications of «onpittels. langttag( .ltd thus(.111he 111111any alge ALGOL and 1:01:1'I:'AN ale discussed briellv. Inai,language. 70.Bolt. .\11)(.1tIt.. and M. E. %Vaidle.Communi-__ 63. I hompson. !lime Bio/ogr. i'hrsi- eating with a (.'omputei. NeuYin k: Camln idge o/t)gr: .11athemaInof Chemist:I,:andSecond- l'itheisity Ness, 1970. Paperback. m I e,,e1 1'111qt%.Minneapolis: Control 1)ala I he book is an elementaty inttoduttion to Cori). Paper bac ks. what a (oniptuer is and how it works. Although (Sour«. of pioblein settings lor teat him or stu- it was vritten for teacluns. itis well within the (ient texts) grasp of juntor high school students. !Item. lour student booklets illustiate a wide 71.Bowles. Edmund A.. ed.Computos in Humau- of :11)Pli(ati°"°Ithe wmPuter iu islir Englewood(dills. N.J.: llcntice- ma t henia t i( :111(1s( ien( e. I tall. (;. p.. Reed. inbodudion lo This collet tion of papers west:tits shot tstir- .Vionciical Analysis.%Valthant. Nlass.: Blaisdell Se-s on the use of WIDIMite" anthropology Publishing (:o.. 1966. and alchaeology. language and literature. and (ea( her releren(c) intisi«)logy. \it .id on«.(1 text thatan be used for (el tain 79.Bulletin of the National Association of Second- topics that ale appropriate011:111101eC117111(11. cos. School Mincipals.February 1970. tar). level. such as eriors 111 e01111)111:111011.teal This issue wittains a number of :uncles on loots 01 an equation. and the Gauss ieduction. the theme -Hie Computer in Education.- One 65.%Vvissinan. K.School BASIL%Hanover. Nil.: by HelenI fughes discusses an experimental Se«nid111y S(11001P11611(:111011S. 1):111111011111Col- 1)101p:tinin computerassisted mathematics in lege Kiewit (:omptuation (:enter. 1970. Paper- whith the students plogramed the computer to back. solve problems in mathematics. grades 11 and 12. (Student text) 73.Bushnell. Donald I).. and Dwight W..\1Ien. The \ language manual. but one mitten for the (:ompitli.).in Anirsiirin Education. Newl'ot k: student 1%.1u) haS//OE (01111)1C1C(1Vell011Csettles. John Wiley Sons. 1967. Papeibatk. ter of algebra. For use's of the 1)arlinouth sys- Shonld be of inlet est to teacheis. Readings on tem. but easily tulapted to any 1ASIC system. the many roles of the computer in education- indiidualiiing institution, assisting in instruc- B. SURVEY OF CONIPUTER-SCIENCE tion and tesearch. teaching the computer sci- REFERENCE MATERIALS en«.s (se«mdaiy s(hool and colleg), and infor- 66. Adams. J. Mack, and Robert Nloon. Anlitho. mation pto«.ssing, duction to Computer Science.Glenview, 7,1.Calingaelt. Peter.Principles of Comlnitation. Scott. Foresman g: Co., 1970. Reading, Nlass.: .\clisost-Wesley Publishing Co., An elementary computescientetextwith 1965. 1)1'i nary emphasis on FORTR \N programing. A lefelente book or generainteiest library, 67.Allen. Paul, Ill.Exploring the Cum /lifer.Read- book for students and teachers. It includes all ing, Mass.: Addison-Wesley Publishing Co.. 1967. introduction to the field of computing :111(1 a A piogianied intioduction to computer haul- discussion of number bases as well as sonic mate- e. Very elemental y. with little emphasis on rial on numerical appioximation. use of the facility. Appropriate for students. 75. Cation, John NI.Caieers and Opporltothies in 68.Atkinson. Richard C., and II. A. Wilson. eds, Computer Science.New York: E. P. Dutton& Computer-Assisted Instruction: Al Book of Read- Co., 1967. '198 CI I \ PTI21: SIN

A sue ey 01 ()ppm 'unities in (oinputer tech This book is .111 inuodtu non to the field 01 nology applied to solving ptoblems in mann- digitaomputt usage,It assumes no pie% tons fat titling. 1:.Inspot (alio!). M16(1111. engineering. (0111ptiur experien«. and is ppropriate for use and many other fields. A fairly «mtpreltensive I) high sthool students. 0 ICW 01 the (ontputing (onninunity. Fotsthe..\ lexandraI.. hontas Iseenan, 76, Count)! Data Cot p. The Teacher-Student A p. Elliott I. ()Igattick. .111(1 0'11 Stenbeig (.o»r- Mom I, to Compute, Thogiamming Com 41)1+. 2 prrlrr Fitt Net% York: (chin vols. Minneapolis: Connol Data Corp..1963. Wiley Sons, 1969. Paperback. An excellent intioductionto winputer 56. Good tefeteme mate] In addition. there em e. The fits' chapters deal with algorithms ale a number ()I settings for possible classroom mid flow(harts.Nlath matical examples ale the. used thtoughout. The It'N1 is otganired mound 77. Crawford Rudd A.. and David II. Copp. intro- flowcharting. with a .supplententaty book on the ductiontoCompute)Thogiamining.Boston: mote widely used languages. and itis appt() Houghton Alifflin Co.. 1969. Taped back. plate for high 5(11001 or undelgraditate use. (Student text) (Based On SAISG reit:truce in pact .\) An inlet esting introduction to inachinelan 84. Gerard. R, W. Compulets and Edit< alum. New guage plograming which teaches studentsto York: ;\1(Graw-hill Book Co.. 1967. mite programs lot an imaginary computer. No A general reference book for cdtuatots which «mtptiter facilities are needed. and fitst-year in( hides It survey of developments in «11111)111er- algebra is the only pretequisite. assisted instruction and leatning, library ili 78. Crowle). Thomas II. Untie/standing Computer). I ion. and administrative I cm' d keeping and pit). New Yotk: McGraw.' lilt Book Co., 1967. Paper- Ledut es. The book is a report of a «mit:tenet: back. held at the University of Cali10rnia to discuss introduction to how a «mtputer works. the Inane of computer~ and education. Readable by students. 81i.Gtuenbiger. Pled. Computers and Communi cations: l'olemd a Computer Utility. Englewood 79.Dainowski, Vine:)S. ComputersTheory and Cliffs. N.J.: PR:mice-Hall. 1968. Uses. W ishington,D.C.: NationalScience Publication of the Computer (:ontinunications "Teachers Association. 1964. A pamphlet written for student use (with a Symposium. 1967. This is a series of articles On teacher comment:it)) in learning about com new directions in computer organi7ationnote. puffers. The short twit includes materials deal- bly 11 computer utilityin goveinment. industry. or the community. ing with what computers are, the development 86.Ilaga. Enoch. Understanding Automation. Elm- of computers, how computers solve problems, lutist. III.: Business Press. 1965. Distributed by and how computers are used. aplinger Publishing Co.. New York. 80.Digital Equipment Corp. Introduction to Pro- A reference book designed specificallyfor gtantining. i\laynard, i\ lass.: Digital Equipment teachers who are teaching or planning to temp Col p.. 1970. Paperback. courses about data processing. An excellent introduction to machine- language 87.Hassitt. Anthony. Computer Pt ogiamming and programing which is well suited for better sec- Computer Systems. New York: :\ cadentie Press, ond:it y school students and teachers. 1967. 81. Piogramming Languages. Maynat d. A «imputerscience textbook for advanced col- i\lass.:Digital Equipment Corp., 1911. Paper- lege students. It assumes knowledge of program. ba( k. ing and deals with fin titer refinements of ma- :\ compact reference for the programing Ian. chine-language programing, monitor systems, gt:.tges FOCAL BASIC, and FORTRAN and computer hardware, and advanced programing. for several assembly languages. The text does 88.I.ohberg, Rolf, and Theo Lutr. Computers at not teach programing techniques, and assembly Iroth. New York: Sterling Publishing Co., 1969. languages are unique to the DEC family of An easy.toread, accurate introduction to com- computers. pute's at work. It relates the different computer 82,Farina, Mali() V. Gomputets: A SelfTeaching languages and their uses;ittraces the growth Introduction. Englewood Cliffs, N.J.: Prentice. in complexity of computers and illustrates the Hall.1969. Paperback. differences in their potentials.

t' 1111., 101.1 OF ELECTRONIC. CONIP1I FRS \ ND CALCUL \ 101:S 199

59.la)%is. I. B. Coniptite)1 1 and Comp:de); 2. Con. .linetitan,\ «mipieli(thke lc% ieu of computet teinporatS( !tool Nlathentatics Seties. Boston: technolog% in ittan :treas. Ilonghton Nlifilin C.o.. 1961. Papdback. 97. Smith. Lugene. and Joseph Hitsch. Compuid Booklets about computers lot junior high or .1latheniainsI. Nlidu est senior high N11104,1 students. Public:16(ms Co. 1966. 1%11)db:wk. 90.Nfontst)nd, Da% id(,. /tow compute, /)() 1 he student text is .1 mo.week unit. for grades Belitiont.(.,thl.:\\*.tclswoithPublishingCo.. Sthiongli 12. on how a «mipitter lorks. The 1969, Papetba(k. material deals Ivith such topics as the binary The book (1111)h:1Si/es algorithm design and s)stem. floating-pointarithmetic. and flow( /LI/ ISII) illusnate how the computer is «Atwitter plograming and history. Thde is a used to sohe ploblms. It was Written to be used tea( her commentary. 1)) high school students lvithabac kgionnd 98.Sterling. Theodor I).. and Seymour V. Pollack. thiough se«,11(1.ar algelna (II 1) undeigradu- comparing and Compute)Sr ieiz eN: Phst ate wIlege students. CoioNe with Pl.I. New York: \I:1(1161 an Co., 91.Nlitipliv. John S. Mons of Digital Conilmiris. 1970. 3 ,ors. 2d ed.. te%. New Yon k: Ilayden Book Co.. This is an undeigiaduate text giving an intro- 1970. ductionto computer science and employing \ good elementaty intioduction to basic coot. I'I.Ias a language. Amer haulate elements and how they Work. 99. Stolitow. Lament e NI, Compute, .4.Nsisted in- ill a slit4lit emphasis 4,11 tionic 5. Usable by stitution.1)ettoit: American Data Piot essing. high ,(hoot students. 1968. Papetl)ac k. 92.National Council of Tea dens of Mathematics. .\11 cs( ellen' overview of CAL (.0mjite).IANiAted hiNtuction and the Troth. 100. Walker. Tot). NI.. and \illiam W. Cotterman. ing of Matheinati(,. Washington.1).(:.: The 4/ihit)odiutiontoCompute,.S.ciente and Council. 1969. Tape's back. :11gmithinie Pimesses. Boston: Allyn R: Bacon. This bookletreports the proceedings()Ia 1970. National Conference on (:()1»pitter..\ssisted In- The hook is intended as a text for an intio stitution conducted at Penn State Univelsity. ductory minse in computer science for students The repoltdiscusses hardware development, of all disciplines. The emphasis is on the analysis softwme de% elopment. and the developing tole of classes of problems and the design of al. of CAI in education. gorithms. 93. - . Computer Far/hues fen Mathematics Instruction. Washitigum. 1).C.: The Council. C. 'MISCELLANEOUS ARTICLES 1967. Paperback. AND PUBLICATIONS .\sin vey ()I computer facilities for possible 101.Arithmetic feather, 1969. classioom use. Sonic sample problems are given, This issue is devoted to applications of the and the results (times and outputs) for running computer i» elementary school instruction. Sam- on different pie«s of haidware (compute's) are ple aiticles: "Computers and Art," by John compated. Should answer many questions for Nlott-Smith; ".\ Teaching Program for Experi- schools just getting started. mentation with Computer-Assisted Instruction," 91.Nolan. licItaidI.. 101011m-6mi to Computing by James I,. Fejfar: and "A Workshop on Com. lloough the BASIC Language. New York: Holt, puterAssisted Instruction in Elementmy NIathe. Itinehatt g: Winston. 1969. mati(s." by Max Jerman and Patrick Suppes. This is a good lower-level undogradnate text. 102.Association for Computing Machinery. "Curric- 95.Post. Dudley L. The Use of Computers in Sec- ulum 68. Re«)mmendations for Academic Pro. ondmy School Mathematics. Newburyport, Mass.: grams in Computer Science." Communications, Emelek. 1970. March 1968, pp. 151-97. A very good general intioduction to (01»t An outline a»cl description of programs and putets for [cachets and admi»istators with no courses iu computer science at the college under- «mtputeraelated experience. graduate and graduate levels. Students may be 96.Scientific ilinerican Ediuns. Information. San interested in this for college planning. Flancisco: IV, II. Freeman $ Co.. 1966. Paper- 103.Business Automation (journal), Hitchcock Pub- back. lishing Co., 288 Park Ave. \Vest, Elmhurst, A reprint of the September 1966 Scianii/ic 60126 200 C11 \ PT FR SIX

10.1.Conzpulei DuAiom (journal). Hayden Publish. III.Doi n.\\valiantS."(annputetsintheHigh ing Co., 8:10 Thhd Axe.. New York, N.Y. 10022. School." i)monaium, Vein tuns 1967, pp. 31-3:). This monthly publication includes attic les on An article describing tin tole ()I the computer all aspects of computers and computet use, well ill ptoblnt solving or (in the xxonls ut the au. written and readable by the non-computer sci- dim) computcr.extencled institution. (El. entist. It is also an excellent source of up-to (late 112.1.,(111(alional Media. Educational,Nledia.Inc., 111101111:aim) on hardware and soltwate des dup. 1015 Floience St.. Ft. \Void),1"Cx. 76102, meats. Montlilx publication that «intains occasional 105.Compuiers (old inlomalion (journal). Betkeley articles on 5(11001 «impute' ac tit !tics. Enterprises, 815 Washington St., Newtonille, 113.1(111«ilional Tohnology (Journal). Falmanonal Mass. 02160. Technology Publications, Englewood Clills. Almost every issue includes good material lor 076:12. keeping abreast of developments and issues in This monthly publication deals xidi tech. the wmptiting field in education, and once a "01()gi( al de' el"Pments. andits wymge in- year :in entire issue is devoted to computers in dude, "au tines ofinquiry and ptofessicitial education. Many articles on sociological. cul practice Which. oiler methods of making teach. rural. and moral issues imolving wmputels and ing mole rational... auunnation. Highly tecommended. 11.1.11,(.. R.''lteiatisemethod, in High 106.Con/piaci:1,mb/ (weekly newspaper). Coinputer. School\ Igebia." .11a1 hemains Teacher, Jam'. world. 797 Washington St.. Newton, ,Nlass. 02160. a) 1961. pp. 16-19. Devoted exclusively to the current world of The article contains some suggestionslor computers.1 laidwai C. Sal W:11 e, and all appli- teachers. cations ale included. This paper is a must for 115.Hughes. J. and K. J. 1:ng%01(1. l(xapawlt: all school libraries: it is motivating material for \ Lea' ning 1)el»onstrat ion." Dalamalion. ,Nlarch both mathematics and science departments. 196$. pp. 67-73. 107. Comfmling Surveys (journal). Associationlor A stmple demonstraticxn plogfattl for explain. Computing Afachinet y,1133 Avenue of the ingtothe uninitiated how a computer can Americas, New York. N.Y. 10036. "learn" through experience. An excellent acti( le This is the survey and tutorial journal of the for students and teachers, ACM. It contains many interesting and tela- 116.Inl(ofa«.. Association for (:onipliting .Machin- liver easy.toead :wicks on current activities cry, 1133 Avenue of the Antelicas, NewYork. in the computer field. N.Y. 10036. 108. Conthwopics ( journal). Associationfor Cont- This is the bulletin of the Af:M Special Inter. poring Machinery, \Vashington. 1).C. Chapter, estCiotti) on iComputerj Uses in Education Reiss Science Bldg., Rm. 23,1. Geoigetown (S1GUE). Always contains a satiety of interest- Washington, D.C. 20007. ing papers. Two special issues (February 1965 and Alarch 117. Journal of 1)ala S()( it:1yof1)ata 1967) contain bibliographies on career lamina- Edtuatots.2,17Edythe St.,lavermote. Calif. tion and films. 91550. 109.Dalanunion (journal). F. 1). Thompson Publi- This monthly publication includes a number cations. 35 Mason St.. Cieettwich, Conn. 06830. atticles of interest to teachers honeIesetch A good sow( e of up-todate information on lepoits to discussions of pogratning languages. lundwate and soft Wale developments -the state 118. Journal of Educalunial Dala P>oo.sing. EdIna of the alt. nequent at tides of interest to teach- tional Systet»s Con p.. 25 Churchill Ave., Palo c15 and students. Alto. (:alif. 9-1306. 110.Dale Processing 31 agazine. North American Poi). The journalis a qualterl publication "de- lishing CO.. 134 N. 13111 St.. Philadelphia, Pa. voted primarily to the publication of technical 19107. information. original research and descriptions collection of duce to seven main of the oigani/ation and operation of data woe- articles on data processing in general. Also wn. essing systems. .\ special issue, Spring 1968, intains a section on new pioducts, some short news cluded duce articles relevant to computer use items, and occasional at tidies pertaining to edu. in the mathematics classroom: "Smiting Cont- cat lona! applications. inuo Insnuction? The Third Phase is Most 1111: ROLE OF ELECTRONIC COMPU \ND \ \ MRS 201

1)111,(111,I))joint O. Pal krt.: "Compute" In Journal. Reti ie 01dinational R, %earth. and struction-a Thre 1)imensional ppioa(11.- by E(//1( at wind /1e%ear( he) (n del terI Judith I. lcluaids. and I he (.0inputr.\ 125..\inerican1 ecletation 01 fulmination hocssing ,Nlatheinatio, I11,01,1101.- 111\ \'1111.1 :u S. 1)01 11. So( ki,. 210 SI1111111il .. N10111 .11e. N.J. 07615. 119.Knuth, 1),,lii \vim!Is.01..dgmithmi. finsis a federation of societies to set ke as a /)ab:mat:on. 0( 'obeli1967, pp. 30-32. national wi(e of the comptunig held. The Tall ex«.11ent article (h.( usss a pillion of and spring joint minputr (mleten(es are note- the Inst( hapter 01/071/1/tenia/llgonthins, worth) I):iteN of these conleirm es ale gkell in listed in pall.\ under File author the .\(:M put >l0atious (see below). introduces the concept 01 an algolithm, using 126..\ssociation for Computing .N1a(Iiiner), 1133 .\\ - the 1..tu Wean a Ig(): (:(:1' as an example, elute 01 the Allied( as, Ne YOU,. 10036. ,\ 120.Lahen/. I)ale E.. and Thomas '1:0111. plolessiona: gimp seeking to "achatne the sci- puler, 101' All Schaal St ienee and eii«., and all, ()I information ino«.ssing- and Mathematio. Januar 1969. pp. 39-.11. immune the bee interchangeif information .1t) ()%eniew that imposes the point of view about the sciences and his of info] proc- that compute", should be used more extensiely essing both among specialists and among the in the sec olds:schools: it suggests some sample public.-Ithas several special intelest gimps setting,. and specialinterest «munittees dealing with 121, "The New Coniptiteri/ed .\g.- Saturday Re. selected :leas. view. 23 July 1966. pp. 15-12. Periodicals:The Journal. Communications, \it excellent set of at ti( les dealing with man). Computing Reviews. and Coniputing Surveys. aspects ()Ithe «miputer rekolution-social ini 127..\ssociation for Educational Data S)stems. 1201 pli(atiolls. edi (amnia' uses, business applica- SixteenthSt.. NW, \Vashington. D.C. 20036. tions. etc. \pplopi late for high school students Seeks to "Immune a greater flow of information and !cachets. among ethic:tuns tegaiding data and informa- 122.Schaal. L. The !high School Mathe- tion plo(essing ideas, techniques. materials. and "wths I thrall'. '1111 ed.. ley. Vashington. D.C.: appli( a 1 National Cotutcii 01 "feathers 01 1:ttlieniati(s. Periodicals: ,IEDS Monitor and .I /;US Jour. 1970. nal (quarterl). This :Illtplilet contains :I section with a bib. 128..1ssociatio1 for Machine TransIathin and (:0111- liography 01 many general or suive books and putational Linguistics. 1755 .NIassachusetts Ave., pamphlets On computers and automation and NW. \Vashington. D.C. 20036. Comerns itself on linear plograming (pp. ,17-51). with significant application of computers to ver. 123. Spei(er. Donald 1), "Computers: Their Past, bad data plocessing, including:methods of textual analy.sis. :distracting. editing. psychological and Piesnt. andFuttne.- Mathematics 'leacher, sociologicalanalysis of inteivie nanscripts, ja111111V 1968, pp, 65-75. group discussions. mass communication. and so Ilistoly of computers: useful lor background inlormation 10r «iniptueroriented teaching. Periodicals: The Finite String (newsletter) and T,anslation. D. PROFESSIONAL SOCIETIES 129.International Federation of Information Pro. 121.Ante" i(anEducationalReseal II.\ssociation, cessing Societies, 11:11) Secietaiat, P.O. 'lox 311, 1126SixteenthSt..NW. \Vashinglon.1).C. 1211 Geneva II. Swit/elland. .\ federation of 20036. Fluis group 0! educational reseal(heis in s0( ieties to set e as all Intel national wice of the chides malty pecmins actively engaged in coin. computing field. It has a technical contznittee on pute,oi kilted pit Alen's, Excellent sou' ces of in. computer science education (1'C3) and a work- 101111ation (intoning lecent uork include the ing group on computer education at the second- olganuations annual meeting and its publica- y level (\VC 3.1). Evel y tln ee yea] s the leder& tions. Of special help is the amtual abstract of lion 51)011501s inteinational computer meet- impels leadatthe )ear's meeting, available ing: the most le«alt one was held in the summer thlough die ()flue is Vashington. This otgalti- of 1971. /ation also has a special interest group for te- 130. National (.01111(11 of Teachers of Alathematics, searchers in mathematics education. 1201 SixteenthSt., NW. 1Vashington,1).C. Periodicals: A»ierican Edmatiomil Research 20036. . aftook,ime

4.,

e). 203

CHAPTER 7

PROJECTION DEVICES

by DONOVAN R. LICHTENBERG University of South Florida, Tampa, Florida

Chapter 7 treats such instructional aids as motion pictures, television, filmstrips, slides, and overhead and opaque projectors. The overview of the kinds of devices and accompanying materials that are available, particularly recently developed ones, should help teachers make wise selections. Also included are suggestions for making effective use of projection devices, hints on the preparation of transparencies, and advice on keeping informed about films, filmstrips, and other media.

A VIDEOTAPE RECORDING SYSTEM includes a television camera and monitor, a videotape recorder, and a receiver. 205

7pRojEcTioNnuicEs

Of all the «mtibutions that modem ter Imo lop .%nother reason for the lack ol acceptance is has made to education. the mate' ials discussed tlidoubtedk the inconsistent qualit. inRams in this chapter are among the 1110511%Cd alld of mathematical content. of many 01 the films also the 1110S1abused.In some espects the% on "lcarket. Vith esPcti to this there has seem to hold the greatest promise for the future. fortunately been much improvement in Ica em This chapter deals with the use of devices that years. Film producer s have begun to pay heed Iwoject a light image cm a select]. Strictly speak- to the fact that :t film on mathematics is not of ing. teleision does not la IIwithin this ( lassifi. much use if itis filled will: ina«tracies. cation. but it has much in common projec- mathematics teachers eje( 111011011 14( tion de ices :111(1is therefore included here. itlreS because they question their appropriate- Vrojection dt%ic.es are usually looked on as ness. They contend with a great deal of convic- insounients for bringing «minion expetien«'N tion that motion pictures are not as appropriate 10 large groups of students who often vary great - ill the teaching Of this subject as they are in the I% in backgrounds and abilities. IVith our bur- teaching of the social studies or science. BM geoning. schoolpopulations.thisfunctionof many or the e:150n% given by the proponents these aids be«)111es increasingly significant. 1 !ow. of the medium for using motion pit 'tilesare as e'er. in recoil years more and mole attention has relevant for mailienimits as 1.01 oel mie.(1. been paid10individuali/ing instruction: and .xs any experienced teacher knows. the prob. tea( hers are finding that projection devices call let of motivation is critical in the teaching of play, a unique role in this aspect of education. mathematics. There :Ire11 :111\ 111111s available also. can (10 lunch to arouse and sustain in- The purposes of this chapter are to give the terest. Good examples ol such films are the olies reader an overview of the kinds or materials in the .%dventtires in Number and Space series. mailable. to disc us. some ways that they 0:m he These nine films le:tture Bit Raird and his pup. and are being used. and to (del some suggestions pets. and tile,deal with topics ranging from for eflective use. We begin our discussion with :1111111(.1h to topology. From a mathematician's motion pictures and Own move on to television point of view the films may leave something to aud to the several means of projecting still ()k- be desired. but any junior high teacher will find itties. that they generate «oisiderable interest in the subject matter. And, like so many other films. MOTION PICTURES they add %aiety to our teaching and provide a dramatic impact not easily obtained in another Although motion pictures intended for class- way. room use have been available since the twenties. The motion picture also affords a war of is not possible to assertthat they have met bringing to studenh in the classroom experiences With it great degree of acceptance olt the par that %conkd not otherwise be theirs. III the film of mathematics teachers. indeed, there are prob- The Mathentwititot and the !liver:: viewers are ably thousands of teachers who have never shown how mathematics is used in analy/ing the titili/eda filmin the mathematics classroom. flow of the Ohio and Nlississippi rivers for the The reasons for this are varied. For some teachers the task of orderilur afil111, previewing it, and arranging for its :.lowing seems too time- I. Published kr ((((I ilui..Ili nun It st. 30 consuming and too complicatedto make it min. (lull, 2. Published 10 Educational 'Testing Service.It;111114 worthwhile. «dor. 20 min. 206 (Ai rER srvi:N

pin pose 011100(1«,111101.Itis unlikel that a has to know what is a%ailable. means that icaclic I(0111d plesclit the ideas contained ill this he must familial with the catalogs put out film as %vell through another medium. 1) (ommeicial distlibutois and filmlibialies. l'Ite most obvious characteristicof motion in( ipal distributol5 01 films in the field pi( titles is. of «Huse. the (cation ()I the illusion ()Iinathcmaths ale listedpleceding the icier- of motion. Although certain mathematical «ni- ences at the old 01 this chapter. (epts sit( II as l0( us are now taught from a static The teat het should also be familial with some point()I iew. thole are still man% aspects 01 of the (0111pielienske 501111es ofIillt 1111011111:V

mathemati(s ill which movement «ffltibutes to 11011. Fin' 1111:111% 1C.11S The UM aitOttaiFlhfl

clarification. This isparticularly true indis. (;fr, plIblished1),'III(' I I. W. (:(111". cessions of the applications of mathematics. In I):111. Was I leader illIts held. t man% good mathematics films. motion is1)10, I Ile Inlhlic.Ition of this guide was dis«nnintied (hued through the ie( 11114111 of animation. This after the supplement lot 1962 %%.s issued. allows the poitia),II of phenomena that are not Copies of the guide ale still available in librar readily observable ill teal lile. In the film Time.. ies. however. and it+, still useful. for example, a rocket circling the earth k shown ,Nlait% films arelistedin The Edmalional to illustrate the necessit)lot OA international Media Index(5). This loot teettvoholle hidex date line. isa compilation of available institutional le .\iiimationis cle%erl% used in, Ile s,yo sources other thanprintedmaterials.\'ideo Budge.%ofKiinigAbrigito show how Euler tapes.films.kilics«,pes.filmstrips. slides. and analiedthe problem of crossingthe sever hanspareluies ale inc hided in the listing,. \'01

bridges in .1 unictirsalwalk.In i'ambil So. time 10 is de% oted emit el; to resources lor mathe- hogomm.'. the Pthagoleall fitment is demon- matics leaching Mont the junior high 5( 11001 level

slimed 1)% nalislormation 01 areas which is upwaid. Volume I (Prt.Achoo/ a),(1 Prim Irv) and effectedthrough animation.Inthe fiscfilm volume 2 (InIermediale) «nimbi listings ()I ma Dis«)%ei ing Solids writs :" animation k used Ie. telials useful ill the tem ling of eleotental) pool peatedl to &Ado!) the formulas 101 volume mathematics. Supplements to the /:,/ton/iotto/ and stir rt. area of prisms. cylinders, pyramids,. Media Index were planned, but none has yet cones. and spheres. ()flier outstanding examples been published. or the use of animation to communicate mathe. Lt 191;7. the NationalInformation Center matical «mcepts can be found in the student forEducationalNledia(NIC:ENI) Was estab- filmsc)Ithe NCTNI series Elemental.% Mathe- lished at the Piliversit% of Southern California. matics for Teachers and Students.' NICEM plans to produce a soles of relerence Aspects Of motion-picture technology such volumes on audiovisual materials. One volume as changspeed photograph; and microphotog- in esenth asailable isits index to Irpnun Edn- iaph% can also be used .1(1%antageousl) in mathe. tational hihtt.. (hi). .\ !proximately Ike-hundred matits films. Students will be fascinated by the closeop shots ofbees manipulating waxin 3. Published Inlaitli.nta ruketsit%.It; color.

.11a1hernati, s of the Honeycomb,' afilmthat .1. PublishedInIntonational FilmIttite3ti.IliImo. sho, how bees minimiie the amount of wax color. 1 'I. PublishedInInteinationalFilin Bowan,Ic+ 111111. needed I); using hexagonal prisms. color. I 1 win. lb. Published In Cenlo 1:(10(aii(mallid.. It; antil.«11or. 15 min, 7. 1bneloped In the Natiotial Council ofl'earliels of Using Holum Piclures Effeclivel ill cooper ation with General Learning Coq). and tlistlibuted by Sitter Bolden Co. II; 111111. (010r. 7 I0 In order to select:ufilm that will help achieve I:1 S. Published In NloothInstitute of Science.Ilinun. a desired instructional goal. a teacher naturally (010r.13 pRolEcriox rwvicEs 107

11111 11C111:I1 1(s Alms alC lisIC(l in 1111% %/111111C. s1(m1(1behot used. aud the sound adjusted. 'rile 111(0 1C(C111 C0111111 ClICIISIVe 111(14.`Xis the \\*henitis time lor the class to view the Mtn. Idarzong Onc(ior% (13). This is a se%elmolume fli(k of the sst ho It is all that should beIle( es. listing ()Ifilms. slides. filmsnips, transpareinies. s,%. and other insult( tiottal materials. Itis, ot ((nose. the lespoo.,:bilit% 01 the teach

.\nother imblhation hi(11 the lc:ache' er to In:Ike stile that ever%( 11;1(1in the (lass is should be lamiliar is Educator., Gulf:e 1, ee seated so that he (an see :1101hear. .\rule of Film.%(6). This guide isiSsIled M11111:111 :111(1 thumb is that 110 child should he seated Ittrthet 11.4% the manv hums thatare available loan from the s(reen than a distant e six times the business. inditsti%. and gmettnnental agencies width of the screen. Run the best way toasses, 11llen a tea( het finds a listing ()I (ilia that seeing and heating (onditions 1s Shill 10 walk seems applopriate for his purposes. itis a good from plate to place in the room. idea to look for an evaluative review of the :\ motion pi( tme should not he regarded as I Re% joys 01 apploximately 1:30 films were a lesson initself.Research has demonstrated published in the December 111113 issue of the the importance ()Ithe beforehand prep:nation .liathenntinA Tem her. Other re%iews have been and the follow-up artiyiues, \laximum benefit puld;shed in more retell) issues of the Alaihr- will be gained front a film orals in leacher maths Teacher, Ihe Irion Eoaluotion Guide prepares the class properly. Students should be published by the Educational Film Library As. given suggestions about what 10 expect in the SO( 1:11 1(111 «1111:11115 0:1111:11 1011i Of Amu. 50 mathe- film. Some teachers experienced at using Hints matics films. Film leies (an also be found find that presenting alistof questions about. 111 the 11111111s11C(1 twice :I 11111111 by the the key concepts to be dealt with is an effective .\merican Library Association. way of setting the mood. It may be necessary If a teacher (bides to 11w a particular film to discuss some of the vocabulary used in Ille :cuditis not available loc:111.%, he should order 111111. Students quite naturally tend to lose in- the film 11011) a distributor «msiderabl in ad. terest if dwy hear 100 many unfamiliar wools. yance of the date for showing. To lit the film The teaclr must «tsequently learnto an- plesentathm in at the proper point i:1 a course tici»ate difficulties of this nature. 01)%imisly requires (awful planning. If alist of questions has been given to the

11.1101 the film arrives. arrangements should class berme the showing of a film, the follow- be tade to preview. II. No teacher should beso up activities would obviously include adis- int)-lent as 10 neglect this step. for the preview cw.sion of these questions. Rut a good teacher serves many purposes. Even if a review of the will go beyond this. Ile will use the film asa 111)11 has been favorable. a teacher might decide springboaul to further learning. Suppose. for upon preiewing itthat the film does not do example. that Alalbenunies of the Honeycomb what he thoughtit would do. Certainly there is shown to a class. This film should lead to an is 110 wa for a teacher 10 make his final de( ision examination of the relationship between %ohmic tegading the use of a film except by familiar- and surface area of threedimensionalfigures i/ing himself with it. The pre%iew also provides andbetweenarea andperimeterofplane an opportunity 10 plan the class activities that figures. The possibilities for extending the will precede the showing of the film and those epis developed in such a film are almost un- that will follow the showing. limited. \\glen a fill))is to be shown. the room and One of the immediate objectives of the follow- theprojection equipment should be readied up session is 10 help the teacher ascertain wheth- ahead of time if at all possible. Elte film shottId er the film was used at the right place and be threaded into the projector. the machine whether itis worth using again. Souk. teachers 208 ClIAPTEI: SEVEN

keep le«)1(1 on index taids of the !Huth the% 1.11111% CI Ira in.lir%I )IIe of the M.1111 01/%1.1( he" 10

use and their ellecikene.,%.IIIthis wasitis 11(1111(311 11+1.' 01 11101 HI, better

possible to build a %altiable caul file in a le inoie(tots are also (apable 01 %topping on .1

yeals.II.tteacher finds that .1 tertain 111111 is single frame %%idiom burn tlamage to the lihn. espetiall worthwhile and thinks he will he us- and the (an be leethed so that .1 stint:me in ing it mei and (mei. he should (onsider tecont- a 111111 Lail be repeated for emphasis and ( mending it, purchase by the [(Hal so 11001 sstem. cation.

Despite the greatimprovement%in16111111 Developments in Projectors plojec tons. Ietc11m development% Ill S111111 equip. The siandaidelassioommotionitiore pro mew 1si11 plobabl have an (nen neater jet tor for a numbet of ;eats has been the Itimn pact. Ve are lefetring- to the development of hound plojettot. Nlan of the newel models ape tartridge plojectots and the improvements in hellthreading (Figure7.1). This featurecer- Smin film.

I: lent,: 7.1 Self-threading MOVie projerior

Courtesy of Bell and Howell molEcTioNDEvicEs 209

Loading .1 tartridgcinojet 101issimid translucentIn:net ial. The main advantage of matter or pushing a filmalit idge into .1 slot this .Aqe1111is thatitlellildieslittle darkening (Figtue 7.2). The lihnis lead\lotimmediate ml the iewing room. 1:eal !Attic( lion de\ it us au. showing. As itis iewcditis rewound iuIlic ailalde .15 Alta( !MIMI% tollmills designed lot tat tridge. The problem, ()Ithirading. adjust Iron! itrojet lion (Figme jug. and rewinding ale noneNistent. Theat- (:.01ridge pwic( tor, use 8-nun lam. 11 hen die nidges tan iu (vat it\rioni tom minutes or film ones were introduced. the 8-111111 film that film tip to dartminute,. seas mailable %vas ()I the It desittned fur home he plospective putt hasel of .1 tatilidge ino um\ iv making. This film felt ninth to be desir ed jet tor has a (Amite t)I several g ins. Muth silent in the wa of picture (inalii. althoughit was and sound inojet lots are being mat Lewd. Some satislat tor) for indixidnal 01 sittall-!anal, \jett- inojectins t)Ithe normal hunt-st leen\ ing.IIII !Hirt. super-8 film Was introducedIA' (I:igtue7.3).while others make the oftear Eastman hi-; film lia-; In:my advantages inojet lion (Figtur 7.1).In Ike le:11-1)10jel11011 mei the 51.111(b1(1.8 furor.lie Chid one being an sIstulll the i)loieeted IW:1111isrefle( ICll from :I .11)1,1oximatel50-10elientlaiger1,itttne mirrmr 01110Illy tear ofa select' made Of Tiw larger :nettis obtained l), decteasing- the

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spate lor sprocket holes and the space between frames. Because of the increase in pictue si/e. theprojectedinageiilunchhi ighte and sharper. Super-8 con,ecitienthsuitable for viewing b groups of a< na) as a hunched people. athantageous Icatic ul supc1-8 lilin isthe1:3 latio of picture c%ichlitopicture heightthe saute as for 16-inin film. This means that supei-ti prints ca be inade hour hint without cutting oilpait of each frame. The teacher can expect to see the printing 01;: -nun velsions ofratan'educational films invionsk available old% in16iiiiii loimat.. \s a matte] ol

hiuutRE 7.3. Fro/a-screen ivpc

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1:1GURE 7.4. Rear projection 1.....0¢RAMMIMOOW.,41' - 7= 1i:()1EcTi()N DEvc:Es 211 la(t. atleast one leading disilibutor adkettises The majorit% ()I the (arilidged films that ale thatit will make 8-min prints of ally title in curl end% akailableareshort.single«)11(epi its library as soon as it has ordcis lotatIcasi loops.'" These ale ideal lot individual of small- live pints ()Ithatfilm. This means that the group instruction. The simplicity of operation motion-pi( !tire budget will suck!) mini) further. of the (.11 tiidge project()) means that almost any, Not Old\ is the 01 1011.11 (011 101' S -nun Mtn, (On- child of .school age (an operate one by himself. Nidetalthless than that 101 16-11)111ih11,. hut .1t)(1 .situe the film rewinds itself in the cacti idge. 11.111d5 Ile\ CI haC to 1011(11 a (.11 tridged an indiNidtial student (an Nicw it as many times filmitshould stayin good shape for many as he wishes. The motion-picture film has finally 11101e showings. become as accessible 10 a child as a library hook.

riG lila; 7.5. Rear -laolecliwz deviceas allachmenl

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edtn ational television station, ott the air in the The possibility of mill/big television for edu- 'Hited State.. There are Wrote than eight Inindt ed tat ional purposes was tecogniied almost howl closed- circuit wtents in educational institutions. and More of these are being inn into operation thebeginningof experimentationwiththe ever) tear. Sex eral states ha .e statewide medium. At the height of the depression in the educational networks.Itis difficultto estimate how early1930s.thetiniversityofIowa.Kansas inanY students ate receiving at least some of their State. and Purdue University producedtrial instruction h. of television. but the nu - educational programs. New York University be- ber tinquestionabh gan to experiment with such programs in 1936. Cells 10 million. Despite these figures. television has not had After the interruption catrsed by World War the profound influence on education that man\ II the pace of de\ elopment of educational tele- enthusiasts had ptedicted loritinthe1950s. vision picked up rather slowly atfirst. butit Few will dispute the statement thattelevision accelerated in the early 1950s. In February 1950. is the most potent. vetsatile. and compiehensive WOI-TV began operations on the campus of medium of coma unication vet devised to. man. Iowa State at Ames. WOI-TV was licensed as a butitsimpact on educationhas nevertheless commercial station. but from the outsetit de- been negligible. This is patic ularly true at levels voted much time to educational programing. beyond the elementary school. In1952 the Federal Commtmications Commk- It is not that students cannot he b% means Sion reserved 2,12 channels for exclusive educa- of television. In the years since the advent of tional use. The number of reserved channels has educational television. hundreds of studies have since been incteased. 1:01-IT in Houston began compared the effectiveness of instruction using broadcasting in June 1953. thus becoming the this medium with that of more conventional firstnon«anmetc.il educationaltelevision sta- means of instruction. Most of the research intion. In the next few eats doyens of other sta- dicates that time is no significant difference tions were established. students learn at least as well from televised in- Coneuriently with the development of broad- struction as they do front ordinary classroom casting stations. experiments were being con- instruction. ductedwithclosedcirctutinstallations. The Why, then, has the potential of educational Pennsylvania State University experimental pro. television not been realiied? Certainly one of grant beginning in1951 and the Hagerstown the factors is the inertia in the American educa- (Washington County) 'Marylandprojectiniti- tional system. Most change takes place slowly. atedin1956 were part icularly noteworthy in Nfany teachers, particularly at the higher levels, the early years. Both attracted widespread at- seem pet willing to conduct classes in the tention and amply demonstrated the feasibility usual manner and stubbornly refuse to even of direct teaching by television. consider the possibilities of television. Commercial networks have also played a role Also hinder log gi eater a( «ptance of television in educational television, In October 1958 the is the fact that the vast majority of the programs NationalBroadcasting Company initiatedits produced so far have not taken advantage of the "ContinentalClassroom-withacoursein medium. The programs are adequateas stated physics.:\year later a chemistry course was above, children learn from thembut most of telecast. andin1960-61"ContinentalClass- them do not generate great amounts of enthusi- room" presented a coursein modern algebra asm among the participants. In February 1967 a during the fall semester and a course in prob- conference on television in mathematics educa- ability and statistics in the spring. tion was held at the National Center for School At present there are mote than four hunched and College TelevisioninBloomington,In- vkolamoNDEVICES 213

(liana. The conletees were inathemati(s edu(a- idemape tecolders. .\ school of modlate si/e tots and teloision peis(nnel. (1.1s these (an now all(»(1 this pie(e of equipment. Ho tape people %imed idcot.tpes andkinescopes of machine (Figure 7.6) enables a school to ie«ml.

tele% ised 111.1111C111.16(5 lessolls 110111.111 1).11N of foilater 1>la\back. progiams that are broadcast I 1W (01100. The% were not impiessed with at an in«mivenient Gine.Italso allows repeat the) saw (15). 'The general win Illsi011ofthe showings for classes that need the extra exposine. group at the end ofthe «Adel cure was that Another significant doelopinent is the open- only afew of the lessons had exhibited any ing up for edit( ational purposes of sonic lie- real neativitv. hi fill' too man) cases a telelesson (piencies inthe upper regions ofthe electro- isproduced simpl transferring a classroom magnetic. spectrum. Onl) One piogiain at a time setting to a television screen. Itis not surprising (an be broadcast oser a gken television chan- that the response to such plOgrallls has been nel, and in the VHF and portion, of the indillerem. spectrum only one channel can he assigned to The inflexible schedule of In Oalk as( television station. In 196:3, the Federal (:(minatinications has been another deterrent to 41 eater ittiliia- Commission opened up :31( hannels in the 2500- lion. This pioblem may eventually be con 2610niegacycle frequency range. A licensee can dueled with the aid of sonic recent develop. be assigned as many as four of these channels ments. Oneisthe production of low-priced and can hence broadcast as many as four pm-

1, \

I.R.URE7.6. 'rape machine 214 SFVFN

grams simultaneously. The Instructional Tele- inn:lest in the lesson. Studies base shown that vision Fixed Service (thisis the term the FCC an enthusiastic attitude on the part of the c lass- uses for the 2500-megacyc le system. also known loom teacher is extremek impoitant. A negative as the 2500-megahertz system)is stillin its ex- orindifletentattitudewillbe quicklx «mu- perimental stage (9), but it looks promising. mimic:liedtothe students. and the televised Often raised as an objection to instructional lessmi will not be effective. television is the contention that the prospective II the lesson guide lists «Alain matclials that learner plays a passi\c tole. The student has no studentswillneed duringthetelecast.the oppoitunity to ask a question of participate in teacher should make cellain that :be mateiials a discussion with the instructor. Students ex- ale mailable and that students have thei»cn- peliced at learning by television have, how- ganiicd before the lesson begins. esei. dis«)5eied that this dillia tilt) can be mei-- The classroom teacher should emphasiie the come by taking (mei td notes and by jotting down impmtance of calelid listening and sicwing. lie questions as theN come to mind. The questions ma) want to give suggestions to the students can(lienbe discussed ina lollow-up session. regarding note-taking. As a matter offact. some experimental studies It is up to the classroom teacher to deterine have noted that a positive iesult of televised in- the eflectixeness of the telex ised lesson lor his structionisthat students feelresponsible for particularclass. During the telecast he should their own lealning to a greater degree than be using his intimate knowledge oil his students otherwise. 'Tice skills of listening, concentrating. to assess the degree to which they tire compre- and taking notes aie inuneasinabl) shaipened. hending sal ions poi lions of the lesson. This will give him sonic idea as to which areas need re- Effective Utilization ofITV inforcing during the follow-up. Some carefully Immulated questions alter the telecastis oscr There is a distinction between instrueilona/ tele- will shed still more light on the degree of com- vision (ITV) and educational television (ETV). prehension. The classroom teacher then does The former ieleis to loimal sc-11001 or college what he can to leinforce. clarity, and expand instruction using television, whereas the latter the ideas presented by the television teacher. term applies toallcultural edneationalpro- The follow-up session should be deemed an in- griming, The discussion here is limited to ITV. tegral and important part of the total lesson. Although there still is certainly much to learn One of the criticisms sometimes voiced about about mi, itis possible to give sonic definite televised instructionis thatitdoesn't take in- suggestions to the classioom teacher who wishes dividual differences into account. Itis true that to utiiiie this medium. every student is presented the same material at The most important thing for the classroom the same rate of speed. But here. once again, is teacher to keep in mind is that he is part of where the classroom teacher makes a contribu- an instructional team for a television lesson. His tion. Itis primarily his responsibility to provide duties aspartof the team differ from those for the individual differences. The methods for of the studio teacher, but they are just as im- doing so are not essentially dilfeient from the portant. methods used in the conve»tional classroom situ- The classroom teacher must prepare himself ation. Some students, for example, are going and his students for the lesson.If thereisa to requite a great deal of individual attention. written guide that accompanies the program, he It is «mceix able that after a telecast the majority should familiarize himself with it and make use of students will be able to carry out assigned of the hints therein. tasks independently while the teacher works He should do everything possible to promote with it few students on an individual basis. PRolamON DEVICES 215

Classroom Arrangement for 1T1 In arranging the classroom for a televised les son itis important to keep in mind (as obvious asit seems)thattelevisionis an audiovisual tool. As with motion pictures. itis not possible to achieve maximum ellectiveness unless every student can see and hear Ivhatis intended lot hint. Placement of the teceiving set. seating artangement. lighting, and acoustics must all be «msidered. 30° MAXIMUM The maximum viewing distance is naturally determined by the site of the picture tube. Ac- «tiding to a definitive study clone for Educa- tional Facilities Laboratories (3). students with normal vision cut be plated at distances from the screen that tange up to 12 times the .1(11,11 Width 01 the televised image (not the diagonal measure. which is the nominal si /c of a tube). Fwvitp. 7.7. Vertical viewing angle This means that lor a 23inch set the maximum viewing distance is about 20 feet. 21'he minimum viewing distance is dependent upon the height at which the setis mounted, and this height depends to a certain extent on the seating arrangement to be used. The set should definitely be placed above eye level, but if %iewers ate to be placed one behind the other the set will have to be considerably higher in order for each student to have an unobstructed line of sight. If a staggered seating arrangement is used, itis not necessary to mount the set so high. Once the height of the screenis established, the minimtn viewing distance should be determined by keeping in mind that the angle of elevation from the eye to the top edge of the screen should be no greater than about 30° (Figure 7.7). Tillie is also a hori7ontal viewing angle to consider (Figure 7.8).If the angle formed by the line of sight and the center axis of the pic- Ftculti.: 7.8. Horizontal viewing angle ture tube is greater than 45°. the distortion of the image usually becomes objectionable. For the viewing of some kinds of materials the distortion becomes objectionable if this angle is much over 30°. It should not be necessary to darken a room 216 (i \PTER SEVEN

(luring a telecast. At times the television teacher ononthallfeasible in expel intent Atm eN. may want the iewe ;sto do some penciland pet intent. paper work during the lesson. This and otdinar) The letnalksinthepreceding paragraphs

note-taking, tequire adequate lighting. Cite on pet twinmainly totelevisioninstructionin .1 thetelevision screen may he a problem. but comentional classroom setting. Planners of wilt this can usually he takenare of by proper (ational facilities should Consider (alefullv the placement of the set in the room. In the opportunitiesforinclividualiiinginsttuction classroom the best location for the set is prob. which are afforded byitleatning (enter such ably in a corner on the side neatest the windows. asthatillustratedinfigure7.9. Atypical There is perhaps not much that can be done booth in this center is equipped with 3 small about acoustics in our older schools, although television screen, earphones. and a telephone some teachers have noticed some improvement like dial. dialing all assigned number a sog when the walls are covered burlap Or similm dent gains access to a lesson re«ilded on video material.Inplanning new schools.thought tape or film. Perhaps when centers such as this needs to be given to the importance of good become «nuttiottplaceand creatie programs acoustics for televised instruction. Carpeting on become plentifulteledsion willfinallybegin the floors has pp-wed to be advamageons and 10 assume a vital role in education.

Itmum: 7.9.Learning center at111rtUniversity of South Florida, Tampa PRO) ECTION DEVICES 217

OVERHEAD PROJECTION the class when using an osctl aced projeo tor.Ileonscduentl) has better «nurol. The modem 05 erhead projector has evolved and itis easier for him to gauge the le- front device used by the military services in ceptivity of the class. training programs eluting World War II. The i.The fact that itis not necessary to dm ken earls models had scleral disadvantages, the fore. the room means that students can take most being that they were large and cumber- notes as the lessonis presented.Italso some. Today version isvet y compactyet means that attention can he shifted to a rugged: itprojects an image bright enough to model. a chart. or other visual aid to be be seen in a lighted room. and itis relatively used in the lesson without changing the inexpensive. lighting in the loom. (In connection with Mechanically speaking, an overhead projetot this. the teacher should remember an int. is a simple device (Figure 7.10).Itconsists of a base and a projection head that is supported by aserticalpost attached to the base. The base contains alight source and a horiiontal projection stage. Light passes through the stage and is tellected onto a set een by a lens.mirror PROJECTION HEAD assembly in the head. Appearing on the screen is the image of whatever is on the stage. This TO SCREEN might be a sheet of transparentplastic with written material Ottit(such a sheet is usually called a transparency), a transparent slide rule, or the like. It has been noted that motion pictures and television have not been enthusiastically received by mathematics teachers. With the overheacl projector the situation is just the opposite. It has won overwhelming acceptance in the past decade. Perhaps itis overused by some teachersthe amhor has been in classrooms where it seems that students arc so accustomed to the overhead that it no longer commands their attention. Some of the reasons for the popularity of overhead projection follow: FRESNEL I.Mathematics teachers have always made LENS extensive use of a chalkboard, and almost anything that can be displayed on a board can be shown better with an overhead pro. jector. It is easier for most peopleto write on the horizontal stage of a projector than on a vertical board. Improved legibility and LAMP MIRROR more accurate diagruns arc the tesults. When a teacher is presenting materialon a chalkboard, he is turned away front the FIGURE7.10 class much of the time. The teacher faces Schemalic diagram of lypical overhead projeclui 918 cal\ PTER \

portantrule:\\lien attentionisto be a grease pencil or a felipped or n)lon-tipped shifted away from the screen, the projector pen. This method should, howes el% be music!. lamp should be switched off.) ered a temporary measure.

1...aterialsNI can be prepared in advance and A lunch more professional looking; and mote used tepeatetlly. The projector is certain!) permanent transparency can be made with the a great timosaver, since it enables a teach- aidof aninfrared copying machine (Figure er to prepare a table. graph. diagram, or 7.11). matelial to be .shown isfirst drawn drawing Of a complex mathematical figure in pencil or ink on a piece of paper. A sheet and use it over and over. of nansparent lihn is placed in contact with this original. and both are fedinto the copying machine. :1 transparency ready for projection Pre paring is ran.s paren cies is available in a matter of seconds. In the past ICWears. hundreds of commercial!) Forreproductioninaninfraredcopying prepaued transparencies have become available. machine, the original must be done in black ink They ate being produced by book publishers. or No.:? pencilthe machine will not copy color. manufacturers of copying equipment. film mak- The 01 iginal must be a sheet of paper that will ers. and man) other companies. Se eral oftlte pass through the machine. This means thatit main som ces of such tanspauencies are listed is not possible to copy fuom a bound volume at the end of the chapter. with the infrared copier. While some of theommerialh prepaled A photocopying machine is more versatile. It materials are eNCCIICIll, III:1111 teachers find that will produce ablack and white transparency the transpaiencies of most saloe to them are from a «doled original and will also cop)- luom theonestheyhave madethemselves. The a bound sohunc. But the photocopying process simplest means of producing a transpaent:),is requires more time. It involves the production to ;, cite or draw on a sheet of cleat plastic with of a negathe. 1)epending on the type of machine.

FIGURE7.11 Infrared copying inachi ne

Couttcay of Minnesota Alining and Manufactioing CompatiV PROJECTIONDEVICES 219

the positke prim is then (Inc loped either by One simply,rubs amoistenedfinger on the heat or in a liquid solution. maigin of the page. If a white residue rubs oil (.0101 (an be added to nanspatendes in se%- on the finger, the color can be lifted.(\lost era! ways. A simple but elfe( the method is to of the betterkno vim maga/ines such as Life :old apply, desiled ((1101' with a [eh-lipped pen. Look ;:e printed On cla;coated paper.)Of Another method utilizes piessuresensitive (d- «ifise. tausilaen(y can be wade only ou«. oled nanspai cut sheets. A tea( her who has these from a single page. The Overhead System (25) sheets on hand (an add color to a legion of a gives full details on color lilting. nansparemy b 0Iluillg bola a (doted sheet a pie( e ()I the size and shape needed. IllusiralionsofTransparencies liy using special film. an image in a single A few examples will serve to illustrate the kinds color on a clear background (an be produced of projectuals which the teacher can make for with ail hanned wpier. The special film mines himself. One type that the author has found very in several dillerem colors.Itis naltiallv more useful is exemplified by Figures 7.12 and 7.13. expensie than tilt that pio(1110 es a black image. A transparency can also be produced in a single color through the so.called (Ham process. This profess requires special dia/o film. an ultraviolet light sonne. and a development chamber containing ammonia fumes. Diazo film has a diemical coating that produces a brilliant color (diaio films arc available inseveral different colors) oil exposure to the ammonia fumes. The coat 11g is, however, rendered inactive. and color will not form in the parts of the film which have first been exposed to ultraviolet light. To produce a colored transparency. a translucent Or transpamit "master- is produced first. This is then placed in contact with the diazo film. When the film is exposed to ultraviolet light, the opaque regions of the master protect the cor- FICCRE7.1 2.I 'ettit-diagra in Ira wpm ettcy responding regions of the dinzo film. These legions thelelore become colored when the [din Each tiansparem; consists of .t pennanent basic isplaced in the container of ammonia fumes, format to which material will be added when a while the exposed legions remain dear. A com- concept is developed in class. The Venn diagram plete description of the diazo process can be transparency in Figure 7.12 can be used, for ex- found in the booklet The Overhead System: ample, to show that product ion,lin Memel, union,andUtilization r) (I;U (;)= ri1;) U r) (:). (25). stepby-step fashion, students can be shown An illustration front a magazine can be trans- that A fl (B U C) is represented by a certain referred to a transparency through acolor-lift gion within the left-hand rectangle and that ',Imes,. As the tenn implies. in this process the (A fl B) U (AnC) is represented by an identical inkis literally lifted from the magazine page region within the right-hand rectangle. The in- and transfored to the film. This can be done formation added to the tanspalency in class is only, if the original was printed on a clay-coated done with a grease pencil. It can easily be erased paper. To determine if the paper is clay coated, with a tissue or soft cloth, and the transparency cii\p-1.FR ',FAT

can immediately be usedto show a different ielationship. .k transparency like that shown in Figure 7.13 can be used in developing the addition and multiplication tables for a 1):tefive numeration system or for a finite mathematical system. Rather thanhastily sketching the table format each time it is needed. it makes sense to take the time to produce a peimanent tanspalency. The mathematics teacher can illfectively utili/e the technique of revealing a transparency a portion at a time. This works quite well, for example, in explaining the solution of an equa- FIcantr. 7.13 tion. All the steps in the complete process can 7.ranspare icy for base-five nnmeralion sysion 4119,96 IAik#V1AO,/1 KNACIV4E011I 1441/SANKI IONIAta404.11 6101361.4.r41in NO 1r FIGURE 7.14 nctnur. 7.16 7' 'airs mrency for Pythagorean theorem Second overlay 411A.AEl / rloyie I TripiumII I. ti,18 Aeonig DELIMATIFYIN Wrokiromm1

Fictntr. 7.1:5 Ficukr. 7.17 Overlay for Pythagorean Iransparency Third overlay ilEXICES 111

be printed on a permanent transpaiency. A discuss the inverse of a function. Again the base sheet of paper (or anything opaque) can then trAnsparenc) can contain the «uffin:tic grid. be used to cover all but the original equation. The graph of a function is then drawn on an As a tea( her explains .1 step 01 as he elicit. the overlay thatis attached to the base with tape explanation Rout the class, he reveals another hinges along the line v = x (Fignie 7.18). When line. This continues until the entice nanspar- the oveilais flipped over. the giaph of the enc is 1111«)%elecl. inverA: appears. Tile revelation technique %%.orks as wellill ,Nlan vitiations of the .11)ove technique can the presentation ofa geometricpool'. Some be used.I low does the graph of v (x): !cache's also use itI() administer all «nnpare tothe graph of y= f(x)? Suppose (loll. It) revealing one test item at a time. they thatthe graph of the later is shown asin set a pace for the student and make itlikely Figure 7.19. Since the absolute value of a num- that he will give some thought to each item. ber is never negative. the graph of v = If(x) consisting of a single sheet. will not contain ally points below the x-axis. A are extremely 'befit!. itis the o%erlay technique than has soldmarry 11131ileill:Ilkiteachers on overhead plojection. B% placing one or more transparencies of top of afirstitispossible develop a mathematical concept ill stages. author uses the transparency shown in Fignie 7.1.1as the beginning phase ()I'a ohs. ussim of a conjecture about the discover)' of the 1))thagmean theoron. Tiles have been found laid outin such apatternhiruinsinthe Middle East. A second transparency isplaced over the first to produce the shading shown in Figure By counting triangular regions, it is easy to see that the area of the square on the hypotenuse of the isosceles right triangle is Fictmr. 7.18 equal to the sum of the areas of the squares on Transparency forinverseofa function the legs. When a third transparency is overlaid (Figure 7.16). attention is focused on a linger right triangle. and it can be seen that the same relationship among the squares exists. Another overlay fixes attention on a still larger triangle (Figme 7.17). The shadings on the three overlays arc in three different colors. Overlays can be used very effectively in developing graphical concepts. When working on the slopeintercept method of graphing, for example, the coordinate grid can be shown on :t base transparenc) with :1 line offixed slope on overlay. The overlay can then be moved OVERLAY op and down to show the effect of changing the y-inteicept. nano:. 7.19 As another example, suppose one wishes to Graph with overlay 222 (.111 xpl-Fik sFvEN

moment's thought will (Out hue one thatthe graph()Ir I (x) he the, :me as the graph ofV =f(x) exceptthatthe Pall below the x -axis 611 be tellected above. 'Phis tan he illmtrated be hinging au methn along the x-axis (Figure 7.20). Fignre 7.21illustrates another tvatouse overlays. This well-knon optical illmion tan be projected tvith the two parallel lines on the base tramp:nowt- and the family Of intersecting lines on an overlay. When the overlay isre- moved. students see that the illusion of bending lines is destroyed. Optical illusions such as this FIGURE 7.20 sometimes help geometry students lewgni/e the Overlay Pipped above x -axis importance of not hunting conclusions from the appearance of a ligtne. For mole examples of teacher-made tianspar- elides. the reader is lefened to excellent articles by Ilausen, Osborne. and 1;nkrick (II: IS: 29) and to a booklet br Krulik and Kaufman (12). See also Chapter -I of The Teaelwr and Overhead Projection. by Schultz (22). If thete is a general guideline for the teacher to follow in the design of a transparency. itis this: Keep it simple. Do not try to crowd too many diagram., or too much written material onto one transpalency. The si/e of ligtnes and lettering On a projectual is all important con- FIGURE 7.2 sideration. A teacher cannot expect to obtain a Optical illusion usable transparency by copying mdinary type- written materialthe letters are simply too small to he seen when projected in a typical classroom. Figure 7.22 shows the si/e of type that is FIGURE 7.22

THIS IS THE SIZE OF TYPE THAT IS FOUND ON SPECIAL TYPEWRITERS FOR USE IN THE PRIMARY GRADES. IT IS RECOMMENDED FOR TRANSPARENCIES. 1.1:0.11....C110N DEVICES 223 found on special typewrite's designedfor use by teaches in the iai grades. The image projectedi in-is higeenough to be seen b .a pet soil W11 11 1101 111:11 %151011 at t111:111(CS 111 1011) 1(11 110111 111C %Creel'. 'These perites ale «msequentlighl ie«ininiended for the I prepaation ()1 nanspaiencies. As intik:lied earlier. the 0% el head In OjeriOr isnotIitnitecl10the projection of material printed()II11:1111):11cm sheets (althougo these trillundoubtedly be the 11111111 1)10iCe111:11i 11 DISTORTED

Ieaidler uses). Tr:011411M 1111C1.5.11"011:ICIO S. slick I :111(1 other 1001:,caul 1/C (11:T1:lye(' Very IMAGE well withthe OVerlIC:1(1.Because of the site of the projected image. itis ptobably easier to explain the Ilse of011C (11. these toolsby weans of overhead projection than in ally 0111C1' Way.

.Screen Placement for Overhead Projection The main consideration in placing the screen for overhead projection conceit's the keystone I:estoning is the image distortion (Figure Funnt.F. 7.23.Keystone allecl 7.2:;) caused by the fact that one edge of the screenis faither flout the projector than the opposite edge. The overhead projector is placed much Closer to the screen than other kinds of projectors. This (combined with the fact that the scicen must be high enough for students to see above the teacher) produces keystoning tin- less contact-me:ism es are taken. SCREEN Keystoning is especially undesirable in mathematics because viewers may not be able to see what they are supposed 10 see. For CX:11111/1C, a squate will not project as a square noless keystoning is eliminated. 9 To mold keystoning. the line determined by the center of the screen and the center of the projection head should be perpendicular to the screen. In 1110S1. ClaSNI*00111S, the top of the screen will have to be tilted forwaid (Figure 7.2-I)in older to achieve this. It is probably best to have a screen permanently mounted in such a position. If a commercial screen cannot be so placed, a piece of hardboard will suffice. Nlany teachers have found thatitpiece of hardboard painted Vitanty, 7.24 white will reflect a very satisfactory image. Proper screenplacemeni for overhead projeclor 224 CI !AFTER SEVEN

OP:\ (ItI'. PIZ() ECTI()N The lighting chit iencyof5011IC of the newer models has, however. been imploved to the ex- With the increasing popnlatity of the ovethead tent that a modetatel darkened loom is ade plojector. the opaque type seems to have lost tptate. some of its appeal. Tasks for which the opaque Other le:tones of the ncwer opaque plojectols plojector %vas ttsuall utilised are now It etinent- are a down4halt tooling sstein that holds pi 0- I% performed with the aid of the overhead Iwo. jected materials in place and a relatively light jet tor.Nevertheless.dune ate thousandsof weight that makes lor reasonable portability. opaque projectors available in our schools. and The model shown infigure 7.26 weighs 33 teachers gill continue to find uses for them. pounds. .\11 opaque projector is just what its 11311Ie impliesa device for projecting an image of The main advantage of an opaque piojector is that materials tan be shown with little helot e- opaque material.. \s ,Itch. it projects light that is reflected front the material being displayed (Figtne 7.25). Situ e the lightisreflected. the plojeted image is not as brilliant asitis for a nanspatency projected on the overhead, and this means thatthe room must be darkened,

L FIGURE 7.25 Schematic diagram of typical opaque projector

FicuRE 7.26 Model of opaque projector

CourfrAYof CharlesBrselerGompan"..."7 PROJECI'l ON DEVICES

hand in epaiation. If a teacher wishes to display (17)listsnearly a thousand titles on mathe- a paper or a page from a book with an overhead. matical topics. a transpalenc) must be made first. But %vith an Allele ate good reasons for the popularity of opaque projector any paper (or practically any slides and filmstrips. The\ areI Ciall1C1: ineN" object less than No inches this k) can be shown pe11.11C. Mid because of their siie they require to the class ah»ost immediately. .`.1)(1the pro. little storage space. They thus can be purchased jeered image is in title color. Exai»ples of pic- b) schools and made readily available, The pro- tures that «mime to the author's mind as he jection equipment is also low in cost.is easily writes this are those lonnd in the Life Science portable.issimple tooperate. and requires Library volume Mathematics and the ones in little maintenance. Ilehnich Tiet/e's Famous hoblems of Mathe- Nfathematics teacheis have found filt»stiips to ma o.s. Such exceptional coloured illustrations cam 1116particular liking because so man) topics be nansleued to a nansparency thiough the in mathematics must by their eery intim e be olomfift process. but in so doing the original discussed with the steps in a certain sequence. is of course destio)ed. Hence. displa)ing them The proof of a geometrical them cm. the solu- with an opayie projectmis the sensible thing tion of an equation, the NiepTstep process of to do. The 'cache)mill also find the opaque the slope-intercept method of graphing, and the projector a handy device when he wishes to explanation of an algorithm all lend themselves display a student's paper. well to presentation by means of filmstrips, Each Still another use that some tcachms make ol frame can he viewed as long as necessary. the the opaque pi ojec lotis the enlaiging and copy teatime' (an anmer questions that misc, and the ing olintricate geometricfigines. lo cop) a strip can, if desired. be backed up to a previous diagiani. the image is projected On a clialkboaid frame. or poster baud. Itis then cptickl) and easily Many motion-picture producers put out film- traced. For a disctission of this technique and strips to accompany their motion picttnes.In °the) useful ideas, the teacher should refel to most cases the lih»st) ip consists ol ceitain frames The Opaque Projector (2.1). taken directly from the motion-picture film. A filmstip of this sort l»ThideS ateacher with SLIDES AND FILMSTRIPS an excellent means of focusing attention on h»- portant points in the motion picture. Slides and filmstrips can be treated together be- Projectors for slides and filmstrips are of sev- cause of Iliasimilau chmacteristics. Commel- eral types. There me those that will poject only ciallyprepared slides come in two sues, 31 filmstrips or only slides, but one of the most by -1 inches and 2 by 2 inches, with the smaller popular projectors is a combination type that size being by far the most popular. Teachers will handle either filmstrips or 2 by 2 inch slides can and do make their own slides. Al) excellent (Figure 7.27). One of the greatest stimulants to referencefor the teacher interestedintrying the use of slides has been the development of his hand at. making slides is Production of 2 x 2 projectors that can be fitted with a cartridge Inch Slides for School Use (2(i). holding a number of slides inproper order A filmstrip is a collection of still pictures on for projection (Figure 7.28). The cartridges also a strip of 35m» film. There are usually some- serve as storage boxes for the slides. A useful where between 25 and (i0 frames on a strip. feature of some projectors is a remotecontrol Thousands of filmsnips are available, and per- attachment that enables the operator to change haps more use is made of them in our schools slides« advance the filmstrip from anywhere in than of any other kind of projected material. the room. The index toi5-mm Educational Filmstrips Some filmstrips and slides are coordinated with CI I \I FR \i

sound recordings.Stu Itcombinations pm\ ide the 'cachet with ,mother means of indi5 instruction..\ tecent de5(.Iopment is a sound- slide sstem utilising a slide holder that has a magnetic sound.dis«qu itching the slide (Figure 7.29). This s% stem assures sxm oni/ationof sound and pictute. Each disc will hold np to 35 seconds of sound. A teacher using the s)stem can ploduce his own 35111111 slides and can do his omit te«nding. New sound can he recorded mei the old. and a slide can he replaced in the holder in a matter of seconds. While talking about sound in «mnection tc ith slides and lihnsttips. xe should mention a promising doelopment that does not seem to fit into this yearbook an) \Acne eke. This is the cassette audiotape sstem. The cassette system employs a reel-toreel unit encased in a plastic container that measures about 2!! b -Iinches. 'Hie big advantage of a cassette is the ease of Ilse. All one has to do is snap one into place and press a button. Sound quality is good, and cassettes ate so durable that the) can be played thousand, Couylesyof Gloflex, 111C01P01 Wed of times before wearing out. FIGURE 7.27 Most of the mathematics andiotapes available Combination type j»-ojecto, for films of .slides in the cassette format at the present time ate

Cowles>.of Eastman Kodak Company

FIGURE 7.28.Projector holding slides in order for projection PROJECTION DEVICES 727

designed to be used for drill a»d teview. They cover such topics as addition, subtraction. multiplication, and division of whole numbers and ISMS I:Ilion:II numbers, inoblent soiling. and elemen. tali 11111111AI theory. Students listen to the tapes and usuall) respond using pencil and paper. Some tapes ate coordinated with study booklets. Teachers find that audioutpes help tore- inforce and Ill skills and find them use- fulin diagnosing weaknesses. The tapes also inesent a way of providing for individual dif- lerences becanse a student cart proceed at his own pace and listen to a single tape as t»any times as he wishes. One other innovation that should be noted is Coto tes) of .11inuesota Mining and Mann/a( lining Company the transparencyLartridgeprojector(Figtne 7.30). This combines some of the advantages F)ca 'RE 7.29. Sound-slide .\stem of the overhead projector with those ofa slide projectm Itprojects by by -Iinch slides that can be produced b) the teacher ill the same manner as transparencies for the overhead. As many as six slides can be made at one time from an 8 by I; inch piece of transparency film. The slides are mounted ill snap-together frames that interlock with one another to form cartridges of its man) as forty frames. As with the overhead, the transparency cartridge projector can be used inanormally lighted room. Slides canbe changed through a remotecontrol switch, so the teacher can remain at the front of the room.

FicuRI: 7 30. Transparency cartridge 1»jector

4 .

Courtesy of Minnesota 411 ining and Manufaelming Company CI I\I' l'FR SF \'f\

SOME MAJOR SOURCES OF PROJECTION EQUIPNIENT AND MA IMIALS Addisses of producers and distributors are listed in the Appendix.

10-110N PICTURE PROJ ECTO RS \NO FILMS rRiP PROJECTORS

I 6NINI PRO j EcToRs Abecutipt, Bell andI towel! Co. AO Instrument Co.. Scientific Instrument Div. Busch Film and Equipment Co. A tidioNlaster Corp. Eastman Kodak Co. Bell and Howell Co. Grallex, Inc. Chalks Beseler Co. Harwald Co. Brumbeige Co., Inc, Kalart Co., Inc. Ditkane Corp.. Audio Visual I)iv. I.-W Photo, Inc. Eastman Kodak Co. NrovieNlite ( :oq). El«) Corp. Paillard, Inc. GAF Corp. RCA. Commercial Electronics Div. Giallex, Inc. Honeywell. Inc., Photographic Products Div. 8.Aim CARTRIDGE PRojEca ORS E. Leit/, Inc. 11st111:111 Kodak Co. ItCltire Projectors, Inc. Fairchild Caniera and Instrument Corp. N hind us Co. Technicolor, Commercial and Education Div. Spindler and Sauppe Standard Projector and 1....quipment Co. OVERHEAD PROJECTORS Victor Animatoguiph Corp. (Div. of Kalart Co.. AO Instrument Co.. Scientific Instrument I)iv. Inc.) Applied Sciences, Inc. vimex. Int. Bell and Howell Co. FILMS Charles Beseler Co. Buhl Optical Co. (There are many more distributors of educa- Graflex, Inc. tional motion pictures, but these appear to be Gregory Magnetic Industries, Inc. the main sources of mathematics films.) Keystone View Co. li NINI Moios Picruluis E. Leib. Inc. Association Films. Inc. Nlath-U.Nlatic, Inc. Bailey-Film Associates Nlinnesotalining and Nlanufactuing Co., University of California Extension 7Niedia Visual Products Div. Center 0/alid Div.. General Aniline and Film Corp. Cenco Educational Aids Projection Optics Co., Inc. John Colburn Associates, Inc. Tecnifax Corp. Coronet Films II. Wilson Corp. Encyclopaedia Bri al1111 Ca Educational Corp. Film Associates of California OPAQUE PROJECTORS Indiana University AudioVisual Center, AO Instrument Co.. Scientific Instrument I)iv. Field Services Dept. Bausch and Lomb Optical Co. International Film Bureau, Inc. Charles Beseler Co. State University of Iowa, Bui eau of Audio- Karl Heitz, Inc. Visual Instruction Keystone View Co. Knowledge Builders LacerLuci Products, Inc. McGraw-Hill Book Co., Text-Film Div. Projection Optics Co., Inc. Modern Talking Picture Service, Inc., SquibbTaylor, Inc. Modern Learning Aids Div. vitoJEcTIoN nEvicEs 229

`)il%etIlindett Co. Collnin ,\ sso( ices. 1 'Hi% ersa I1:.(1 in at ion and Visual At ts aColonial Hillis. In( . (:titic nlunt Nlateriak Corp. 8.1\1(:\ItIlLII)(11) 1:11,\I, Curtis\ ludic) Visual Materials All(110Ikttal (I)11. of .\ nglophoto. Education:II Audio Visual. Inc. Ltd.) dutational Plojetticms. \I is 1:ilins I let hert NI.1:.11, ins Co. En( lopaedia Britannic aFclut at ional (:ot p. I tem (.1 and Amu, jaw, clop.tedia Ilritannica Eck:cation:II Cot p. ,:e Gale Flottse. modem Learning Aid, Filmstrip Hottse National I ilnt Boald c)I (::t nada 1:11nI lauds School Service. Inc. National hist' in tional NI( Graw.1 I ill Book Co.. TeNt-Filet 1)k. Pifiles 1)11()1()gral)lii( A1)1":11i()11' Cu. Uniersal 1:.clunation and \Istial Arts Number Hillis Popular Scien( e Publishing Co.. Inc.. Audio. 1:11.NIST1tIPS Visual Div. Statile\ Dowmar (:o. Societ for Visual Ed in at ion. (:etno Educational Aids St ipes Publishing (:o.

SELECTED REFERENCES

I. Mown. J. \V., R. B. Lewis. and F. Ilarcleroad 10. Goldon. George N. ChkszooniTelevision. New ,1-1"In lion: Media and Allethods.8(1 ed. New Fron le)S in /7T. New York: Hastings House. York: NI( Graw-llill Book Co., 1969. 1970. 2. Carnegie Commission On Educational Television. 11.I lansen. Viggo P.New Uses lot the Olrhead Publit /I Moglam for Action.New Projector." Alai/zonal/esTearlie, October1960. York:\ von Books. Bantam Books. 1967. pp. 67-69. 3. Chapman. Dale. IncDesign Jo) E (Educationa0 12.Krulik. Stephen. and Irwin I:autumn. noteto 7T: Nanning lot Schools Tlevision. New Use the Overhead Mojectt» in .11athemalus Mu- York: Educational Facilities Laboratories. 1960. tationWashington. D.C.: National Council of 4.Diamond. Robed NI., ed.,1 Guide to liktinz Teaduls ofItitheinatits. 1966. tzona/ 74enc,ion. New York: Mc Graw.11111 110(11.; 13.Learning Dirertmy. 1970 -i!.New Yolk: West- Co.. 1961. inghouse Learning Cm p., 1970. 5.Educational Media Council.l :dttrallonaIMedia I 1.NImphy, Judith. and Ronald (toss. /etaning by Index.I4 soh. New York: Craw./ liltRook Television. Newyolk: Fund for the Advance Co.. 1961. ment of Education. 1966. 6. Edmalms Guide to Free Films.30th annual ed. 15.National Centel for School and College Televi- Randolph. Wk.: Edtuatms Ptc,gless Smite, 1970. sion. "TelevisioninNI:ahem:1M,Education." 7.Film alms Guide to Fire ilmst,ips.22d annual Irithmetir Teacher.November 1967. pp. 59&- ed. Randolph, Wis.: Edlleators Plogiess Soviet!, 602. 1970. 16.National lamination Center forEducational S.Ez ks(m. Carlton \V. II.Fundamentals of 1'etteh- Media. Rev. ed.Index to ló-mm Educational mg with ,Iudiovisua1 TechnologyNew York: Films. NewYot k: McGraw -Ilill Book Co.. 1969. Macmillan Co.. 1965. 17. /m/exto3i-innEducational Film. 9.Federal Communications Commission. Commit- strips.New York: Al(Graw-Ilill Book Co.. 1968. tet for the Full De elopmeut of the Instructional 18. Osborne, Alan R. "Using the ()vet head hojector "TelevisionFixedSet vice.'YTS. Instiuctional in an Algebra Class. Math(nzatics Teacher, Feb- Television Fixed Service (2500 J1Iegaheitz): What ruary 1962, pp. 135-39. It Is. . How to Plan.Edited by Betharr Coop- 19.Reid, J. C., and 1). W. MacLennan, eds.Research er. National Education Association. Division of in Instructional Television and Film.Washing- Educational Technology. 1967. ton, D.C.: Government Printing Office, 1967. 230 I p.I.ER

20.S.n.tilt.r. Paul. ,I //i3tml ofI n%1 netional Trelr 21,VisualInsunclion 411i()pulitefret nolog,. NewVotk: Book Cu.. 1968. le(PH.twin; l'inietsity 01l'eN..t., 1960. 21,s(111,01,111, (.(1, 1,:(1 Ij'elevnion, 25. -- The Ove,headS43tem: Minim lion. I - The eV Ten Yeae3.Slant.d. Institute plem en la I ion and rtilnatimi.thin]: t?iihei,iiv lor (.0nununication IteNearcli. 1962: \Vashington. ((ITeN.a.,. Govoinnent Printing (Abu.. 1965, 26. hod:teflonof 2 2 Slides for School 22.Schulte. \10r1:011. The 7'emhe, and (lvei head ,Rev. ed. ersii NO. hair( lion.high-wood Cliffs. N.J.: Prcliti«..11a11. 9. A.. and G, I, Siuller, Mu/top:ma/ 1965. Materials: Their Nat air and I '3e.lib ed. New 2:.lInkric 11:0 mon. "Using ilic Overhead Pro. Votk: Ilarper k Row. Publisher.. 1967. jec torinTeaching (.4.01Nelty."Ale:then/Wks 'reacher. C) iolier1162. pp. 502-5. 8.USING MODELS AS INSTRUCTIONAL AIDS THE VOLUNIE 01: THE GRAVEL PILE is appioximately I cubic ard. its shape is almost that of a right circular cone. The pile was formed by filling the small steel box (inside measmements 6 by 6 by 6 inches) shown at lower left 216 times andetupt- ling the contents over the center of the base of the pile.I lenue the volume of the pile should be 27 cubic feet. or I cubic yard. 1(216 x 6 x 6 x 1728 = 27.1 The box with faces marked off in squale inches has the shape ofa cube 12 inches on edge. Its purpose in the picture is le help the reader make «miparisons ()Isire. The height of the boy k 60 inches. The "2" inside the target on the vertical ruler indicates that the height of the pile is 2 feet 2 inches. If the «mesliaped pile could have been topped (i.e., put in the lorm of a perfect cone). its height would be 2S inches. The radius of the circular base of the pile is -10 inches. If the very smallerror incurred bv taking 28 inches as the heightofthe cone is ignored. the volume of the pile, computed by using the formula f/=;',7rr2h, is 27.11 cubic feet. which is approximately 1 cubic yard. [(4 x x .10 X 10 X 28) 1728 = 27.111

LIM 154111,311Wer ;11.1

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9-9- ,, ) . s t *11-,` / "4; 'S I ^.4"6,,.i1.55 9I A " sla ??..f. *.! g , '

e 4# '41 t 0; A ,* - 4,4" II. ' " Ri4421.9iy.S4;"/:1::* 4) Z? 2, .7. '4"` 4Zr.A`k -,:'99,%1, * l'APVIA.t441)-19 itt . * . .1,): "- . / . 7,f;., : e 1.., t' IS.414,4str:'141'. # **-#. '4Lig 4,?47;n1-,'4;f1 ..- 49,1 10" re= xr (..,;.". ; ' 94.'4, 'r' `Tri,ttP.-<''."4` ;;i .9 ;PIN."' Z':%: V.;.)' t 41.7) 49,1. ,:' iU

119. 1# . <.k 2". 11:11: ..0.,-... # 4

AIN o-- ....4...4,-., .. . ,'Z 9 44,,Z4-''''P': #...,,,q A :, . ,, .,,.# ..,,,.. . .241. '9 r -9 .94 -`..?! °A ' .4.1.` +.9_-:4`. --Zi .,9 t,ItSCI''.1'.(Pt 419 '9" ..., -9 '/91,7j9i4Oki. .:-. " 1*-,itiS; )3`.'-`;:7#41W.' .: ', ":9:911 ...b, 4. `'.;',4", 1. '4' ,="7"--404 MI, . c- . LI...... -. 00.1", 24,1144.,:n* .:' -AY!-i AWAS. _173141- 233

CHAPTER 8

USING MODELS AS INSTRUCTIONAL AIDS

by DoNovAN A. jouNsoN University of 'Minnesota, 'Minneapolis, Minnesota EMIL J. BERGER Saint Paul Public Schools, St. Paul, Minnesota GERALD R. RISING State University of New York at Buffalo, Buffalo, New York

The principal concern of Chapter 8 is the role of models in the teaching and learning of mathematics. There are illustrations to show how models can be used to promote discovery, to enhance class discussion, and to encourage independent investigations by students. There are also illustrations to show how models can be used to generate interest in new topics, promote enjoyment of mathematics, build appreciation, improve retention, and teach applications. Suggestions are given for making effective use of models, promoting student production of models, building a school model collection, and selecting models for purchase. 235

8. usING MODELS AS iNsTRucnoNAL AIDS

Teachers and students of mathematics have used mist uses graphs to find themost efficient pro- models as instructional aids since ancient times. duction process. And the psychologistuses a flow Using models helps to enlinge the totality of sen- chart to display processes in learning theory. sations and to improve the quality of sensations Because much of mathematics is concerned received by the learner. These two phenomena with abstractions. there isa special need for contribute to imptovement in perception, which models in mathematics. ;Mathematiciansrecog- is basic to improvement in learning. 'lire this when they createnew mathematics. Models may be nmde of paper, plastic, wood, Elder's development of certaintopological metal, or other kinds of construction material. properties of geometric configurations isa good They may be in the form of threedimensional example of a mathematician's use of modelsin objects; or they may be pictures. diagrams.or creating new mathematics. Euler soughta solu- graphs. The word model is used in two ways in this chapter. It is used in the usual sense to refer to concrete representations of mental constructs or ideas. For example, a geometric figureisa mental construct; and a model of the figure isa concrete representation of the mental construct. The word model is also used in a genericsense that is, to refer to the various kinds of instructional materials listed below. I. Demonstration devices (largeconcrete representations. including charts) 2. Manipulative devices 3. Computational devices 1.Instruments for measuring and drawing 5.Kits G. Games and !mles 7. Science apparatus S.Realia (ordinary objects)

THE ROLE OF MODELS IN THINKING Models are constantly being used in thinking. Consider how scholars in different fields use models to solve problems. The geologist uses maps and charts to locate a new ore field. The chemist represents atoms with models made of little balls and connectors. The architect designs a building by using a scale drawing. The econo. Pylluzgoras making a model CI IAPTER EIGI IT

tion to the Kiinigsbeng bridge problem (Figure Ro OF 101)FLS IN IF 8.1). The challenge in the problem was to find TEACHING AND \RNIA'(.; :1 pedestrian path that would cross each of the OIm.vri vmATics seven bridges exactly once. Euler identified those features in the concrete Thisse( liondes( tibes nine geneial %%.1%, in situation that were essential to the problem. which models can be used b% the tea( het.InI he These he represented in a diagram called a net- leainer. orby both.hi each ease the point of work (Figure 8.2).In this network points rep. view presented isillustrated with examples of lessonIca huuk tie,.

Models Provide a Selling (or Discovery 01 Concepts Consider hose students can dist.mer Elder's lot- Hilda for polyhedra. First, each student makes sketchesofpiane networks and teem ds the number of %en k Cs r, edges E. and legions hi for each network as shown in the table.

Plane Number ofNumber e)INumber of NetworkVerde es Edges E Regions I? 0 A 3 11

B 5 11 5 1 FIGURE8.1 I) Ill 5 A 8 19

Alter dimmeli1114and hukum:111 Pim hugthat the Iohnula /: I" E -z; 2 holds for plane iwt- works the students are given several models of polliedra (1:igure S.:1)to examine, or they :ter asked 10construct their own out of cadboad. 'This lime the student:,record the number of %entices. edges. and Il«., ;insteadofregions). The formula = 2. which isalmost D identical to the lormula Ism plane netwolks, is FIGURE 8.2 lound to appl% in thieedimensional spate. stn. (10115 maybe helped to prove the formula b resent hind and arcs iepresent bridges. By using studing the ploof curtained in the book What this network as a model Euler discovered sev- /.c Math/ turiiirA? Im Courant and Robbins i):1). eral generalizations. Later on he extended these When the students ale satisfied that the foi generaliiations to networks(in-..ohing regions, mula + I' F2 holdsforallpoly Nedra. %ertices. and arcs) that had no physical counter- theteathei In esents rats modeistillat pol pats, In this way topology developed into a hedral model, but dillerent Isom the mdel, mathematical strit:-.ene that was completely in- int-sentid (milieu in tii,ttit has .t hole thioughit dependent of bi idges, networks, or applications (Figuie8.1),Substitutionofthe numbet 01 to reality. laces. vertices. and edges9fthe new model in the USING NIODELS AS INSTRUCTIONAL AIDS

"proven" formula leads to a contradiction. A lively discussion ensues. Can the same statement be proved and disproved? An important aspect of the scientific method becomes evident. Stu- dents can (I)'eject the "proven"' formula, (2) restrictitto eliminate contradictory cases, or (3) extend it to include the new case. Eventu- ally, the students recognize that the formula F V E = 2 is true when properly limited but is only a special case of the more general formula F + VE = 22H, where 11 is the number of holes in the polyhedron. Successful discoveries prompted by manipulating or just contemplating physical models give students a sense of independence, a feeling of activeparticipation, anr: the exhilaration of success.

FIGURE 8.4 238 C:I IA VIE!: EIGI IT

Models Can Be Used 10 Focus Allen lion 3.\Vital expression (an be used to teptcseni on Ideas nun .1 re under Discussimi the volumeof the cube: 1.Flow manydimensions does :tline seg. Flow models (an be used to locus attention on men' hate ??a square? a cube? ideas that ate nuclei discussion is perhaps best 5.!low manydimensions :tie suggested 1). illustrated vith a description ()I a lesson. each of the following algebraic expres- A hearty class discussion involving far-teach. sions: X. x:. s ing ideas can be started by showing students a Using the chart helps students fonts attention chart like the one displayed in Figure 8.5 and on the possibilit. of 'elating the measures listed intoning the following- line of questioning: below on the lett with the algebrai( expressions I. What does x represent in each figure? listed on the tight. 4,.What expression can be used to represent Length of a segment the :tea of the square? Area of a sprite Volume of a cub? FIGURE 8.5 Using the chart also helps students focus attention On developing- a correspondence between the algebraic expressions x. X. and x' and familial one. two.. and threelimensional figures. Pun SEGMENT let us proceed with the lesson and a few more questions. k.flow many dimensions does the algebraic expression x' suggest? 7. Can a geometric fignre have mote than duce dimensions? SQUARE xi 8.Does the drawing of the cube in the (hart have duet; dimensions? 9.Is the drawing of time cube in the chat ta twodimensi(mal figure? X 10. Is the drawing of the cube in the (h :nt a twolimensional teoresentat ion of a duce. dimensional figine? The last lew questions should give the leader a hint of the reason for starting this lesson with it chartratherthanwiththree.dimensional models. At this point in the discussion. the class should be trade to consider the question: "Who can ex CURE plain lime to build a threedimensional model of it Four-dimensional ligCp If the question draws a blank. then the time is at hand for the leacher to unveil threedimensional model of a tes- sera( (Figoi e 8.6). A R.sserat is also referred to as a fourthdimensional analog of a cube or as

:1 hyper( tibe, Representing a tesseract with:tthiec.dimensional model is analogous to representing at cube USING MODELS. \S INSTRUCTIONAL AIDS 239

ipated. Such demonstrationse ically p1oolsof the special cases. The technique recommended is the zone as described aim% e--1 lass disc ussion mo tivated by questions that refer to figures ejne- semed by ciiagrams or threedimensional models. The first diagram in Figure 8.7 shows a segment of length a joined to a segment of length b to form a segment of length a b. Certainly. (a by =a b. Inthe second diagram in Figure S.7 each side of a square is composed of two segments whose combined length is a 1 b. The diagram

FIGURE 8.7

FtcuttE 8.6 a 4I

(a+b)--.1 with a twoIimensional drawing. Oneway of reineseming a tesseraci in duce dimensions is by, joining corresponding tertices of an upper cube and a lower cube with parallel oblique rodsas shown in Figure 8.1i. Once the imaginations of students make the leap to the idea ol fourdimensional spate, some eager student can usually be counted on to raise the question 01 how to give concrete representation to afigure in fivedimensional space. To keep the imaginations of students in orbit. students mat need to have a clue to help them with the new pi oblem. A useful clue can be git en by having- suidents focus attentionon the key to the solution of the problemthat is, by asking them to count the edges joined to one vertex of the tssetact and to do the same thing for each figure displayed in the chart. Consideration ol the abstract idea of ,r- dimensional space is onlya step away. Using- models to help students focus attention on ideas that are under discussioncan lead to sonic very sophisticated mathematics. As a sequel to the lesson outlined above, class discussion invoking demonstrations or special (aNeS 01 the binomial theorem might be anti- 240 CI I APTER EIGHT

shows how a square may be separated into two smaller squares with areas (12 and b2, iespective- l)ind two Iectangles. each with area ab. Thus, the diagram shows that

(a-f- by = a=+ `Lab +1).1. The dissectible model shown in Figule S.can also be used for this demonstration. The bottom diagi am in figure 8.7 shows a cube with each edge composed of two segments that have a combined length of a 1- b. More inipoltanti), the diagiam shows how a cube with edgea + bcan be sepal ated into the eight three-dimensional figures desclibed below. Icube with volume ti 3 rectangular prisms, each with volume (121) i 3 iectangular prisms, each with volume OP Icube with volume b3 Therefore, the diagram shows that (a + = as 3a2b 3ab: b3. Coup toy of LaPinc Scientific Company FIGURE 8.8

FIGURE 8.9

Courtesy of LaPinc Scientific Company USING MODELS AS INSTRUCTIONAL AIDS 241

For some students a three-dimensional dissetible model is mole helplul with this demonstration than a diagram. Figure 8.9 pictures a dissectible plastic model that is available commercially. Students can also make a model like this. (See Figure 10.20 in Chapter 10.) Figure8.10 shows a tesseractwith each edge separated into two segments having lengths a andb,respectively. The entice tesseractis separated into the fourteen subspaces described below.

1 tesseract associated with a' I hyperprisms, each associated witha3b 6 hyperprisms, each associated with allb2 -1 hyperplisms, each associated withab3

Itesseract associated withb' Thus, the tesseract in Figure 8.10 can be used to show that (a ± b)l = a' + +01) ±6a -b= + dlab3+b'. Focusing attention on ideas that are under discussion is one of the most important functions of models. The examples described thus far illustrate uses that have been carefully planned. FIGURE8.10 However,thisparticular functionof models sometimes operates dramatically even without preplanning. Here is an account of 411 episode that actually happened. A geometry class was asked to prove that the volume of atriangular pyramid equals one- third the product of its base and altitude. The class had available a chalkboard diagram similar to the one in Figure 8.11. The usual way to do this proof is to start with the triangular pyramid A-BDF,construct a prismACE-BDFon the base of the pyramid, and then show that planesACF andAardissect the prism into three pyramids with equal volumes. The crux of the proof is to show that there are two pairs of equivalent D pyramids. (The wordequivalehlis used in the same sense as in Cavalieri's theorem.) A few -1 students in the class were able to do this proof with the aid of the chalkboard diagram, but FIGURE8.11 most of them were unable to identify and keep track of the pairs of equivalent pyramids that enter into the proof. No amount of talk by the 242 CIIAI'TER EIGHT

teacho helped, eithei.l-Iowelca,when the teach- anguku pi ism yields Chace p)ramids that ale not erintroducedadu ee-dimensionaldissectible congruent. This last difficult) cannot. of course. model of ato pi ism (Figine 8.12). the be complete!) acnumed. even b) using a concrete proof came into focus for most students in class. dissectible model ofatrian, prism. The One reason students had difficult) with this difficult) is inherent in the pioblem. So it is sug- proof isthat the chalkboard diagram put a gested that when students aie asked to considci premium on selectke attention. Inhere was sim- the 1» oof about which we hale been talking the) ply too much in the diagiam to keep track of. be gken an opportunity to examine a three- Using a II» cc-dimensional dissectible model en- dimensional model of a cube that is dissected abled students to focus attention on fewer rela- intothree «nigHtenl pyramids (Figure 8.13). tionships at a time. The proof that the volume of each of these Another season students had difficulty with pyramids equals one-thiid the product ofits this proof is that the usual dissection of a tri- base and altitude is both easy and cons incing.

FIGURE 8.12

Courtesy of Leine Scientific Company USING MODELS AS INSTRUCTIONAL AIDS 243

Conzlesr of La Pine Scientific Company HGURE 8.13. The models of the three congiuent pyramids can be put togethei to form a model of a cube that lust fits inside the cubical-shaped box.

Models Provide a MCalS for Making difference C D, the product C X 1), and the Independent Investigations quotient C D for each object. By comparing theresults thesecomputations, Students can find a rational approximation to of students should be able to see that the quotient C 1) the value of r by comparing- the circumference is constant for all circles and is close to 3.14. and diameter of each of several circular objects (See the table.) such as a tin can, a plate, and a hoop. One way of finding the circumference of a circular object The fact that rr is approximately equal to 3.1-1 is to toll it like a wheel for one complete turn can be %et ified by computing the perimeter of ;t and then measure the distance traveled. An- man)-sided regular polygon inscribed in ,t circle °diet way is to measure the circumference ea- of unit radius. Archimedes found the value of

ly by wrapping a tape measure around the object. 77. to be between 3-14 and 3p,t by computing In call )ing out in% estigations, students should pet imeters of 96-sided regular polygons inscribed record in a table the diameter 1) and circum- in and circumscribed about a circle with unit ference C for each object and then compute the radius,

Diameter D Circumference C Object in Inches in Inches C I) CxD C4-1)

Tin Can 3 9.4 6.4 28.9 3.13 Hoop 26 82 56 2132 3.15 Plate 10 31 21 310 3.10 244 CHAPTER ElGlIT

A formula for the area of a circle can be determined experimentall)b)thefollowing procedure. Cut ;tcircular disc of radius r into an even number (16 Or more)of congruent circular sectors (Figure 8.11).Fitthe sectors together as shown in Figure 8.15. The shape of the resulting figure is somewhat like a paal- lelogram. The sum ofthe lengths of the arcs along the bottom (or along the top) equals hall the circumference of the circular disc. Since the radius of each sector is r. the height of the figure is approximately r units. Therefore, an estimate of the area of the figure shaped like a parallelo gram is onehall the circumference of the dim times the radius or rt.:. Since this figure has the same area as the circular disc. itis reasonable to accept A = vr2 as a formula lor the area of a circle. A formula for the surface area of a sphere can be obtained experimentally by using- a grape. fruit. Slit the skin of the grapefruit from each pole toward the equator, peel it off. and make Courtesy of Yoder Instruments FIGURE 8.14 an outline of the skin on a grid (Figure 8.16). Also draw a circle with the same radius as the grapefruit on the grid. Next. count the squares in the interior of the region bounded by the outline of the grapefruit skill. 1 he number of squares in this region may be considered an approximation of the area of the grapefruit skin. FIGURE 8.15 USING MODELS AS INSTRUCTIONAL AIDS 245

. 11112gNIMMNINDa I 4alail Alh. ' , lligiallIMIIIIIIPrANNllir ARM, IIIIIPASIRIMMIEWAIIIIMUM111, IlaaVAIIIMMINIMUNIIIIIMPlaiVaa7..D= I ; 1 I I : NW 1 flI 11'1 'AIMILIr; I I ! 1 ; 1 1 ; NI111/1/1 1 AV lia 1 I I 1-1.7' 1 . 1171111111111111/1111 ' ; I I I ; r , I V --11_: (I 1 I I 1 7.71-1 I 1 I winiunumal 771111 +III.1 inl_ til +1 , 1 I t 1 im a 1 i 1+,11 11+1 ; 7 7 7 MINIMILIIIIII11111111111111 I I : I I 4' I 71I ! : i

I I I ; 1 t I ; t i : t t 1 I IUIIINV la IT KVAI___!_i_I I I I Val ISIVA 111011111111M1 I I 1 I I 7 ft, -..-- i 11111111111111111/111k1 III NW t 1 1 r nu IIIMPINallafaillalifill:/allll 7 I IllaaaillVIII KIM HIMIMilialKIIIIIIPAallik I -17 I "WM I 7 i NIENSUP21 I Nur/ 7 7 'Ms P LI -1- MN ;--i-1 1 14:1-1-Ii1--i-, ----1 1 , 'I 1 r . .

FIGURE8.16

A comparison of this area with the area of the measure of the area of the surface covered. If circle that has the same radius as the grapefruit the work is carefull.) done, a comparison of the suggests that the area of the grapch nit skin is lengths of the Iwo pieces of cord should show that about foul times the area of the circle. Thus it the :ilea of the hemispherical surface is twice scents teasonable to accept A = lirr' as a formula the area of the circle. Thus itappears that for the area of a sphere. A = 27;r2 is a suitable formula for the area of There is another interesting way to determine a hemisphere; and this leads, again, to A =41;1'2 a formula for the area of a sphere using con- as a formula for the arca of a sphere. etc materials. Divide a croquet ball into two The result of the following experiment can be parts b.) sawing it through the center. Coyer the used to obtain a fonnula for the volume of a hemispherical surface of one part with a spiral sphere. The three plasticmodels shown in of venetian blind cord (Figure 8.17). Then cov- Figure8.19areconstructedsothat D =H er the flat surface of the other half of the croquet ball with a spiral of venetian-blind cord (Figure 8.18). The length of the cord needed to cover the surface in each case may be considered a

FIGURE8.17 FIGURE8.18 246 C:IIAPTEI: EIGHT

for all three models. II the cylinder is filled with .\ more sophisticated way of obtaining a lot -

water and the Oleic put inside and forced to 111t11.1101 the %ohmic of a spite' e is suggested b the bottom. the water that remains in the (:)I the possibility that .t sphere can be dissected into inlet will ills( fill the cone. This result suggests spliel 4.11p%Iatnicls (Figtne 8.20). Lithe' an cm- the following derivation of a formula for the any,e 01 a giapehuit can be used .1, .1 model. 'Hie volume or a sphere, -...:xpressed in terms of the radius.

V=19:if--I7rn 9 3 2

V =;D1)'3 7_ 4 19 =trn' 6 Let = 9r. where r represents the radius of the sphere. Ficuttr. 8.20 Then = development of the formula relies on the follow- 3 ing assumption:A sphere is a polyhedron with an infinite number of small faces; with these facesas bases and the center of the sphere as the common vertex, (he sphere can be separated into an infinite number of pyramids.These pyramids all have the same altitude r, and the sum of their bases is the area of the spherethat is, A = 17r=. Since the volume of a pyramid equals the product of its base and altitude. the 'oltinte of the sphere is 1r.117):, Or :1

,Models Can Be Used to Provide for Individual Differences Students differ so drastically intheir cultural experiences, in their ability to deal with abstract ideas,intheir interests, andintheir mathematical backgrounds that it becomes necessary to tailor instruction to provide for these differences. FIGURE8.19 The slow learner needs concrete representation. He needs to make and deal with objects because he usually reads poorly and has difficulty understanding verbal instruction. In the case of the slow learner, there is no substitute for experiences with concrete materials. The abler student usually progresses rapidly Courtesy of Ideal SchoolSupplyCompany and independently. Many of today's accelerated USING :\IODELS AS INSTRUCTIONAL AIDS 247

1)1'01.0;11115were \ elOpeti with the gifted student agtam by counting squares in the inteliots. The in mind. Vitrot tunately. acceleration has often average student can purbably dis(Over rotundas meant limitedinstruction,lengthenedassign. for the areas of polygons A. 1). E. and F but may ments. and complex lest1111teriai Theresult need to count squares to find the areas of poly- has beenrtIst111 Iloll. hostility. and I ejet lion of a gons R and C. The abler student should be Itunrc in mathematics. Even for bright students encouraged to explore the possibility of obtain- instruction needs to be enriched with experi- ing a formula for the :ilea of a polygon in terms ences that (let clop interest, arouse curiosity, and of the number of lattice points On the perimeter build appledation. Experieives with couttete and in the interior. The table contains the per- materials often enhance attahunent of these goals tinent data for polygons A. B. C. D. E. and F. better than do direct appeals to abstract ideas aml formal assignments. Perimeter Interior Polygon Points I' Points /Area A I;oween the slow loaners andthebright studentsthereisahuge group of average A 12 S students who also need varied experiences to I1 9 ensure learning. I-1 1 10 I) A pegboard on which the pegs tepresent a S 10 8 ulnae latticeis an example of it model that I; 9 6 .1 tan be used to find areas of polygons in several dilferent ways depending on the ability level of By examining specific cases, the discerning stu- the student. The diagram in Figure 8.21 shows dent should be able to discover the area formula, how elastic thread cat' be used to locate the =.121) + I I.wrnis isPick's formula. The vertices and boundaries of polygons on a peg- formula holds for simple polygons under the boad that has the desired construction. condition that vertices are latticepoints. The The slow learner tan obtain approximations proof of the formula leads to some rather ad- of the areas of the polygons shown in the di- vanced mathematics (18).

FIGURE 8.21

A E P1110 JII 0 ')-18 CI I.\ I' MGM°

,1 I 7 Models made b folding paper are another / Lind of model that (an be used atdifferent ability le% els.II %vaxed pap(nr is used. the row!,

suold 0111lemiii and can he projected on :1 m1(21111)% an (net head ptojc( tot. This tel cife(tive in helping the slow learner learn basi(ideas about pcipencli( ttlar lines. parallel lines. (on( utrent lines. angle bise(tors. segment bisectots. :111(1 «mgittent Iigtnes. same tech. nique is elle( tive in helping the avelage student dis(mer plopetties 01sc.con(lat)lines ofnil. angles. diagonals ()I a patallelogram, (hords of a circle. and central angles and :iis of a circle. The abler student ()lien, finds real excitement infolding paper to obtain teptesentations of envelope., ()I parabolas. ellipse.... and hypetbolas. FIGURE, 8.2:3. Each line remesent.s a crease and then investigating pope' ties of tlice c urves farmed by folding paper .so that a point 01 (Figures 8.22. 8.23. S.21). For further infornia the cirele that has 0 as «.nter coincides with tion about paper folding, see l'oper Folding for point F in the intoim. of the chute. Twenty- the Alathettittlits Glass 1)) Donovan Johnson (35). eight lines me shown in the diagram. The set of all such lines is the envelope of an ellipse

with points 0 and 1 as foci. Each line is tangent to the ellipse.

F1ct1iu8.24. Each line represents a crease formed by folding paper so that a point of the circle that has 0 as center coincides with point F in the exterimr of the circle. Twenty- seven lines are shown in the diagram. The set of all such lines is the envelope of a hyperbola with points 0 and 1: as foci. Each line is tangent lb the' hyperbola.

FiGuRE 8.22. Each full-drawn line represents a crease formed by folding paper so that a point of line DIY coincides with a point 1: thatis not in line DIY. Thirty-seven full- (haw?: lines arc shown in the diagram. The setof all .such lines is the envelope of a parabola with focus F and directrix Each line is tangent to the parabola. USING MODELS AS INSTRUCTIONAL AIDS 149

AlodeLs Can 13e I "Ara to Genera/r' 11:11:111111055111 g 01115 256 11.111eN. These ex- ihiercA/ ia a .Vein' To /lien periments are equitalent in the following sense: The number of balls expected to fall into each Sitnlents lime bum est.inmatlic- inati(al topi(s that :tie related0) theegttla of the nine (hantiels at the bottom. (minted 1111 01(10 Isom either end, corresponds rspe(ti%ely it 111111111)111 ale not pail ofit.I oldie', (an to the number of times 0 heads, I head, 2 heads. (.11«,mage itu the! (letelopmentof S11( 11 1111C1*- Csls 1)1 1)110 big suggestions for doing indepnd- :11111 so 101111. ale expected to 01 C111' if $ loins :ire wssed 25(it ent'noir( Is. .1131 M' 1(I.)In 11(1111.1 1011. leal 11(1 s 5110111(1 seek 10 1,111We 11110 es1 101)1(s 111at ma 11Ise( omplelek 11c 10 s111(1e111 s. 011e 1%":1 \ of doing this is I() stall with a familiar idea and (11(.11 expanditto a new topi(, ing students manipulate familiar (om tete material, FIGURE 8.25 is au example of this I( httu tie.. \11otier tray to stimulate inteiesti11 new topi(s is to tisc a Coutto of liatrourt Brace lOvanovich, not el (let i( e. Consider how speriments int oh ing tossing of di«.. or thintibta(ks (at) help students be- ((nue interestedin plobability. Students often bemme so .11).01 bed with exrel intents like these that[het illingh keep le«nds ofresults for 100 losses, 01 et en I.000. Re«11,1 keeping of this Lind (an be 1)51(1 to intiodit«. stub ideasas a sVil01 1111 pOss1111C (mu onles. a distri:nu ion, a graph ofa disilibtition. and so forth. I'llese ideas lead nattnalh to (lel initions of events as sets of ottuoines. 11111111:111%exaltisit e and hide- pendent mem.. and empiri(al and theoretical In oba its. The simple espci intent smentioned above (an be cillaiged 111)01111.11111 1W :ad or :1 device known .Is (,:hut's Quin( mix. A modern adapta- tionof Gallon's (let i«..(aile(1a I lexstat,is 1)k:tined rignie 8.25. Approximately 250 small steel balk ale trapped inside the devi(e. The .", picture shows the position of the balls after be. ing pouted down the (hmlets hunt the reservoir al the top The distribution of iesults obtained is an applosimation of.t binomial elk- tribution. which in 11111) is an appmximation of .1 notinal (lisp ibution. Thus. the I lexstat may be used to introdme important .111(1 engaging ideas lion' plobabilit%.

Itis interesting to note the parallel between pouring 250 balk down the channels of the Ilex- 250 CILW1Ut Elcirr

Courtesy of Winter Herrmann Courtesy of Winter lki: mann FIGURI: 8.26.Reflection board FIGuRE 8.27.Rotation board

Courtesy of Winter Herrmann Courtesy of Whiter Herrmann FIGURE 8.28.Half-turn board FIGURE 8.29.Reflection and translation board USING MODELS AS INSTRUCTIONAL AIDS 251

Another topic in which student interest can The metric system has been adopted by 82 be generated bl using models is geometric trans. Of the 135 nations of the world. Britain and formations. The animation !maids pictured in the United States are the only highly in- Figures 8.26. 8.27, 8.28, and 8.29 can be used dustiali/ed nations that continue to use to advantage in helping students peiceive what the inch-pound system. and Britain has an- nounceditsintentionto convert to the is meant by such transformations as reflections. metric system. [63, p. 3] rotations. half-turns. and translations. The following excerpt from the forewod of Logic is still another topic in which student the same publicationindicates whytoday's interest can be developed with the aid of models. students should have adequate ()ppm tunity now Shown in Figure 8.30is an electronic logic trainer that can be used to clarify the basic to gain familiarity with the metric system. postulates and theorems of logic and set theory Emphasis in this public:10ml has been given and the meanings of the operations "and," "or," to the development. growth and current status of the metric system in anticipation and "not." The logic trainer shown in theil- that the United States and the automotive lustration can also be used to demonstrate what industry may move to expanc! the use of is meant by Boolean algebra and switching metric units in the inreseeable ;noire. [63. circuits. p. 2] Arousing interest in a topic can be critical Since the United Stales lives by flitinch. to the future of our culture. Consider this start- pound system. teachers of mathematics do not ling statement. which is contained in the open- have a feeling of urgency about teaching the ing paragraph of the Ford publication World- metric system. "Fhe icsult is that the topic often wide Use of Measuring Systems. gets cursory treatment. But the topic is import-

FIGURE 8.30 FIGURE 8.31

FUNCTION

VARIABLES

:SHY LAME/LACTRAINER

''''. e..,t 'Ai- *" . ... 4. rrt;...E. rir 4, .i. kf . ...., t'r'-',.. Courtesy of La Pine Scientific Company Courtesy of LaPine Scientific Company 252 CI I. PTE It EIG I IT

ant. and itis likely to become mote important. The packaged products shown in Figure S.32 so the present lukewarm interest in the metric are pal tof the contents of a kitchen cupboard s)stem should be heated up. To achieve this. the that one of the miters I aided. 8) actual count. following equipment should be a.ailble lot use 30 out of 90 items found in the cupboard were b) ewer) meditnn-sired class group: a half-dozen labeled in both English and metricunits. The meter sticks, a metric licit:id measuring- set. a English and medic weights or volumes printed dissectible liter block (Figure 8.31). and several on the labels of the packaged prodm is shown in inexpensivebalancesthatalecalibratedin the picture are displa)ed in "boxes." Presenting grams. Procurement of this equipment. however, a collection like this to students is certain to is only a start. The really useful materials for tu erate interest in the metric system. Students firing up interest in the metric system will come who lune an ()ppm utility to examine such a col- from a collection of 'calla that can be obtained lection will no longer think of the metric s)stem from the kitchen ctipboald. the grocery stole. simply as "the other s)stem they have to know or the gasoline filling station. in science class.

FIGURE 8.32

1 3.25 oz.' 92 Tn. 16 fl. oz. .473 lit 8 oz. 227 gm. 1WED Gerber ti MEA RICHEDFLOUR OAT004 i46 =4:4 IlEINZ7 I i VINEGAR ". k L:N!a zoia flIMMEN,._ 4 .IdW lip., ,,vss Baas ti

1-* -1,111111111.''"-- k 7,1 _dig! SI II iiit a/r. SMUCKER.S tit RSLEY 10P FLAKES Schilling CREAM of , .. mUsHR001" 'SC %S 111".. 21136 ant USING MODELS AS INSTRUCTIONAL AIDS 253

:11odels Can Be Used to Promote to controlled adaptation of learning materials En joyment of Mathematics inprogramed instruction. For :t further discus- Most students are intrigued by games, punks. sion of the analogy between games andpro- and novel problems and willingly spend time otamed instrtyt ion. see -Games and Proorammed 011 them. The resulting learning experiences can Instruction- by Allen (I). be both challenging and entertaining. Teachers may wish to construct their own Games contribute to a student's enjoyment games. but there are pleat} of well-constructed of mathematics because the} ate fun.:well. games available horn connneicial prochicers. One structuted game also helps students leant mathe example of a versatile and creative game is matics. Folloxving the rules of a game is not Equations (Figure 8.33). This game is designed unlike pursuing a programed insti action unit. to prmide stimulating and entertaining oppor- Ordinarily the rules of a game promote actke tunities for placticein addition. subtraction. participation and pro% ide for immediate tein- multiplication. di. ision. exponentiation. and the forcement of leaning. Usually. there is also a square root operation in various number bases. cern.in :C.,cowan«. for individual pacing. Games Punks contribute to:tstudent's enjoyment that are structured so that the leaner encounters of mathematics becausethe}ateintriguing. increasingly complex situations asth.. game One example of an exciting puzzle involves the progresses have an added feature that is similar Soma cube. This is a cube cut into seven pieces.

, Ar -44

.111001110

EQUATIONS

Sr- vj

,02-* *.,

14, FIGURE 8.33 *V' ep 44,

Gourley of IVFF 'N PROOF 254 PTEI: EIGHT

no two of which ate alike (Figure 8.3.1). The puzzle isto put the seven pieces together to login a cube and to make other shapes. The pietCs of this puzzle ate easily constructed gluing together cubical (minting Ithals. For a tulle'description ofthispuzzle,see Mal/re- magical Models by Cundy and Rollett (11). Another puzzle is the oxbow puzzle shown in Figure 8.35. This puzzle is somewhat like a game, because there ate rules to observe. The object of the puzzle is to exchange the mai bles on the two sides of the oxbow using a minimum number of legal mot es. A marble can be mot ed either one or two spaces in one mote, but no more. If the marble in hole D is moved into the open hole E, this leaves the D hole open. Then either the marble in hole C or the marble in- hole F can be moved into the open hole D. This process is continued until the marbles on the two sides are exchanged.

FIGURE 8.34

FIGURE 8.35

--07 417=7.:7,. USING MODELS AS INSTRUCTIONAL AIDS 255

Students also Iind enjoyment in. mathematics by working on intriguing problems, or just con. templating them. Rademadter and Toep delightful book The Enjoyment of Alathematics (52)is filled with exciting problems and their solutions. Listed below ale the titles of NeVeral sectionsthatale particularlyrichinsolved problems that (an be tested with the aid of models. "Traversing Nets of Curves" "On Closed Sell.Intmeeting Curves" "The FourColor Pioblem- "The Regular Polyhedrons- "Producing Rectilinear Motion by Means of Linkages" "The Figure()IGreatest Area with Given Perimeter" "Curves of Constant Breadth" The section entitled "Producing Rect:linear Motion by Means of Linkages" contains a number of inviting problems that suggest the use FIGURE 8.36 or production of models. Nance Bier's cell is one ()I several 11111:ages that ale described. Pe:ma:Bier's cell (Figure 8.36) consists of a hinged rhombus ADCE and two nods dB and BC that are hinged to opposite vertices of the hombus. The two rods AB and BC are equal in length and are hinged together at B. Because of the symmetry of the linkage, B, D, and E are ahvays collinear. By applying the Pythagorean theoremit(an be prayed that (BD) (BE) = (IL 1)= (AD)r. =11.,. a constant. Therefore. D and E are said to be inverse to each other with respect to a circle with center B and ratlitti k, where B is referred to as the center of inversion. It can also be proved that the image of a circle that passes through the center of inversion is a straight line. Thus B will travel a straight line if D is constrained to move along a circle which passes through B. In Figure 8.37, bar DU has been added to the Peancellier cell. and the positions of H and B fixed so that the distance 1311is equal to the length of bar D11. As 1) moves it will follow the FIGURE 8.37 arc 1)13. which goes through point B, the center of inversion. This will force E to move along a straight line perpendicular to BFI. 256 (:1\ P'I'ER EIG I IT

Models Can Be Used to Build Appreciation for Mathematics The following statement by Benjamin Frank- lin gives an indication of the importance, he placed on both the practical and the aesthetic aspects of mathematics. It would be well if they [the students] could be taught everything- that is useful and everything that is ornamental; but art is long and their time is short. It is therefore proposed that they learn those things that are likely to be most useful and most otna- menta 1.1 If students are to "learn those things that are likely to be most useful and most ornamental,- they need to have experience in making sub. jective appraisals of the utility,beauty, and other aspects of the subjects they study. Such appraisals may be quite informal and will probably involve feelings. To buildappreciationfor mathematics, students should be presented with situations or given opportunities to participate in activities in which they are likely to develop positive feelings for some aspect of mathematics. The examples that follow illustrate how models can be used or how they function in helping students develop such feelings. FIGURES.38 As a prelude to work with standaul units of measurement, students may be introducedto such units as the fathom, cubit, palm, digit, and spun and invited to make measurements using their own limbs as units of measure(Figure 8.38). Students who engage in this activity usually develop a sincere sense of respect for the utility, convenience, and precision of theoncliu- ;try foot ruler and yaulstick. Afathematicians ate always talking about the beauty of mathematics, but few seem able to elaborate on what they mean. One mathematician whom we queried thinks beauty in mathematics refers to the elegance of a proof. He considers a proof to be elegant if the results contain an element of surprise or if the techniques used are clever.

I. Thic Ammon appears in The Proce.sc of Education, by Jcinine Bruner (10. p. 4). An example of a proof that has at least some of these qualitiesisLegendre's proof of the P)thagoreau theorem. This proof is unusual for its brevity. There ate only three steps. This is certainly clever. Following are the steps of Legendre's proof and the at compan)ing diagram (Figure 8.39), as excerpted nom Loomis's book ThePytha- gorean Proposition(1'1, pp. 23-2,1). The proof attributed to Legende is contained in a discussion of the shortest possible proof of the Pythagorean theorem. The equations in the first two steps can be obtained with the aid of the diagram. a:y = Ii:a a= = hy. b:hy = h:b b= = h= hy. Courtesy of La Pine Scientific Company Adding these gives h= = a= + b=. FIGURE8.41

While the last paragraph was being written, a visitor came into the office. She spied the ellipsograph and the full drawn ellipse pictured in Figure 8.11 on the office blackboard. (The ellipsograph was held in place with rubber suction cups.) She contemplated the situation for a moment and then asked: "You mean you can Bdraw an ellipse with the device fastened to the blackboard?" "Yes," was our reply. "But how FIGURE8.39 do you know that the figure is an ellipse?" she For many students of geometry, a three-piece IValltedto know. Hurriedly we sketched the set of overlay models made of plywood, card- dotted lines shown in the picture, labeled points boat d, or plastic will be needed if theyarc to see with capital letters, and then took up a proof. the similar triangles on which Legendre's proof Let d(PR) =a,letd(PQ) = b,and let 0 repre- is based. An appropriate set of models is shown sent the measure of the angle that the crossbar in Figure 8.10. makes with the positive x-axis. Then

= cos 0, and Y =sin 0; a b and x= y= -I- 7. = cos' 0 -I- sin= 0. a- b- So x2 y2 + =I. (I' 1)2 "That's elegant," said our visitor; then she ex FIGURE8.40 claimed, "No, it's beautiful!" 258 c11.\1,1 -ER

Students, and adults,too, me usuallyfas- Detailed instructions for building the models cinated by the many ligtues and forms of ge- shown in the picture may be bound in the book ometry that lend themsch esto the arts and b) Anderson. Space Concepts through Aestheouz- sciences.PicturedinFigure8.12 me string ry (2). models of stellar polygons constructed within A Major Function of Models is Their a cylinder. String models of this kind can con- Positive Effect on Retention ceivably be constructedwithin the walls or ceilings of buildings by using wire or other 11 you were asked to recall the most memor- nhtunidis. consuucting three.dimen s ion al mod- able mathematics lesson diningour education. els like these helps students de\ clop a real apit would piobabl) be a lesson invok ing the use preciationforthe beauty of geometric forms. of a model, a transpaiency, or a film or a lesson

FIGURE 8.42

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()king an experiment.Itis not likel)that (ailing the student an oppouttnit) to make it would be a lesson iikoking a let:tole on an measurements with a )aulstick will usually en-

abstiac tthcotctu. I.eaining based on wok %%id) stne his Icaming and lemembeling that :3lee( models of leat wing that Icsults110111 experiment, and I aul arc equkalent measm es ol length. isusually temembered long alter the leatning Ilaing the student lotto a model ola square period ends. 1:d with 9 square-shaped corktiles.each 1 Students olten have difficult)learning and loot on edge. k all thatis needed lot lementbeling mensuration equkalents. The set hint to learn and temembet that 9 mptare feet

below is tpical 01 those we Itme in mind. and I stymie sand .ne equkalent ineastues ()I area. Only the volume equivalents inthe last line give students much rouble. but unloruni- 12 inches I loot awl) this tumble is usually persistent and dune I square inches 1 squaw foot are plenty ol adults who do not know that 27

1728 cubic inches Icubic foot cubic leetand I cubic said are equkalent meastn es ol Voltinte. I 'ere is a was of teaching this lact When there ale no ph)shal telerents to toll so that it will be lemembered. Ilave each student in class xvith. students can have difficulty learning and

cut out13 cardboard models or squales 1 foot temembering thatthe eastuesin each line on a side, as a homework assignment. The nest ale equivalent. However. if a studentis given day have each student a foot ruler. a picttne of a square foot ruled thehis6 cardboard into squaw inches. and a cubical box one foot squares to make a model ol a cubic foot using masking tape to join the edges. Then have the on edge withfaces itiledinto square inches. -ancients in the class build a model of a ct,oic he can ee- the equivalent measures and make saioI. Experience indicates that if there me be- associations that will assist him in learning and tweet) 20 and :30 students in class. every student tentembering them. Another way oIassisting soonrealises the snalent ill learning and lemembering that that27 isthe magic number. Students (and teachers. too) who participate the measures in each line ale equivalent is1.)) ha\ ing hint make cardboard models of squares in this activity rawly forget that 27 cubic leet and I that toe one inc!an a side and then use these cubic )ald are equivalent ineastnes of models to lied the dimensions and :teas oI volume. objects in the classroom. IIthe last line in the To illustrate how to use physical relerems to foregoing table proves troublesome. have the help sttalents remember what they leantwe student make a model of a cubic loot with deliberately picked examples in which remem- bering is a Matter of practical Cottcerri. However. 1728 blocks. each one inch on edge. The student who does this is not likely to Iorgei that 1728 we MOH jtist, as easily have illustrated our point ol view with other models. cubic inches and I cubic loot ate equivalent measures. If a student learns the »leaning of similarity Let us consider how concrete releients might by preparin g. a scale drawing of a Iloor plan, he be used to help students leant and remember islikely to use the idea of sh»ilarity inthe another set of mensuration equivalents. future. If a student constructs his own slide rule (Figure `!13), he is not likely to forget that slide rule scales are logarithmic scales. Ifa stu- 3 feet 1 yard dent explores the variationoftigollonletric 9 square leet I square yaul ftilitiolis with a device such asa Trig-Aide (Figure 8.4-1), he will usually be ableLO recall 27 cubic feet I cubic yard the variation of each function. 260 CI IAPTER El GI IT

FIGURE 8.43.Slide rule made with meter slicks

FIGURE 8.44

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.; USING MODELS AS INSTRUCTIONAL AIDS 261

Models Gan Be Used to Teach I."The super market has started selling Applications beef stroganoff in a box that has tape- /olds for bases and rectangles for laces. The teaching olapplicationsiscomplicated What's their angleto take my mind oll by two factors. First ol all, most students do not the calm iesr "The landscape 1»a» just dtlinped a load have backgrounds in Ivhich significant applica- of black dirt on our lot. I-le says it's live tions have meaning. At the same time, students yards.' How doI know his truck holds have atendency to become bored with the fiveyards.? "How do you find the area of a flower ordinary. bed that has the shape of an ellipse?" Neither factlessens the teacher's obligation The first question points the way to a host of to teach applications. What is to he done? One application-type problems about boxes used by suggestionisto make applications catchy or distributors of food and (»het oducts.' Circular even offbeat. Unfoitunatel), thinking up off- c)linders and tectangular solids arc still the most beat applications isnot easy. Occasionally i»- popular shapes of boxes usedfor packaging sph ado» comes hom unusual questions asked I)) products, but frustums of circularcones, frus- students and grown-ups. Following arethree tums of pyramids, and right prisms of all kinds such questions: are being used more and more (Figure 8.45).

2. In this discussion the %void box is used both to convey its usual meaning and in a genetic sense when meletring to canons, cans. containets. and so on. FIGURE 8.45

0.61 I e s. 262 CI I.PTElt EIGHT

Font teen boxes are identified below in the column on the left. .1 description of the shape of oath is given in ge0111e1.11( al language ill the column on the I iglu. Ha% ing stock:ins give sat (11:M11111011S can be an exciting learning activity.

BOXES 1)ESCR IPTION

I.Corsage box 1. Right hexagonal (nonregtdar) prism 2. Camembert soft-ripened cheese box 2.Right cylinder with a semicircular base 3.Iodized salt carton ItRight circular cylinder

1. Timex watch box I.A convex solid formed by a closed rectangular prismatic surface with a frustum of a rec, tangular pyramid enclosing each end

5.Cottage cheese carton 5.Frustum of a right circular cent:

(3. Strawberry box (3. Frustum of a regular square pyramid

7. Talcum powder can 7.Right elliptical cylinder 8.Seth Thomas metronome box 8.Frustum of a regular square pyramid

9.Dobbs hatbox 9.Right octagonal (not regular) prism

JO. Lady Esther face powder box 10. Right regular octagonal prism

11.Easy window points box 11. Right tegolar triangular prism

12.Orthodontic retainer box 12.Frustum of a regular octagonal pyramid mounted on a right regular octagonal prism

13. Gold Label Shamrock cigar box 13.Right trapezoidal prism H. Hormel chicken cacciatore dinner box H. Right rhombic prism

Having a box collection available makes it is the total area of the outer surface of the possible to pose all kinds of application-type box? problems. The following are suggestive: 1.Flow long is the piece of string used to tic 1. A salt carton has the shape of a right up the box shown in Figure 8.46? Allow circular ;:ylinder (Figure 8A5). The 12 inches for the bow. diameter of the base is 3inches and the 11" 4"-« height is ii?! inches. To what height is the carton filled when it contains 2 cups of salt? What isthe length of the longest spoon that can be hidden in the carton? 9. A strawberry box that has a capacity of 1 quart has the shape of a frustum of a regular square pyramid. If the edges of the top and bottom bases are .1 and 3 inches, respectively, what is the slant height of the box? 3. A box of bake cups (Figure 8A5) has the shape of a right regular hexagonal prism. If the perimeter of each base is 14 inches and the height of the box is 5 inches, what FIGURE 8.46 b

(

Firm 'E 8.47. The piclure shMeS Iwo idenacal devices. The device al the lop is nailed to a 7(.01. The bouom device is welched horizon- Pull, and held againssl the wall.

Now let us look at the problem of the person la)man who asks how Ilied the area of an who has doubts about his landscaper's veracity ellipse usually doesn't know calculus. or about the capacity of his truck.It is easy to One way of finding a plausible formula lor give instructions telling how to compute the the arcaof an ellipse. without appealing to %ohmic of a truck box in cubic yalds; but doing calculus. is by using .1 rectangular piece of elastic this will probably not satisfya skeptic who is with sticks fastened at the ends. A deice of this «mlronted with apile of dirt that has been kind with a circle and it square drawn on the dumped On the ground. The problem is that he elasticis shown in the upper part of Figure 8.17. probably has no notion of the siie of ;tpile of The square is circumscribed about a circle of dirt that contains live cubic yards. As a matter nadirs b. The al ca of the circumscribing square of fact, very few people do. That's why the ques is.1 x b x b. and the areaofthe circleis lion comes up. -,Txbx b. Here is a suggestion that deserves being tried. Bystretchingtheelastichorizontallythe Obtain one cubic )ard of dirt and put it in a pile square istransformed into a rectangle(lower that has it shape like that of :t right circular cone. part of Figure 8.7).It follows that: Then build a tent that has the sin and shape At ea of square > Area of rectangle. of the pile. This tent can be set up inside the xbxb 4xaxb. school or on the playground to exhibit the si/e The picture also suggests that stretching th 2 ofit cubic yard of dirt. elastic transforms the circle into an ellipse and The person who asked how to find the area that: of :t flower bed that has the shape of :tn ellipse Area of circle 4 Area of ellipse. was interested ill the cost of fertilizing the soil. 77xbxb--)z-xax b. Finding- the area of an ellipse is not, of course, Therefole, we surmise that the arcaof the an offbeat problem if one knows calculus. But a ellipse shown in Figure 8:17 iszab. 264 ci 1.EREicon.

NI)%vlettis «o.idei some applbations mat be alittle mole Ina( ti( al than th(s, ,(lilted thus lam.l'i( lined in 1:ignic SAS isthe Itolanipe ineastiring wheel. an instillment measuing distam es. This instillment was used .11 1)tilles Intonational Airpoit when the tin%va%, twee 'hued. The (it( tunIcremc 01 the Itolatape

%Owe! is 1 leo: its diameter is about I:)in( lies. ()tie Of the %%theelis calibrated to !tin II.

and the other side to 2 inch. Attached to1 he instrumentisallautotnati( «mutel to le«)1(1 Ievoltnions. 1)istames ()I all kinds. imlding the lengths ()I cline ls :111(1 at( s of ( bele,. (an be meas tiled quiekl) and at( in:0cl% with this instil uictit. l'sing itis as (Nast as %alking. To measure angles. students (an the mull in. Nimmons as the seNtalit(Figure S.19)or field protractor. \\ilt the aid of the Rol:nape:Ind an appropriateangle.measuringinstItiment.sm. dents (an undertake.11(11 it IL's as finding the areas of parking lots and floorsof huge buildings. finding the heights oftall lmildings. and making maps. The speed. «01%enien«.. and Conttesy of Rolatape, Inc. act twat')of theItolatape in measming fail long disian(es Will add tiet ex(itentent to Ira(- Elcu Itr, 8.49 ticalproblems invol% lug the meastn emelt'of distances.

S U G GEST! 0 NS 1:01: l'SING NIODE FEE l'IVELY 1 low and when to use models are matte's that involve tastes and prelerem es.Itis. theteloie. inapproptiate to ptopose a haul.and.htst guide to be followed by all teachers. hei au.. how ever.smilesuggest butsthatseem generally applbable.

First of all. the paiticular needs ()Ia( lass should be %vaulted for (nes that indi(ate whet het or not models are to be used. \Viten juiciest lags. when student participation wanes. when students fail to make the right associations. itmay be time to use a model. The model may be a chalk box, a coin. a pie(e of string. it lamp shade, orit (ommercial clevbe that will add needed leality to the idea beilq; discussed. However. when Nill Comics) of roller losountents t'SINC MODELS AS INSTRUCTIONAL AIDS 265

dents I espond immediate') to ditect appeak to ple guide sheet is included latet in this chapter. abstiact ideas. when students are eager to spend .\ bash releren«r tool in the .teat()I insum lar,ge blocks of time ono hallenging problems. tional aids in mathematics is the pamphlet A when students want to explore new ideas inde- Guide to the Cm: and Plea nr, mat of rrachtng pendentl., thenitmay be timeI() withdraw Aid lot allaihmaio %. It% Beige!' and Johnson (6). Models (hal MC being used Or omit some whose This 1)111)14 aliOn lists NeVCII hays 10 follow in tiA: has been planned. .k teacher turbo is sensitive using teaching aids ellectivel.(6. p. 5). Th.se to the needs 01 Ins N111(len1:, Cites ca re- apply especially to the use of models. Itil attention to what works best lor him will I.Select the proper aid In We in ureter to soon lean when models should be part()I the alluierthe deieel ()hie(liar.Select cisely total applom It to .t lesson and It hen the) should how the tvealth of materials that is a%ail- be trilltheld. able. or deveiop precisely the type of item that will suit your purposes best. Secondly. adtante planning should precede 2.Prepare vourself to use the teaching aid the use ()Imodels. Plans should be made well that has been selected. Become familiar ahead of 1Ann:1110ns so that the best nuxleb, lull techniques for making effective use can be pun-tired and time will be available for of the item that you plan (0 use. If it is a film. preview it: if itis a dynamic de- the teacher to familial ize himself with the models vice or model. familiarize yourself with he intends to use. the manipulations 111)1)11 IIN as Thild. strategies lor using models should be a !canting aid depends. If prepaatoty developed. Nlerely holding up a model before a materials mu It as study guides or desrip- tive literattne are available. consult them. (lass or pointing to a model testing on a shelf is 3.Prepare the eh/strawy,. Check 10 insure not enough.If necessary. teaching techniques thatall needed materials are cm hand should be modified and the strategy for content and in "running older- before the l:tss development( hanged in order to incorporate meets. desirable uses 01 1110(1CIS in a lesson. 'Ills means. -I.Prepare the class. .kcquaint students with the purpose of the learning aid. what of course. that models ate to be incorporated may be expected of it. why it is being with other instructional aids such as bulletin used. what they can hope to learn from boaul displays. films, chalkboard diagrams. text- the experience. and how they should ap- books. and so on. \iten models are used fre- ply the information which they may gain. Often awrittenlearning guide quently, using them becomes spontaneous. and placed in the hands of students increases they become all indispensable part of the total the effectiveness of the learning experi- instructional piocedne. ence. Fourth. there must be a willingness to use 5.Use the leaching aid selected in the most effective manner possible. Relate the aid models without insisting on total justification to the lesson in i»ogress and to the sym- before attempting to use them. bolic representation thactrillbe used Fifth, an effort should be made to involve stu- later. G. Provide follow-up activities. l)iscussions, dents in discovery activities. Whenever possible readings. reports. projects. tests, and te- it nuxlel should be put in the hands of each stu- peat performances ate needed if maxi- dent so that an idea that is under discussion can mum learning is to esnit. Whenever pos. be explored by him. sible. provide opportunity for application ()I' the information learned. Sixth, guide sheets should be available for use 7.Evaluate the effectiveness of the aid. On by students who are investigating ideas inde- the basis of your experience with the pendently. Prot iding guide sheets helps a maxi - learning aid. it is wise to make a notation mum number students make discoveries and regarding its usefulness. A card file system should be employed so that over it formulate generalizations (hiring the time that is period of time you will know what the a ;1 ble for independent investigations. A sam- best aids for each lesson are. 266 (:I LAPTER EIGHT

THE EV TO EFFECTIVE l'SE The quo...slid u,ct ol models is :male that NIODELS mathemati(%is .111:11"11';1( 1%Obit%I I ILOmakes WIli.AL %%-ilkthe realtV1/11(1 thiough plosi(.11 Iit 1,:ey10die(LIVC 11NCOi1110(ICIN INunder- teptesentations that ale apploximations ol al) standing the connection between aodel and Strati ideas. Fur CX:11111/1e. ht the mathemat ks it depict,. ICNthe st ring model inFigitte 5.511 as an amp( ituation 01 One teat her uses. models um elle(titeit. Itypell)(11041 and the (lumber lint in rigine and with confidence. Anodic! :cachet in a NIIIII N.51 as an appioximation 01 a (art potuleme I:11"N1111:11i011.t-arcely lists models: and. when hearten a set ()I points and a so 01 numbers. I It he does. his manner is hesitant and Ittisno% 11(nv (an this dillerem r l e explained: One explain! OW..these:11/1/10Xilli:11101IS InilletC1O1) lilt" idea, 111/i11C11 IA 1lie mouths. lionisthatthe lust leather muleistands the hiN Ii) the a in whit Ii mathr- conitectiim between a model :milthe mathemath iansIC:lit lit V mathematit s. matics it depicts and the other doesn't. he% Inake ". esse apptoxintalions abst tad idea,. ca(-11 less von( rev. than the one before. That is. the% FicuitE 8.50 develop hieraohies()Iabstractions IAtaking successiv steps :twat !tout tealitt. Enlet's Betel. opulent of tupohin t. (leurihed cattier. is an 11111%. um ion ()I' this principle. I laving student, al laliqe the sectoEs or a circular dim in the form of a -pantllelogntin- to help them find a fonnula lot the au:a of a ( io It is an illtiqration of the sante principle. The principle involved is one ()I ttansi Lion limn tlw tutu vete to the abst rat t. In the caw of the mathematician this transition is usually spoilt:Mt:MK ittil it is nol always so101" N111(11:111,.. CXECI)1 perhaps rot lilt try bright. The) trot e. the teacher mat have to (10 some prompting- to get students to make the desited transition. The teacher who mulct:stands the (mute( lion between COMtoy of!Alm: Scientific Company models and the mathmaths tltey depict will tecogni/e when insights ;term. to individual stn.

FIGURE 8.51

2 3 4 ! I I > 4 5 6 7 8 2 2 2 2 2

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Coin Iesy of ldcal School Supply Company CSING NIODELS AS !NY I'RUCTIONAI. AIDS 267

ditts. Ile will know %viten to move students :ma% Ilw stud( 1s hat c mot cd int to .1 mote .11),11.111 hunt tottetete models and towatd the abstract ltel and au- -tatting to Compute sums 1)% 'hint;

letrl. Ile will also .cube W11e 11.1Ni Ittletti mathetnathal ...emetic CS. Tile te.11hr1 walk, 110111

Adtliii011.11 %V01 k with ((mom models in older to to &.I. .111(1 ott.INio11.1 .1>1o. .1NI 11(1C111 10 wadi the abstract fetel and will make needed Itt dietk his t omptuat ion. -1°Ite teat It. pm% ision on an Indic ideal basis. Belot. is .1 (le el is spotting weaknesses and allowing icgtession NI I iptinnof a lesson thatillustrateshat to the use ()I (otutete models 1)% students who teat her might do when helping .talent, make still need this assistance. This tedutique rites the transition hum the (Onttete to the abstract. each student the %cult its he needs to make the seeittli-grade ;:a(her of remedi:1 mathe desitccl nansition at tarions stage., 01 leatiting as math_ '.as been using paper (itle (moms (Fig lie times 11(1111 the (ott( icle 10 the abstia( t. tn 8.52) for :.everal dal:. to help students learn how to add tational numbers. Each audcnt ha. USING GUIDE SI I EFI'S IN 111:Ill ilk overt :WI Of tilt le c 111011b anti Ica, Merl CAI:I:11N.1 0 EXPEI:INIENTS 111011 Nilltple addition problems'. Now \VITII NioDEis To help the individual Nimient otgatti/e his ellous when usitignu)del for cxperinlcut :u ion. it open ploc, rtiiirtti to :apply him with:I sheet. Doing No Win usually imptove the student's cham es of making hoped-for discoveries and humiliating generali/ations daring the I:mum: 8.52 pet iocl of lime that is mailable for experimema- ti nt.Following is a destription ol a unit dim illustrates the techniutte. The purpose of this unit IN twofold:(1)to IC:1(1 students to diseover and state getterali/ar ------lionsfor the telationships that exist between t 1 s's, itteastn CS of angle, that illteINCC1 :1 ell 1:112':111(1 measures of the intercepted arcs. and (2) to have ------students !note the genetaliiations the}state. The guide sheet presented in this section is or- gani4ed into seven cases. F.ach cube invokes one general way in whit!' an angle (an intersect a (it cle. The %vol.!: ol the unit can be started with a motivational activity that is designed to set the stage for individual expetintentation. One way or starting is to have each student make drawings or all possible ways in which au angle call intersect a circle. Alter each studenthas had time «) make drawings of all intersections that occur to him. he should be given an opportunity to compare his set of drawings with a Nei displayed on a screen with an oerla.ad projector (Figure 8.53). Courtesy of Franklin Publications, Inc. 268 CHAPTER EIGHT

INTERSECTIONS OF ANGLES AND CIRCLES

1 2 3

8 9

10 #°"

FIGURE 8.53 USING NIODELS AS INSTRUCTIONAL AIDS 269

The ovedlead visual (an be used to oIarif. ghat or in the exterior of the tin le. The 'impose of is meant by "an arc intercepted by an angle.' the protractor is to measure angles formed in Students need to It.ne .1 clear mideistanding or the bars. III different positions the bars 'einem:1u %dna this explessioll means in order to experi- radii. holds. secants. or tangents. ment On their own. Formulating a definition o an The guide sheet that follows should be avail- sere as a N11111111;11 MA% it) for the intiodut- able to help the students in making apploptiate tor) moth:Ilion:II phase of the unit. (An arc)l manipulations Iith the c ircle de. it C. making and a (-hole is inten epted I). an angle if and mil) if re«noling ineasinements. and lonunlating gen- each side of the angle contains at least one end- erali/ations. After the student has formulated a point of the att. and the arc. except for its end- gem:rah/alio» for one of the cases specified iu points. is contained in the inlet iiii of the angle.) the guide sheet. the tea( her should ask hint To give each student an opportunity to ex- whether there are conditions under which his pt. iweut his own. he should be pro. ided .rich generaliiatitm does not apply. This question a click device like the one pictured in Figure calls for a proof. which may be the next step. $.5 I. This device has a circle marked off in Of course. there :ire variations to this procedure. degrees of art rot measuring inteicepted arcs. Some teachers prefer to hare students complete Thiee plastic liars and a small cirt Lila'. protractor all experiments outlined in the guide sheet first are attached to a pill that can be moved to and then attempt woofs of the generali/ations points in the interior of the circle. on the circle. they have formulated.

FIGURE S.54

Courtesy of Sargent-Wckh Scientific Company 270 ( :1 LAIIT EIGIn

(:E:I 1)E SI-I Err Pal po.ce: To find the iclationship between Ihr drgire meamo of au (1114( that illIciArttA a tircle and Ih degree measures of the arcs intercepted by the angle-.

(:ASE I. The angle has its vet tex at the center CASE 2. The angle has its %atex On the circle. orthe circle.

Directions: Directions: I.Set the device so the bars intersect at the !.Set the device so that the bats intersect at center of the circle. a point on the circle and each bar intersects 2. Choose one of the angles formed by a pair the circle ill another point. or ball and read its degree measure from 2.Choose one of the angles formed by a pair the small circular protractor. Record this of bats and lead its degree measure from measure ill the table below. the small circular protractor. Record thii 9.Read the degree measure of the arc inter- measine in the table below. cepted by the angle yon chose. Record this 9.Read the clew cc in.Nisine of the arc inter- measure in the table below. cepted by the angle you chose. Record this .1.Repeat steps 1-3 for additional settings. measure ill the table below. I.Repeat steps 1-3 for additional settings. Setting Angle Measuie Arc Measure 60° Setting Angle Measure Arc Nfeasnre li 90° A 150 120° 90° 1) 150° C 60° F. lb° 1) 90° 120°

interprerarion: I.I low cle..s the degree measure of each angle ttipare with the degree measure of its /flierpreiaiion: intercepted arc? I.!low does the degree measure of each angle 2.Write a sentence describing this relation- comp:Ile with the degree measure of its ship. intercepted arc? 5. What 50111c:esof error may account for 2.Write a sentence describing this relation- minor inconsistencies between your data ship. and the relationship you stated? 9.Write a formula for the relationship. t'sING Ntoplos INsTm.rioN Alm 2 711

CASI:. 3. 1.11C:111:41(7 is 101.111C(1 1.11 a tangent and a CASE I. The angle is foi tiled by two se' ants. The %e( ant. The vertex of the angic is on the circle. Vertex of the angle isin the extetior of die

third/WM: 1)h-crawls:

I.Sct the de he so that the bats intersect On 1. Set the devhe so that the bans inteisea in the citle. Use one bar to repo.sent a secant the exterior of the circle. Use two or the and a :Aloud bar to lepresent :ttangent. bats to 'epic:sem secants. 2. Time ate two anglesfor each setting. Record the degree liwaNtoc of the angle Ite(01(11 di . degree measure or cart angle knitted 1W the two secants inthe table and the degree ineasine of its intercepted below.

an in the 1:11/1e :1. Itc«)rd the degree measure of the two hi- Iter:11 steps I:111(1 2 for additional settings. tt:i:pied arcs in the table below. I.ltepeat stein 1-3 for additional settings. Setting .thgle Nleasatie c Nleasiti Angle Smaller .\rLager Ale Setting NU:a:sore Measure Measure 120 i; 150 . 30 50° (: 131 (: 19 1) 60" 1) 105 7.5 InIrprlation: harpretation: I.flow is the degre treasure of etch tingle I.! low is the degice IlleaStIe of each angle telatrd to the degr«. measure of its inter- related to the degree Ille:INIII e or its two ill- cepted arc? tettepted titN? 2.1 1'rite a Sell WIRT deribing nits relation- 2.kVriie a senten(e describing this relationship. ship. P \Vile a formula for the relationship. 3.U'rite a formula for the relationship. 272 (:11AvrEit Eicirr

CASE 5. The angle is formed by a tangent and a CASE 6. The angle is formed b two tangents. secant. The vertex of the angle is in the ex- The vertex of the angle is in the exterior of terior of the circle. the circle.

Dirccl ions: Directions: 1.Set the device so that the bars intersect in 1.Set the device so that the bars intersect in the exterior of the circle. Use one bar to the exterior of the circle. Use two of the represent a secant and a seond bar to bars to represent tangents. represent a tangent. 9.Record the degice measure of the angle 2. Record the degree measure of the angle formed by the two tangents in the table formed by the tangent and the secant in below. the table below. 3.Record the degree ineasines of the Iwo 3.Record the degree measures of the two in- intercepted arcs in the table below. One tercepted arcs in the table below. arc is a major air and the other is a minor Repeat steps 1-3 for additional settings. arc. 1. Repeat steps 1-3 for additional settings. Angie Smaller ArcLarger Arc Setting Measure Measure Measure Angle Measure of Measure of A 30° Setting MeasureMajor Arc Minor Are B 50° A 60° 75° Ii 30° D 15° 1) 100'

F Interpretation: 1.Flow is the degree measure of each angle Interpretation: related to the degree measures of its inter- I.I low is the degree measure of each angle cepted arcs? !elated to the degree measures of its inter- 2. Write a sentence describing this relation- cepted arcs? ship. 2. Write a sentence describing this relation- 3. Write a formula for the relationship. ship. 3.ll'rite a formula for the relationship. USING NI)DELS AS INSTRUCXIONAI. AIDS 273

ANOTIIEI: GUIDE siiErr Since thc bars of the device pictured in Figure 8.5I ate marked oll in centimeters. the deice can CASE 7. The angle ha, its %ertex at a point in be used to help students discover. state. and prove the interior of the circle but not at the tenter. generaliiations for relationships «ificerning nos of%ariotts segments oftwo inteisecting chords of a circle. of two secants (ham] from the same external point. and of a secant 1111/1 a tan- gom drawn fromt he Nallle external point. Ac- cordingly another guide sheet can be otgani/ed around the following three caws:

CASE I. Two chord.,.at intersect in the interior of the circle

Directions: I.Set the crevice so that the bars intersect in the interior of the circle and each long bar intersects the (bele twice. 9.ChOOSC one of the angles formed by the two long bars and read Its degree measure 1111111 the small CiretlIn protann.. Record CASE 2. A tangent and a secant drawn to the this me:isn't in the table below. circle from :t point in its exterior Read the degree meastne of the ate into- /ewe/I by the angle you chose and the de- glee meastne of the arc intercepted by the opposite Rec/w! -hese two measures in the table below. I.Repeat steps 1-3 lor additional settings.

Angle Measure ofMeasme of Setting Nfeasure One Arc Second Arc

600 1; 2!F CASE 3. Two secants drawn to the circle from 108') a point in its exterior I)

init/pry/a/ion: I.How is the degree measure of each angle related to the degree measures of the two intercepted arcs? 2.Write a sentence describing this 'elation- ship. 3,Write a formula for the relationship. 274 (.1I.1P l ER Cd I

STUDENT-MADE NIODELS 'FOOLS

As has aheady been indicated severaltimes, 11111,11 pens Methanit al thawing kit making- a model, either as part of a labor atory Carpenter's rule Metal shears Carpenters square N)Ion point pens lesson or as an independent project. can be a Centering squat e Paint blushes meaningful learning experience for the student. Compasses Paper cutter To assist and encourage students to make Coping saw Paper putt( It models. space to make them should be available. Drawing board Plane In addition. an assortment of inexpensive con- Eyelet punch Niels Gluing clamps Plotrat tor struction materials and simple tools should be I I annum- IZaror blades and holder provided. The list below represents a minimum I land drill and Scissots of construction materials and tools that should assorted bits Sc t ewdriver be kept on hand. and saw Soldering it on Knife Vise MATERIALS 1o(1(1 rasp Adhesive tape Pins common AN an example of a student-made model con- Adhesive Iva.. Plaster of Paris sider a linkage in the form of a parallelogram Aluminum foil Plastic chips Balloons Plastic foam that is made with folly strips of stiff cardboard Balsa P last ic rings joined by brass paper lasteners (Figure 8.55). Beads Plastic sheets The eyelets make each vertex a movable joint; Bolts and %ringnuts clear and colored therefore. the model can be used to represent assorted sins Plastic tape many (lament. parallelograms. With the aid of Caldboard assorted colors this model and a protractor, the student can Cellophane tape Plastic tubes Co lot ed chalk Plywood observe that the sum of the meastues of the Colored corrugated Pocket mirror angles of a parallelogramis 360 degrees, that cardboard Pushpins and map pins opposite angles have the same measure, and that Construction paper Rubber binders consecutive anglesare supplementary.Using Cork panels Rubber cement elastic thread to represent, diagonals enables the Sandpaper (:rayons student to see that either diagonal separates the Draperycord Soda straws Dralving paper Spray paint puallelogram into two congruent triangles, and Elastic thread Stapler and staples that the diagonals of the parallelogram bisect peltassorted colors String each other. Fiberboard Stytofoant balls and Students can test their observations concerning Fishline weights blocks a parallelogram by using- a linkage that has the Glue Tackboard Golf tees Thumbtacks shape of a quadrilateral that is not a parallelo- Graph impel 'rilesasphalt. ceramic. gram (Figure 8.56). The sum of the meastnes of Letter stencils plastic the angles in the figtne represented by the new Let let s for mounting Title board model is again 360 degrees, but opposite angles Nfarbles Tongue depressors do not have the sante teasure, diagonals do not Masking tape "roothpicks .Modeling clay Wax paper bisect each other. and neither diagonal separates Nails. brads, tacks Wire thefigureinto two congt centtriangles. By Paint mesh stitching elastic thread through the midpoints of Paper fasteners kVooden dowels consecutive sides of the quadrilateral model, stu- Pegboard and pegs 11rooden Venetian.blind dents can see that the elastic thread represents a Pickup sticks slats parallelogram regardless of how the linkageis l'ill boxes Yarn uansfortned. If students are asked to compare the lengths of the sides of this parallelogram with USING' NIODEIS AS iNsTRucrioNm. Alps 275

The model also has various other mathematic alb significant leatmes. Iluwoer, the point we wish to make is this: The design ploblems en«mn tered by a student who builds a model like this compel hint to think about mathematical details that may not have occurred to him before. The student-made model pictured ittFigure 8.58is wade 01sheeta lum intim. Except for exlreme cases. Tiler" nimgies of eer) shol'e FIGURE 8.55 can be repesented withit. The student who constructed this model had only the usnal two dimensional chawings II 0111 a geometry textbook to guide him. This model represents a remarkably creative effort. Building it required a great deal of knowledge not only of mathematics but also of metal working.

FIGURE 8.5(i the diagonals of the quadrilateral, they ha% e oppot mit). to discover an important generalira- tion that can be pioved as a theme'''. The linkage that has been described is one that every student of geometry should be able to make. Construction or mole elaborate models FIGURE. 8.57 should probably be undertaken as special IMO- jects. Such projects can be very rewarding to the student. In designing and constructing a model. the student has an opportunity to add depth to his understanding of ideas with which he is al- ieady laminar or 10 discover ideas that are new to him. Figure 8.57 pictures a model ofthe most general kind of parallelepiped. The model in the picture is made of Walnut Veneer. Each face has the shape of a parallelogram. The student who made this model wanted to be able to show that a plane can be "passed- through a pair of diagonally opposite edges. The model is built so that the representation of the plane through a pair of diagonally opposite edges is removable. FIGURE 8.58 Courtesy of LaPine Scientific Compass FIGURE 8.59

The opening sentence of this section(". .. Displayed in Figure 8.59 are some of the con- making a model, either as part of a laboratoi y le, tents of the Sage Kit with illustrations showing son or as an independent project, can be a mean- ways in wInk.h the student can use the kit to ingful learning experience for the student.") build models. Auompanying the kitis an in- is no empty claim. Building a model is a mean- struction booklet that soles a purpose similar to ingful learning experience for the student. One the guide sheet suggested earlier. reasonisthat students are basicallycreative. Displayed in Figure 8.60 are the contents of I: \ell when a student lacks the needed skills, he a distance-measui ing instruments kit. Three de- still likes to make things. Fortunately, this char- \ I( Cs can be built with this kita lange finder, a acteristic of students has been lecogni/ed by pi o- parallax%jewel ind an opticalmicnouneten. duce' s of luso uctional aids for mathematicsso Figule 8.60 also shows how the iangeI hide' there is on the market today a variety of kits can be used to find distances. Budding the three containing partially assembled models. Assem- devices will vise the student the thrill of cleat- bling models with these kits gi es the student the ing soinething; usingit(especially the range feelingof accomplishmentthatcomes from finder)will make him wonder why the device cleating something. In the process, he can leain works; and prol big why it works will make him important mathematical ideas. think about exciting mathematical ideas. PSI NG NIDDEIS ASI NSTRUCIIONA L AIDS 277

Movable Mirror Object Wooden Base Fixed Mirror

Mirror Holder Pivoting Arm

Masking Tape 116k (Used As Scale)

Nut Paper Clip (At Each End) (Used As Pointer)

Comioy of ,11maleqer Saenlific Company

Disl""c""i"""s"ri"g use, "" "hie° is viewed ("1"1 Ihe liA"(1 mirrot as shown in the right-had diagram. The movable mirror is then rotated wilh the frivoling arm unlil an image of Ihe obfeelran be seen hi the fixed mirror. The image and oiled are aligned. Tlu' in,situmeniis odibmied by marking On a sirip of masking lupe on ihe base the posilions of the paper- poinier for objecis of known dishi('es.

conclude this section I))mentioning, One NirtleliOn Material is cardboard. Aftel cutting out other way in whir h teachers can encourage stur the patterns. the cardboard should be folded dents to build models. This is leany a variation alonp, the dashed lines. Edges ma) be joined with of the kit idea. Supplying students with a scale tape. When the three pieces of the completed drawirg of a pattern for a model, or with the model are placed together so that vertices with pattern itself. will often be sufficient to get. them the same letter are to,gethei, the resulting model started. should look like the one in Figure 8.1 1. 'Ihe model that can be made with the patterns displayedin Figure 5.61is a three-plece 3. This et of e' igiams is an adaptation of those shown in Mathematics laElementary School Teachers, by I Men sectible triangular right prism.3 A suitable con. 1.. Gaistens and Stanley It. Jackson (20, pp. 67-71).

4 (.11p EIGHT

ut.11.1)1m; I.For what purpose (an this model be used? (1) Can it be used to inotkate learning? .\ model colle( lion is not born. It simply glows. II)Cali it be used to awaken student ill. How does a model collection start?It usually let est? starts spontaneouslywhen the lea( her disco% ers ()(:an it be used to cm ich leanting? oinc bjc( t that is useful as a model and de( ides d)(:an it be used to help students clis«)% er to keep a «nicept? r)(:a11 it be used to help clarify an idea? One wa' of making a model collection glow is f) Can it be used to speed tip corn hull a itoutishiiit it with useful realia such as exciting- lion? looking boxes. pieces of pip., lanipshakles. and so g) Can h he used to teach an algorithm? It) Can it be used to help students soh e On. Adding student-made models is another way a pioblm? of making a model collection grow. Students are 1) Can it be used for practicing a skill? usitall% pleased 10 be asked to contribute their 1) Can it be used to help students graspa inoclek,. A %void 01 caution is in order here. how. spatial relationship? ever. Having students make models simply to k)(:811it be used to help students make enlatge a school's «dlection is not re«nninencled Innisitions front the concrete to the al). stract? as end in itsell. "lime are several reasons for 1) Can itbe used for ale( reational ac this. First of all. the purpose of having students tkity? make models is not always served by having them »t) Can it be used to illustrate application concentrate On the sort of craltsmauship looked of a mathematical idea? lor in a model thatis to be putill a school's 2.In what kinds()I'instructional arrange- collection. Second!), students cannot he expected ments is this model to be used? to prcxhice the %.aliety of models that a school a) As a demonstration aid with large or meditimsiied class groups? needs.Finally. students (being students)are II)For laboratory work? ordinarily not able 10 produce many of the e)For independent investigation by NI In' models that they themselves needill()icier to dCilth? learn ma t Itenia tics. (1) For displays or exhibits? 3.With what subject; in mathematics can Collecting realia and acnimulating !..nclent- this model be used? made models are helpful schemes for building a I.Is the mathematical idea that this model model collection. However, neither scheme. nor depicts significant? even both together, are adequate means. Thet e- 5.Is the construction of this model faithful to fore. the purchase of commercially pioduced the mathematical ideaitis supposed to models should be anticipated as the most reliable depict? way of' building a model collection lor a school. Is this model laige enough so that details pertinent to the idea depicted are clearly Itis best to begin by acquiring a few basic isible? models that are versatile and slimly and which 7.Is °Iiis model well designed and well con- serve purposes that cannot be met by using pic- structed? tures from a book. chalkboard diagrams, obje( is S.Is this model sturdy enough to be handled ill the classroom, or student-made models. ()nee by students? mai led. the collection can be enlarged year by 9. Can this model be used to do something year. that cannot be done as well or better ill some other way or with some other in- 13efol e purchasing a model the teacher should structional aid? ask hinisethe nine major questions listed be- Itisunlikely that any model you purchase low. For penetrating appraisal. the subques- will have all the qualities you wish it to have. lions might be considered. Hence your selection must be tempered by your USING NIODELS AS INSTRUCTIONAL AIDS 279

judgment of the total contribution it is like'; to NIODELS ARE AVAILABLE? make to your instruction. NVe have left mail last the matters of cost and To use models effectively. a teacher needs to be sun age. Cost should be commensurate with the aware of what models are available. To keep in- use expected of a model. if a model is versatile fo' med about the availability of new models it is and likely to be used a great deal. then a si/able helpful to lead professional join mils in which expenditure of money would be justifiable. flow. articles about models and advertisements for ever. if a model is going to be used only once. commercial models frequently appear. Visiting extravagance should be ',avoided. Nfodels that exhibits of manufactiners and distributors of depict the same idea are available today at (hi- models at conferences and Inaintaining an up. lt:tem levels of cost and quality. The expected todate catalog file ate also helpful. use of a model should be a (lett:mining factor To assist teachers in selecting and purchasing in making a proper choice. models. a compilation of the usual catalog names In building a model collection for a school. it of models onsidered useful as instructional aids should be kept in mind that models need to be stole(' when they are not being used. In a large school. models can be stored in a mathematics : resource center.lit a small school, itis likely _ . . that the enthe model collection of the school will have to be stored in a single room. The roller-mounted combination storage cabinet and demonstration table pictured in Figure 8.62 is one example of a piece of equipment that Gill be used for this purpose.

.. `,."

C72

FIGURE 8.62

Colote of !.aline Stientiftc Company 280 CI 1.PTElt EIGHT

in mathematics is presented at the end of this \VIIAT IS TI 11.1 1:1-11:RE OF NIODELS? ( hapter. COMI)11:1IIVII is part of a laiger list pit-paled b% a (lass ill mathematics education .11 Elie ['umbel of ((minim ial modtls .1%ailable lot the Unkersity Of Minnesota. Nlembeis or the [cachets 01 inailieniati(N has ill( leased th .1111.11i- class eNainined thanLunn di.n.ibtoors. ( alb illI etent vear....111(1 indications ale that an Goalog, in preparing their list.' ever-incre:.sing supply will be available in the The ;:ompilation that appears at the end 01 future. this chapletis (ngani/ed into [went' caiegolies The necessity rot imne students 10 lea'u more .14«niling- to the mathematical topits and «Al- inathematits 'note efficiently JI:IS been the ono,, (epts shown in the box. .\ model considered ap- or numerous expel internal plogiams. One of the piopriate lor use %yid' more than one topic is chief conceit's of these programs is indhiduali/a- listed under each topic for which it is suitable. lion of instruction. To this end many of the In addition. the «mtpilationlassifies each model plOglalle (liNt()% CI.11I1 labol.11o1).1p- acwrding 10 the eight types .mtlincd at the be- proarlies. Nlodels pla an important role in such ginning of the ( bapier (demonstration devices. approaches. manipillathe (Ie%ic es. and so forth) and lists one Progiains for students in 11001S. for or mole dist! ilmiors for each model. mlents %%ill'special learning- disabilities. and for culturally deprived students stiess the importance of providing 11011vellml means of com- NITIIENIATICAL TOPICS munication. Stich communication is an impor- AND CONCEPTS tant function of models. I. SON RecentlymathematicstextbookpliblisherS Numeration S%siems. Number Sstems have begin' ploducing materials packages to :lc- 3.Plane and Solid (;eomeiry companv the textbooks they publish.!t seems I.(:(undinale Gollieny almost certain that this practice Will lead :o ex- 5.Nleastii emelt' panded use of models1)y. teachers or mathe- 6.Probability and Statistics matics. 7.Formulas Finally. continued governme:u support foriit- 8.Logic noyatiou in ethic:Ilion will undoubtedly provide 9.R.:11 1o. Proportion. Pei (Tillage continued impetus lor die development of more II).()iterations and Their Properties new models and for wider use of models. I I.()pen Sentences 12.Graphs 1:I. Relations and Functions I-1. Logarithms and Exponents StTNIMARY 15.Trigonometry Ili.Linear .\Igebra Iiithis chapter we have described the role of I 7. Topology models in teaching and leauting mathematics. IS.Transformations made various suggestions for using models effec- 19.Calculus tively. and attempted to give sonic guidance in 20. Computer Concept; the selection and procurement of models. If you ale not now using models in your tea ching. we trust that you will ill the future. And if you do. I. (:;11:1101:: used tte:t: obtained flout the Saint Paul Pub. lic School's Mathentatici Itesoinet Center. %%inch keeps :111 we are confident that you, like thousands of up toll:tie lily of t:tlalogc of (Usti itnitors. Helen Hughes other teachers of mathematics, will rarely want of 10,15011. te

NIODF.I.S FOR TWENTY 1:VEI I NITICA TOPICS AND CONCEPTS

Nlodels are (1.1ssilted in two ways in this compilation: (1) mulct the topics andconcepts Ior which the ate «msidered useltd as instruLtional aids and k2) .11tonlingto t)pe as identified I)) the Lode system Listed %vitlt cad! model is the name ()I' one or mote distributors. The fullnames and addiesses 01 distlibutors are listed in the Appendix.

1)Deist ottstratiott Devices KKits NINlanipulative Devices C; Gaines and Pttilles CComputational Devices SScience App :uatus I Instruments for Nleasttring and 1)rawing RItealia

I. SEFS

At ra y Board ience Reseach Associates Electronic Logic and Sets Demonstrator land I loll -on Set Nlaker Webi:r C:ostello Nlinne-Bars Judy Nlodern Math Demonstration Boaul (;ainco NIodern Math Symbols (;:tnco, Geyer, NASCO, Science Research Associates New Nlath Charts lush NASC:O. Nystrom, \Veber Costello On Sets (game of set theory) 'N PROOF Series ofmillChants Walker Educational Book

2. NUNIERATION SYSTEMS AND NUMBER SYSTEMS Abacus (IS. 15, and 23 reeds) D M C 'Tuttle Base Blocks Ideal Base Conversion Chart D Base-Five Place Value Chart D C Ideal. NASCO Ba.,e-Ten Counting Frame Science Research Associates, Weber Costello Binary Cotinter I)M C Sargent;Welch Binary System Fact Cards and Slide Rules Bomar Chinese Abacus M C Sargent;Welch Circles lot. Calculating Mactions D M C Franklin, Ft eyer Direct Reading Ball Abacus I) M C 'IItl-TIX Expanded Notation C;auls D NASCO Fraction Chart D M Exton, Ideal Fraction Finder AIN1 Fraction Inlay Boards (squates. circles) D M Judy Fraction Kit (teacher and pupil) I) M Ideal Fractional Parts(squares, circles) D M Bowmar, Ideal, Instruct°, Weber Costello Giant Dominoes M C NASCO 282 C.:1 IA VIVA EIG1 l'1'

2. NUMERATION SYSTEMS ANDNUMBER SYSTEMS (Continued) Hunch al Chart/Boat d D NI Ideal. Instruct°, JudyNASCO Integer Slide Rule M C Bowmar Kinesthetic Numeral Cards M Instruct° Magnetic Numb-R-Ods D vI C Yoder Magnetic Numerals D M Instruct° Nlodern Computing Abacus .' NI C Ideal Multiple Base Abacus D M C Weber Costello New Math Charts D C Instruct°. Instructor. Nystrom. Weber Costello Number Line Rule M Sigma Number Lines n Bowntar, Houghton Mifflin, Ideal, Instruct°, Instructor, Judy. NASCO, Weber Costello Number and Picture Dominoes M Weber Costello Number Relation Blocks M NASCO, Weber Costello Open End Abacus Kit D M CK Houghton Mifflin Place Value Board D M Ideal. Instuct°, NASCO Place Value Charts (student and teacher) D M C Ideal, Instuct°, NASCO. Weber Co,tello Place Value Indicator D M Bradley Place Value Tab Rack D M Judy Projectablc Abacus D C Tweedy Rational Number Line D Ideal (ptinted chart and pegboard) Real Number Game D WIT 'N PROOF Square-Root Chart D Geyer

3. PLANE AND SOLID GEOMETRY Angle Prism I S Cella) Applied Geometry Experiments Kit M K LaPine Archimedes' Theorem Model (cylinder, D Viking hemisphere, and cone for filling) Beam compasses M I Charvoz, Lictz, Math Master, Post Burns Boards D M Ideal, NASCO Cavalieri's Theorem (cylinder and oblique D LaPine prism set. cone-cylinder-hemisphere set, pyramids set) Chalkboard Beam Compasses D I Sargent-Welch Chalkboard Drawing Instruments D M I Bowmar, Geyer, Ideal, Lano, LaPine, Lietz, (T-square, compasses, ruler, protractor) Math Master, Math-U-Matic, NASCO, Post, Sargent-Welch, Weber Costello, Yoder Chalkboard Drawing Instruments D I LaPine. Post, Yoder and Templates Chalkboard Graph Charts D Geyer, LaI'ine, Math Master, (rectangular and polar) Math-U-Matic, Sargent-Welch, Weber Costello USING NIODI-S AS INSTRUGMONAL AIDS 283

Calkboatd Graph Panels Claidge. LaPine. Nlath Nlate. \Vebei Costello Chalkboad (flap!) Stencils I) Lano. LaPine. Nlath Nlaster. (tectangulai and polar) Cleat T anspattit Afarkable Globes I) 1.itcjithat Coaxial Clinclets limsected by an Sat gent-l1. el(lt Phi (:one with Circular Section D Salgent NVeldi (:ones and Conic Sections I) NI Lano. Saigent-Welch Cone. and Clinclets I) NI LaPine. Math Alasir Cone vit Pataboli( Section and I) LaPine. Viking Dandelin's Sphere (:one of Two Nappes with IIperbolic U LaPine. Sargent-NVelch. Viking Section Cone 01 Two NappeN with Hyperbolic I) LaPine. Viking Section and Two Spheres of Contact Configurations CI \hull Nlodek I) NI Salgent-NVelc%. Science Related Materials. Viking Cube and Pyiamid I) Sagent-IVelch '1)attdelin's Cone D LaPinc. Sargem-\Velch. Viking 1)emonstiation Conic Section Curves D Lano Scaled for Ilse with a Pegboard Dissectible Binomial Cube D Lano. LaPine. SagentAelch, Science Related Materials Dissectible Prism Set I) NI LaPine Dissectible \Vood Cone D SargentW elch EightOtants Model Viking Ellipsoid D Viking Fillable Hollow Plastic Geometrical Bodies D Viking (cube. sphere. Cones, cylinders. prisms. pyramids) Flexible and Adjustable Polygons I) M I Geer. LaPine, SargentAelh. VisX Flexible Plastic Templates Showing AreasI) LaPine of Circular Cylinder, Gone, Frustum of Cone FoldOut Uncial Area Set D NI LaPine. Math Master Fundamental Locus Demonstration Kit I) NI K Viking Geoboard (square and circle) NI C:itisenaire. Sigma Geodestix Kit IC Geodestix Geometric Models Construction Kit. K Bradley Geometric Solids and Surfaces D M Yoder (39-piece set. wood) Geometric Solids (large plastic) D M Ideal Geometric Transformation Animation D LaPine Boards C;eoetry Charts D Ideal. Instructor, Judy. Viking Geometry Charts (eleentary) D Ideal, \Valker Educational Book 284 CHAPTER EIGHT

3. PLANE AND SOLID GEOMETRY (Continued) Geometry of the Circle Devices (measure-D M I La Pine, Sargent-Welch ment of angles formed by chords, tan- gems, secants; ratio of segments-of chords, secants, etc.) Hemisphere, Cylinder and Cone D LaPine Hemisphere with Spherical Pyramid D Sargen t-AVelch Hemisphere with Spherical Sector D Sargent-Welch Hinged Layout Model of Cube and D LaPine Pyramid Hyperboloid D Viking Instruments for Determining -a. D M I Geyer, LaPine, Sargent-Welch, Bowmar Inverse Square Law Demonstrator D Sargent-Welch jigged Geometric Shapes D M Weber Costello Locus Construction Devices (hyperbola, D M Geyer. Lano, LaPine, Math Master, Yoder parabola, ellipse) Locus Kit D STAS Mathematics Design Maker D I NASCO Miniature Geometric Solids (27-piece M Edmund set, solid plastic) Moby Lynx Construction Kit K Geodestix, Kendrey Models of Polyhedra D Cenco, LaPine, Math Master, Sargent- Welch, Science Related Materials, Viking, Vis-X Multi-Model Geometric Construction Kit D K Yoder Multi-Model Geometric Construction Set D M Yoder Multi-Purpose Pegboard D Bradley, Ideal, Limo, Yoder Oblique Pyramids D M LaPine, Math Master Pantograph D M I Charvoz, Geyer, Lano, Lietz, Math Master, Sargent-Welch, Yoder Pal a boloid D Viking Parallel Rulers D M I Charvoz, Geyer, Math Master, Sargent- Welch, Yoder Perimeter and Area Demonstration DeviceD M Ideal, NASCO Perspectograph M I Yoder Piometer I Geyer Plane Figures Set D Bowmar, Lano, Math Master, VisX Plane and Solid Geometry Charts D Viking Poleidoblocs M Math Media Polyhedra Construction Kit D M K Geodestix, Hartley, Lano, Math Master, Science Related Materials Polyhedron Construction Kit D M K Limo, Math Master Prism and Pyramid Sets D Lano, LaPine Pythagorean Theorem Demonstration D M Bowmar, Cenco, Lano, LaPine, Device Math Master, Sargent-Welch, STAS, Viking Regular Polygons Set D Math Master Rotation of Plane Figures Devices D LaPine Sectioned Circular Cylinder D Sargent-Welch USING NIODELS AS INSTRUCTIONAL AIDS 285

Sectioned Prisms (four- and six-sided) D Sargent-Welch Segmented Sphere D M Sargent-Welch Single Octant Model D Viking Slated Globe D M Geyer, LaPine, Math Master, NASCO, Nystrom, Sargent-Welch, Yoder Solid Geometrical Figures (wood) D M Ideal, LaPine, Sargent-Welch, Viking Solids-In-Solids D LaPine Solid Shapes Lab K NASCO Soma Cubes G Parker Space Geometry Lab Kit K STAS Space Spider Kit D K Cooper, Math Master, NASCO Speed-Up Geometry Blackboard Stencils D I Speed-Up, Yoder Sphere, Cone, and Cylinder Set (metal) D Sargent-Welch Sphere with Great Circle Sections Showing D Lano, LaPine, Saigent-Welch, Viking L'Huilier's Triangle Spheres with Removable Lunes, Segments,D M LaPine Spherical Cones Spherical Triangle D M LaPine, Saigent-Welch Spirograph 1 K Kenner Student Thawing Instruments M I Charvoz, C-Thru, Geyer, Lietz, Math Master, NASCO, Post, Sargent-Welch, Sterling, Yoder Student Geometry Drawing Template M I Academic Industries, Speed-Up, Yoder Three-Dimension Graphing Device D Science Related Materials, Viking Transparent Cones and Conic Sections D Lano, LaPine Transparent Models of Geometric SolidsD M LaPine with Sections Trisectible Triangular Prism D Sargent-Welch Twixt G Minnesota Mining R: Mfg. Co. (3M) Variable Cylinder-Cone String Device D M La Pine Volume Demonstration Set D M I Lano. NASCO, Viking

4. COORDINATE GEOMETRY Chalkboard Drawing Aids D M I Lano, La Pine, Lietz, Math Mast.:, NASCO, Post, Sargent-Welch, Yoder Chalkboard Graph Chart Stencils D Cram, Daintee, Math Master, (rectangular and polar coordinates) Sargen t-Welch Chalkboard Graph Charts D Claridge, Cram, Denoyer-Geppert, Geyer, (rectangular and polar coordinates) Lano, LaPine, Math Master, Math-U-Matic, Nystrom, Sargent-Welch, Weber Costello, Yoder Chalkboai d Liner D I Math Master, Viking Dissectible Right Circular Cone (sectionsD M Lano, Saigent- \\'elch, Viking circle, ellipse, parabola, and hyperbola) Eight-Octants Graphing Model D M Viking Ellipsograph D I LaPine Geoboard (square and circle) D M Cu isena ire Geometric Transformation Boards D Gilmer Herrmann, LaPine Graph Stamps (rubber) IM Geyer, LaPine, Math Master 286 CIIAPTER EIGHT

1. COORDINATE GEOMETRY (Continued) Hyperbolic Paraboloid (string model) D M Math Master ypocycloid and Cycloid Models D Geyer, LaPine Logarithmic Graph Chart D Denoyer-Geppert Map Projection Models D Farquhar, Math Nlaster Moto Math Demonstration Device D I Yoder Movable Vector Adder D Lano Multi-Purpose Graph Board D Lano. Yoder Pantograph D M I Charvoz, Lano, Math 'Master. Sargent-Welch, Yoder Parallel Rulers D M I Charvoz, Math Master, Sargent-Welch, Weems and Plath, Yoder Pegboard Grid D M Math Master, Weber Costello, Yoder Radian Protractor M Math Master Slated Globe D M Cram, Denoyer-Geppert, Math Master, NASCO. Sargent-Welch, Yoder Solids of Revolution (ellipsoids, D M LaPine, Viking hyperboloids, paraboloids) Templates for Drawing Conic Sections D I Charvoz, Lano Templates for Drawing Ellipses Charvoz, Post, Yoder Three-Dimension Graphing Device D M Science Related Materials, Viking Variable Cylinder-Cone String Device D M LaPine

5. MEASUREMENT (DISTANCE, AREA, VOLUME,ANGLE MEASURE, ETC.) Arc Measurement Device D M Sargent-Welch, Vis-X Area Kit K SchoJI Service Assorted Equal VolumeSolids (1cubic D M Gamco, Houghton Mifflin inch each) Blocks and Trays M Bradley Board Foot D Vis-X Bubble Sextant Geyer, LaPine, Sargent-Welch, Weems and Plath, Yoder Calipers (vernier, micrometer, regular) D M LaPine, Sargent-Welch, Yoder Centimeter Grid Paper C Houghton Mifflin Centimeter Number Line Houghton Mifflin Chain Tape I Post, Yoder Chalkboard Beam Compasses Sargent-Welch, Speed-Up, Yoder Chalkboard Drawing Instruments D M I Bowmar, Ideal, Lano, LaPine, NASCO, Post, Sargent-Welch, Yoder Circle (30-inch circumference) D Vis-X Cone, Sphere, Cylinder Set (metal) D Sargent-Welch Cube and Pyramid Volume Set D NASCO, Sargent-Welch Cubic Foot D NASCO, Sargent-Welch, Science Related Materials, Vis-X Cylindrical Cup 1 Sargent -Welch Demonstration Micrometer Caliper D Sargent-Welch Demonstration Vernier Caliper D Saigent-Welch Dissectible Cubic Foot D M Ideal USING NIODELS AS INSTRUCTIONAL AIDS

Dissectible Liter Block 1) S Cenco, Lano, LaPine. NASCO, Sargent -Welch Dry Measure Set (pint, quart, peck, bushel) D Ideal, NASCO Finable Hemisphere, Cylinder and D M Cenco, LaPine Cone Set Geometry Demonstration Set for M K C Viking Calculating Area and Volume Gravity Protractor D I Geyer, Lictz, Sargent-Welch, Yoder Inscribed Angle Measurement Device 1) NI Cenco, La Pine, Sargent-Welch Isometric Protractor Gra phiCra ft Leveling Rod with Target 1 Boger, Cenco, Geyer, Post, Sargent-Welch, Yoder Liquid Nleasure Set (gallon, quart, pint, D Ideal gill) Map Measures I Lieu. Measuring Disk (for determining 7) M 1 Sargent -Welch Measuring Tape I Post, Sargent-Welch, Yoder Meter-Yard Comparison Chart D Bradley Metric System Chart D Sargent -Welch Model for Demonstrating A = Trab Viking (area of ellipse) Number Line Ruler M Sigma One-Inch Cubes and Half-Inch Cubes D M Vis-X One Square Yard D Ideal, NASCO Pattern Dial for Drawing Regular M I Sargent -Welch Polygons Perimeter and Area Board D M Ideal, NASCO Piometer I Geyer Planimeter I Charvoz, Sargent-Welch, Yoder Pocket 'Transit M I S Post, Sargent-Welch Projection Vernier D M Sargent-Welch Proportional Dividers M I Post, Sargent-Welch, Yoder Protractor (radian measure) M Math Master Radian and Circle Demonstrator D M Sargent-NVelch Range Finder, Parallax Viewer and I1 1 Sargent-Welch Optical Micrometer Rectangular Geoboard D M .Freyer Rolatape I Post, Rolatape Semicircular Blackboard Protractor D I Geyer, Post, Speed-Up Set of Boxes with Different Volumes D Ideal, NASCO, Science Related Materials Sextant (student) I Cenco, Sargent-Welch, Yoder Spherometer I Sargent-Welch Spring Scale D I NASCO, Sargent-Welch, STAS Thermometer (centigrade or I S NASCO, Sargent-Welch, STAS Fahrenheit) Transit Level with Tripod I Berger, Cenco, Geyer, LaPine, Post, Sargent-Welch, Yoder Trip Balance Scale S Sargent - Welch, STAS Trisectible Triangular Prism D Cenco, LaPine, Sargent-Welch 288 CI IAPTER EIGI IT

6. PROBABILITY AND STATISTICS Coin Tossing Machine D M La Pine. Math Master Dice M Lano Dissectible Binomial Cube D M Cenco. Lano. I.:Wine, Math Nlaster. NASCO. Sargent-Welch, Svicticv Related Matet ia Is Percentage Protractor M C Math Nlaster Probability Demonstrator D M Cenco. Geer. Harcourt Brace jovanovich. Lano, LaPine, Nlath1aster. NASCO Probability Experiment Kit K Lano Probability Kit K Edmund Probability Slide Rule D M C Rolatape Probability and Statistics Lab D M K LaPine, STAS Probability and Statistics Lab Unit D M Math Master Random Motion Graph Paper D M Science Related Materials Roulette Wheel (regulation) D M Lano Student Slide Rules D M C Compass. C-Thru, Geyer, LaPine, Math Master, Post, Sargent-Welch, Sterling, Yoder Topologic Nomograph for Arithmetic C Schur Mean or Average

7. FORMULAS Burns Boards (pupil and teacher) K Ideal, NASCO Circle (30inch circumference) D VisX Circle Area Device (sand filling) D M Lano, LaPine, Viking Circular Disk Divided into Sectors M Yoder Circumference Demonstration Models D M STAS Cone, Sphere, and Cylinder Set D Math Master. NASCO, Sargent-Welch Cube and Pyramid Set D NASCO, Sargent-Welch Demonstration Conics D Lano Dissectible Binomial Cube D M Cenco. Lano, LaPine, Math Master, NASCO, Sargent-Welch, STAS Dissectible Binomial Square D M Bowmar, LaPine, Math Master Dissectible Dimension and Capacity Models M Viking Dissectible Pythagorean Theorem D Bowmar, LaPine, Sargent-Welch Demonstration Model Dissectible Trinomial Square D M La Pine Equal Arms Balance M Harcourt Brace Jovanovich, Math Media Flexible Lateral Area 'Templates(cube, D M LaPine p)'ramid, cylinder, cone, frustum of a cone) Geometry Dial (formulas) C Viking Geometry Dial (33 formulas) C STAS Interest Computation Board M C LaPine Indirect Variation Board (xr---h) MC LaPine Percentage Board (P = R B) MC LaPine Percentage Protractor M C Math Master USING MODELS AS INSTRUCTIONAL AIDS 289

Perimeter Aiea Board Ideal. NASCO Rack and Pinion for Determining D M LaPine Right Triangle Models (30.60-90 and D M Lano, Math Nlaster, Post 4545-90) Transmobiles (principles oflevers, S STAS pulley, gears) Trisectible Triangular Prism D LaPine. Sargent-Welch

8. LOGIC Brainiac Kit K Berkeley, Lano Dr. Nim G Edmund Electronic Logic and Sets Demonstrator D Lano Electronic Logic Trainer D M LaPine Hypothesis Machine D M NASCO Kalah Game G Kalah Logical Blocks M Herder Tac-Tickle G 'N PROOF Tic -'True -Toe Game (3-Dalso called Gangler-Gentry "Quad") 1.171:1:: The Beginners Game of Modern G 'N PROOF Logic WFF 'N Proof: The Game of Modem LogicG 'N PROOF

9. RATIO, PROPORTION, PERCENTAGE Circle for Calculating Fractions and D M Freyev Percentages Decimal and Percentage Board D M Ideal Decimal Place Value Cards D M Ideal, NASCO Fraction Discs M Bradley, LaPine Fraction Disc Calculator D M C LaPine Fraction Equivalents Board M Creative Playthings Fraction Finder D M C AIM Fraction Inlay Board (squares, circles) D M Judy, NASCO Fraction Pies, Cubes, Blocks, etc. M Creative Playthings Fractions Game G Bradley Indirect Variation Board D M LaPine Inverse-Square-Law Demonstration Device. D Sargent-Welch Magnetic Fraction Circles D Instruct% NASCO Markab le Plastic Demonstration Boards D M LaPine for Ratios and Fractious Math Wheel D Bradley, NASCO Pantograph M I Geyer, Lano. Math Master, Sargent-Welch, Yoder Parts Imparter D M Exton Proportional Dividers I Compass, Post, Sargent-Welch, Yoder Rubber Inset Fraction and Area Boards D M Physics 290 CIIA ',TER Elotri-

10. OPERATIONS AND THEIR PROPERTIES Addition Cubes Creative Playthings Addition-Subtraction Slide Rule NI Judy Array Board I) M Science Research Associates Chinese Abacus I) C Satgent-Welch Circles for Ca lc...ating Fractions I) Freya Combinations Card Game Creative Playthings Computing Abacus NI C Ideal. NASC:0 Cross Number Puzzles G C Ideal Cuisenairc Rods Cuisenaire Direct Reading Abacus I) Edmund. NASCO. Electric Math Quiff Game (elementary) Edmund 1-1Vin (card game) G Exclusive Nlaguetic Numb-R.Ods I) M Yoder Matrix Cards for Building Number M C Ideal Patterns Nlodern Alath Demonstration Board I) Math NIaster Moto-Math Abacus Parts D Yoder Multiple Base Abacus M C \Veber Costello Napier's Rods D Bowmar, Ideal New Math Charts D Nystrom, Walker, \Veber Costello New Math Number Frame Ideal, NASCO Number Facts ,Matrix I) Ideal Number Frame D M C Creative Playthings, NASCO, Sargent-Welch Number Frame to Demonstrate D Ideal, La Pine Distributive Property Number Line Ruler Game°, La Pine, NASCO Number Lines D M Bowmar, Cameo, Ideal, La Pine. NASCO Number Relation Blocks M C \Veber Costello Pattern Boards Creative Playthings Projectab le Abacus D M Weber Costello Relationship Cards D C Bradley, Ideal, NASCO Spinno I) M Weber Costello Symbolic Bead Material for Multiplication M Creative Playthings and Division TUF (mathematical sentence game) G Avalon Ilill \Vinning Touch G Ideal

II. OPEN SENTENCES Basic Facts Cards D Scott Eoresman Equal Arms Balance D M Harcourt Brace Jovanovich, NASCO Equation Cards D M Houghton Mifflin Equation.; Game G WIT 'N PROOF Flip-Flash Combinations M Bowmar Grevtn. Than Less Than Chart D M Instructo Mathematics Equalizer Balance D M Math Media USING MODELS AS INSTRUCTIONAL. AIDS 291

Moto Nlath Demonsttation Board and Yoder Accessories for Graphing Multi - Purpose Graph Board D Lano New Math Relationship Cards D Ideal, NASCO Pan Balance and Weights D M Math Media TUF (mathematical sentence game) G Avalon Hill \VFF 'N PROOF WIT 'N PROOF "What's the Numberr Cards D Franklin

12. GRAPHS

Centimeter Grid Paper D C Houghton Mifflin Chalkboard Graph Charts (rectangular D Claridge. Cram. Denoyer- Gcppert, Geyer, and polar) Lano, Math Master, Math-U-Matic, Nystrom. Sargent-Welch, Weber Costello, Yoder Chalkboard Graph Chau Stencils D La Pine. Math Master. Sargent-Welch (rectangular and polar) Chalkboard Graph Panel for Plotting D Math Master Trigonometric Curves Demonstration Conics D Lano Drafting Machine with Drawing Board M I Charvoz, Post, Sargent-Welch Eight-Octants Graphing Device D M Viking Flannel Board Graph Set D M Instrncto Logarithmic and Semi-Logarithmic D Math NIaster Graph Charts Movable Vector Adder D Lano Ntunber Line Paper D Educational Supply, Ideal. NASCO Pegboard Grid System D Lano, Math Master Rnbber Stamps for Small Graph Charts I Edmund, Geyer Space Spider K Edmund, Geyer, La Pine, Nfath i\laster, NASCO Three-Dimension Graphing Device D M Science Related Materials, Viking

13. RELATIONS AND FUNCTIONS Chalkboard Graph Charts (polar and D Lano, La Pine, Math Master, Sargent- rectangular) Welch, Weber Costello, Yoder Inverse-Square Law Demonstration D Sargent -Welch Logarithmic and Semi-Logarithmic D i\fath Master Graph Charts Logarithmic and Trigonometric Functions D LaPine, Math Master, Sargent-Welch Charts Moto-Math (demonstration board and D Yoder accessory materials for graphing) Multi - Purpose Graph Board D Lano 292 ctlArrEttEIG1 IT

H. LOGARITHMS AND EXPONENTS Analog Computer Kit D K Edmund, Lano Circular Slide Rule M C Charvoi, Compass, C-Thou, Post, Sargent-Welch, Yoder Demonstration Analog Computer Sargent-Welch Demonstration Slide Rules D Claridge, Keuffel and Esser, Lano, LaPine, Math Master, Pickett, Post. Sargent-Welch Log and Trig Functions Charts D LaPine, Math Master, Sargent-Welch Logarithmic and Semi-Logarithmic D Math Master Graph Charts NomographsGetting a Line on C Cuisenaire Mathematics Precision Slide Rule M C Charvoz, Compass. Crawford, Keuffel and Esser, Post, Sargent-Welch, Sterling Projection Slide Rule D Charroz, Math Master, Pickett, Post, Sargent-Welch, Tweedy, Yoder Student Slide Rules M C Charvoz, C-Thru, Geyer, Lano, LaPine, Math NIaster, Post, Sargent-Welch, Sterling, Yoder Tyler Slide Rule MC Weems and Plath

15. TRIGONOMETRY Ambiguous Case Demonstration D Math NIaster Angle Mirror D M I Yoder Bubble Sextant S I Weems and Plath, Yoder Chalkboard Graph Chart for Plotting D Math Master Trig Curves Chalkboard Graph Charts (polar and D Crain, Geyer, Lano, LaPine, Math Master, rectangular) Math-U-Matic, Sargent-Welch, Weber Costello, Yoder Clear Markab lc Globe D Farquhar Demonstration Slide Rules Claridge, Keuffel and Esser, Math Master, Pickett, Post, Sargent-Welch Dial-A-Function M C Math-Aide Dial -A- Trig M C LaPine, Viking Drafting Machine with Drawing Board D M Cllarvoz, Post, Sargent-Welch Full-Circle Protractors M I Ccnco, LaPine, Post, SargentWelch, Wabash, Yoder Hypsometer with Clinometer and 1) M I Yoder Graphic Sine. Cosine, and Tangent Table Leveling Rod and Target D S 1 Cella), Geyer, Math Master, Post, Sargent-Welch, Yoder Level.Transit with 'Tripod D M S ICcnco, Geyer, Math Master, Post, Sargent-Welch, Yoder Log and Trig Functions Charts D C LaPine, Math Master, Sargent-Welch USING MODELS AS INSTRUCTIONAL AIDS 293

Map hojeetion Unit D Farquhar, Sargent -Welch Plane Table. Tripod. and Alic lack: M I Cenco, Yoder l'tecision Slide Rides M C Charvoz, Compass, Crawford, Geyer, Kenai and Esser, Pickett, Post, Sargent-Welch, Sterling Projection Slide Rule with Trig Scales D Charvoz, LaPine, Pickett,Post Protractor Triangle I Post, Weems and Plath Radian and Circle Demonstrator D Sargent-Welch Radian Protractor I Math Master Range Finder, Parallax Viewer, Optical D M I Sargent-Welch Micrometer Ranging Poles M I Celia), Post, Yoder Schacht Dynamic Adjustable Triangle D M Sargetit-Welch Steel Measuring Tape I Cella), Geyer, LaPine, Post, Sargent-Welch, Yoder Student Slide Rules M C Charvoz, Compass, C-Thru, Geyer, Keuffel and Esser, Lano, LaPine, Math Master, Pickett, Post, Sargent-Welch, Sterling Transparent GI id Globe D Farquhar Trig-Aide (student and teacher) D M Brooks Trig-Check M C Math-Aide Trig Circular Slide Rule MC Fullerton, Post Trig Demonstrator, Unit Circle D LaPine Ttigonometric Functions Device C LaPine, Math Master Trig Tracer (student and teacher) D M Math Master Trigtacker (student and teacher) D M Lano Tyler Slide Rule M C Weems and Plath Unit Circle Device for Determining Line D M LaPine, Vis-X Values Vita= frig (student and teacher) D M Math-U-Matic

16. LINEAR ALGEBRA Moto-Math (demonstration board D K Yoder and accessory materials for graphing) Movable Vector Adder D Lano Multi-Model Geometric Construction D M K Yoder Set Multi-Purpose Graph Board D Lano

17. TOPOLOGY Clear Markab lc Globe D M Viking Dissectible Sphere D M LaPine 294 cAtAPTEI: EIGHT

17. TOPOLOGY (Continued)

Five Regular Solids D Lano, LaPine, Math Master, Saigent-Welch, Science Related Materials Geometrical Solids(dissectible or D LaPine, Viking flattened) .\fap Projection Models D M Sargent-Welch, Viking Slated Globe D Cram Sphere with Great Circle Sections D LaPine. Sargent-Welch. Viking Showing Elder's Triangle 18. TRANSFORMATIONS Area Transformation Boards D Gfinter Herrmann, LaPine Moto-Math (demonstration board D K Yoder and accessory material for graphing) Multi-Purpose Graph Board D Lano Pantograph D I Charvoz, Geyer, Lano, LaPine, Sargent-Welch Reflection, Rotation, Half-Turn D Gunter Herrmann, LaPine Translation Boards 19. CALCULUS Coaxial Cylinders Intersected by a D M Sargent-Welch. Viking Plane Differential of a Function of Two D Viking Variables Model Differential of Volume in Spherical D Viking- Coordinates Model Eight-Octan is Model D Viking Single-Octant Model D Viking Solids of Revolution Models D M Math Master (ellipsoids. hyperboloids, paraboloids) Spherical Shell Model D Viking Volume of Revolution Model D Viking 20. COMPUTER CONCEPTS Analog Computer Kit D K Edmund Binary Computer Cards and Kit M Science Related Materials Binary Counter D M Sargent-11relch Binary Teacher M Math-Media Brainiac D K Berkeley Brain Teasing Flip-Flop by G Edmund Computer Simulator D Cenco Computer Trainer D Arkay, LaPine, Math Master Demonstration Analog Computer D Sargent-Welch Digi-Comp D M Edmund, Sargent-Welch Digi-Comp Game G Edmund Dr. Nim (digital computer game) G Edmund Model Digital Computer D C Sargent-Welch Modern Abacus D Edmund, LaPine USING MODELS AS INSTRUCTIONAL AIDS 295 BIBLIOGRAPHY

1.Allen. Layman E. "Games and hogrammed IS.Fisher. Alan A. "-Elie Pegboard: A Useful Aid in sti uction."Ailthmetic Teacher,March 1965, pp. Teaching:\ Iathematks." A rithnieticTeacher. 216-20. Apia 1961. pp. 186-88. 2.Anderson, .James T.Space Concepts Ilnough 19.Gaidner. "Logic Machines." Scientific Aesthrozneby.Artesia,N.Mex.:Aestheometry, A :helicon,Mach 1952, pp. 68-71. 1968. 20.Gaistens. Helen1...and StanleyB. Jackson. 3.Beiger, Emil J. "Principles Guiding the Use of Mathenzatics for Elementary School Teachos. Teacher. and PupilMade Learning Aids."In New York: Macmillan Co., 1967. Emeq;ing Mactices in Mathematics Education. 21.Giles. Richard P. "Building an Electrical De% ice Twenty-second Yearbook of the National Coun for Use in Teaching Logic."Mathenzatics Teach- cilof -Teachers of 'Mathematics. \Vashington. ez.Mardi 1962. pp. :03-6. D.C.: The Council. 1954. Pp. 158-70. 09.Giossnickle. Foster E. "Wise Use of Materials 4. "Using Concrete Materials as Teaching and Devices."Bullein of the National Associa- Aids."Bulletin of the National Association of tion of Secondary School Principals,May 1959. Secol:dary School Principals,Nlay 1954, pp. 127- pp. 122-25. 38. 23.Haggerty, John B. "Projects Make Mathematics 5.Berger, Emil J.. ed. "Devices for the Mathematics More Interesting.'Arithmetic Teacher,April Laboratoi y," a depai tment.Mathematics Teacher, 1901, pp. 172-75. October 1950 through October 1956. 24.Ilaiard. William J. "Curves of Constant Breadth." 6.Boger, Emil J., and Donovan A. Johnson. A Mathenzatics Teacher,February !955, pp. 89-90. Guide to the Use and hocurement of Teaching 25.Ilendeison, K. 1;. "Strategies for Teaching by Aids for Mathematics.'Washington, D.C.: Nation. the Discovery Method."Updating Mathematzes, al Council of Teachers of NIathematics, 1959. November 1958 and April 1959. 7.Boys, Charles V Soap Rubbles. 3d ed. New York: 26.Hendrix, Gem ude. -Learning byDiscovery." Dover Publications. 1959. Mathematics 'reacher.May 1961. pp. 290-99. 8.Brandwein. Paul F. "The Visual Aid: A Cerebral 27. Hobert. 1Iarriet B. "Balls and Clay as Teaching Hook for, a Concept." AudiovisualInstruction, Aids in High School Mathematics." InMulti- November 1958, pp. 233-34. So:so!), Aids in the Teaching of Mathematics. 9.Biumfield, Nina Oliver."ModelsforMath." Eighteenth Yearbook of the National Council or Audiovisual Instruction,November1958,pp. Teachers of Mathematics. New York: Bureau of 235-37. Publications, Teachers College, Columbia Uni- 10.Butner, Jerome S.The Process of Education. versity, 1945. Pp. 245-52. Cambridge,Mass.:HarvardUniversityPress. 28.Hess, Adrien.Mathematics Project Handbook. 1960. Thinking with 'Mathematics Series. Boston: D. C. 11.Bruyr, Donald I..Geometrical Models.Portland, Heath R: Co., 1962. Maine: J. Weston \Valch, 1963. 29.Iless, Lindsay L. "Models Depicting Some Groups 12. Cameron. A. J.Mathematical Ente, pilses for of Nlovements."School ScienceandMathematics, Schools.New York: Pergamon Press, 1966. November 1958, pp. 585-92. 13.Courant, Richard, and Herbert Robbins.What 30.Holden, Alan, and Phyllis Singer.Czystals and Is Mathenzatis?New York: Oxford University Czystal Cowing.Garden City. N.Y.: Doubleday Press.19.11. & Co., Anchor Books, 1960. 11.Candy, 11. Martin, and A. P. Rollett.Mathe- 31. Jenkins, Jen. "Teaching the Concept of Cubic matical Models.2d ed. New York: Oxford Uni Nleasure through the Use of Manipulative Aids." versity Press, 1961. Mathematics Teacher,October 1956, pp. 489-90. 15.Davidson, Patricia S. "An Annotated Bibliog. 32. "Teaching the Concept of Perimeter raphyof SuggeFted NlanipulativeDevices." through the Use of Manipulative Aids."Mathe- Arithmetic Macho.,October 1968, pp. 509-24. nzatics Teacher,April 1957, pp. 309-10. 16.DminRankin, Peter, and Raymond Sweet. "En- 33. Johnson, Donovan A. "A Design for a Modern richment: A Geometry Laboratory."Mathematics Mathematics Classroom."Bulletinof the Na. Teacher,March 1963, pp. 134-40. tional Association of Secondazy School Princi- 17.Fish.Dorothy. "Benefits Derived from Mathe. pals,May 1954, pp. 151-59. !unties Projects."Mathenzatics Teacher,Novem 34. Gauzes for Learning Mathematics.Port. her 1959. pp. 568-69. land, Maine: J. Weston Walch, 1960. 296 (:11A1'TER EIGI IT

35. Pape, Folding far the A. 7ihrMaiieS 19. Osborn. Roger. "The Use ofModels inthe ChM. Washington. D.C.:National Council of Teat hing oflathentatits." Arithmeinleacher. "Teachers offatheinatits,1957. janum y 1961. pp. 22-21. 36.Johnson. Donovan aml Ilenry W. Syer, eds. 50,Peaty. J. F.. and K. Lis. Exi"limrnis in "Aid, to Teaching." a deli: t intent. M al hem a In s Mathenzaths (StageI. Stage 2. and Stage 3).3 Tem her. Febuaty 1948 through May 1954. vols. Boston: Houghton Mifflin Co,. 1967. Jones. Phillip',. "111 Early Work on Mechanical 51.PI )Iya . 0 ge. Mathenzai zeal .1leiltads in Suer+ Devi( esfor Drawing the Conic Sections."In St:11114nd. Calif.: Stanford Link eisity.1963. Midi/Se,/ (my lidsin Teaching Mathremi.ties. s2, Rademacher. Hans. and Otto Toeplit/. The En. Eighteenth Yeal book of the National Council of jayment al Mathemalus: Selections from "Madre- 'Feathers of Nfathematits, New York: Bureau of mains far Inc Arnaldo." Princeton. N.J.: Prince. Publications. Teachers College, Columbia thii. ton Univosity Press. 1957. eisity. 19,15. Pp. 273-79. 53.Rising. Gerald R. "Teach for Undetstanding the 38.Kenna. I.. C. Understanding Alathenialhs with Squate Root Algolithins." Professional Growth Visual Aids. Paterson. N.J.: Littlefield. Adams k for Teachers. Mathenzath s, junior high ed. 1-2, Co.. 1962. February 1965. 39.Kennedy. Joseph W. "3D String- Models for 51.Ross.AlvinE."Mobile GemiteticFigures." Math." ,4 z«liaviszwl Instruction. May 1956. pp. Mothernalus Teacher. May 1958. pp. 375-76. 80-82. 55.Shuster. Cat IN.. ancl FredI..Bedford. Field 40.Kennedy. Leonald AL Alodels for Alathemalhs in the Elementary School. Belmont. Calif.: Wads. 11*(0.7 in Mathematics. East Palestine, Ohio: Yod. wolth Publishing Co.. 1968. er Instrument Co., 1933. 1. Mutt/. Marguerite. "The Mathematics Labora. 56. Simons. Lao Genevra. "Historical Material on tory:.1 lieaningful Approach to Mathematics Models and Other Teaching Aids inMathe. Insnuction." Alathemalws Teacher.larch 1963, Illa tics.- In Mullz-Sensory Aids in the Teaching pp. 141-45. of Maikenzatie,$.Eighteenth Yearbook ofthe 2. Lewis. 1.::zice. "The Role of Sensory Materials National Council of Teachets of Mathematics. in Meaningful Learning." Mathematics Teacher. New Yolk: Bureau of Publications. Teachers Col. lege. Columbia Ilnivel city. 1915. Pp. 253-65. Apt il1956. pp. 27.1-77. 13.Lieber, Hugh G.. and Lillian R. Lieber. Alt's. 57.Sister Nlaigatet Cecilia. C.S.J. "Mathematics PIO Ints. and Logic. 3(1 ed. New York: W. W. Nor. Mathematics Teacher.November 1961, ton Co.. 1960. pp. 527-30. 4. Loomis. Elisha Scott. The Pythagorean Proposz- 58.Steinhaus. Hugo. Mathematical Snapshots. 3d lion. 2d ed.1910. Reprint. Washington. D.C.: American ed., rev. and enl. New York: Oxford National Council of Teachets of Nlathematics. University Press, 1969. 1968. 59.Syer. Henry 1V. "Sensory Learning Applied to 45.Mock. Guidon D. "The Peri y Movement." Math. Nfathematits." In The Learning of Alathenzalies: emaths Teacher, March 1963. pp. 130-33. Its Theory and Practice. Twenty-first Yearbook 46.Moore.EliakintII."Onthe Foundations of of the National Council of Teachers of Malin:. Nfailinaii(s." Mathematics Teacher, April 1967. Illa S. Washington, D.C.: the Council 1953. Pp. pp. 360-74. A reprinting of the 1902 address. first 99-155. published in Science. 1903, and later included in 60. Tuittey. BillyI.. Gromelly Teaching Aids You .1 General Survey of Progress in the Last Twenty. Can Make. Portland. J. Weston Waldi, five Years, First Yeaook of the National 0)un. 1958. of Teachers oflathematics. 61.Wenninger. Magnus J. Polyhedron Models for 7. National Council of Teachers of Mathematics. theClassroom.Washington,D.C.:National dIfithiensaty :lids in the Teaching of Madre. Council of Teachets of Mathematics, 1966. moires. Eighteenth Yeatbook. New York: Bureau 62.Wilkading. Maigatet F. "Stimulating Interest in of Publications. eachers College. Columbia Uni- Junior Ifigh Mathematics." Mathematics Teach- versity. 1945. er. Maich 1950. pp. 197-201. 48. Niven, Ivan, and H. S.Zuckerman. "Lattice II'mldwide Use of Measzorng Systems. Fold and Points and Polygonal Area." American Afathe. the.\ letricSystem, Report no.2.Dearborn. matiral Monthly. December 1967. pp. 1195-1200. Mich.: Fold Motor Co., 1967. 9.MANIPULATIVE DEVICES IN ELEMENTARY SCHOOL MATHEMATICS BLOCKS CAN BE USED TO ILLUSTRATE SET OPERATIONS SUCH AS UNION AND INTERSECTION

"1 299

CHAPTER 9

MANIPULATIVE DEVICES IN ELEMENTARY SCHOOL MATHEMATICS by ROBERT L. JACKSON University of Minnesota, Minneapolis, Minnesota GUSSIE PHILLIPS Caseyville, Illinois

Chapter 9 explores the types and uses of manipulative devices that arc appropriate to the teaching and learning of elementary school mathematics. A classification scheme is developedso that types of items, as well as individual items, can be examined. Suggestions for use of devices are presented, related research is examined, and evaluation techniques are detailed. An extensive listing is given of sources of different kinds of devices. The chapter concludes with a bibliography grouped into general references and nine special classifications. 301

9. mAmpui...vrivE 1)EVIC:ESIN EI.F.NI ENTAla SC! I 00 I. NIATHENI:VIICS

The impact of technology on the educational I.Formation of positive attitudes toward roces,, and. in particular, on the learning of t hematics mathematics has been amplified by new knowl. 5.Understanding of the logic al structure of mathematics edge about the nattne of the learning.process. Utili/atio of mathematical conceptsto the of children. andthe stuctme of discover new generalisations mathematics itself, as well as by new mathemati- 7.Ite«tgnition and appreciation of the role cal content. \ \'e have learned more about how of mathematics in society children learn. think, and behave, and at the S. Development of study habits es,ential for same time have been able to integrate this knowl- independent progress edge into classroom activities and curriculum 9.Demonstration of mental traits suchas creativity, imagination. curiosity. and planning. \ \'e have learned more about inathe- 'i5. nali7ation. inaticai structure and have discovered ways in A manipulative device is considered to beone which this Nirnet me can be made understandable that utilises inintaily the senses of sight and to children. We have introduced new mathe touch. Implicit in this statement is the under. matical content and eliminated outdated content standing that sensory contact with objects in the so that objectives of mathematics instruction teal world :s something more than a casual "look c an be mote readily attained. These changes «ll» at" or "feel" interpretation by thesenses; for to bitted with technological growth point to the fact manipulate means to handle or to use. Thus, the that now. mow than ever before, educators recog- term maniptdative dcvi( es refers to those co- nie the importance of the child's active partici- crete objects which, when handled or operated pationin the learnin. process. Furthermore. by the pupil and the teacher, provide the pupil these change, have given rise educational with an opportunity to attain specific objectives took that enable its to attain goals more teadil identified tinder the broad goals listed above. than ever before. Time seems to he wide agreement among The educational tools with which we are con- mathematics educators and psychologists that cernedin this chapter ate «mumify called sensory learning is fundamental, or at least basic, mattipd/aftve devices. Itis the purpose or this to all other learning. There is no question that chapier to examine the nature and tole of mani- in the early stages of learning, modification of pnhuive devices in the teaching of elementary behavior in a young child is largely dependent school mathematics. on his ability to perceive his environniew, and In order to mulerstand the role of maniptda. perceptions fotmed by young children arise from live devices in the learning of mathematics. it is the use of their senses. Since children can only necessary to keep in mind the main goals of sense that which exists in the real world, itis mathematics instruction and to examine how importantthatthey have available court etc such devices contribute to the attainment of these objects with which to carry out explorations. goals. Sobel and Johnson (29) describe the pur- Perhaps the very young child who in his quest poses of mathematics instruction as follows: for learning picks up small objects to eat them I.Formation and understanding of mathe- is not unlike the nine-yea -old boy who takes matical concepts. fact,. and processes an electric clock apart to sec howit 2. Development of skills in computing with works. accuracy and efficiency Although far different in maturation and in 3.Development of a genetal problem solv- established knowledge systems, the two are learning technique ing by manipulation. Syer's scholarly discussion 302 (.1I \I' FI;

of sensory learning points out that moil% ation. c an stuf mathematics mote satisfactoril% memo'i /at ion. mat hematifa 1tin del NI and i ng 5. %vitenac lI (1111(1 has abundant opportimit% ploblem soling skills. and attitudesall impor- to manipulate suitable plisical objec is.1 1 21 tant facets of the learning process and overall The learning them fistsPiaget.Bruner. and goals of mathematics Insti 1101)) sen- 1:lavell agree that an operationis something sot, learning (35). Van Engel, empliasiies the that may performed either internall% with role of sensory learning in con( ept formation sYntimIN rein csent hit; "materials or extern"11.) with %viten he states: theac mai mate% lats.I:miller.theplobahil Iteactions to the world of concrete objects itof perlorming internal opetations a« titatel% are the foundation from \chic!) the structure and «insistentl is in( rased repeated expel i- of abstract ideas arises. These reaction's are elite with concrete materials. Thus there appeals termed. reorgani/ed. and integrated so that they become even more useful and even more to be strong support among educators and ps- powerful than the original response. [37] etiologists for the position thatthe manipula- Dielles's research with stIctural materials has 'ion of it device or devices should pi e( ede the led him to conclude: requirement of abstracting an idea or model. In short. it %%linkd appear that manipulati%e devices The olganism Scow; to wish toexplore and manipulate the enviromitent.It does are essentialinthe instructional progiam for this. prestiniabl;. for the purpose or being elementary mathematics. able to predict how the environment is going to respond. Nlatilematies learning would probably be no exception to this. but a pre 1:ESEAI:C1-1 ON MANIPULATIVE. liminary groping period is notably lacking 1)FA'IC:F.S IN F.I.EMENTAIZ.). SCI in most mathematics ,sons . ... So play. NI:V11-1 I.:MA*1'1GS INSTIUICTION it seems. should be regarded as an integral part of any learning cycle. "Alathemat kat Some support and direction 101 the tiNe 01 ma- play can be generated simply by providing nipulative devices (all be 10111111111 %MIMI\ IC- children with a huge variety of constructed mathematical materials. [6] SC:11:11 projects. However, nnuh of the research evidence is inconclusive. Sole 101111(1 that using It examining Piaget's theory of concept for- :t%%trice, of materials does not produce better mation for implications for teaching. Adler comes results than using :1 single devi«- if both ploce- to this conclusion: d tires are used for the same amount of time (30). Physical action is one of the bases of learn. If more tittle ing. To learn effetively the child must be a is spent, achievement improves. participant in events, not merely a spectator. regardless of what Or how many 111:11erials todevelop his concepts of 'mower and used. space, it is not enough that he look at things. III attempting- to determine the effectiveness He must also touch things, move them, t111.11 them, put them together. and take them ofdevices on the achievement offirst -grade apart. For every new concept that we want Hasliman found very little difference the child to acquire, we should start with in achievement between groups using expensive sonic relevant action that he ran perform. commercial materials, inexpensive commercial 12] materials. and teacher -made materials (Ili). \Viten Gagne: points Out 5111:111 (1111CIVIICCS 111 achievement were found. Instruction needs to be fundamentally based on the stimulation provided by objects and they were in favor of tile group using teacher. events.... Objects and events are the stim- made materials. uli from which concepts ale derived. [10] Regading the use of materials by teachers. The sometimes controversial Cambridge report arvin found that teachers with mathematics makes a strong- case for manipulative desires methods courses use more manipulative devices when it states: than teachers Without such courses; she further The conclusion is inescapable that children found that pictures and symbolic materials are DI:VIC:s IN 1.1.1:NIENT \I:1'(:,11001. m R,N 303

used mole fre(Iuent!) than .1(111:11 inaniptilatke should possess. .Nlost authors agree that.1 good de% be, (I7). She also established through an maniptilatke &nice should extenske (pwslionuaite that liequenc ofuse I.be elevant to the mathematical content 01 maniptilatke materials appeals to he a con with a desirable otu«nne ill mind: IIil)111111q factor in :whim einem in eleinentar exploit as many senses as possible: ma 1 henta i( 3.be durable. it, durability being ((unmet' In defining the tole of tionalaids. surate with its (ost and anticipated usage: Eidson iepoi ied that it is generalb, ac«.pled that 1. be constructed so thatits details ale ac- proper use of aids is based on the same frame- curate: work of principles as all good teaching proce- 5.have high nlards of craftsmanship so (Imes (7). His stud) showed that most ellectise that parts nie not easily broken or lost: it:stilts ale obtained by placing major emphasis he attractive ill appearance: on the use of devices to obtain data rather than lor 7.he maintained e.tsil and .tt a reasonable Nupplementat )2 activities. Ile cautions users to re. cost; member that aids never teach mathematics and S.be adaptable to the school facilities (con. that the tole of the teacher is paramount. An- sideringmobility and convenienceof other wattling is given to teachers using inanipti. storage): la tive devices by Sole when he stales that manipu- 9.be simple to assemble: lation of any device must not be (onfused with 10. be flexible :111(1 lwve varielY of uses: the learning of mathematics (3(1). 1:ttlteniatics II.be simple to operate: is made up of ideasnot materials:it can he 12.he huge enough to be emit) risible to all illustrated with deuces and has application to pupils. if used for demonstrations: ( oncrete sitnations. but it must be under- 13.either involve a moving partor parts Or stood as a system of ideas. be something that is moved in the process The findings of RV) researdiers, Swick: and of illustrating the mat hemathal principle Spross, give strong support to the desirability ()I involved. using intiliisensory aids in teaching both «unuo. In the final analysis. the devices that are effec- union and reasoning (3(i: 32). In additio1, the tie and ellicient hi helping the child under. use of aids and devices was found to stand the mathematical concept to he learned the attitude of second. and third.gradeis Toward are the devices that should be used. mathematics. Adkins and Sitddeth concludedthat there SOME GENERAL GUIDELINES FOR seems to be a tendenc to the 1110Ie instructional THE LSE OF NIANIPULATIVE DEVICES materials in the primary grades for the purpose of discovering relationships, for mod% ation. and Each manipulative device has its OII palliCt11:111 for inlItiencing attitude (I: 33). set of instructions that should be followedif This research gives credence tothe hypo one isto achieve maximum effectiveness front thesis that manipulative devices are useful in its use. There are, however, it number of general the attainment of the goals of mathematics guidelines that can be helpful to the teacher in instruction. the selection and use of manipulative devices. The following ate suggestive: SOME CHARACTEItIST1C:S OF I.Choose the device that best suits the pur- GOOD NIANIPULATIVE DEVICES pose of the lesson and will help attain the objective. Many attempts have been made to prepare lists 2. Become familiar with the device and the of eharteteristiis that good manipuhttive devices techniques for making effective use of it. 304 (:1111`11:: NI NI.

3. Corte late the opelations depicted by the -I.Number !maids device with those to be done either with 5. Gauls and chat is pencil and paper Or oil the elialkboald. Measurement cleric e:, I.If possible, arrange for each chilid to have 7.Nlodels 01 geometric relittionships his own device. I f this is not possible, make 8. Games and pui/les certain that each child has the opportunity Special computational choices to use the device many times. Try to aerate all atmosphere that encourages rather than forces the use of devices. DENioNsTRATI()Nitomths 1:reate opportunites, as soon as possible. :xxDrvich:s for each child to become progressively less 'There are man) desirable teaChillg-le:1111hIgeN dependent on manipulative devices and periew.es in an elementar) mathematics program MOM derlIdelll on using symbolism and which, for one reason or another. cannot be sabst ract ions. provided effectively by having each child work 7.Ilave the child stop using a device when individual!) with his own set of materials. Small he is ready for a higher, more abstract level group and class instruction call make all impor- so that the device does slot become a crutch. tant contribution to the total growth pattern Of 8.Use a "math caddy" or movable storage hildren. 1.0 ace miplish these ends and stillpro- unit designed to hold and properly display vide chilchen with real-world referentson which (1011( es ill order to increase their useftthiess. to build abstractions demonstration boards must Nlanipti lathe devices will not teach mathe. he available to the teacher. In:mks.Itis their use by the teacher and his The demonstration boards most lamiliar to guidance 01 pupils' use that (lett:mints the use- elememar) teachers arc the flannel board and fulness of devices in learning. the magnetic board. both types of demonstration boards can serve the same 1111101011. the CLASSIFICATION OF flannel board is perhaps the more widely used NIANIKILATIVE DEVICES becauseit can beI eadily constructed by the teacher. Advocates of the neignetic Wald prefer The presently vast and ever-growing supply of the durability and workmanship qualities of a manipulative devices for use with elementary commercially built device. Stieltz describe:, spe- children rules out the possibility of presenting a cif ications and coustruction procedures for both detailed discussion of all available devices. Since flannel boards and magnetic boalds (3.1). most of the devices currently available are mud- The great strength of these devices is their purpose devices and can be used to objectify versatility.Most of the concepts taughtin many mathematical concepts, it is necessary to elementary school mathematics call be effectively selecta classificationscheme thatwillkeep presented on these boards. overlapping of discussion at a minimum. For this Generally. the flannel hoard is a 2 by 21 foot reason we have chosen to bypass a classification piece of hardwood covered with flannel or felt. scheme built entirely around the mathematical If a piece of sheet metal is insetted between concepts to be taught to pupils. For the purposes the board and the Ilaintel,it can also serve as of discussion is this chapter, we shall consider :t magnetic board. It is propped on an artist's specific manipulative devices under the following easel or placed on a chalkboard tray so that categories: it gives the appearance of a blank picture. Ac- I.Demonstration boards and devices companying the board are flannel or felt cut- 2.Place value devices outs that are held in position by the cohesiveness 3.Cohomed beads, blocks, mods. and discs of the materials. \\IC \1.1.\'"1!111ilt .s.3..)14\:;/(1 NI Ali :)S `10011 Lixtv, \ SOC

all() w.:11 Jo uopvticalolimp qiucoci 111:1 (1 o:)110 ronrinalippti aJpolinc c! paelop,op p:nristit Aci alp Nutmoneg 110p1mgc II! mipplio :New 01 alipu xptio 1015 s)! .10.1 .11 itoolm 111 11.)131:hopum inxtpllti.) sop 51:m ,(311) elopAap aApning iunoD.) 1)311:3% III 1: DI,/,11,1!111;r: II: alp ..1311.)1:31 oral .101 ay tiegat:/ntionaW painacamlal et alp 1(oc p311:34: :)11 1:.11 1311111:1j 1)110(1 111 3111 %1311131 )opol *(131:Ipi1!s 1):»11a3 uotioinigns- s)Dig 111:1 oet JO 1101!p1)1: 1: I 0)1 11a) 111: -lc:3331111 poelop.n)t) 211!opp11 .(q .11:1)0.1. tit: ."111!0l1 111).( 111 'x.1010 5111.1 AJOP: NI Pimp: "iltiopattioc imp linnvitelDc .1o11)1:3 111:t1) 2iqu!of 1101101: x111. ponaele11:11 nolploi.) om:ti ;),991:)1No qapp11113001010 01 I c pupped Sip ;)cluni pontigel apin asp non1:3)910111:)1) p.11:0(I 111 2111111.10j .1131p at(int ),;)t),,Itti I papery! cillac.(ett) Jo ca11cn1:3 HMO N31.1010 1)111: 111 113 '.11111x1 010,1 pm: pLIDADc 1)0 .1)1 cateriali).% 211PVi41111V 11:.q11:111;)1111:111 nt,to .latuttnic liunnotp I :olemp: )111m cDom) 0.1 stalpo II! an) eInalli I -111A1 1111!11:3 Sin cl:ill:out palm)! Pm :Pp )v 3:)111.411 .p.vg supp:),1 1)111: i uoctng Amp II: 11a1,.:1:16.(111 AIN linopoploc :»1:3)91111 1ow:1100J JO un!so1.11);) 1111.1 op 110. Tipp )! ,:.c1:A) 11D.Ipipoj 311:111 C1311p0l01 It 3.11 -011:3 no alp 1x11111:11 p31:0(1 fs:1:;;;:ti c.Ia imp sem :1(11 Set lin111310_, cSILLIn (01) gel 1110)13) no alp puncli h).11:0(1 v ppv 11:11m 011 110. acodeltic 3q q:.%) 411!op II! 'aunalzi 1.6 3.11:)p I al cu.%) oc 11:1 W11!)1:;! tcappnactio.% listrealit). J'iSetwilektliMENNIMNIIIIIIPZ,I aim I 110.1110111 I ppiom non:.%) pip1 :.loj 31111A1 Items 1 An:c 4tioponeetc :)93 tlilipoin 11 c1;A1 I a.)1:1,1 aog getepu no alp (1)31:0(1 0(1 poS acodellic N1:A1 ,_oc)) .11111 eo1(il:1343.1 :osimelcald (ivas I A01 Anent oiam apg) :Itam 0,mt

01!(111:.1 11141 si: 1 palpn:A% 1 Ans t W11tm0111 )ctif op!%1110 Sul poluell :),:ntoj ictiAl 0p 1105 )coetc mg) :ostodcold 3a1pouv,. Liigget:3 c;),k nig 1011 )clif atm 1!q011:3 az)11.1 :).lam 'olio :):).1111 sgerpm ;midi ao.1111 10111 ciptepu 110 :Pp p11111:11 1).11:0(1 aIII41:1) lam Oq I))1111:A 01 ol. .1.1i1111111 .6111 11:a.. III: :)111 :;)410(1%)111 Nal(11:1 1 I III 011.41 prn omi citerri lipine) aanp snepp:3 -nomm ",111! MU 1100c alp :)D.1111 sgeppm Yttiolo);:A) ;nin9t:i 7,.6 ;: a.)11;).1 I)111 punoj awn II! A111 imp31:2 packing 0111! alp Imp 01 I Awl egeign3 II:a 5111:111 an: -*7V4M4111.1111111111111.11.11111111Wa& Amu :;o110(1colij e`101.1 01) 110S ,!0111 :acnoelcald 1 pantile%) [...maw ;PORI ic.nd aump et.w1 swigri '.1111!)1:a pun Pow Damp aloul s1e1p:3 polool LI1J111 oc )1:11) .1).1).a an; cl!qq;:i 1u!);:a 7.'""-N 't sp, awn .00)c pm: .11:11111!c cDito (Ilan ap!Ao.tel 11:13plop Ip!A) sa3I1al.1:)(1xo II! 1.111,11,19; 01 star pm: 01 :II lugun Jo Poors11) ems u! I: -U0) a31:(apl man) 3oj 1.1oA II! tion!ppy lions sopoic a1)!A01(1 cpppinvoclelo .10.1 DIp planieloFmat) Jo ropiu11min:111 saouomos iptic alp ono Imp 9 pownsnin oln`ti!:i 7;11 .1.1.)11A) )n) itopimaelo logiu.(c sanDwit1 ails up:tu 110031: JO on) .(30)c (..11m.rEp. \I\1

The .ante strategc used crith the addition ol whole numbets can be used to ilhearate the addition of rational numbei N. II tlty t irt ular legions shown at the leltin Pigott.,.:1 and .1 Ivor sent a stand:11d unit. then the addition ol mid A (.111 ht. Icines,111e1 b. the diaglani .11 the right in either (Iglu' t..\ !though the addition is complete. itis ttecessarto find an (tittered pair of numbets which can he used to tepirsent the stmt. The apptoptiat ordered pair to use in :3 this 0:1wis (5.I;). Peskin suggests the use of a flannel !maid at FiGuRE SL3 an even mole advanced level. in telating figmes geometrically and algebraically so that the multiplication of a binomial I) a binomial can IL. isualiied (.1 1). Other items in the category or demooNII:ilioo howds which ate rapidly gaining- in popularit ate pegboaids and geoboaids. :\ laity pegboalds ate so constructed that they can set ve as hunched boamds. Geolmaids (Figme 9.5) are partinlati) helpful when developing the concepts Of pc.- rinteter and at ca. Squares. triangles. Iertaigles. quadrilaterals. pentagons. and other polygons can all be easily shown on a geoboard musing elastic. and Raskin. in Nippon or the use or physical objects as models for concepts. FIGURE9.4 suggest the use of a pegboard a model for developing an uncleistanding of rational numbers 09). Kennedy NlIggeNi". :IC Wit Iona! uses for this dam, or mate' ials (21). As teachers begin to use :lemonstration devices with children. a new itniveise of cieative ideas will open up. These will result itt stimulating reaming experiencesforbothchildt enand teachers.

DE,ficEs

Two features ol a place value system of numeration are base and position. or place value. Before at child catt comprehend place value. he need,: to) have au onderstanding of a single or ones FIGURE9.5 loth and also all undeistamicling of base number. Courtesy of Cnisenaire Company of America, lite. To help him gain this trnlerstnding, manipula- m Nip( T DFAics IN ELENIN I my scifooL \ I !IFNI \ I R.., 307

tit L. devices that pimidt. istiali/ation are most helpful. NIallipulatke divitt., 111.11 can he used to leach place laltie ate innuctotts: the include diliciem tis 01abactise,. place %Atte boaids. pocket ...... (5. and slit ks 01 (mint-Is %%hi(can be sot led ,. off,,s L. and gimped. L.L.-,7.-- do. t-!--. -- M _\ !Um experience 101- children ma. be gimp- r" ON; L.L. L- L1-... HotoRios II *.3! ing couniti, 01 bundling sticks 10 exhibitIten. i 1..11,-1. 10 tens ,fiq 1 h bundled to slum .1 hundred. ONES II a boat(' similar of the one shown in Figine 9.6 is aailabic.hildten can he asked to 'cwt.- sent dilleient numbets. hi the illustration. the VICI;KE 9.6 number 133 is tepiesented at lower right.. The poi ket chart_ shown in the bakgtound can be used in the same way. as hell as to help explain tenanting ill addition and suboaction.

Consider the ploblem that k illustrated: 1.1 19 31. It can he seen that the sum is represented b the 5 tens and 1.1ones. lilt1.1 ones equalsI ten and ,1ones. and 10 oncs can be gicuipd together and placed trills the tens. showing a sum of 61. Such at example makes meaningful the addition ol I tett when the child shows his lvork like this:

1-1 19 3i

(1.1 FiGuut.: 9.7 Vertical abacuses. both the openend and the dosed types. are also adaptable to exercise., in. %dying- place value. numeration. addition. and subtraction. The abacus shownillFigure 9.7 shows the number 113. The sum of 255 and 143 is Hillstromd by lite Nhown in Figure 9.8. The necessity for regrouping- the beads when the number of ones exceeds 10. the number or tens exceeds 10. :mil so on. becomes readily apparent io children.

Fiturttli 9.8 308 CI IA PTER NI N E

Man) boaids ale adaptablelottank %%id] decimals. The boat(' NhOWn in Figure 9.9 uses small SqUareS that children can place on the proper pegs. Thus in the top low the number illustrated is1102.510. The number illustiated in the second row Iron) the top is 2011.111. Most vertical abacuses (.u1 be used xv hen %%oil. ing, with bases other than ten. Nlultibase «m xci tci !mai(Is like the one NIUM n in Figme 9.10 ale also available. In the illustration. the movable comet ter is set to show base-On ee numerals. The base-three numetal illustiatedi., 10010a The use of the abacus lot teaching place value has been the subject of man) lepotts dealing with this class of manipulative devices. Hamilton suggests that the form of the abacus used with FIGURE 9.9 children should parallel the mental structure it is hoped to create (I1). Thus those abacuses which consist of beads representing fives as well as ones and any abacuses with bead rows placed horizontally would be inappropriate. He suggests using the vertical abacus in one of several forms: string- or wile frame. open spike. or with markers moved behind a shield. Clary prefers *A- two dillelem lormsa Nhoebox abacus that has .7,-A string sewn through the box and uses colored buttons for counters. and a spool abacus similar to the spike type with spools placed over dowel pills mounted on a board 17). Cunningham provides detailed instructions lor the construction of a "traditional- abacus and a

- simplified openend version, both using wooden .. forit6' frames and coat hangers18). One suggested innovation lor vertical wireframe abacuses using ' -EV 0107. toil large colored beads is to make the tenth bead in each column the same color as the bottom nine Me, beads in the column to the left. e- @aisle Other suggestionsFortheconstructionof &WI abacuses are made by Smith, Peterson, and Woe Ralunlow (62:59:61). Nechin and Brower. Spross, and Sueltx recommend the open-end or spike abacus as the most useful type (58; 31; 3'I). Not only were children motivated to improve Ficuiu 9.10 computational skills in their experiments, but pupils could see the decimal system of numera- Photos on this and the following page, courtesy of Respon- sive Environments Corporation tion as a system based on logic and order. NIAN I PU LATI V E DEVICES IN EfJ MEN'I'Alil SCI M ATI I ENIATICS 309

A hand-tally counter is suggested by Hooper COLORED BEADS. BLOCKS. RODS. lor teaching place value (5). This «muter is ANI) 1)ISC:S similar to the devices used by adults lot counting such things as the number ol customers at a The many differenttypes of coloredbeads. store and to the devices used on elevators to blocks. ais, and discs that are now available can be used in a variety of ways to help develop indicate Boors. An advantage noted. pat ticularly for pupils in grade I, was the improvement of concepts in numeration. logic. and geometry. audiovisual and kinesthetic responses. For years, colored beads have been used in the primary grades to help children learn about Bridgers describes the use of a block box lor place value workan open-top box with three number. Children lane strung beads to illustrate layers of blocks insidethe first layer containing a number. stringing live beads on a cord. for example. to illustrate the number live. The little IOU cubes. the second layer 10 block's. each equal in volume to 10 cubes of the first layer. and the plastic markers shown in Figure 9.11 and the counting pole shown in Figure 9.12 are typical last layer consisting ofIblock equal in volume of some of the "extras" that are now part of to 10 blocks of the second layer (.16). He feels that bead sets. the uniformity of sire of the cubes and the lack of differentiation between colors are strong advantages. FicuRE9.12 Other .suggested devicesfor teachingplace value are popbead Reck laces, by Swenson; pop. side sticks, by Weyer; place value charts, by Peterson; and pupilesigned computers, by Rab- inowitz (63: 66: 59; 60).

FIGURE9.11 2 110

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3 Exinnfiri 6 iumumme f r r2 FIGURE 9.13 FIGuRE 9.14 Blocks ol all sot Is Wad descriptions are presently available. These can be used for counting 20' 20 purposes and for showing the one-more relation- I' I119 18 ship as in Figures 9.13 and 9.14. 18 II 11:4 Counting ladders are helpful in teaching addition and subtraction facts. If a child is learning I1 115 It I. that 7 I I, he may first arrange his blocks I I113 1 I)1 as shown at the right in Figure 9.15 and then 12 12 remove -1blocks and place then' with the 7 JO blocks. Ile can then read "I I- on his numerical 1 1 1:1 ladder. Similarly. ithe wishes to show the sub- 8 II Ira( lion 1 1 I. he will first place his blocks as Ii Li shown at the left in the illustration and, after ell I temoving 1 blocks. read "7- on the ladder. While I II 2I 2 childien can do and have done this type ol a activity work without the aid of devices, the MU aj- novelty of the ladder increases interest and pre- vents counting- errors that often occur because a FIGuRE 9.15 child is careless or because he is unable to count correctly. Itis generally agreed that blocks are important in the play stage. Many of the colored block sets now available are being used at the primary levelfor sorting blocks into classes according to shape, site, and color. Some sets are merely simple nesting sets (Figure 9.16), and children build according to size; but much work can be

FIGuttli 9.16 Photos on this page, courtesy of Responsive Environments Corporation \NIPITIA FIVE DEVICES INI LNIN I \RN st1100L ,N1,\1111,N1 \ I IC, 3 1 1

done with striation (placing blocks or other ob- . \tthe intelmediate leNel,achild ma) be

jects in order den.' mined by some in operty or gken objeos with I(Unclean shapes and I hametclistic smh as size or shape). dilleient «dors, as in Figure 9.20. and :asked 10 Consider problem in which a child is gken a».angt these objects so that no two having the objects like the ones shown at the top of Figure same shape or color are inthe same low 01 9.17 and asked to artange them in order from column (Figure 9.21).II16 objects pose too smallest 10 Largest. as in the bottom of the nista.' much of a problem. l(,t the child begin first with lion. ()I he might be asked to lank them in older 9 objects. using old) 3 (Ullman shapes and 3 ,1ze. as in Figure 9.18. later he may be asked different «dors. lo pailtwo sets 01 objects matching them I)) size, the smallest with the smallest, and so forth. as is done in Figtne 9.19. 0

Ficulut 9.17 AAAA

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FIGURE9.18 FIGURE9.20 FiGuRr.9.21 DAOR FIGUR19.19

G CB AAA/R/

B 0 312 IAPTER NINE

Typical of the many interesting games that develop logical thinking is the following. which emphasiies the meaning

A childis selected to be "it.'He then chooses a block Itont a set of blocks and hides it behind his back in one kind. Sup. pose. for example. he hides a ,teen. c ircular. Hat block. Ile then must give the class One(' clues using the \void no!. Ile may. for ex. ample. give these clues; block is not red. It is not square." "It is not yellow." He then asks a child to try to guess which block he has hidden.IIthe second child guesses incoiretly. the first child must give Courtesy of Cuisenaire Company of America, Inc. another clue about the block and ask anoth- FIGURE 9.22 er child to guess which block he has hidden. After each incorrect guess, thefirstchild must give another clue. After a correct res- ,;,,, I. '''''.,..' ponse. the child naming the block becomes ,.. and gets to select a block to hide. ,. . , ..% it This also makes an interesting team game if one point is counted against a team for each clue that has to be given.

Rods are particularly useful in building trains to illustrate ideas such as the commutative prop. FIGURE 9.23 crt) of multiplication 01 whole numbers and to develop multiplication facts (Figure 9.22). In the primary grades, a child might build a train showing that 2 X 3 = 3 X 2. Unit blocks can be used FIGURE 9.24 to show that 2 x 3 = 3 x 2 = 6. (See Figtne 9.23.) The uses of beads, blocks, rods. and similar objects are almostlimitless. Teachers should not hesitate to use them in situations to develop '0 0:,40.-'0, J110 logical concepts as well as to develop notions ir *,;.0, 0' of counting. Often children develop new uses ;... 41 're for them. One ingenious student used pegs and a pegboaid as a hundred board to illustrate two types of symmetry (Figure 9.2,1). Designs showing line symmetry ate. of course, easily constructed and offer other possibilities. The influx of colored rods during the past few years and the development of mathematics programs built completely around such rods have stimulated ninnerous research studies to investigatetheireffectiveness. Most ofthese studies have involved the Cuisenaire rods and Courtesy of Responsive Environments Corporation the associated program. Fedon used Cuisenaire NI A NI IT LAT! VE DEVICES I N ELENI ENTA Ry sct 00f, m xi. II \ 313

rods with one group of pupils in gradeIand In addition to colored rods. many types of an eclectic approach with another group (70). blocks ale advocated as useful in teaching ab From test results he found an apparent trend strict concepts. Lucas used attribute blocks with in achievement that favored the group using the first-grade pupils in an attempt to show that the eclectic approach. Ile concluded that the intro- teaching of classification, seriation, operations, duction of color notation as a basis for the sym- and comparison with colored blocks is analogous bolization process delays the successful transition to teaching the same things through set abstrac- from mathematkal structure to corresponding tion (73). Ills conclusions indicate that children mathematical symbolism. Passys findings were trained in an attribute-block program conser%e similar (76;77). He made a study of the use cardinality better than traditionally-taught chil- of rods with third-grade pupils. Nasca reports (It en and show a superior ability to conceptualize that the rods as a variable had no effect on the addition-subtraction relations, but do not learn achievement of secondgrade pupils as measured computational procedures to the same extent. by a traditional standardizedtest(75).In a Additional uses of this class of materials are study of two groups of third-grade pupils, one described by Stern and Bridgers (79; 116). having used the rod approach during both first and second grade and the other following the N UMBER BOARDS regular curriculum, Lucow concluded that the rod approach is effective but that there is some Teachers have found various forms of number doubt as to its general superiority over other boards effective for representation of abstract methods of instruction in current use (7.1).In mathematical ideas.The boardsshownin contrasting the rods with the traditional pro. Figure 9.25 allow young children to represent gram in England and Scotland, Brownell found the concept of number in several ways. Each no striking differences in the pupils' progress board has a raised numeral at the top, over towanl maturity of mathematical concepts and which children can place the appropriate nota- understanding (68).Aninterviewtechnique tion card. There is also a vertical channel into employed by Howard showed that teacher re- which the appropriate number of plastic cubes action to rods was favorable and that rods hold can be placed, and the raised outlines below promise as a supplement to current methods FIGURE. 9,25 (72).Hollis.working withfirst-grade concluded that as much traditional subject matter was learned by using rods as by traditional teaching techniques, Studies seem to indicate 10 that high-ability pupils appear to benefit more 3 from using rods than low-ability pupils (71; 72; 7.1), although Callahan and Jacobson, in an experiment with retarded children. indicate that results achieved by pupils using rods were definitely better than might have been achieved inthe same time without rods (69). The evidence. although inconclusive, seems to indicate that children do need the help of structured manipulativematerialsingainingabstract number ideas but that flexibility of approach rather than one particular physical structure should be employed. Courtesy of Responsive Environments Corporation 314 call \ l'TER MN!

the nunulal p10% hie childten Ividi an oppoitu- nit to place c tibes in .1 panein. The raked outlines on the !maids suggest all "odd and een giouping. Boards of'his nature give children experiences ill :tssoc iating objec Is. ntimeials. and numbers. In addition. the number plopert of rqui,alent sets and pattern ielaConships can be obsehed. Other !maids 01 a similar nanne have raised outlines ill -domino- patterns lotea., emgn it ion. AnotluT important concept for primary (161- (hen involves die ordinal 1)101)('1y 01 number. Un(IerNIanding of this property is enhanced by having children build and order sets. The ap- FiGuit v. 9.26 patanis shown in Figine 9.26 is a typical device that can be used for this purpose. Sets with from one 10 ten members can be built. The illustration suggests that there are ten seigtrate baseboards. but there is lean). only one. The baseboard shown in the picture has been gromed to giw the int- piessionthatthere are tenlittlebaseboards. Since there is only one baseboard. it serves asIt control hi the ordering process. A variation of this device that uses multicolored cylinders and a wooden frame serves the same function (figure 9.°7). Activities with number trays C:111 be effective substitutes for number line activities. especially in the early stages of learning when manipula- tionis desirable. Placement of cubes in inter- FiGuRE 9.27 locking trays can be used 10 illustrate counting, adding. subtracting. multiplying'. and dividing numbers. The cubes in the tray in Figure 9.28 show an alternating pattern of odd and even numbers.

s y ujelop.ps1,61.11,41,* Perhaps the number board in most «ninnon use is the hundred board. Its flexibility permits concrete representation of counting. the four fundamental operations, the meaning of place value nouttim in the decimal system, and pattern analysis. Although available commercially in a variety of forms, number boards can be easi-

FIGURE 9.28

Photos on this page, Courtesy of Responsive Enahonments Corporation NI.ANIPULATIVE DEVIt:liS IN 1:1,1iNIEN-1 1IZ1 ',C,II001, NI VI III:NIA 315

ly constructed by pupils or teachers. Two coin. sent MOON have lines across themone line mercial vat iations arc illustrated in Figures 9.29 means one distinct pair of fa( tors: two lines mean and 9.30. the number has two distinct pairs of factors. Volpe! suggests that the simplest homemade A somewhat unusual use of the hundred board board is one made of plywood with nails or pegs in the intermediate and tippet grades is shoun in the( enter of each square (83). Val iations in Fip,tne 9.31. The black beads are supposed tan be ellected b using spools. curtain rings. to represent prime numbers; but one mistake has rubber washers, or number cards with the board. been made. If the board is used as :howl' in the Osborn's variation of the hundred board is a display, children can be asked to correct the cardboard chart on which each numeral has a placement of the black bead so that all prime circle or square above it(8 1). Circles represent numbers ate tepresented by black beads and all odd numbers; squares represent even numbers; composite numbers by white beads. black circles and squares represent prime numbers: and white circles and squares represent composite numbers. Circles or squares that !epic. FIGURE 9,30

04-104did4irQQQ-4-04.0 $6 4-04004:04;0404004iO40 Qiii4SQQQ1040. 404.0 QQQ404444;04:04.10 CSQOid4;04;04;04i0CIOQ 4144.0104604i0C/4004040 040410' 4.S4i0QQQ4id4.0 41;0*taiO404.10Q4;4Q4'd Q4;d4d4:0410' QQQCBOQ 44d4^d4-d4iO4)4r4r4;0 Courtesy of Responsive Environments Corporation

000000000 FIGURE 9.99 `0000000000 0000000400 O 000000000 O 000000000 O 000000000 0000000000 O 000000000 O 000000000 Ficuttr, 9.31 0000000000 316 (:li.WTER NINE

CAI:DS ANI) (:11A1:"1 S Gauls and( hartsforleaching mathematics ha% e retained their poptilaiir through the .% ears. They are appealing to tea( !leis not only be(atise of their perm:men( V but also be(ause of their adaptability to individual and group histitiction and to various methods of instruction. Mall) teachers feel that the use of cards and charts as twodiniensional representations or the physical work] serve:, as a bridge between the concrete Ficuitv. 9.32 and the abstract (Figure 9.32). /2 This bridge is often referred to as the "senti- concrete" level. It is not known whether le:lining is enhancedI))using cards and( harts. The -complete lack of research on this problem per. inits the teacherIt)select or rejectactivities involving these materials as he sees lit. There are many uses ()I' cards and (halts that ale unrelatedto the concreteseiniconclete:11). one strati secpience. Demonstration cards can be used to relate numerak and words that name numbers (Figure 9.33). Expanded notation cards (an be used in conjunction with a flannel board to show the meaning Of plate value (Figure 9.31). At the two manipulative level. notation (-aids can be used with a 10 x 10 tray. In Figure 9.35 a child has represented 3.1 concretely by placing 3 ten bars and I four bar on the tray. The coirect notation cards. "30" with a ".I" card placed over "0" in three the units column. are shown at the bottom of the illustration.

Courtesy of Responsive Enviionments Corporation FIGURE 9.33

EXPANDED NOTATIONS 3 2 58 i+50 + 8

X10;+ -1-:(5x10 s+ , FIGURE 9.34 8 3x103) + (5x10 1+, 8 NIANipi: LAT! vE DEviciis IN ELDIEN-r It) solo°, NI1.11ENI Vries 317

by pupils as a telerence when it question relating onthli jars to this fact o« urred. 100 .\s the chats is e\panded. chilthen cats 01)Ser\c Girq Its Gel, ION 'e I.V 910.- dere:oping patterns. One important use of patterns is in introducing the zero facts. in which at least one addend is /en). the identity element in addition. Since these sums are often difficult to objectify. chilthert can predict the appropriate stint by : naiyiing the pattern (Figure 9.37).

+0 1 23456789 0

1 2 3 1 4

thirty four 5 34 r units 6

Coultesv of Responsive Envilonments Corporation 7 FIGuRE9.35 8 Drill or flash cards remainVOIMIar devicesfor promoting memoriiation of basic facts. It is im- 9 portant to remember that drill on basic facts should never take place before the child under- FIGURE9.36 stands the nature of the operation on which he 1 23 is 10 plat lice.' Other uses of cards, for building +0 4 interest as wellaslearning- concepts. are de- 00 1 23 set." )ed in the section On games. 1 1 2345 78 Two other important uses of charts that de- 6 serve mention at this time ate pattent analysis 22345678 and enrichment. One type of chart that can be 3345678 used in pattern analysis activitiesis a matrix addition chart.. preliminary use of sucha chart 445678 is to record addition facts as *,hey are developed. 5 678 For example. when the fact 2 3 5was developed in the rabbit story, this fact could have 6 789 been tecorded on the chartas illustratedin 7 89 Figure 9.36. Thereafter the chart could be used 8 I. See the discussion on dial and repetitive practice in The Learning of Alathematies: Its Theory and Practice, 9 Tentpfirst Ycaibook of the National Council of Teachers of Mathematics (Washington. D.C.: The Council. 1953). FIGURE9.37 318 cumiTFR NIX!

TI After the ( hart has been «mipleted. ( hildi en . (an pursue .4 piton seal( lilot interesting iela tts.. 11A-1- - tionships sin 11 as impel tit s of the addition op -."?.t -.'; eration and so on. (.11.iits lot ()Owl opriations (an be pity( !lased or prepared. Chalk (an also 1' fr.tr- Ire used 10 examine chant( terisd( 11011(1V( 1111.11 1111111(T:111011 :111(1 111111e matheinati( al ssienis. Flie use of (harts for enrichment is illustrated Ira1:igines 9.:;S and 9.39. Flit e two (hails (an .0- Ire used to help (1111(11 en gain in imileistanding 1 r 4.1 of the histoli(al de%elopment ofnt.Ithcmrti(S. .^ .111(11111.1.in101'11. (.1111(2.1(1to an ecition of (mews being learned. ()they spe( ili( suggestions foi the use of (aids and (-hails are made Deans. 1)1asin. In ;laln.uu. ... Machlin. ()%erholser. and %Villianis cola tesy of Pm d moio company pi 1 90). FIGuRit 9.38. In the first quarter of the fourteenth century King Edward II established an NIKASCI:FNIENT DEVICES inch to be the length of three full, dry barley- corns from the center of the ear, laid end to Yardsticks.metersticks.chalkboaid(Impasses end. and protractors. balam es. and individual student riders. ( ompas..es, and protractors should be at ail. able as stand:n(1 classroom equipment for use in developing ineastiren,ent con( epts. kir example. pupils might be given a number of foosplare .774 ...... - co,.**4 . 1. pie( Cs of (aidboaid and asked to «)%ci the Ilooi . -- ---;-:-.$44141.144-10"."`"""*" with these squaw shapes. Aftei finding the ale.' 1..14. .-347,:asporArrattql of the door in Ibis fashion, they (an be asked t a*, lt r to find ilie dimensions of the I him using a %.d I .4 ;. slick and ihen asked io( ()mimic the area of ilie 11(mr I)) using die lormula A he. situleni :vim has had such expelieme is not apt to con- fuse a linear unit willi a squale mill and will Ira:e a g,00d idea of area. JP' Sets of materials like the Lake and Islands Board shows) in Figine 9.10 ale :nailable and serve a similar purpose. The set shown in the picture contains unit cubes. unii Hat 'pieces. and a large pie( e Of inasoniie on which irregular shapes are palmed. Com elms in% olving boili Co, les of Fold Motor Company :111(1 V0111111C 1/C (ICVe101)(.11using such a set. FIGURE 9.39. During the sixteenth century M.- ;\trundle wheel (Figure 9.4 I) can be used legal rod was determined by lining up the to measure distances. placing the wheel so left feel of sixteen men as they came out of thatthe stalling- mark is down and pushing church on Sunday morning. the wheel along- a marked line. distances can be DEVIGIs IN Eu.,NII \ I Ry NcliooL 319

Iiicasonal In muffling the 'idled. 11) thing a ttmulle vItel the child has opportunity to gain an intuitive idea of the relation between the diameter and citctittilerence Of a citric. hob Ions that in\ olve finding distances b) using a o !iodic %died :.o. asik I.11:111/ed soled the

Set?,ofchoices lor meastning capacii are Photos on thi% page, fOU? ten of Itetimomre l',111.110,11nent% aailable in Mall) (Ili .11:11)C. and sites. One Cot 1)07a/ion such setis shown in rignie 9.12. Other types ate discussed in the section on models of geo men iselm ionships.

FiGintv. 9.41,

FuuntE 9.40 Ficintr 9.12 320 CI 1.\l'EEI: NINE

BalanteN such as the one shown in Fignie 9.43 tan be used lor %ariotts purposes. In the primary glades the de% ice can be used to illustrate. for example. 111:117 6 10. as shown in the piettne..\t a later stage. the balanteall he used with problems imoking le%ers. In the latter (.se. the illustiation «mItl be iniel 1/1 Vird as showing that ( I X 7) ( I 0) (I x :1) (I .\lore than one iveight (an he 111111;4 on a peg. Two iveights plated On the I; at at the left ivould balance three ieights placed on the -I peg at the right. shut- 2 < t; .

To find the height of a building. NCI111)i Ile device at some convenient distance from its base. say 50 feet. Sight to the top of the building Photos on this page, courtesy of Itlponsiee lin,,ironmentc Corporation through the soda straw and locate the interscr. NI v'o'ters IN Et.E \lEN S(:IR)01., 321

lion of the plumb line and the 51) (loot) line on theightliand ,(ale of die graph chart. The dotted lines on the graph ch:11 Iindicate whey: to lead the measm of that part of the height 01 the building that is above the devit e. Since the 'number pointed out On the bottom of the graph ( hart is 23. the height of the building is 23 feet phis the height of the device.

Other appropriate devices for fteld work activities include Ilie angle mirror. the alidade and plane table, and the transit. Shuster and Bed- fooldescribe methods of constructingthese devi( es :111(1 suggest activities for their use (91)). FIGURE 9.45 The results of a limited study by McClilitie 5110W a preference by kindergarten childien for the use of a unit type or measure such as a cubical counting block. rather than a rod or a tape. to obtain ()vend1 length dimensions (9.1). Bonnie reports 011:11 experiences Withyoung- can actually use the unit of ineasine involved. children seem to slime that ideas of dill:C.(1111101- thus beginning atthe conci etc level (93). He Ni(nial space or volume appear to develop more suggests that measultIllell tell Vir0111111CIIIS rich easily than ideas of area (91). He suggests 111;11 Withconcrete materials be provided by the to establish a film concept of volume. an activity first.grade teacher in older that pupils can solve such as packing solid Ivooden unites into iectangu- e yday in-s( hoot measinement problems. Park- lar containers be used. Similar activities can he er proposes the use of a in oject to motivate pupils used to develop a firm concept of ;ilea by consid, in making- various measurements around the evil g the amount of space in a plane legion (area) school (95). Such aproje( Itends to increase rather than the amount of space occupied in pupils" skillillusing actual measuring tools. three dimensions (v(thime). A suggested activity Swart reommends the development of it Inca- is counting the number of objects ()I'identical smement laboratory that would include mate. siie needed to cover a plane region. rial to provide experience ill measmement (97). In a study of beginning first-grade chilthen's He lists some materials available for less than concepts. Mascho conch:tied that measurement ten dollars and describes how such a laboratory should be taught in situations in which pupils could be used. 322 CI IA ['TER NINE

NIODELS OF GEOMETRIC RELATIONSHIPS -4111.111_ Geometric models include the familiar wooden rubes. cones, cylinders. and spheres and, in addition. many other sophisticated wooden. metal, and plastic models. Soule 1110dds such as the One shown in Figure 9.46 Lan he filled with sand or water and used when studying volume. Hat blocks such as those shown in Figure 9:17 can he used to help pupils %isualize work with areas and volumes and to make designs. Plastic blocks similar to those shown in Figure 918 are useful in design work, comparison of areas, and in working with rational numbers. Geometric puzzles such as those shown in Fig- ul e 9.49 provide interesting enrichment material.

Ki Angle boauls such as those shown in Figure 9.0 are helpful in providing visualization in working with angle measure.

FIGURE9.46 FIGURE9.48

FIGURE 9.47

Photos on this and the following page, courtesy of Responsive Environments Corporation NIAN1PULATIVE DEVICES IN ELENIEN l'ARY SCI1001. NIATIIENIA 323

FIGURE 9.50

FIGuRE

Johnson dem ribes a huge number of geometric terns designed lot ease of cutting and pasting concepts that can be illustrated by paper fold- (I 10). Woods and Hoff feel Olaf experiences with ing (I01).Included are the basic geometrical mathematical models lot geometties of more than constructions. the building of three-dimensional three dimensions should be pl ON hied at the ele- paper models. and the formation of regular poly- mentary level to reduce the difficulty pupils have gons by tying knots in strips of adding machine with ideas im oh ing more than duce dimensions paper tape. (1 I 3). In two articles on geometry inthe primary Combinations of pegboauls and string have grades, Vigilante notes that at this level geometry been suggested by many. Smith suggests using should be approached informally, using a theory pegs and tubber binders with the pegboard to of instruction based on multisensory experiences ptmide asuccessful mediumforportraying (108: 109). These expel fences should provide op. geometric ideas (106). Hewitt used a length of portunitics for exploration, manipulation, and enetian-bIind «d to fashion geometric figures development of spatial perceptions. Smith con- and then tied knots at intervals ofone foot, to curs by stating that children require things they obtain a meastnement tool to enable pupilsto can handle. see, feel, work with, and produce find the lengths of the sides and diagonals of poly- (105). gons (100). Using colored thread and a pegboard The construction and use of three-dimensional or heavy cardboald, Knowles had pupils con- models is a recommended activity. Buck suggests struct geometric (ism es using only straight lines the building of regular tetrahedra out of D-Stix (102). Major outlines a different, axiomaticap- (99).Black describes a plogramusing large proach to geometry inthe upper elementary wooden solids to help students find various shapes grades, an approach based on rings and strings in their environment and discover basic proper- used with a pegboard (103). ties of shapes (98). Wahl provides a set of pat- 324 CI PTEI: NINE

FIGURE9.51

Cline stiu hing. as shown at the 1dt in Figine can obsene that the medians, altitudes, and angle 9.31. is alwa)s popular with pupils and in o ides bisectois of an equilateral n jangle all coincide. anexcellent opportunity for cieative design The model can be easily adjusted to show that work. At the present time, many mechanical this is not the case if the triangle is isosceles or drawing do ices ale a\ ailable lor purchase in scalene. Such manipulation results in a sasing MOICS and in the toy sections of depat- of teat het time because the geomen it figures do ment stoles. The Sphograph and the Magic De- not have to be constructed on the chalkboald. signer are two (le\ ices that children can use to Other types of (des ices lime also been suggested. create did' own designs b) using the colored-ink Uncapher (des eloped an "Object-a-Scieen" on pens and the rotating puts inc Ruled with the which geomen figures can be represented devices. different arrairgements of light behind a translu- Nfeial and plastic models that incorporate the cent plastic mice!) (107). Walter cleated Mirror use of elastic ate useful in illustrating geometric Cards to pi m ide a means of informall) obtaining propositions. The one shown in Figure 9.31 can geometric experiences that combine. spatial in- be used to illustrate that the medians of a tri- sight and play (Ill). Richards reports many angle ale concurrent. Since this model can be ad- possibilities of using 'Finkel( )ys in constructing justed to show an equilateral triangle, childien models of geometric figures (101). NI NNIPI/I, \TINT DEVICES IN ELENIENT \RN' SC:1100I. NIATIIENIATICS 325

GA NI ES AN I) PI:T/.1.ES lte changing stt tic [me of the elemental), classtoom bow .1 teachercc meted. cloginati( 01 ienta IlollIo our of a child-mitered, learning labora to! here !canting is itilutt.sting. clijophle. often Ion. has «)ii.i(leral)1 ARAM the tole of games and pui/les. In days gone by. games and pti/zIes ieset cd lot kincletgat ten and early Ili »nary glades x%itli the exception of their use at oils r tirade I" cis on the PI c(edlilg 41 holiday as .1 -something to do" 11(6\ it). They were not iexcd as an iiItcgt.11 pain of the (Intl( tiloin 'sere seen by (cachts and administrators as a de\ ite employed by 'seal !cachets incapable of structuring !tlegitimate lesson. Today a strong case can be made lot including games and ptiiiles as .1 incons of attaining desh able outcomes of ui.itltcm.itic, inst tu. t ion. They seem to be especially x% ell-suited lot ptogiams of (hill .111(1 Prati(e, P10\ iding for indi' ideal dif" fct enc es.building desit able attitudes, and encouraging problem sok ing. Although commercially prepared games are be«)ming mote and mole popular and their immix:is ate rapidly increasing.thecleat ivenatureofelementary teachers and pupils piovides sources of ideas lor FIGURE 9.52 conn»eicial companies. Typical of leache -made games is .NIathi»agicland (Figure 9.52).2

$.Each player begins atthe place marked b feel ive: "S1:111.- 1'1:1(11 player chooses one card. and To give practice in bas, facts of addition and subtraction at the primary level and multipli the player with the card showing the great- cation at the intermediate level est basic fact starts the game. The cads drawn ale then buried at the hot win. The /11 a l cried s: play is then from left to right. Four toarkers:'s bite cards containing basic -I.Play begins whet the first player draws a facts without answers: geometric shapes: red white card. Ile computes the fact and moves cards containing' basic facts without answers: "tracks" for addition and subtraction. and his marker to the nearest numeral that designates the answer. (Example; 2 +.1 = (i.) multiplication: and the board The player moves his marker to the first Rules: "6." I. The game may be played by 2-1 players. 5. The numbers in the primary track arc ar- 2.Moves are determined by drawing a white ranged in sequence from 0 to 9 and then card. begin repeating the same sequence. Inter- 2. Candy akkunen, Judy Kaplan, Peggy Mahan. and Ruth Alverson, four former students of the authors of spersed in the track are frames containing this chapter. ate responsible for originating this game. geometric shapes. The numbers in the in. 326 ( 11 \I III

termediate level arc multiples of a certain .\popular commercial game thatprovides

number (e.g..3:3, 6, 9.12... .). These pupils with acti%ities involving addition is doini- numbers are not in sequential (tide' so that noes, Figures 9.5 I and 9.35 illustrate two of man) children cannot compute them by checking variations of the domino game. Practice is pro- the board. vided in this case with fractions. decimals, and 6,Play continues with each person drawing a percents. white card and moving. his marker IQ the nett se(pleIlte of limbers until one player teaches Mathmagicland. 7.If a player chooses a card containing a geometric- shape instead of a basic fact, he moves his marker to that shape directly and follows the directions there, if any. 8.If a player's marker should land on a frame already occupied. this pla)er must choose a red card. give the basic fact. and then wove Iris Miter backward to the nearest numeral that designates the answer. 9.A variation at the primary level is to have facts with answers greater than 9 and have the children move according to the figure in the tens place instead of the ones place. Figure 9.53 illustrates the use of a puzzle situation to help young children match sets of oh . jects with numbers. Flout& E 9.59 FIGURE 9.53

.c 4 %) 2 -`) I ")CV°

FIGURE 9.55

Photos on this page, courtesy of Responsive Envitonmenis Corporation "FIVE Ill:VIC:ES IN FI.D1 FN\ItY SCI 1001. NIATI 1 F.NI \TICS 327

:\11excitingclic1 game that combines the ability to compute with the ability to sore prob. 4 lents is 1

3. For a solution to the Instant Insanity pluile see ''A FIGURE 9.57 Note on Instant Insanity," by 'I'. A. Brown, Mathematics Magazine,September 1968. pp. 167-69.

vo.7.31RED IBLUE GREEN 1111111111111111 WHITE FIGURE 9.58 328 CHAPTER

There are many sources of information on available lehen needed have been doeloped types and uses of both «miniercial and teacher- (Figure 9.(i0). Students should have ampleop made games and punks. These include Calvo. pornmitv to use desk calculators. Deans. Dohler, 1 laggerty. Inbody. Johnson, Jor- The extensive use of the number line in mod. dan. l': l ker. Ituderman. Shur low. and Timmons ern mathematics programs has drawn new alien. 11.i -25). don to such devices as the addition subtraction slide rule. Figure 9.61shows an inexpensive SPECIAL. CONINITATIONAL DEN'ICES demonstrationaddition subtractionsliderule constructed with two yardsticks. To operate this Computational devices have been used by men slide rule. locale Elie first addendOn die bottom throughouthistorytoincreasethespeed. rule. Slide die top rule 10 the lightso that the accuracy, and efficiency of mathematical calculi. index (icy() end) of the role is aligned above the bons. The ancients found the abacus to bean first addend. Find the secnul addend on the top effective device for doing- calculations, and the rule and lead the number on the bottom rule widespread use of the abacus in the Orient today thatisaligned with the second addend. The confirms that it has continued to be an effective number indicated on the bottom rule isthe calculating device. Shown in Figure 9.59 isa sum. For example. the slide rule pictured in modern version of the abacus, designed for class. Figtue 9.61is set to add 8 and 5. The S is 100111 Use. clued on the bottom rule and the top rule is The historical significance of the abacus asa moved to the right so that the index is opposite computing device justifies the use of class time 8. Now 5 is found on the top rule and the sum for exploration of its use. of 8 and 5. which is 18. is found on the bottom In the culture of twentieth.century America rule. To subtract one number from another. align simple desk calculators that do not require years the two numbers and read the answer opposite of training to operate and are nearly always the index. The difference. 17 9. can be found in

FIG taut: 9.60 Fictokt-: 9.59

Courtesy of /deal School Supply Company Coui lesy of Clot r Business Machines company NIANIPLII.ATIVE. DEVICES IN ELFNIENT \RY SCI 1001, NIXI*1 III NTIC. 329

1 7 3 5 6 7 8 7 1. 11 18 11_1:_it 17_26 21 22 73 23 . 25 26 77 A z4

1 tt S 6 7 6 ? 3 ' 3 10 11 77 13 11 '4 14 1' 1v. 14 't 7" '; " ^p '1 1 31 3:)3 33 ' 3$

FR: t um, 9.6 I

Figine 9.61. A slide rule made with prdstiks tan also be used foi addition and subnadion of (el- lain rational numbers. Ordinary loot rulers ate :pp opt laic lor making addition-subtrat Lion N ide tides for use b Another example of a computing device is called Napier's rods. This de% ice. which is an adaptation of the lattice method or moicipika- lion. musk's or too Fmk and 311 111131(22: iod. The rods are usually It:fel-red to as the 0rod, 1-rod. 2-rod. and so on. Pictured in Figure 9.62 are the:; -tod. the 6 rod. and the index tod. A tont- plete setof rods is pictured in Figure 10.3 hi the next chapter. The rods may be in the form of rectangular prisms or rectangular cardboard snips as in Figuie 9.62. The top cell of each todis labeled by the numeral that names the rod. This numeral also represents the fast of two factors of a product. The remaining cells on each rod are labeled by numerals that represent the products of the first factor and the numbers Comlesy of Ideal Sillool Supply Company from 1 to 9. The diagonal slash in each cell FlGuttF 9.62 separates the tens and units digits of each product. find the product 3 x place the index is more than one digit in a diagonal column, the rod next to the 60d as shown in Figine 9.62. numbers represented by them must be added. In Then locate 3 on the index 10(1. and in the same the present example, the first digit on the right horizontal row lead 1/8 on the 6rod. This is of the deshed product is 2. The second diagonal

the product, IS. column contains twodigits. 1 and . Since To find the product 7 x 36, lead the numbers I + = 5, the second digit of the product from in the horizontal low to the left of the multipliel the right is 5. The diagonal column on the left 7 on the index rod. Eachdiagonal columnin the contains a 2. which is the third digit of the prod- row representsonedigit in the product. If there uct. Hence, the product is 252. 330 (;IL\h"I'IMINI

The special calculator devices described thus designed %%1101 allot% students intelesting prate far ate for use With the decimal system of nu- Ike periods on basiclac's. The Tabletamer. a

meration. Except for the desk calculator, each of whiled( bulk11.11 (Tomes with :1 flashlight the devices can be adapted for use with 'tondo i, batten.is one stub de% he. The basic mal numeration systems. tamer device is ph-tilled in the upper leli.hand It is not important for the child to memori/e part of Figure 9.63 It tan be used b students the basicfacts of addition or multiplication to practice multiplication facts. 'l'o check the when studying nondecintal numeration s)stents, prodtut. of two numbers. the student plugs in butitis necessary for hint to have these facts the lefthand elemode below the pair of num. available in exploring- and discovering- charac- bers to be multiplied and plugs illthe right- teristics or such numeration systems. A set of hand electrode below :t possible product. If the pondeeinial Napier's rods or a tiondecimal aba. answer is mrie(t. the (irctiii is completed and a (us can be most helpful in handling computa- green light Hashes on at the top of the board. tional situations. Plastic oeerhtes (all be used to (II:Inge the basI0 Numerous forms of circuit boalds have been device so that it (an be used to plot isle piacti(e

Comiesy of AIM Indultriec. Inc.

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461;4', ,.660 1/Ch. 0 .,)40s s,0"/ /, c 1 k I,0. 0 7'" /,, e 0\\ 'adf%\' azs*e+.a ," 0 4, s', -% % / t04s/!.ae e '% \ ...a.k%'44.`' e .0.,°A. ./ 0 .1 ./ hase'S "V 9"..-4), ) 46,44 1 7/, 04.',/sjfal k le\ 144,7s: \ stilts"' ,..p \t/14,, 6'41,. ).'"0,1,7've.tit .1\ % % \4i-A \ ,`'.>, .0, 2, s %/ ;../,. 66e1i % 1, % , "% Ne . % se e \ s -14° -z its,a s sa, ',% p , k k \ S404 1\ 6.14 , s e," 51/$41 '' A ',% '.4 \ 1 cr ^ sy.a /9 fl, \ S _Ad' p \ A .115, 41 %:' Ap lk '11 '1 , kf," ,l% e4,.1. 60 4. 4' 4°11 1* r 1 e ..L o- 1 k, ... 4 Jr Nfi %,,,,.*--s1 ...01)- 1) 4,,* ov 4 '6 Ss -.vs, 1.rsp.. st, 4e % stia a it A \ 4,s4:$44, 7 to \la 00,0 4 e \ 's 4,' 1 s 1411s fi* 01 Jj..0,00314 a0* ko,* % 40-k 1,-0.4 14- 's lest a- AD I1K211 to Al 'Is 10.71115.. 192121 11919119 11 1 16 11 19 12

1 111 1 111 I 1 112 10 1 1519 10 11 1 1

241 14411 iiP010010

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1 7 3 4 6 11 91011 12 1314 1511 1111

19 20 21 22 23 24 25 26 2121 79 3331 3233 3435 36 EMNINSMNKUMM011010/MOMMIMMOlain

FIGuRE 9.63. Pictured at the upper left is the bask Tabletamer device Jo' practicing multiplication facts. At the upper right is the 5(111W device with a subtuntion oveilay. An addition overlay is shown at the bottom. 7'he same (levier is used for all three kinds of facts. \TI VI:, DEVICES IN ELENIENTAItV SCI 1001.1.\ Ti I FNI \ TICS 331

-10 9 8 7 6 5 4 3 21 0 1 2 3 4 5 6 7 8 9 10 k I I

E II I I I I I 1 111111 2018161412108642 0 2 N.N.1. 6 8 10 12 14 16 18 20

1 I I I I %si I I I I 10987654321 0 1 2 3 4 5 6 7 8 9 10

VIG Olt r 9.6

with addition and subtraction facts. re!ation among the coordinates of the three Another special minputational device that has imelse0 Hon points dote mined h'aline that wide application in science and industr) as well crosses the three scales is B + I-1 E. The dia- as in the classroom is the nomograph. It allows gram inFigure9.6.Iillustrates the example the user to do rabid calculations by reading + 6 = numbers from scales drawn on graph paper or .\ multiplhation and di% ision nomograph can notebook paper. The diagram inFigure 9.6 I be produced rather easily. Obtaina standard illustrates 3 nomograph that is designed to add slide rule and copy the (:. A. and I),s( miles fr0111 and subtract integers. the slide rule on a diagram as shown in Figure It has three parallel wales. B. E. and 1 I. Each 9.65. If you pia( e a straightedge across the three male is perpendicular to the line that includes males in the diagram. the points of intersection the /en) point of each scale. The B and I-1 males ale related by the formula C x I) A. To mul- ale the same distance from the E scale. The tiply two !lumbers, locate one number Ott the C length of the unit segment on the B scale is the stale and the other on the I) scale. The line same :is on the II scale. The unit segment on the joining these two points willc cos; the A scale at E scale is half as long as on the B and 11 scales. the point representing the product.

FIGURE 9.(i5

2 3 4 5 6 7 8 9 1

2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

0 V I I 2 3 4 5 6 7 8 9 332 CI I \ 111 R I \ I

Tile Ilse ofthe.11):1( lls as a «nnputational CONCI J'SIONS 1:1A DI N(. device has been le«numended Its mans' audio's. I. FSF RCI I I lamillon e\plains it., use in addition and multiplication ;I 1)eN'aillt extends its use tonutl- rhe eN:1111111:111011 01 the litelantle and lescault lil:li(.ttion(19): and Smith dis( 'ism., addition in each of the ninelasses (4 aids seises to lent- lot «. the «nninents made at the beginning of the sith the aba( te. t(i:2). Jenkins pieselits a unique chapter. The use of manipulaike des it es in sun \idea olthe abacus as aI 01111)01ing tool tea( hint elemental \s( !tool s is an (13(i).I leak des( vibes how the abacusan be a«pted pro(edine and appeal, to 1w suppoued used in the tipper elemntar:g:ades to suengthen undeistanding 01all the °pet mime,. including thcolists mid cdtu Aims alike.1'550 dangris should Iw looted in passing. The lust %Yolk with de( imals and common liactions (:r3). dangel is that of a teacher's wholesale a( ( eptanc e 0.11(.1 suggestions lot thing the aba( us as a com- of one all - purpose de\ it e. Most ploponents of putational de\ i«. ate presented (:tinningliam. die use of manipidatise deices agree that a l'eleison. and Spro,.,Iti:I-10: 31). These slew:, :,e perhaps hest sommaiied be 1;ernstein. sho wide sariel t(ul(let i( es s110111(1 be used. The second danger :irises front the belief that points out that the ads amage of the aba( us as a devices leach niathentati:s. Devices thelibelVeS computational deviceisthatpupils (an use (10 not teach mathematics. Itis their use tinder visual. lac tied, and kinesthetic sense, (-I). the guidance of a wise teacher that determines The number line as a computational device their elle( tiseness in facilitating pupils' learning. has frequently been discussed in the literature. This means that itis necessary ;or the teacher to and most classroom teachers are familiar sith it. haw a thorough lac kgiound in the l.dagogical Ashlock.(:Iark. Cochran. Coon.Overholt. :is well as the mathematical grin( iples invoked Schinickrath. Sganga. and Shaw all suggest ways in the the of manipulative aids. in whi(li the number line (an be used (126: 128 Thus the question does not appear 10 be one :10: 139: VI:1.1(i). A note of caution regard- of using maniptilatise devices or not tisini.4 them. ing the introduction of the number line too early Rather. the crucial questionis which devices is given (113). Ile suggests that much should be used with particular pupils in teach- experience with «uurete objects must precede ing certain (incepts. 011 this question more re- any attempt to deal with number symbols. as the searchisneeded. since existing research gives change from objects to sInbols on the number little direction. line appeals to be difficult for many children. A ptipilmade slide rule is suggested by Cram- EVALUATION OF \1: \N 1 IT 1..V1.1 lick as an effective computational device (13.1). DE\'ICES The advantages he lists include pupil interest. cal 1 ,s-osei to out-of-school ac Os hies. an introduc- Es el)lesson in mathematics requites some sort tionto meastnement in physical science:, and of esaltiation plan to determine the extent to quick estimation of reasonable answers to a which the objectives of the lesson have 'ken problem. attained. Lessons that involve the use of mani- Kreitz and 1:101Iioysuggestusing Napier'N pillat is e devices ale no exception. Tea( hers select rods 10 stimulate niftiest in multiplication and inmipulatise des it es be_anse the) feel that the to compare placement of partial products in this devices call make a major contribution to the and the conventional methods of intiliiplication It uning pro( eSs. the do 01 not cannot (137). be determined unless evaluation is planned. As ()dietforms of computing- devices are de- :t general guideline for evaluation. the teacher scribed by Hyde and Nelson and by Lawlis should state his objectives clearly in terms of (135: 138). pupil behavior. N1 \NiPV1ATIVE DEVICES IN ELENIFN s( :110 )I. 1.11C% 333

Malli1)01:0ke de\ 110We%Cl. prcscill a %pc- 17. Gm this(lesIte be ustd if die dis( (net dal 1)1(11)1c0. Silk(' UL\ (le\ it es tequire si/- u(Iiswhy% edtote.nIi the in ..ble Cs01100 01 1000(1...( 11001 ,\g ems (:10001 :hed: :t1looll0 pm( hase.test. .111(1 perhaps dimaid OUTCOM ES .\ND PURPOSES devi«..s. itustments in inaniffitlati5e de- 18. Can this device be used to moth:fie .111(1 hes ate 1101Unlike illes1111C111s iu leN1110k.S awaken or broaden interest: leX,11100k Selel 110115. 5( 11001s generally adopt 19. can it be wed to reinforce (1;110:111\ (.1110(e(1111(.. de\ clop esaluatke cri 20. Can it be used to imprme (ommuni(ation: lel ia. and 111(41 examine the books On the market 21. Gm it be used to (larif) (on( effis? to 1111(1 the sok, thatbest Its the ( rite' ia. A "2. Can it be used to help pupils see telation- similar poxdute should be 1,Ilowed to justify ships? a heav inestment iu 111:11111/0131he (IV\ 1(C.N. The 28. Can it be used to help pupils de%clop skill 10110%011g suggested evaluation lobe (an sole in using methods of reasoning: as such a ( ritcrion for school sy tons: it (an also 21. Can it be used10 hIP pupils de` clop a was he used 10 help the individual teacher make de- of thinking: ( isions about indkidnal items for Ids classroom. 25,Will the use ()I the devhe enlaigc ro(aludaies% Evaluation Form for Manipulative Devices 26.Will the use of the device (IC\ C101) pupils' DATA a II cda ion of mathenuni(s? I.Usual name of the device 27. Can the (Ignite to be Used Io deC101) 9.Photograph, drawing. or picture of the inkiest? device Can the device be used to enhame pupil 3.Verbal description initiative; I.Commercial sources Can the device he used to awaken creativ- 5. Approximate cost ity or to arouse imagination? 6.Uses that can be made of the device 30.Can the device be used ill making summa- ries or in l'eViCVillg work? AVAILABILITY AND USE 31. Can the device be used as a guide to show 7. Can the device be made by a pupil? steps in a process 01 procedure? 8. Can the device be made by the teacher? ,39. Can the device be used for of 9.Is the device designed to be used primarily yet reution and enjoyment? by the teacher for demonstration purposes? 10.Is the device designed to be used primarily by the pupil? otri.00K FOR THE FUTURE I I.Is the device designed to be used by both The increased development and use of mathe- teacher and pupil? matics laboratories in the elementary school call 12.Is the device designed to be used by some- for a classroom setting that contains a multitude one other than teachers and pupils? Of manipulative devices for teacher demonstra- 13. If the device is to be used principally by the tion and for pupil experimentation. Without teacher, will he be able to onerze it easily question, projects like the Nuffield experiment and understand its use? in England' and numerous related experiments 1-1. If the device is to be used mainly by the in this country will have a heavy impact pupil, will it withgand wear? on the schools of tomorrow. They call for a complete 15. Can represcnuative learner behaviors bean- ticipated when this device is used? -I. See the Nuffield Foundation Mathematics Project series 16. Can its'effectiveness be judged or measured? (New York: John Viley k Sons). 334 (:11:11)TEI: NINE

resnucturingof the methodolog)of tontent pation by prk Ate enterprise 1.111 aWakell interest in mutation. a( tompanicd changes in school in ho al and state gout 'intents to in ov ide gicatel orgaiiiiation. Further impetus for the labor: tor) financial sitppoit to local schools. setting has come hom matelials developed 1), Mole pschologists hate directed their atteil institutions and companies likethe Learning lionto problems ol human leaining. As this (.entel and (:leatke l'Ia; things. both of Prime- iifteiest continues 10 glow. new inhumation on ton. New jerse). %chi(liae Lau:hill) Assembled the leaining pro( CNS Willill( 1 e.INe the ellideniy kit, of materials lot pupils to -le ;uu mathemati- Of the pupil who learns and the teacher who cal (0111 epts Iluottgh ex pet 'menial ion. guides the learning. .1.s pointed out in the be There is a glowing willingness on the pail of ginning or the (hapter. knowledge of the learn- Iotal school sstems to inovide Imam i.tl suppoit ing pi oc es, has led toe( onimenclations lot gicat. for die purchase ol manipulative (ievices. The er use of manipulative devices. impetus provided Ithe lederal go% manumit teachers hale demonstiated evei %cid'the enactment()Ithe Nacional1)elense in« easing willingness to experiment with new Education. \(tin the late Miles. followed I)) the tedinnities. strategies. methodologies. and loll Elemental and Se( onda Education. \(tof tent. The climate of the yeais prior to 19CM was has ante mei to lotal school (Us not faumable to experimentation..\s elementai

tt it is which .11e 110 budgeting monies on .1 teaches betame bettei:mill:tinted with newel egulaibasis Iol equipment and inrp105ement progiams and methods and began 10 undo:stand of Fat Hides. the changing goals and expected outcomes of a Learning- needs of pupil, Inane resulted in the 5(11001 mathentatits program. their interest in doelopment of .t valiet of ape( iali/cd progfams. t hildtcn and then (lushes to do the best Lou them .\ mong these are l'ioject I leadstai t. inograins for heed the teat hers flout the (hail's of stagnated Ilie disathantaged. And programs for inogiams and teaching techniques. Elenientar slow and gifted learners. .\II of these (-allfor teacheis have not only become capable of carry- increased INC of manipulatke de% it es. The Na- ing mu lintimlEt:se:itch and experimentation t nial (:(min i1 ol Tea( hers of \ fat heinatics. in its with manipidaike des it es but 11.11e demonstiated ploiet lor development of special unit, of intheir interest and desire to do so. s 1 act ion Iol slow !camels. typifies these CHOI Is. The role of manipulatk 0: de% ices in the elemeil- Pile apploach used with pupils ha% ing pal tic ttlai tatschool mathematics ( lassioom of the Intuit- lc:lining difficulties k hugely explimemation looks bright indeed. with manipulative devices. The involvement of bilge business cuter prises in die education field offers three main advantages. First. the development of new technology will have a duce! path to the educational establishment. The lag between these technological advances and their educational uses will be reduced to a minimum. Second. the educational institutions.lecogniiingtheir needsfolnew technology. have direct aces to companies with highly skilled personnel who can create the tech nologial advances needed. Third. such partii.

5. See Expo ione's in illathematiral Ideas, 2 cok.each withTeaching Parhage, ed.AinoldM. Chandler(Wash- ington. Tht Council. 1970). 'MANIPULATIVE OEVICES IN ELENIENTARY SCI 1001. L\ !ICS 335

SO r RCES OF NIANIPULATIVE 'MATERIALS FOR ELEMENTARY SCII0OL MATHEMATICS

The (ompanies whose names appear in the list below piodineor disiiihme manipulative devices that are appropriate for elementary school matliemaths instruction. Thehill names and addresses of the companies are given in the .1ppendix. The (lasses of materialsmailable from each company me indicated bl x's accouling 10 die sorting scheme used in this chapter:

I.Demonstration boaidi and devices feasinenient devices 9.Place value tlevi(es 7.Models of geometric relationships 3.C:olored beads. blocks. rods. and discs S.Gaines and piriiles . Number 'maids 9.Special complitational devices. 5. Cmds and chaits

z vi ::: . :- 07.

:., =Id =:- bF: l'12.01)1.1(:E12. OR DISTRIBUTOR F45 ....-..- Luc; 7.0 9 6 9 ABC: School Supply. Inc. x x x x x x x x x :\cadentic, Industries. Inc. x The Advancement Placement Institute N Acro Educreional Products x Aestheomety. Inc. x :11 \I Industries. Inc. x Arithmetic Clinic x At ithmetical Principles Association x Associated School Distributors, Inc. N. N N N N N N N N AV:11011 Mill Company x M. C. Ballaul Company x licckleyCady Company x x x x x x x x x BenGProducts. hue. x x Omitting L. Bete Company. Inc. N x Book-Lab. Inc. x

Stanley Bowmar Company. Inc. N x Milton Bradley N N N N N N N N N Bremner Mtiltiplication Records. Inc. N C:adaco. Inc. x Caddy-hider Creations, Inc. x x Cambosco Scientific Company. Inc. x Carr Plastics. Inc. N x Cenco Educational Aids N N Champion Publishing Company x Childcraft Equipment Company. Inc. x x N Robert R. C:lamage. P. A. x 336 CHAPTER NINE

PRODUCER OR DISTRIBUTOR

(5 7 9 Claridge Products and Equipment, Inc. x John Colburn Associates, Inc. Cooper Mothers Company Cooperative Recreation Service, Inc. x Corbett Blackboard Stencils x

C:leative Playthings / Leaining Center X X x x The (:-Thrti Itu ler Company x C:uisenaii e C:ompany of America. Inc. x The Curia Company x Custom Fabricators, Inc. x Daintee Toys, Inc. x x x Dana and Company, Inc. x The Denny Piess x x x Ralph I). Doncr. Mathematical Puiilcs Dyna-Slide (Sec Science lelated Alaterials) Eckel and Ballard x Edmund Scientific Company x Educational Aid Publishers x Educational Development Laboratories Educational Fun Games Educational Playthings x x x Educational Supply and Specialty Company x x x x x x x x x EduKaid of Ridgewood x x x Encyclopaedia Britannica, Instructional x Materials Division Encyclopaedia Britannica Educational Corporation E.S.R.. Inc. x ET A School Materials Division Exclusive Playing Card Company x Exton Aids x Fearon Publishers x Foi tune Games x Franklin Teaching Aids x x x Frederick Post Company x Hans K. Freyer, Inc. x Ganico Products. Inc. x x The Gangler-Gentry (:ompany x Garrard Press x x Gel-Sten Supply Company, Inc. x x x x x x x x x Genius Supply Company x GeodestiN Geyer Instructional Aids Company x NI \NIPULATIVE DEVI(:ES IN ELENIENTAItY SCI lOol, NI %THEM VI-1(:S 337

r- ,d; PRODUCER ORDisTRI BuToR

Ginn and Company x x N G\ \` School Sopply Specialists x x N x x I lall and McCleary Company x J. L. Hammett Company x x X N N N X X X I I:11 COUR Bra( C Jovanovich. Inc. x x x x x Nliles C. Hartley x Hayes School Publishing Company, Inc. x x I lelbeg Enterprises. Inc. x I Icrder and Mei der x x x !loll. Rinehart and Winston. Inc. x x x x

Houghton Mifflin Company X x N I lubb,ud Scientific Company x I ludson Pt ()ducts x Ideal School Supply Company x x x x x x x X x !MOM x

Instructional Aids. Inc. X X Instructional Nlaterials (:ompany x Instruct() Products Company X X x jael onda Nlantiracturing Company X dell', . \i'11hIlletie GallICS

The 1 tidy COMpay x x x x x x Kala. Inc. x

The Kendry Company N. Kentvorthy Educational Service x x Kindrey Nlantilact tiring Company x Kohner Brothers, Inc.. Trvne Game Division x Krieg GaIlle.s x Krypto Corporation x Lanco x. Lano Company x x x LaPine Scientific (:ompany x The Leaning (:eer x x x x x X x Little Red School House x x x E. S. I.owe (:(mpany. Inc. x 3I) MagnaGraph (:orpoiation x

Mainco School Supply Company x x x X x x x x N. Math Nlaster Labs. Inc. x x x x x x x x x Nlat Nledia Division. H and M Associates x x x Nlathaids Company x Nlathematical Pie, Ltd. x x N N X X X X X Nlidwest Publications, Inc. x x x x Miles Kin>ball COnipally x hineNota Mining and Manufactm ing Company 338 CI L\ ',TER NIN

> PRODUCER OR. DISTRIBUTOR n 9 Alodels ol Industry. Inc. Moyer Division. Vilas Imlustries Limited x x .NIcGralv-Ilill Book Conipany. Educational x x Games and Aids Nasco Mathematics x x x x Nilty Division, Si. Regis Paper Company x x x x x x x F. A. Owen Publishing- Company x x I

Pacific Coast Publishers x . x Palheys School Supply Company x x x x x x x x x Physics Research Laboratories, Inc. x Plaway Gaines Playskool Alantilacturing Company x The Plymouth Picss 'I'he (liar les T. Powner Company N Psychological Service (luau t um Corpora lion x Responsive Environments Corporation, x x x x x x Learning- ,Nlaterials Division Si. Paid Book and.Stationery (:ompany x x x X X x Saigent-Welch Scientific Company x x x School Material Company x x x x x x x x x School Products Company, Inc. x x School Service Company x x x x x x x x x Science Prodtio ions x Science Related Materials. Inc. x x x

Science Research Associates. Inc. N x x Science Seminars. Inc. Sciences Materials Center Scientific Educational Products Corporation x x Scott. Foresman and Company x x x Scritchlieldlantifacttiring Company x Selective Educational Equipment (SEE), Inc. x x x x x x x Self-Teaching Flashers x Sigma Enterprises. Inc. x Skool-Aids Corporation x

The Speed-Up Geometry Ruler Company, Inc. X Standard Education Society, Inc. x STAS Instructional Materials, Inc. x x x Steck-Vanglin Company x x Summit Industries Systems for Education. Inc. Tal.(:ap. Inc. Teacher's Aids NIANIPULATIVE DI VICES IN EI.ENIENTARY SCI1001. MATHENIATICS 339

it JU Dc; PRODUCER OR DISTRIBUTOR. OJ yW

6 9 Teachers Publishing Corporation x x Teaching Aid x x x x 'Pouch, Inc. x Fern Tripp x 'RIF (Sri Avalon Hill Company)

Charles E. Tuttle Company, Inc. x x United Chemical and School Supply Company x x x x Viking Company Vision Incorpotated x x Vis-X Company \\Tabash Instrument Corporation x Walker Products x Walker Teachim2; Ptograms and Teaching Aids x w:1,14 Laboramies, \Veber Costello

Webster Division. McGrawHill Book Company x x Webster Paper and Supply Company x \Veldt (See Sat gent-Welch Scientific Company) western Publishing Company, Inc. 'N PROOF

\V. 11- Company' World \ Ga 1,. NI. \Vright Company x Xerox (:orporation. Curriculum Programs x x l'oder Instruments x 'Lima Educational Materials

SELECTED BIBLIOGRAPHY

For the readers greater convenience, entriesin this bibliography have been separated into several groupings, as shown below:

A. General Relerem es. entries 1-38 F.Cards and Charts, entries 81-90 II.Den ionmnition Boards and Devices, entries G. Nleasurement I)evices, entries 91-97 39-42 H.Iotlels of Geometric Relationships, en- C.Plat e Value Devi( es. entries 13-66 tries 98-113 D. Colored Beads. Blocks, Rods. and Discs. I. Games and Ptinles, entries 11.1-25 entries 67-79 J. Special Computational Devices, entries 126- E. Number Boards. entries 80-83 ,16. CI IA PTFli NINE

A. GENERAL REFERENCES nsti uct iona 1Materials." 1 n Distinctionin Arithmetic.Twentfiltlt Yearbook of the Na- I.Adkins. Bryce E. "A Topical Listing and Ex- tional CouncilofTeachers ofNlathentatio. planationof SelectedInstructionalAidsin \\::shing")11. D.C.: The Council. 1960. Pp. 22,1- Arithmetic."Diswrlalion Abshacts19.no. 7: 1609. 1958. 1(i.Harshman. 1 lanlwick Wilton. "The Effects of 2.Adler, Irving. "Mental Growth and the Art of Manipulative Matetials on Alithinetn Achieve- Teaching." ArithmeticTeacher. November1966. ment of FhstGrade Pupils."Dissertation Ab pp. 576-8.1. snails2 :3. no. I:150. 1962. 3. Andetson. George R. "Vicual-Tactual Devices 17.Harvin. Virginia Raines. "Anal)sis of the l'ses and Their Efficacy: An Experiment in Grade 8." of Instructional hfaterials by a Selected Croup Arithmetic Teacher.November 1957. pp. 196- of -Teachers of Elementary School Mathematics." 202. Dissertation Abshacts25, no. 8: 4561. 1965. 4.Bernstein. Allen L. "Use of Manipulative De- 18.Hauck, Eldon. "Concrete hIaterials for Teach- vicesinTeachingNfatheinatics." A r ithmetic mg Pet centage."Arithmetic Teacher,Decent- Teacher.lay 1963. pp. 280 -83. ber 1954. pp. 355-57. 5.Bruner. Jet otne. The ProcessofEducation. 19.Heard, Ida Mae. "Using a Math Caddy to Store Cambridge. NI ass.:fiat vard UniversityPress. and Display Manipulative Devices." Arithmetic 1960. Teacher,October 1963, pp. 355-57. 6.Dienes. ZoltanP. "Some Basic ProcessesIii - 20.lien?. Pauline. "Manipulative Devices in Lower volved in Nfathentatics Learning." InResearch Grades." ArithmeticTeacher,November 1957. in Mathematics Education.edited by Joseph M. pp. 214-16. Scandura. Vashington. D.C.: National Council 21. Kennedy. Leonard M.Models for Mathematics of "Teachers of Mathematics, 1967. Pp. 21-34. intheElemental), School.Belmont,Calif.: 7.Edison, William P. "The Role of Instructional Wadswor th Publishing Co.. 1967. Aids in Arithmetic Education."Dissertation Ab- 22.Lerch, Harold 11.. and Charles T. hfangruni 11. stiacts16, no.I I: 2095. 1956. "InsnuctionalAidsSuggestedby Textbook 8. Raven, JohnII.The DevelopmentalPsy. Series."Arithmetic Teacher,November 1965, pp. chology of Jean Nagel.Princeton. N.J.: 1). Van 543--16. Nostrand Co., 1963. 23.Macy. Murray. "The Effectiveness of Rept esen- 9.Fox.Ni:11'1011 W. Manipulative Materials in tative Materials andAdditionalExperience Intermediate Grades."el rithmeticTeacher, Sitnations inthe Learning and Teaching of April 1958. pp. 140-42. FotnthCrade 'Mathematics."Dissertationelb- 10. Gagne, Robert M.The Conditions of Learning. shorts 17. no. 3: 533. 1957. 2d ed. New York:I loll, Rinehart k Winston, McIntosh. Jelly, and Philip Peak. "Materials of 1970. instruction." InThe Continuing Revolution in 11. Glennon. VincentJ., and C.'11'. I Ittnnicut. Mathematics.Reprinted fromBulletin of the IVItatDoes Research Say about Arithmetic? National Association of Secomhoy school Prin. Washington, D.C.: National Education Associ- cipals.Washington, 1).C.: National Council of ation. Association for Supervision and Curri- -Teachers of Mathematics, 1968. Pp. 155-66. culum Development, 1952. Out of print. 25.MaltiSensmy Aids in the Teaching of Mathe- 12.Goals for School Mathematics,Report of the matics.Eighteenth Yearbook of the National Camblidge Conleience on School fathetnatics. Council of -Teachers of Mathematics. New York: Boston: Houghton Mifflin Co.. 1963. BureauofPublications,TeachersCollege, 13.Grossnickle, Foster E.. Carlotte kluge. and Wil- Columbia University. 1945. liamhletmer."InstructionalMaterialsfor 26. °Amin, Roger. "The Use of Models inthe Teaching Arithmetic."InThe Teaching of Teaching of Nlathematics."Ailthinetic Teacher, Arithmetic. FiftiethYearbook of the National January 1961, pp. 22-2,1. Society for the Study of Education, pt. 2. Chi- 27. Nagel, .Jean.The Child's Conception of Num- cago:University of Chicago Press,1951. Pp. ber.London: Routledge Kegan Paul, 1952. 155-85. 28.Schott, Andrew F. "New Tools, Methods for 14.Ilamilton, E. W."ManipulativeDevices." Their Use, and a New Curriculum in Arithine- Arithmetic Teacher,October 1966, pp. 461-67. tic."Arithmetic Teacher,November 1957, pp. 15.1 I ardgrove. (Induce Ethel, and Ben A. Stteltz. 204-9. m \ TivE DEVICES IN EL :m EN I' \RV ..,c11001. m \ t IC's 341

29. Sobel. .111(1 1)0110% \.101111S011. \n,11% M. Perkins.(:;nol,and Nan()Ilanson."11()m sis of Illustrath r lest Items." InEvalualion Alan) \Va\s?"lizthinclu Tea(ho. Match 1968, .11alhenialu%. relit)sistliYearbook 1/1 the p. 277.

National Council of Tr:idlers of Mathenialics. II.Peskin. \nu 8, "Crouton( Itepirsoitalion If \Vashington.D.C.' I he (.011111( il. 1961. Pp. B11101111.11 1)%1111011111:11^1.:11/1)1111011 St1 lulh 71- nielu Tou her.January 1968. pp. 10-11. 30. Solr. Da% the l'N ofMaterials inthe 12.Such/. Ben\. "Counting l)evi«'s andFilch l'ou king of \lithini(.- /)m(tation Ai's/mit% Uses.Inthmetz( 7'«z(h(9, Februan 1931. pp. 17. no. 7; 1517. 1957. 25-30, I.Sptoss. Pan i( ia .\I. "Considerations in the Selo tion ofLearning Aids." :bithmetieTeacher. May1964. pp. 350-33. C.PLACE VA I E DEVICES :(2. \ Studs of the Elle( tof a Tangible 13. \dkins. Julia I "An Ilistorical and Anabik al (:onceptttaliid Pisentation of Mithintie Setts,1 ow aw,. the Knotted Cord, the fing- on \chir)'emnt in the filth and Sisilt Cradc." ers, and the Abacus.Dicerlalzon .11m1)ach16. /)iNseitatitm ilbctuzetA 23, no, 4; 1293, 1962. no. II: 2083. 1956. 3 :(.Suckled'. Nlaitlia ( :ooper. \stI listori(al Sti)e) 11.Battingariner.1:in get y. -What Can l011 DO with of the 1)ewlopment of the l'se of Fushun-lion il an Egg Canon:" ArltionetuTeacho.May 1968. Mato ials for the 'Teaching of Ai itlimeth in the pp. 156-58. Elementary Gtades.1, Recorded inSelected IS.Beckmann. Milton W. "New Des ices I :hind:or Publi(ations." 1)ANeyhition limba(ts 23. no. 10: \nanny' ic." Touho,October 1960. 3822-23, 1963. pp 296-3(11. 31_ Such/. Ben\"Counting Devices and Then 16. Bridgels. Ika)mond 11.. Jr. "Easily Nlade ,Minh Uses." .,'Inthmetu rewher. February 1951. pp. wok Aids.".Iiithm('litTeat/o.1)1.1(1111w, 25-30. 1963. pp. 307-8. 35. Ser. Henn, "Sensor)Leaining\ milledto 17, Clan. Robot C. "Tea( king Aids lor Element:it y NI:talon:11N s." InThe 1.ewning of .11nlhemal School \rithinic," l)thmethTouho.Febru. Its (uuI I'maethe.Twent).first Year- aly 1966, pp. 135-36. book of the National Cousuil of Tea(iels of 48. Cunningham. George C. "laking a Counting Nlathematics, \Vashington. 1).C.: The (:ottluil. \ !mots."Inihnicti(Teach ml,Erbrualy1967. 1953. Pp. 99-155, pp, 132-31. 36. Swi(k. Dana 1. "The Value of MultiScitsol)' DeVatili, Vert.. " Abacus and Mtiltipli. L :lining \ids in the Teaching of .\ritlintokal (ation," .1rithmetitTeacher. 1956, p. 65. Skills and Problem Solving:\ Npen'sliental 30.Fiewelling. Robot \V. "The Abacus and Our Study." /)iscr)talion.11),1,arli20, no. 9: 5669. Muest015:' Toicho.Feln nary 196(1. 1959. pp, 101-6. 37. Van Engel'.I lolly. "The l'onnation of (:otr. 51, _ _ Ike that us as an Arithmetk Teach. (opts."In The I.ea of .11alhenuilies: lic iitgDevice,ArithnieticTeahe,. N(manbcr Theo) curl Fwenty.filst Yearbook of 1955. pp. 107-10. the National CouncilOrV:1(11(1sof 32. Ganegno. C. "New Developments in Arithmetic mat s,\Vashington, 1).(:.; The C:otnieil,1953, Teaching in Britain.",bithmetie Teaeho. April Pp. 69-98. 1956. pp. 85-89. 38.\Villiams. J. I). -re:idling \iiihnicti( by (:"' 53.Healy. Daniel J, "Ancient Devices in reaching (roe Analogy.I. Miming Devices:IL sum. Nbulcrn Arithmetic:4.11ilhougu Touhev.April ittral Systems: Ill. Issues and Arguments." Mu. 1960. pp. 206-7. rational /?esea(ch, 1:ebruary 1961. p. 112; June 54.Hooper. Barbata "An Expenment withI land. 1962. p. 163: l'ebnialy 1963. p. 120, Tally Comae's," Ailthmetie Teacher. Novell' her 1955. pp. 119-20. B. DENIONST1tATION BOARDS AND Jamison. King \V.. Jr. "An Experiment with a Base Abacus." DEVICES Variable ArithmeticTeacher, Fbrulny 1964. pp. 81-84. 39. Cunningham. Ccolge S..and DavidRaskin. 36, MacRae. Ilene' R. "A PaceValue Caine for "The Pegboard as a Fraction Maker." Inthme first Gladels; A Teacher.Made Device." A?ith lie Teacher.Ahodl 1968. pp. 224-27. meth Teacher.November 1957. pp. 217-18. 342 (:II \P FR NIN

57.Nlaer. Louise " Hie Scat bac us or Scaisdale non,tothe Cuisndire Cattegno \ bac its." AnihnzetstTeacher. December 1955. I rnInuent Teas her. Nos ember 1957. pp. 191-95. p. ri9. 73. Stanlel.I he Ella t of Amilifite 58. (111111, and Roben Brower. "The \h,1 Blot 1,Training on Childten's Delelopment ol Nel% Use lot an OldI ool." ithmetzt (mu( epls." /)ow:fa/ion 111,11(111% 27. Tem herDetember 1959. pp. 31.1-16. no. 8: 2100 \-2101 \. 1967. :)9.Pete' son. \ Va ne, -MI111(%111011-.1 Fle:,11 Look:* 71.Ltic(m. II, Vestingthe Cnisenahr habit:u 7 rather, Nlay 1965. pp. 335 -35. N1(701401." Authnzetzt'I eatlrer. November 1963. 60.Rabinowiti. Frederick R. "Building 'Comptit pp. 135-38. els' for Nondec imal Number Systms." ,Inihme 75,Nast a, Donald. "( outp.n.ttm Merit, of .1 \I. nip titleashes. October 1966, pp. 192-93. uLttise .\pploach to Se«onclCiade\ 61.Ralmilow. Hatold F. "Understanding Dilleient hill:moil 'lens lies. NIalch 1966. pp, 221-26. Number Bases.',Itzt/ozzetzt Tenche). Ma 1965. 76,Pass, Robert \ lbert. "The Ellett ol (atisenaite pp. 339-10. Nlatei Iteasoning and Conlon:16(m." 62.Smith. Rolland R.Fhe TenTen, (muffling Asa/it:relitTeacher. N(rvembet. 1963, pp. 139- Frame." .1 rit h meth Teacher. November 1956. 10. pp. 197-200. 77. "1 eon Do Cuisenahe Nlaterials in a 63.Swenson. Esther J. "Pop Coes an Idea." :rill Nlodified Element:11v Nlathematics Plogtaiit . \I Teather. October 1961. pp. 125-27. Ie(t the Mathematical Reasoning and Computa 61.Tuttle, RuthII. "Commas? Yes. But . . 1401:11 Skill ol ThinlCoad (:hildten?" /)is,: I rithmetit 'readier. February 1958, pp. 25-28. Maim tN 21. no. 1506-7. 1963. 65.Upton, Frames Caroline. "AnExperimental 78.Itochnan. .,1:tines1. "Equal Time." .1u/hmetzt Studs ol Teaching Aids at the FitstGrade Les 7'etzeizer. Nlay 1961. pp. 312-13. e17 1045 Beearch p. 91. 79.Stern. Catherine. "Elie ConcleteDevicesol 66.WeY" PoPside:. rithmetis Sti \ Teacher, Teacher. April 1967. pp. 312-13. \ pri 11958. pp. 119-30.

I). COLORED BEADS. BLOCKS. RODS, E. NUMBER BOARDS AND DISCS SO. Fisher. Alan A. "The Peg Board -a 67. \ E.E.. and I,. Royal Nelson. "Begin- in Teaching Nlatlietnatics." ,;ItithrtzelitTeach- ning Number Experiences and Structured Ma. er. April 1961, pp. 186-88. terials." .I :aliment Teat her, October 1963. pp, 81.Osborn. Jesse. "The !lunched hoard." .1lithrm 330-33. tit Teas her. Nlarch 1954. pp. 51-55. 68.Brownell. \ Vahan) A. "Arithmetical Abstnn t ions 82.Shaw. Dula II. "Utihmtion ol Teaching Mate -Plogress toward Maturity of Concepts under tiarain Fist (trade Nlathentatics." Arithnzeth Differing Programs of Instruction." iisithnzetu Teacher. Jannary 1963. pp. 37-11. Teacher, October 1963. pp. 322-29. 83.Volpe!, Nlain C. "TheI Iumlied.floard." 69. Callahan. John J.. and Ruth S. Jacobson. "An z thnzetu'readier. De«inber 1959. pp. 295- Experiment with Retarded Chilchen and Cui 301. senahe Rods." ArithmeticTeacher, Janumy 1967, pp, 10-13. F. CARDS AND CHARTS 70.Fedor. John Peter. "A Study of the C:ttisenahe- Cattegno Method as Opposed to an FA .\1), SI. Edwitia. "Gimping-al) .\ id in Leaining ploach for Promoting Growth in Operational ultiplicat and Division." .1)n/one/it Teat h Technique and Concept Maturity with Fitst r, January 1961. pp. 27-31. Guide Children." Dissertation Abstracts 27, no. 85.I)rasin. Lillian Packer. "The Forgotten Level: 3771.\-72A. 1967. SemiConclete Nlate) ials-a\'alttableBridg." 71.Hollis, Loye Yvonne. "A Study to Compare the ,Irithozetu Tachrt. November 1957. pp. 211- Effects of Teaching FirstGrade Mathematics by 13. the GuisenaireCattegno Method with the Tra- 86.Ingraham, Elisabeth."Flash-tabs.".Inflom111 ditional Method." Dissertation Abstracts 26. no. Teacher. AI» it1965, pp. 289-90, 2: 904 -5. 1964, 87.Nlachlin. Ruth. "The Use of Oerlay ( :Ross." 72.Howard. Charles F. "British Teachers' Reac- :Is ithnietit Teacher, December 1961, pp. 433 -35. NI \ NIPC1. \TIVE DE\'ICFS IN F.I.FNII.N 1 \R1 11001, \ 1111.N1 \ I I(' 343

SS. Nli(11:110. Nlatt. Versatile Number Ittin Maim, Jam( , I. -Ittngs and Sump,.'' Itzthme ner.- Teae het\pail1961. pp. IS2- hr '/'rat he:. 1966. pp. 157 -60. 55. 101.!tic haids.PaulineI.. I iukettot (40111(1). 0.111,,ker jean S, -1:1(m (:tart, lor the Ele I tt ih m et reTeat het. 0(11(ber1967. pp. 165-69. mental Grades.-bah/Hein"l'eae het .N.11 ent !tin int( (Iinlleadtm..!% inGagman nom the her 1966. pp. 719193, Inth incite'real het ashinglonI).L.:NA- 90, WW1, CatherineNI. -The Functionof coamil of I c.alleisofNiallienuti(s. Chalk in dierithineti( Plogrant."hithmetit 1970. Pp. 60-61. 'rem her. 0(tuber' 1955. pp, 72-76. In), Smith.LewisB. "Grown). 1'e, -hut 11(m:- I tt th Tear hetFein tial) 1967 pp. 1-59. G. I .\SURE TENT DEVICES 106, "PegbitaidGeomeny.".1 n th met te 91.Butane. IlatoldN. -TheCon( ept 01W.A." Teat her.\PI.'" 1965. pp. 271-7. Reprinted iu IttiltuteittIeat hetNlat(lt 1965. pp. 233-13. /:eatinittsin(;eametorItem the habit:rile 92.Da is. 0. I...jt,. 11:11barl (..11110% :111(1 Cato111 Teat her." \'asItinglon.D.C.: Nation.11 Conn( il Ctigler. "I he Catarrh (t1Preschool (:hilellett's 01 I ea( het. of NI:latent:1th 1970. Pp. ,56 - -59. F.Intilial it% withNleastnement."Anthraeng 107 Cli(Apher,ChesterL. Jr."1 he'Ohje( Tyra he:0(tober 1959. pp. IS6-90. Sc reen': Nla(hine lur 'rem hing Flententat 93.Nlas( ha. Ceolge. "Familiarity withMeasure Nlathemati(s.- ilhoteli(Teag het .0( tuber mew:\ Sind) 01 Beginning FirstGrade Chip 1965. pp. 162-65. (km', (,0m hishmei''earner.\plii 10S, Nkhob,j.-Geonleil) 1961. pp. 161 -67. (:hildmi: Insider:111011s.-With:tient 'feather. 91.N1( (Minh.Joan. I:indelgat tenChild 0( tither 1967. PP. 15:1-59. 1(19. NI eastn estap.' ill:mein'reacher. Jannal y \\it)Gil( =vent Geontity in the 1968. lip. 26-29. I'tiut :ur Glades:A ithutelie Tem her.0( tuber 95.Pat ker.1 lelen C. "Teadring Nleasurement in 1965, pp. 150-51. ReprintediuI? caeli ngs in Nleaninglid \Vay.,Iithmetit 'reacher. .\inii ( ;roman ;It amthe ill:a:elle Tear het." 1960. pp. I91 -9S. \Vashington. D.C.: National Coup( it at Teach 96,Shuster. Carl and Foe(' Bedlonl.Field froth ers of Mathentati(s. 1970. Pp. S7-91. 110, in.11 a thema te%.. New \meri(an Book \ %%dd. I. Stoessel. "Easy.to.Paste Solids.". Co., 19:15. met ie Octobet 1965, pp. 1(3S-71. \\'alter. 97,Swart.WilliamI.. \Labor:wilyPlanlot' Nlarion.'',111 Example of ham mai Tem hing Aleasmentent iu Glades ..hith. Geontei): Minor Canis.".1 )1 thm et ieTeachet mein Teat hetDe(eunber 1967. pp, 652-53, October 1966. pp. 18 -52 Reprinted ire Ncad- Iii., s (;eliiiie it yft ora the "At i 1 buten( Teen II. MOI)ELS OF GEOMFIRIC et." %%-:(sItingion. National Council 01 REI.VIIONSHIPS of \I athematic,. 1970. Pp. 25-29. 112. -Sonar Mathematical Ideas Involved 95. Black. JanetNI. "Geomen) Alive in himar) in the Minot' (:atds.-rlritlnsrlie Teacher.Feb. Classtainns." .1t11 /one/it Teen her. Fein nal} ruaty 1967. pp. 115-25. 1967. pp. 90-93.ReprintedinlicatlingN 11 :3. Wood,. Dale. mid \\'illiam E. Ilull."htuoduc Getman v(tintthe l butyricTeat het.' ing Models l(t. ,V=1)itnensional (;eantett in the Washington. D.C.: National Omncil of Tea(k Elementat S(hot)1.".1n:burette 'rewrite) amt. (.1, of Mathew:ilk s. 1970. Pp, 9-12. my 1966. pp. 11-13. 99.Buck. CAtat les. "Ge(tmetry for the Elententar School."rithmetic Teacher. October 1967. pp. I. GAMESAND PUZZLES 160-67, 100.Ilewit t.Ft antes. "Visual Aid for Geomen y." 111. Galva.Rohm C."Plato-aNumbetPla«. A till inane Tem her,March 1966, pp. 237-3S. Game. ,Itahmeti< Teacher. Nlay 1968. pp. 467,- 101. Johnson. Donovan A.Paper Poleling for the 66. thema in s Class, \'ashington. National 115.Deans. Edwina. -Games for the Early Grades.- coutio of Teminas of Alathematics. 1957. A ri Motet te ''rather,February 1966, pp. 110-11. 102, 1

Riddles Natile111:1111 I oil/Hurl:4' 132.Eleuelling. Robot I he \limits and Out Teat he:, No ember 1963. pp.,150-52. \ t'slons.-bah/nein each0. Vebual 1960. 118 II:13;34011Y. 101113 1 "131a11-:01 Ain ient Comte 01 pp. 101-6. Nlatilellialltal Skill.-loin:mth Teak he,. Nla% 133,Giline%. Ehomas "I'ses and \buses)1 the Num. 1961. pp. 326-30. her Line." . l u1 hmrtic. Teacher. Nc)%embei 1961. 119,Inbod%. Donald. "\rithinetic Baseball.liith pp. 178-82. meth Teat het . No)%ember 1962. pp. 390-91. 131.(.1andich.fat "Slide Rules lor the Upper 120. Johnson. Dono%.111 .'t. "(:otnitteicial ( :awes lor Element:iv% (.lades." liahmelit Teathet. Feluti- the \ritlinteti( Class.-rithmetic Teathet, ,o) 1958, pp. 29-33. Match 1958. pp. 69-73. 133.II%de. Da%id. and Nhn%in N. Nelson. "Saw 121.loidati.I)iaiia."Tick:1 ack.Fottr.-I uthmetu 11u)se Egg Cations:- Atithmetu Tem het Teacher. Nla 1968. pp. 131 -33. %ember 1967. pp. 578-79. 122.Pat ker.Ilelen."Seesaw(:am.- ahmette 136. Jenkins.01%ille."Lail%andthe \ bac us." Teat he). No% ember 196:3. pp..1-19-50. It:Mulch( Teacher. 0( loiter 1951. pp. 21-21. 137.Ktehi. Helen Marie. and Frances Flounno%. "\ 123.Ittuleintan. Hatt) I). "Nu:En:Vac:roe.- .1)ith me ht Teacher. No% eirdier 1965. pp. 571-72. Ilibliograph% of Ilisun hal Nlateriali lot Use in Mithineticill the Intermediate Grades." ,hilt 121. Shuilow. Harold J.Ehe Game of .11/III meth Teat lie?, October 1960. pp. 287-92. meth 'Feat lu.r.:\lay 1963. pp. 290-91. 13$,Lawns. Flank."Let'sAdd \ ittomat i( 311%2' 125, Timmons. Itobert.\. -Tie:Fac.Toe-a Nlathe Arithutetu Teacher, Nlatcll 1965. pp. 221-23. statical Game for Grades 1 through 9.- Arid/ 139.(herholt, ElliotD. "Flom Number Lines. to meth Teat he, . October 1967. pp. 306-8. 2.1) Spa« Concepts." hill:meth Teacher, Feb. roar v 1966. pp. 107-9. J. SPECIAL DI:ANC:ES 110.Peterson. \I'mne. "Numetation-a Flesh Look." bilk:near 'ratchet. Nlay 1965. pp. '335-38. I26.\shim k. Robert B. l'he Number Line in the III.Pratt. Edna M. "A Teaching Aid lot' Signed Printaly (3tacles."lilthmetit Teacher, Februal% Numbeis.- Teacher. Nmeniber 1966. 1961. pp. 75-76. pp. 588-89. 127. Beckmann. MiltonV. "New Devices Elucidate 112.Rahmlow. Ilatold F. "Understanding Dillerent \ ithinetic." hit/meth Teacher. October 1960. Number Bases." Iiiihniehe Teacher,lay 1965. pp. 296-301. pp. 33910. 128. Clatk. John R. "Looking Ahead at Institution 113.Schinickiath. Eleanor. "Learning hoar a Num. iu\rithinetic.- Apithmetus Tenchet, Decembet her Line." Authmeth Teacher. November 1961. 1961. pp. 388-91. pp. 500 -5(11. 129. Cochran, Beryl S. "Children Use Signed Num. 1.11. Sganga. Francik, 'I'. "A Bee on a Point. a Line. bers.".1thhicielit Teacher. November 1966. pp. and a Plane," A tithmetitTeacher, November 387 -S8. 1966. pp. 519-52. 130. Coon. Lewis IL "Number-Line Multiplication 115.Shaed..1lice Evelyn. "Using the Number Line ha' Negative Numbets." .hithmetit"l'eat het . to Ilelp Childien Understand Ft actions." Nlatch 1966. pp. 213-17. meths 'leacher, November 1960. pp. 370-72. 131.Deans. Ed%%itia. "Grouping-an . \id in Leatning 116. Shaw. Dota II. "Utiliiation of Teaching Mate- NIultiplication and Division." Arithmetic Teach- rialsinFirst.Crade Nlathentatics." rilloneth er, January. 1961. pp. 27-31. Teacher, J :utuy 1963, pp. 37-11. 10. PROJECTS,EXHIBITS AND FAIRS, GAMES, PUZZLES, AND CONTESTS 7

114-

; 347

CHAPTER 10

MATHEMATICS PROJECTS, EXHIBITS AND FAIRS, GAMES, PUZZLES, AND CONTESTS

by VIGGO P. HANSEN San Fel nand° Valle) State College, Northridge, California v.' SAMUEL L. GREITZER Rutgers University, Newark, New Jersey EMIL J. BERGER Saint. Paid Public Schools, Saint Paul, Minnesota WILLIAM K. MCNABB Skyline Center. Dallas, Texas

This chapter describes how projects, exhibits, fairs, games, puzzles, and contests can be used in a school mathematics program. Suggestions are given for the preparation and evaluation of projects and exhibits. Illustrated examples are presented of projects made by students of varying abilities and interests. Following the chapter there is a list of sixty-seven project. topics with references for each and a list of sources of mathematical games and puzzles.

A STUDENT PROJECT as it. appeared on display at the Annual Meeting of the National Council of Teachers of Mathematics held in Minneapolis 349

10.NIATIEMATICS PROJECTS. EXHIBITS Am) FAIRS. GANI ES. PUZZLES. AND CONTESTS

Every student who studies mathematics is ca- Stinimariied below ale the major steps that pable ol creative activities. Such activities include teriie the development of a mathematics beginning manipulative experiences by the pri- project: mary student. independent investigations by the I. The student defines the problem or topic college-bound student, and other exploratoryac- he is going- to incestigate. or describes the tivities undertaken by students at %allot's levels model Or display he intends to build. of a school mathematics program. 2. The student extends himself beyond his The kinds of cieative activities in whichpro- iegular assignments to collect pertinent in- lessional mathematicians engage and the excite- formation for a paper or 10 procure con- ment the) experience in working at the homier:: structimi materials for it model Or display. of mathematical knowledge can be a pall ofel- This step ma. itivole the use of the library. ely student's mathematical education. Mathe- and in die case of a model or display. the matics projects. exhibits.fairs. games. puzzles. SIII(lellI may need to IINC materials Or equip and contests arc ideal!) suited to foster thecx. mem lioni the industrial :tits.crafts. Or creativity. commercial deign Intent. 3. The completed project is given recognition and evaluated. Exhibits and lairs provide TI-1 I: NA-1'I' RE OF A opportunities for displaying projects and MATH F.M:\ TICS PIZ() J EC:T judging their merits. If a project is put on display in a department stole. hotel. library. A mathematics project is an individual or small museum. bank. airport. or other public group activity that incolves the piepaation of place. the community also plays a part it: a paper. the pioduction of a model or display. die evaluation. Or 11 combination ofthese mu activities.:\ project may begin with a student's desire10 CX. 1)101C it101)1( or idea ill gleamr depth than regu- PIZEPAIZATI()N PIZ1)JF.C:T lar Glasswork permits. or it may be assigned by Undoubtedly the most important factor in getting the teacher. .1gain. a mathematics project may outstanding projects from students is the atti- be 'elated to a student's work in science or tude the teacher toward piojects. A teacher some other sc subject. Occasionally a project who is energetic ill A111111:1111114 StndCnls 10 start may grow out of a student's ont-of.class hobby. projects. IescH11.0 eft!!in providing- needed assis- In some respects a mathematics projectre- tance. and supportive of students' efforts will sembles a scientific investigation. It begins witha usual') hate students %vim develop significant statement Or recognition of a problem. It con- projects. tinues with collection Of pertinent information There are a number of techniques a teacher and. if a model or display is involved.pro( lire- can use to motivate students to start projects. unlit of needed construction materials. Aproject One technique is to pill completed projects On is completed when the student finishes writinga display and invite students to examine them. If report. demonstrates the use of it model. or ex- no suitable collection of projects is available. an plains a display. (As usually conceived.a displa) effective sub%tiittle isit display of photographs is a thematic arrangement of models. illustra- and drawings. or the teacher might show slides tions. and printed messages thatare combined to that depict projects produced by students in for- project an idea to an observer.) mer years. This chapter contains nearly fifty (:11A1'ER TEN

I/1(101C% 01 different projects. These pictures are 111M one refers to the project pi( lured opposite part 01 a (Olier1100 01 3.31) pictures that are on the title page of this chapter. file in the Mathemati(s Resource Center Of the My pioject. entitled -The Odds Are with Saint Paid Public Schools. the ouse.- four-sided Citing studentslists of topics with ar(0111. displa. The 101)1111111 houses the main bolt references is another useful technique. or the display. Three ()Iits lour sides open I:eft:retires for sixty-seven dillerent project topics to reveal three different aspects ()I the laws of C11:10CC ;IN applied to gambling: random are given at the end of this chapter. selection. dice. and poker hands. The (caner Still another tednatille is to take students on third contains a dm ire that deals "'poker field nips to possible sources Of ideas for projects. hands."' using (lice for (ands. This can be operated b the observer. The bouom 1111111 Such sources, might include a bank. a computer seites :IS a base.1 aliellyid l0 show Niaills- conapan. an industrial museum. factories. farms. til.111 E11.11. ille 0(1(1N ale indeed Willi the fisheries. and so forth. Trips of this kind give house. students contact with applied mathematics. The only materialsI11:1(110) buy for the project were two lolls of silver pipe-wrap Collecting and maintaining a life of ideas for ping tape. which cost S5c e. and four projects through which students can browse is 21-inch dowel rook. itchin diameter. one of the more compelling motivational tech- tvhielicostitnickel apiece. This came to niques a tea. 'ler can Use to get students started. other materials (playing calds. dice. poker chips. (al(Iboad. etc.) are from Ideas for projects should be filed aecording to the household. subje/ Or topic(e.g.. algebra. -;eonietry. num- A I a kohl, Rillrr ber sstems. probabilit. etc.). Listed below are Soh,l'unl,innsoln sour( typical items that might be included in a project idea folder: project is a display (ailed -Dressing Up1:itlieinatics.' The display contains a I.Questions to be investigated Mobitisdressalongwith :1 pamphlet. A hiicrdescription or ontlio of a possible -Dressing Up Mathematics.' which explains pr oject the chess's construction. Also included in the display ale a Klein Neale. a Mill)his A (ilde sketc Ir Or drawing of a model strip. the book fix /during henurlirs on 1. A progress report of an unfinished project Your Own by II. Glenn and Dono- van :. P)11"0". and an article entitled or a report of an attempt to pittance at Itiniali vs. the Conveyor Itch- which project that didnt materiali/e explains how Paul used the odd properties of the 5. Names Of people %dm may have ideas strip it a conveyor belt. I became interested in this project when 6. Newspaper clippings that contain the germ reading the part on topology ill Exploring of an idea Of project. ;llnl/rrrrrrrlics on Your Own. Since I also enjoy sewing, the chess idea was perfect. Perhaps the most effective way of motivating Although my project is quite involved. it to start projects is to give them an op. cost_ tile only about SW. which ryas for the portimit to read 1.e:slum's or projects written by netteri:t1 iu du dress. All the other articles mete either borrowed or taken from home. modem:, who completed the projects. Itt::stinte:s or Rv doing this projectI got a new dress. a this kind should ordinarily be brief To be useful. new pattern for other clod and a WI)11- a re:sunk: should contain a description of the clei fully exciting experience. I :1150) got a project. a clarifying photograph or drawing, better grade and all understanding Of one- sidedobjects. Objects like this. either in list of the references and materials used. at state- theory or material substance, aen't always ment about the r()s1 of materials and brae.part cvliat they seem. Such things may be thought on the project. the names of advisers consulted. out in the mind but are surprisingly different in solid form. and suggestions for improvement. Below are two Wendy Johnson examples of rc.:stiiiiC:s prepared by students. The Lit rh irld Af inn atom NI NTIIENIATICS PROJECTS. EXHIBITS AND FAIRS, CANIES. PUZZLES. AND CONTESTS 351

FIcria: 10.1. -Dressing Up Alathematio" di3plav 352 CI IAPTER TEN

Establishing ten t ad% e deadlines for each of the with an opport nits to explore a mathematical following stages of development of a project will concept while seeking xv.ts of expressing and usually promote St cad y progress tolvald its Com- illitstating it. pletion once a suldetu has started: In addition. projects sense to stimulate interest I.Finishing the preliminary reading in mathematics. to widen and deepen a student's Preparing an outline of the proposed understand ing of mathematics. and to give the ploject. 01 a diagiam it the proje( t inutdces student a feeling of rele%ance through actie pat- a model ticipation in the learning process. Projects also

1. flaving a conference with the teacher have other. more Specific. values. I.Concluding the poject. 1)oing a project helps a student leant to ap- Time should be pro% ided for the student to pteciate the importance of defining- a problem work On his ploject by making available un- plecisely.It is not uncommon for a student to committed modules of instructional time or by disco% e that:t problem ma need to be rede- setting aside certain days of the week. Permitting fined after working on it for it while. a student to work on his project alter he has Nfoteover. doing a project helps a student learn 101111/1eft('Ills S daily assignments or instead of how to use the library elfectheh. how to obtain working on his assignments. holding special ses- information from sarious sources. and how to sions after school. and. of course. permitting all approach knowledgeable people. work to be done at home ate as in which ad- If a pi oje( t insolves the production of a model, ditional time can be ntade mailable for projects. doing the ptoject helps the student s isnaliie the If models or displays ate involved. a work- concept that forms the basis of the model. The bench or sheets of Nfasonite to cover desk tops process of constructing :timxlel assists the stu- should be a% ailable. It mad be possible to an ange dent in bridging the gap benceen the teal world with the industrial arts department for needed and the world of abstractions. super% ision so that students can use hand tools 1)oing projects gives able students opportuni- and 'toweltools. Similar airangements can be ties to work beyond the le% el of regular classwot k made with the science, crafts. and commercial and offers less able sttidents opportunities to pc:l- depart men ts. imn' at levels commensmate with their abilities. To ensure success Cot each student. the tea( her Doing projects that imolve applications helps should gise hequent etimmagement and make students tecognile mathematical properties in ( onstuctke suggestions when the need arises.. the world about them. The student Avho under- student's enthusiasm is often in direct ptopor- takes a prt/iCd. soon Ieann> that there are more lion to the enthusiasm of those who help him. If places than just books in which to find informa- it happens that a student's project is beyond his tion about mathematics. capabilities. either he should be tactfullyre- Projects help teachers make betty. cnaluations dile( red to another project ot subsequent phases of the mathematical potentialities of their stu- of his project rescheduled to pandit ad% isers and dents. Teachers who !egularl) ha% e stutIvn ts make consultants to become involved. projects are occasionally a n1:17Cd at the high level of achievement of co lain students who ate not T1 1 F. 1:0I.E 01 PRojEurs IN THE successful in other kinds of learning activities. LEARNING OF NIATEMATICS Projects stimulate other projects. Preparation of any project usually generates ideas for related Teachets of mathematics have long- recogniied ptoec is thiough refinement. associati(m. exten- the value of projects that capitalize on literary. sion. teision. or correlation. graphic. dramatic. and mechanical abilities of Students. teachers. and the community have a students. Doing itproject pro% ides the student tendency to react Imorably toward projects and MATH ENIATICS PROJECTS, F.XIII BITS AND FA IRS. CAM ES. PUZZLES. ANDCONTESTS 353

to support saciother in producing them. In local school exhibits. When a student's project is some communities engineers and other profes- exhibited, his name should be prominently dis- sionals willingly assume responsibility for work- placed. This gives the student recognition and ing with students on highly technical projects provides an incentive for other students. that require equipment and facilities to which Mathematics fairs also serve to give students only these people have access. recognition by inoviding publicity for students A student who completes a project may be who produce outstanding projects. This stimu- called upon to defend his scholarship and crafts- lates competition .Ind encourages studentsto be- manship in flout of his peels or before a group of come imolved in challenging projects. judges. This gives the student an incentive to A major problem in concluding a fair is that prepare a report and to learn to cope with the projects must be given relative rankings and nervousnessassociatedwith making anoral awards made accordingly. The difficulty is that presentation. criteria for judging different kinds of projects A spirit of creativeness can be nurtured by are not universal. Pon example. it makes little encouraging students to do projects. Theie are sense to compare a scholarly paper with a work- instances where mathematics projects started by ing model that illustrates a mathematkalcon. students while in school enabled them to make cept. The two pojects represent different kinds impoltant contibmions to mathematics when of student ins okement and different final prod- they were still young. ucts. with merits that are impossible to compare. Finally. students who piepare projects experi- One solution to this problem isto separate ence a pride of accomplishment. a spirit of com- projects enteied in alair into ciao ent cate- panionship. and t he exhilaration that collies with gories. A twit al breakdown of projectscan be creating something original. For many students made b. first separating all projects into grade- preparing a matheniatics project is later viewed level or subject -au ea gioups and then dividing by them as one of the most satisfying experiences the projects in each of these groups into the fol- of their education. lowing categories: I.Paper only i \III HITS ANI) l'A I RS 2.Model or display only 3. Every student who makes a sincere effort to pro. Paper accompanied by a modelor display. duce a project should receive some kind of recog- With this set of categories it is possibleto estab- nition. The extent to which a student feels his lish evaluative criteria that are valid and fair for the different creative efforts are lecognind will often influ- kinds ofprojectsthat students prepare. ence his future achievement in mathematics. There arc many ways of giving recognition to Below is a sample set of criteria that can be students who produce projects. Certainly judi- used for judging projects at a fair. This set is by cious praise should be gken for good work. For no means definitive and may be modified to meet atproject that replesents a sincere effort but local needs. may not have turned out very well. commenda- I.Criteria for a paper only tion for the effort put forth shouldaccompany a) The mathematical topic should be sig- tactful suggestions for improving the project. nificant. Exhibiting it project in it school display case b) The background investigation should or in a store window is an acceptable way of be thorough. giving recognition. Drugstores. banks, hotels. and c)Illustrations and diagrams should be other commercial establishments usually welcome sharp and huge enough to show detail. an opportunity to provide window space for (1) The exposition should be clear. 354 ClIAPTER TEN

c) The topic should be developed in a cal. Criteria used in judging these two impels logical manner. should reflect the different emphases. Students /)"che paper should be neat and weld should know before they begin work what criteria organized. trill be used to judge their projects and what the g)References should be listed. relative point values are. Criteria used in judging h) The student should be able to answer projects are the behaviol al objectives that apply orallyquestionspertainingtohis to projects. project. Before any mathematics lair is held. a team of Criteria for a model or display only judges should be selected and a meeting held a) The matheinatical principles depicted to discuss criteria forthe various categories of should be significant. projects that me likely to be entered in the fair. b) The mathematical principles depicted To promote unifounit) in judging, practice ses- should be readily discernible. sions in judging should be arranged. For ex- e) The quality of workmanship(wood- ample. judges can practice by judging pojects working. electricalwiring, art work, exhibited in former ;eats. But 110 matter what etc.) should be adequate for the pur- set of criteria is adopted. or how dedicated the pose of the project. judges. there will always be some honest dif- d)Research of the literature pertaining to ferences of opinion. Thisisnatural when a the model or display and the mathe- creative effort is being judged. matic:al principles involved should be t easonably thorough. EXANIPLES e) The student should be able to demon- STU D ENT PROJECTS strate the model and answer questions pertaining to its operation. There is some topic at every level that can be The display should be attractive and the source of a project for some student. well mganized, and the student should Elementary school students can prepare proj- be able to explain the theme of the ects to illustrate mathematical concepts ft om dil- display. cnt strands of the elementary school mathe- 3.Criteriafor apaper accompanied by a matics progTam. Such projects will usually be model or display manipulative and concrete. Indeed, no written a) The paper should satisfy the criteria report may be involved. At the junior high school listed for "a paper only." level projects usually tend to be somewhat more Ii) The model or display should satisfy the sophisticated; howocr, it is still appropriate for criteria listed for "a model or display low achievers whose reacting and writing abilities Only." ate poor to produce projects that are primarily c) The model or display should reinforce manipulative and concrete. At the senior high and/or illustrate the topic dealt with school level projects will usually be more ab- in the paper. stract and less manipulative than those under- This set of criteria is not exhaustive. nor does taken by students at lower levels. it specify the weight. or point value, to be as- There are few restrictions on the kinds of signed to each criterion. The relative weights to topics that irill motivate students to start proj- be assigned to the different criteria will depend ects. Almost any :Ica% ity in which students en on the subject or grade level of the project being gage may be expanded into a project. judged. For example. apaper on consumer The topic a student chooses for a project should! mathematics might stress applications, whereas a be appropriate for him; howe% er, appropriate- paper On analysis would tend to be more theoreti- ness is ter) difficult to determine, especially for mATHF.m.vrics pRojEcTs. Eximin.s AND FAIRS. GAMES. PUZZLES. ANI) CONTESTS 355 highly capable students. It is possible for a junior high school student to get deeply involved with probability and for an able elementary school studentto become completel)abso, bed with computes. In short. if a student has an interest and some ;Wilk). it is difficult to determine \ IS .111d N% hat is not an appropriate topic for him. In the commentary that follows. examples of studentprojects arepi esented undet \411 topics. A broad\ alio)of skiffsis shown in conjunction with Varying degrees of depth and undeistanding of mathematics. All of these plc+ ects ha\ e been presented at affairs sponsored by local schools. at mathematics conferences. at con- ventions. or at science fairs. g:=2 Computational Devices and Computers Over the centuries computational devices have changed from pebbles to computers. Variations include manipulative, mechanical, and electronic devices. (The word device is sometimes used to FIGURE 10.2. Chinese abacus Icier to models that have movable parts.) All of these can be projects for students. Abacuses and Napier's bones provide interesting research topics for students in the elementary school. Workable devices are easily constructed and require only a few tools, modest craftsmanship. and a minimum of materials (Figures 10.2 and 10.3).

.."

Ell3if5Q 78&IINDEX 1 iaiaAtirdglIFIVIDMIIMII 2 AdIAMNIM- Meatairgi 3 covormormailLTOSIMI 4 _41Xro AMIWOMPSWOM 5 s4 6 iiiv.AiNadidirMIM FIGURE 10.1 o 7;4 128 5 II. 61N1 7 Napier's hones 2 0 8 cr47 2 8

8 2. 1 356 CI IA PTER TEN

Figine 10.1 pictures a mole sophisticated pioject. The device shown in the picture is a. battery- operated analog computer that can be used to A C solve proportions of the form= B D a 12. Figure 10.5 shows still another computer proj- . 9 is ect. This one is a display pioduced by an eighth- A grade student "to help," as he put it. "the aer- age person understand and icaliie the basic operations of a computer." Figure 10.6 shows a computer project that a 4 3 student suit ted in ninth glade and finished when 0 .2 he was a high school senior. The machine shown / 7 a in the picture can do c% cr) h ing ft on) simple 4 9 io

FiouRE 10.. A n o g. computer FIGURE 10.5. Computer tlislay

;71", -17F49-41, ELECTRONIC

IVA. 1414.4. &m., 100 1011040 10 10 I1010) W:70 194°41'5 10001tat scgl 1100 SAO.. 0.

1.°10111°101r) .i.r0111F. tat I

CENTRAL PROCESSOR MATH EMATICS PROJECTS. EN I-1 I BITS AND FAIRS. GAM ES. PUZZLES. AND CONTESTS 357

additiontoplayingtick-tac k-we. The system includes a central processing unit, card reader, and remote terminal. Programingis achieved by using a machine language designed by the student who produc ed the machine.

Estimating the Value of77 I 2 ,3 4 6 7 8 9 Computers have made it possible to obtain the value of 77 correct to thousands ol decimal places. Even so, estimating the value of r b) calculating the quotient of the (it( wrIerence divided by the diameter for each ol a numbe:r of (lac' ent cir- cubit objects continues to be an informative ip 12,3 4 5 6 7 8 9 10 11 1 project among students in the upper elementary 12 1314151617181920212223 2. grades (Figure 10.7). 14 25 26 27 28 29 30 31 C-D=34 111111111Mmix'; notiRE 10.6 Sophisticatedcomputer built by a high school Ftowt. 10.7.Finding an approximation to the student value of7r

naimaz- nfic STOIC AGE AF! 4.7.03 I0It'r. CAM: : J:c.: D.-. 7 ari: Li .2 ssanv131.,;!131,41:11:41.%:' AITAIRCIC 13%' otailiorerfpgrrk?' ;,.; ',q:ez414 -71%3C113 SAIL 1.0r13UFAi Ai! 358 CHAPTER TEN

Making Measuring Instruments Projects involving die production and use of measuring instruments range from the simple to the sophisticated and can be counted on to kindle interest among students from the lower grades through the high school. Making a micrometer that can be used to measure the thicknesses of objects correct to ' of an inch may sound like a complicated project, but it is really quite easy. In fact, students in the upper elementary grades can pursue this project successfully. Figure10.8illus- tratestheidea. The homemade micrometer shown in the picture consists of a bolt and nut with a pointer welded to the nut. The bolt has X 20 threads to the inch. This means that one so' complete turn of the pointer moves it vertically of an inch, one fourth of a turn moves it Flcukr, 10.8. A homemade micron, ter that vertically of an inch, and so on. can be used to measure the thickness of ob- Making a spherometer that can be used to jects correct to 1180 of an inch measure the radius of a sphere is a challenging project for students of tenth-grade geometry. Figure 10.9 shows two models. One is simply a iefinement of the other. Each model consists of a coffee can, open at one end, and a pencil. The pencilisperpendicular to the plane of the closed end of the can and fits through a hole at its center. A scale is marked on the pencil to indicate how far its lower extremity is above or below the plane of the open end of the can. The operational principle of these devicesis similar to that of commercial spherometers. Making a scale that indicates the capacity of a circular cylindrical tank in horizontal position is a fairly sophisticated project even for students of high school trigonometry; however, the project can also be adapted for work with volumes FRESH SOIlt v,o OaCIAD in grades 5 and 6. Figure 10.10 shows a circular cylindrical tank (tomato juice can) in horizontal position with a scale that indicates the volume of the tank for various heights, correct to tenths of a cubic inch. There is a hole at the top into which liquid can be poured. The piece of glass

FIGURE 10.9. Spheromoiers NIATHENIATICS PROJECTS, EXHIBITS AND FAIRS, GAMES, PUZZLES, AND CONTESTS 359

FIGURE 10.10. Circular cylindrical tank in horizontal position zuith a gauge on the outside for indicating volumes

tubing on the outside of the tank is bent and sealed in at the bottom. The exterior part of the glass tubing is aligned with a diameter. Stu- dents in grades 5 and 6 can make a scale like the one shown in the picture by pouring measured amounts of liquid into the hole at the top and marking the height of the liquid for different volumes. Another way to make the scale is by computing the volumes of right cylinders that have segments of circles as bases. Applications of Mathematics in Science As already stated, frequently a mathematics project is related to a dent's tvork in science. Fig- ure 10.11 pictures asy-tomake periscope. The mirrors are set in parallel planes, thus making the entering rays of light at the top parallel to the sight line below. Producing a device like this and writing a paper explaining its use in contrived situations makes an exciting project for students in the upper elementary grades.

FIGURE 10.11. Periscope 360 \ PTER 'TEN

Figte10.1`? pictures an apparatus that a student produced to investigate the behavior of springs stretched b) weights. The three springs have the NalliC spring constant and identical lengths(when 1101 stretched). The weights shown in the picture are 1 ounce.I: ounce and I; ounce. tespectively. Findinga formula that relates the average force applied in stretchinga spring, the length of stretch. and the springcon. start involves a great deal of precise measuring, computing, formulating. and checking. Figure 10.13 pictures an inclined-planeapl>a- ra designed by a student of trigonometry. The device can be used to relate the variables in the equation F IV x sin 0, where 0 is themeasure of the angle of inclination of the inclined plane, IV is the weight of the load, and F is the down' ward component of the load measured along the inclined plane. The magnitude of Fcan be read

411/1.11!1

FIGuRE 10.12. Apparatus for studying properties of springs

FIGURE 10.13. Inclined plane NIATIIENIATICS PROJECTS. Exiintrrs AND FAIRS, GAMES. PUZZLES. AND CON PESTS 361

FIGURE 10.19 Two cycloidal Tracks and one linear hack. Any Iwo w all Mice of the halls can be released simullancously.

directly from the scale at the top of the inclined plane. The gravity pointer on the gi id makes it possible to estimate the value of sin 0 without using a table of sines. A device like this is useful is the brachistochone, or curve of fastest de- for solving and checking a variety of problems scent. The cycloid isalsothetautochrone, or involving the inclined plane. curve of equal time. If the two steel balls resting The cycloid curve has several interesting ap- at different heights on the two cycloidal tracks plications in science that may be investigated by are released simultaneously, they will reach the means of projects in mathematics. Such projects bottom at the same time. sometimes lead to construction of novel devices, Figure 10.15 pictures a frame composed of two Pictured in Figure 10.19is a device with three adjacent arches of an inverted cycloid and a tracks on which steel balls can be released. Two pendulum bob held by a string suspenled from of the tracks have the shape of a half arch of an the point where the two arches meet. When the inverted cycloid(from the cusptothe low pendulum swings, the suing wraps itself. around point). The third track is an inclined plane. If the cycloidal arches and the bob traces a cycloid. the two steel balls at the top of the device are Since the cycloid is the curve of equal time, the released at the same instant, the ball traveling period of the pendulum is independent of its down the cycloidal track will reach the bottom amplitude. Various other demonstrations are before the one traveling down the straight track. possible.Projects involving investigations of It can be proved using differential equations properties of the cycloid make a strong appeal that a cycloidal path under the action of gravity to capable students. 362 1.\PTEIZ TEN

111aking Models of 'How-Dimensional Figures One way of getting students to do mathematics is to encourage them to do projects that involve the production of models, and perhaps no subject offers a greater range of possibilities than three-dimensional geometry. Checking the formula for the volume of a right circular cone is one such project. An appropriate model can be made by pouring a known volume of sand througha funnel. as shown in Figure 10.16.

FicntE 10.111. Cveloidal pendulum

..... 5

1 11. 4" ...,,,-,'",),.t.-- ,- .44, '.- A izttzlek*i4 t`-?ilkil, it.[..... 7,,qygt,#tz.t...... 44,411 ---'N.

`i FicuRr, 10.16 V t.lcr' lll . - - A device for mking a model of a .. , Kv. right cinttlar cone and checking the fm-ntala for its volume 44:7 'f4ete"- NIATHENIATICS PROJECIS. EN illurs AND P.11Its. GANIES.11.7.7.1.ES. AND CON FES'IS 363

A student need not have a high degree of inventiveness in older to carry out a project that results in a pleasing. meaningful model. For impact in the classroom. compete a commercially constructed model of a cone of two nappes with the model shown in Figure 10.17. Shown in Figure 10.18 is a threedinvsional model that has implications for calculus. The model illustrates the use of right circular cylin- dersin making successive approximations of the volume of a hemisplune. A student who attempts to characteriie successive appoxima- tions of this kind will find himself face to face with the theory of limits.

FIGIna. 10.17. :1 coneof two nappes

FIGURE 10.18. Successiveapproximations ofthe volumeof ahemisphere

`11111.-1411411=11101.-. -aide"

a 364 CI IA l'T El: TEN

FIGURE10.19 :1 board that illustrates the making of a per- sective drai..ing

Geometry and Art Typical of many projects the perspective boald shown in Figure 10.19 bridges two disciplines. The creator of this project WaS a capable student of art who combined his knowledge of theedimensional geomen e with his talents in art.

Giving GCOMeirical Interpretations to Algebraic Expressions Asking students to give geometrical interpretations to algebraic expt essions can lead to challenging projects. The student who produced the dissectible cubical block in Figure 10.20 used it to give a geometrical interpretation to the proof of the following special case of the binomial theorem: (ll + + :la2b 311/l 4. i Figme 111.21 shows a dissectible model that was produced in response to a more imaginative assignment. A student was asked to show by means of a geometrical interpretation that

Y)(x v)(2.T y) = 2x' 2xy-7 V.

FIGURE 10.20 365

FIGURE10.21. The volume of the black box is equalto the product (x y)(x y)(2x y)and also equals the sum of the volumes of Ike six while blocks. FIGuRi-: 10.22 Graphical Rpresculation of Complex Pools an Elmo lion The remit Yearbook of the National Coun- cil of Teachets of Nfathentatics containsa chapter by I lowaul F. Fehr entitled "Graphical Rep. lesentation of Complex Roots."' Figure 10.22 shows a thre'Aimensional model of the graphof v x*: that is picuned in the chapter by Fehr. The graph has two branches-one real. one complexboth of which are parabolas. In the model. the real brattish bounds theupper plane region: the complex blanch bounds the lower -bi plane region. The graph is obtained I)} solving -a the given equation for x to Obtain x = 2 -4- and substituting teal values for v. For y the resulting tallies of x air real. but for y

I.alti Sensolv Aid. in the Teaching of Mathematics, Eighteenth Yearbook of the National Council of "feathers of Nlatheinatics(New Yolk:Bureau ofPublications. Teachers College. C:olintibia University, 1915).hl). 1:10-38. 366 (:11APTER TEN

(the Yet tical axis in the model) but tequites .1 sense or regoi.o it.. Tile cle.ohe hIst in( is complex nombet plane for the representation of of students in later glades find satisfaction in

Avalues (the 1101i/Oill.il plalle With teal axis a designing- mec hanical devi. :s to draw c Imes that and imaginal. axis hi). Fot 1 the teal blanch cannot be thaw n tt ith sit aightedge and c ompasses. of the 0:wit lies in the plane determined by Figure10.2:1 shos a device for drawing a the Y-aNis ttaNis (this tueatts that teal values sine c tiny. The device shown in the philtre was of x lie on the itaxis in the complex plane). For invented In a student of secondyear algebra. ti < the wmplx branch of the graph lies Ititilt into the device is a scheme for detenuin- in the plane that is perpendicular to the aaxis lug line tallies of the sine film lion met hatticall). at the point 2 = Ilt. since for all real %alms of Pushing or !tuning the long driver rod hori- less titanI the real pat t of xis 2. I :t simple /ontally rotates the two wheeis and rates the stutter to determine that the compicx braocil pencil to trace out the c :'e. imermis ilic rompicx plane at (2 2i. 0) and The student who produced the device shown that 2 --2i ate solutions of .1x + = 0. in figure 10.2.1 also had in mind a scheme for Projects involving- hybrid realcomplex coon. tracing a sine curve. As the pointer is rotated dilat ::Stelus such as the one described are a a sequence of lights is mined on. challenge to good students. Figure 10.25 1,tctures the construction of a device that can be used to thaw a parabola. The Geometrical Constructions and white strip with black dots and t.te white thumb Curve Drawing tack tepresent.respectively. the directrix and Projects 111V01% ing geometrical constructions and focus of a parabola. If the piece of chalk is held curve drawing prot ide stimulating experiences tam against the string that is looped over the for students of nearl all ages and :OAR. levels. top of the to jangle and the base of the triangle Drawing simple designs with a straightedge and is mot et! horizontal!t along the dire( trix. the compasses satisfies elementary school students' piece of chalk traces a parabola.

Fimmt r. 10.2:1. Device for drawing a sine curve PROJEC:I'S. ENIIIBITS AND FAIRS. GAMES. PUZZLES.AND CONTESTS 367

FuautE 10.24. Light travels along the sinecurve its the pointer is rotated.

FIGURE 10.2. Device for drawinga parabola

---- r ------II- CI IAPTER TEN

An Optical Method 01 Showing Conic Sections II a lighted floor lamp with an openentled t itt ii- lar cylindrical shade is placed next to a wail. one may leadil) observe a In iglitl) illuminated region with a bountlai) that approximates the shape of a hypetb01.1. The student who made the deice pictmed inFigure10.26triedto make this plinciple precise. Ile placed a small bright light in the centei of .111 Open-ended c)linchical tube. thus causing a conical beam of light to be projected from each end. If the cylinthical tube is held so that its axis is paiallel to a wall. the resulting section is a hypo bob. The other conic sections can be formed by vary- FintntE 10.26. A cone(of light) ofIwonappes is cut 1))' ing the angle between the axis of the cylindrical a plane, producing a hyperbola. tube and the wall. A range of problems inolv- ing ploperties of the diffeient conic sections can be investigated with the aid of this device.

Reflection Properties of the Parabola and the Ellipse Projects involving- reflectionproperties of the parabola and the ellipse ale good swim for students who have difficulty choosing a topic for a project. Only a few suggestions are needed to help students get started making displays of parabolic reflecuns. Demonstrating the reflection property of a parabola is a little harder. One boy constructed the "pool table" shown in hgure 10.27 todemonstratethisproperty. The cushion has the shape of a parabola. All channels ale parallel to the axis of this parabola and the pocket is at its focus. A ball rolled as indicated in Figure 10.27 will bounce oilthe cushion into the pocket. The elliptical "billiard table," on which a ball started from one focus will always rebound through the other focus, can be used to demon- grate the reflection property of au ellipse, The device shown in Figure 10.28 can be used to explain why the ellipse has this property. The string looped aroundtwo nailsisthreaded

Flonut:.10.27.Device for demon.qraling the reflection laopertvof 11parabola 369

FIGURE10.28. A tangent of an ellipse atany point makes equal angles with the focal radii.

through a vertical plumb that holdsa straight wire in hori/ontal position. As the plywood disc is rolled the plumb traces outan ellipse. and it can be seen that the timgent to thecurve at any poini. makes equal angles with the focal radii.

Exponential Curves

Curves of this kind and their applicationsace a rich source of projects. Thecatenary is one such curve. It has the shape of a freely hanging chain , or flexible cable (Figure I 0.29) and b-iks decep. Lively like a parabola. which is the othercurve shown in the picture.

FiGuRF 10.29. Coln Innison of a calenmy (chain) and a pa?abola ( black curve) 370 Gilt\ PTE It TEN

Fundamental Principle of Counting Figule 10.31 shows an electrical device that grew out of the same student assignment as the The projects pictured in Figures 10.30 and 10.31 wooden «nnbination lock. Of the 1.331 possible started with an assignment to a second-year alge settings of the duce switches. exactly 1will turn bra class to collect illustrations of applications on the light. involving permutations and combinations. One student came across an illustration of a combination lock that is supposed to have existed around PrObabililY I560.'I lis disco% el) led to consnttction or the hodin 6°11 of a probability machine is always a wooden combination lock shown in Figure 10.30. 1)01)111.11 pi oje( t. The movement of the shot toll- Flie lock that is illusnated has one fixed ()Uncle' ing down .1 board and the resulting approxima- and row movable cylinders. Painted on the out- tion of .1noimal curve never (ease to attract side of each movable Llindet ate eight snipes or ape( tators and mouse ex( itement (Figure 10.32). dificrent colors. Of the 1.096 possible alignments In the device shown in the pictine the configura- of stripes. exactlyI opens the lock. tion of channels just below the reservoir at the top was created by gluing hexagonal ceramic 2. anklin W. Kokomoor. Mathematics in Minian A1- (111:3(New York: h ma( e1 lall. 1912).p.483. tiles to a piece of plywood.

pfr

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FIcatitE 10.30. 1Vooden combination loch M ATI IENIATics pRojEurs. EX I IIBITS AND FAIRS. GAMES. PUZZLES. AND CONTESTS 371

FiGuRi: 10.31 Electrical co?nbination device

FIGURE 10.32 Probability machine 372 TEN

The pens) board in Figure 10.33 shows a con- penny boatel con espond to the the sets of "like- nection between the expanded form of terms inthe above expansion with apenny + "head- for h a»d a "tail- lor I. The sixteen for a = I. 2, 3, 1 a»d the possible outcomes of tows inthese arrays are exactly the possible tossing tt pennies. Consider the last equation on outcomes of a single toss of four pennies. The the penny boat d. By the binomial theorem (he arrays on the penny board correspond to (I/ + I)' = + .1//3/ lily/- -1///3 -1- /1. the following. events: Shown below is an expansion of -1- 1)' writ- Getting 4 heads ten without using exponents and without com- Getting 3 heads and I tail bining "like- terms. L.Getting 2 heads a»d 2 tails -F- = Wain 4- Nall -I- ±11111 + 1111 Getting I head and 3 tails 1-Oath /h/h ///// Getting' -I tails -1-had/ + h/h/ ///// From this and a delinitiott of probability it fol- hh/h //h/ //h/ lows that Pr(L1) = P(2) =7-14, Pr(E2) = ii /hid P(E,) = and P(E,) = ÷ huh If you tarn the book to look at the picture bot- The five rectangular arrays of pennies in the tom side up, you can see a histogram that ap- right -hand member of the last equation on the proximates a normal curve.

FIGURE,10.33.Penny board MATIIENIATICS PROJECTS. EXHIBITS AND FAIRS. GAMES. PUZZLES, AM) CONTESTS 373

Trisecting an Angle The problem of trisecting an angle comes up for discussion perennially. and any hint that such a construction is not possible with an unmarked straightedge and compasses only serves to challenge sonic student to attempt the impossible. The skillful teacher will know how to 'Mirect efforts of this kind into fruitful projects involving a history of the trisection problem. proof that a classical construction is impossible. and pro. duction of trisection devices. Pipit e 10.34 illustrates the use of a trisection device that was pro dneed by a tenth-grade student of geometry.

FIGuRE 10.34 The "tomahawk," a .simple angle trisector

Non-Euclidean Geometry Non-Euclidean geometry is another topic that is rich with possibilities for projects. Figure 10.35 shows a crude model of a pseudosphere, a surface that has constant negative curvature and that can be generated by revolving atractrix aboutitsasymptote. The asymptoteofthis surface is a straight line (the axis of revolution) towhichallmeridians(curvesofshortest length) are tangent at infinity. The model shown in the picture was produced by a student of second - }cat algebra. lie used the wire frames of two lamp shades which to him suggested the desired representation of a pseudosphere when fastened together as illustrated in the picture.

FIGURE 10.35 Model of a pseudosphere 374 CI LApTER TEN

Map Projections The topic of map projections provides a range of possibilities for projects. by students of varying ability levels. Among the easier possibilities areprojects involving models thatillustrate gnomonic projections (Figure 1(1.36) and cylindrical projections (Figtne 10.37). The major pob. lents involved in making a map projectionare these two: plotting points of a globe on a de- velopablesurla«.and(haracterizingimage points in terms of peimage points.

121GuRE 10.36 Gnomonic projection: a projection of the surface of a globe from its center onto a plane tangent al the "north pole"

Ft Gum: 10.37 Cylindrical projection: a projection of the surface of a globe front its center onto a cylinder tangent 10 the globe al its "equator"

Sundials A study of sundials can be an intriguing project. and an intensive study involves some sophisticated mathematics. A typical sundial consists of two parts, a gnomon for casting a shadow and a dial on which the shadow falls. There are several kinds of sundialsequatorial, hot izontal, vertical, declining. and reclining. The name of each is

1 :3 identified with the position in which the dial is set. The construction of a sundial depends on the latitude of the observer as well as the intended position of the dial. The sundial pictured in Figure 10.38 was produced by a student of trigonometry who lived in Saint Paul. Nfinnesota (latitude 44° 57' 19"). It PROJECI'S. I' WIWI'S AND FAIRS. C.AIES. PUZZLES. AND CONTESTS 375

was «mstructed to be used %id) the dial mount- determined) and tin c I2 o'clock line. L is the lati- ed in a hori/ontal position and is therefore re- tude of the observer. and B,, is the dew ee measure fen ed to as a horirontal sundial. .\ Ithoughit of the hour angle (15 th»es the numbei of hours is not evident from) the picture. the gnomon is between noon and the hour «nrespon(l hig,to perpendienlar to the dial and intersectsitin hour line h).' the12 o'clock line. The angle formal by the A sundial indicates local sun time. Mean solar upper edge of the gnomon and the I2 o'clock time (an he computed from local Still time by lin: is -15°. Since the measure of the angle be. using a correction known as the equation of tween the tipper edge of the gnomon and the time. plane of the dial should equal the latitude of the point of obsemation. the sttulent's error was Making I)esign.c mil2' .11".In the Northern Hemisphere the The beauty of mathenlatics is often reflected in 12 o'clock line must point north. designs based on mathematical curves or figures. The position of any hour line it on a hori/on- Students frequently attempt to express the beau. tal dial can be determined by using the formula ty of mathematics by means of projects involving tan.11,,=slit I. tanBh. curve stitching or drawings. Even for students Referring to the formula. 4,, is the degree mea- who may be poorly motivated in mathematics. sure of the angle between the hour line h (being the fascination of a design is often sufficient to generate enthusiasm for artistic expression (Fig- ures 10.39. 10.10. 10.11. 10:12. 10.13. and 10.11). 3. Carl N. Shuster and Pied I. Bedfold. /grid II'mh in Mathematics (New Yotk: Anna ican Book Co., 1935; dis. tribunal by Yoder histittm('nts. East Palestine, Ohio). pp. 88-89.

FIGURE 10.39.Polygon of eighteen sides and FicuttE 10.38.Horizon tat sundial all possible diagonals 0

V III III . 111.111111Ell M.:11111i mrI ail.11 =IN 1.11.111 I=imiMI MO EMI NI I. I= 111 mi a II III: NI111111 N011111111 INE1111111 1E111 I=IN se =MNI. wil =1mil =RNmoo mI I =ENmil=ma MON

1111111 iiiill 111 1111 .olIll 1111 IIMMEN111111MEI mi 1III =MaMI MillEMI =1 s E = . .1 w I = imm - IN s. I=- =

1.11111 11111;11111EN Mi..111.- III mi--monolog MM.=Ea M'MENO.I 1111111.Rill II I 111111 IN II I .11111 .1 1110/11111 1 1 11111 I 1 001 MIN1111 IN OM 111 AMIMI mt, =Er. . 11111 . 378 clI :\PTER TEN

..111111=asimo.._ 11. A Historical Project Constructing a right angle with a piece of rope that is separated by knots into 12 parts of equal length fascinates students because of the simplicity of the construction and because of its historical significance. The girl who produced the display (Figtne 10.45) showing that segments with lengths of units,. 9 units. and 5 units can be sides of a right triangle probably learned something- of the history and lore of mathematics as she researched her project.

:4 0. {. ; 1 ;.' 71.' 4. 771''4"-':* . !', Cr L. . L -' ,- L, ee...6.7 FIGURE 10.45 2... L . Lt.1 Laying ont a right anglethe stmy of the "rope stretchers ...al L.. t .J....J....,e- L.r-.. a...

Theorem Demonstration Models Many easy projects involving theorem demonstration models arc possible. The symmetry and reg- ularity of the circle make it a favorite topic for such projects. The model in Figure 10.16 "magi- cally"' demonstrates an unusual theorem about angles inscribed in a semicircle. At least one student thought so. To "show" that every angle inscribed in a semicircle is a right angle. he cut a semicircular slot in a piece of plywood. slipped a loose-fitting bolt through the slot, and stretched a rubber binder ;won't(' the bolt and the two flails at the ends of the diameter. The PytImgorean theorem is another favorite topic for projects involving theorem demonstration models. Two projects that illustrate intuitive approaches to this theorem are shown in Figures 1 10.17 and 10:18.

FIGURE 10:16 All angles inscribed in a semicircle are right angles. IIEMATICS PROJEcrs. Eximwrs AND FAIRS. GAMES. PUZZLES. AND CONTESTS 379

MATH ENIATICA L GAN1ES AND PUZZLES A matbematiial game is a mathematical') based activity that in\ ()hes a set of "legal" 'tunes, operations. 01rules, and usual') some kind of competition. Most mathematical games are de- \ ised for two Or snot e persons on teams. but man) of these can also be played by one person; and some games ate intended Cot one person. as is solitaire in cards. Some mathematical games are puzzles. Kasner and Newman consider a pw..-.1e to be "an ingenious game or problem.' The structure of a mathematical puzzleis characterized by some "internal relationship between component elements of the puzzle."-. Creators of puzzles use clever ways of complicating a basic internal relationship by elaboration, restatement, addition of distracting details. and so on. The distinction between games and puzzles is not always clear, Ficntt--.10.47 and the expression ";;:nnes and puzzles" is often used in a collective sense to refer to both. Solv- itig a puzzle means disco:Acting the internalre- lationship that exists between the components of the puzzle or in'se'tting a scheme that will expose the relationship. Games :Hui puzzles can make a valuable contribution to a mathematics program. For example. they can be used to: 1. Add interest to practice periods 2. Teach mathematical vocabulary 3. Teach mathematical ideas 'I.Help studentsdevelopeffectivestudy habits 5.Provide motivation when beginning a new topic 6. Provide for individual differences 7. Promote the development of a positive attitude toward mathematics S.Help students develop intuition in solving problems FRantE 10.48 .1. &old Kasner and .jame:, R. Newman.Alathematics and the Imagination(New York: Simon & Schuster. 1940), pp. 157-58. 5. Ibid. 380 C.1 1APTER TEN

9.(,ke students an opportunit) to exeicise single disc at a time and without ever placing a imagination larger disc on a smaller one. The game as pro 10. Salm Ilari/e or re% lew 8 unit. posed by M. Clans (Lucas) in 1883 involved eight xposme to games and pit/ides should oidi- discs. but fewer tau also be e(i. The manipula- narily be for brief perio i '...C.ii e must be exer- tion involved can be simpliCed 1)) using dots On ( isedthat the ex( iteme. i': genera.ed does 1101 a sheet of paper instead of pegs. and coins in- (Hillis(' student attention." .1. game or :1pil//le stead of discs. (all. or coarse. be Ilse(';1.:, the basis of a lesson. An interesting example Of a puzzle is one that Games and pu//les (.111 be cleated by teachers involves arithmetic operations in which letters and students: howeer. a great many are Mail- are used as numerals. The challenge in such a able nom tommociai solo.(us..\ nip to an puzzle is to determine the digits that correspond major dealer ol games .111(1toys will convince to the letters. Below are several examples of this teachers and students of the wide variety that is kindof puzzle: available. A list of names of pioducers and dis- ME AT SEND nibutois is given at the end of this chapter. E AT MORE The binary system of numeration serves as the ME A MONEY basis for a number of good games. One example E A is the well-known game or Niiii. The strategy for ME A (AB) x (AB) = CDDD winning with this game involves practice with I S mathematical operations. WEEK Another game that involves the binary s)stem is the Tower or Hanoi. This game consists of E V E.TALK TALK.... three pegs secured to a base %%itli a number or D I ,D (lists ol dilletent diameters placed On one peg ill Magic squares are another kind of puzzle. Va- older ol si/e. with the smallest on top (Figine ious construction methods can be found in books 10.19). To win the game one must tansler all on recreational mathematics. Those with an odd discs ham one peg to a se( ond peg by moviii,g a number of entries are usually easy to construct. Students enjoy constructing magic squares for FIGURE 10.49 special occasions such as birthdays and anni- versaries. Figure 10.50 shows an example of a

G. Walter William Rouse Ball,Mathematical Rey cations and Essays(London: Macmillan 5 Co., 1905), p. 107.

1 14 4 15

8 11 5 10

13 2 16 3

12 7 9 6

FiGuitg 10.50. Extremely magic square MATHEMATICS PRO .1 EGIS. EXHIBITS A ND EA I RS. GANI ES. PUZZLES. AND CONTESTS 381

magic square invented by BenjaminFranklin. Consider. for exanqile. this problem: "Finda In this square 'various combinations of four en- number. given that thesum of the number plus tries have :t sum of 4. For example, theentries the stun of its digits is 73.- at the lour cornets add to 34. It is interestingto Probleins iinolving identification of Pythago. find just 1r ,w many combinations of fourentries have a sum of 34. rerun triples are still another source of puzzles. The following example is illustrative: Ceitain problems from elementaly number "In what right triangle whose sidesare a Pythagorean theory can be presented to studentsas pilules. triple is the sum of the legsa perfect square?" One such problem involves determininga number by having a student followa few easy hi- Puzzles that involve manipulation of objects st! action,. For example. have the student selecta have a strong appeal. :\ simple puzzle of this kind

positive integer between 1 and 60. Then give is one that tequires the separation ofa tangled instructions for the following: set of wires into component parts. Various puzzles I. Have him divide his number by 3 andtell of this kind are on sale attoy stores. A fairly you the remainder. Slippose this nennninder sophisticated wire puzzleisthe Chinese ring is a. puzzle. Solving this puzzle involves theuse of 9.Have him divide his number by 4 and tell the binary system of numeration. you the remainder. Suppose this remainder Dissection puzzles have always enjoyed great isb. popularity. These range from simple jigsaw type 3. Have him divide his number by 5 and tell puzzles that are suitable for use at the primary you the remainder. Suppose this remainder levelto dissection demonstrations of the Py- is c. thagorean theorem at the secondary level (Figure Now divide 40a + 455 + 36c by 60. There- 10.51). mainder will be the number the student selected. This problem is an application of the Chinese remainder theorem. An explanation of the solu- Triangle tion can be found in books on number theory. Number theory problems involving Diopiran- tine equations can also be presented as puzzles. 3

1'\ 4 2

Fu ,tune. 10.51 Given aright triangle with squares constructed on its sides, itis possible to dissect the two smaller squares into five pieces that fit on thelargest square. 382 C11 \ vrEit i-EN

Separating .1piece of caldboaul that has the mathematics class should be inn igning enough drape of a quadrilateral into lout pieces by cut- to engage students' intents( and just hard enough ting along lines joining the midpoints of oppo- to pi oc ide a espectable challenge.. biblioglaph site sides gives us another example of a dissection of games and pilules is given at the end of this puzzle (Figure10./52). The challenge inthis chapter. puzzle is to anange the four pieces so that the A mak)! problem in using games andpuzzles resulting figure has the shape of a parallelogram. in a mathematics class is the limitation of time. Puzzles imolving mosaics and tessellations are A skillful teacher willtryto use alter- school similat to dissection puzzles. These call for com- bouts or flee time (lutingunscheduled modules. bining differently colored tiles to form desig- In the case of competitivegames. using a timer nated figui es 01 to «mer the plane. The them y of helps keep games from taking too much time. tessellations may be found in a number of texts. Playing only parts of games is another way of and articles on tessellations ha% e appeared in making eflectke use of time that is ailable lot many periodicals. competitive games. The chessboaul can be used as a basis for a number of good puzzles. There is. for example, I.ATIIENIATICS CONTESTS the "Eight Queens" problem, which involves placing eight pieces on the board so that no two Mathematics contests can be an integral part of are in the same tow, column. or diagonal. This a school's mathematics program. In some school, pr ()blew can be solved by using determinants and contests are conducted within a given class or matrices. There is also the problem of occupy- subject area with-winners mox ing on to compete ing every square of the board by making continu- in district or regional contests. ous legal moves with a knight. Frequently. school teams arc formedfor the Sources of games and puzzles are so %ailed that purpose of challenging neighboring schools in a the obsen ant teacher can constantly hnd new contestinvolt ingcompetitivemathematical ones. Games and puzzles selected for use by a ngames.In conducting contests like these it is im-

FIGURE 10.52. The figure on the left shows a quadrilateral with the midpoints of opposite sides connected by segments. If the four smaller quadrilaterals are cut out, the pieces can be arranged so that the resulting figure has the shape of the parallelogram on the right. m \Tics PROJECTS. EXHIBITS \NI) FAII:S. GAMES. PUZZLES, AND CONTESTS 383 pot min that rules and criteria be established inteested in m.,thematics. regardless of abilit). beforehand and adhered to strictly. Thew must A teacher can do se% eral things to help prepare be a judge or team of judges to serve throughout students for a mathematics contest. For example. the contest. Awards for winners must be an- old tests can be made available. A well-stocked nounced and appropriate recognition given to library on many differenteas of mathematics all participants. Local school and commercial is%alttable. and constant encouragement isa newspapers are generally willing to publish the must. resultsofawellorganized contestinvolving As in the case of pojects. recognition should mathematical games. be given to all students who compete. whether A more sophisticated kind of contest is one they are winners OE not. Contests can adda strong thatinvolves administration of cleverly con- positive influence to a mathematics program. structed paperand-pencil tests to large numbers of students meeting in well-controlled testing sit- SUMMARY uations. Contests of this kind ate sometimes The challenge to teachers of mathematics is sponsored by local school districts. to prepare citiiens who will be able to identify and Perhaps the most prestigious examples of con- solve mathematical problems that may not have tests involving papeandpencil tests are the Na- been encountered before. Since itis difficult, if tional High School Mathematics Contest in the not impossible. to predict the kinds of problems United States and the Olympiads of eastern that future citizens will face, teachers must be Europe.' concerned with the development of skills and at- Contests involving paper-and-pencil tests are titudes in sURICills that will give them confidence generally suitable only for the better students; to cope with new problems. Mathematics proj- 110WC% en. other activities such as games. talks, and ects. exhibits. games. pilules, and contests can be seminms can be included in a day of contests. used to help students develop such confidence. and these usually have a somewhat wider appeal. Itis possible to make contests part of a mathe- 7.I I os% el II.. GI liven. School Mathematics Contests: A Wpm I (Washington. D.C.: National Council of Teachers matics field day with something for all who are of Nlaihematics. 1968). 384 CIL\ liTER TEN

REFERENCES FOR SIXTY-SEVEN PROJECT TOPICS

Listed below are sixty-seventopics for which relerence, ate included. References for the di f- 1 eren t topics follow the list of topics.

I.A116:111. Computational MC1.110(15 36.Curve Fitting 2.Measurement of Time 37.Trachtenberg System of Rapid Cal. 3.Codes and Ciphers culation 4.Computers 38.Algorithms and Flow Charts 5 . Curve Stitching 39.Abacus 6.Numeration Systems 40.Prime Numbers 7.Nomographs 41.Perfect Numbers 8.Measurement and Approximation 49.Figurate Numbers 9.Map Projections 43.Integers as Ordered Pairs of Natural 10.Linkages Numbers 11.Magic Squares i\farkov Chains 12.Paper Folding 45.Repeating Decimals 13.Paradoxes and Fallacies 46.Golden Section 14.Optical Illusions 47.Hyperbolic Functions 15.Approximt.tion of Tr 48.Continued Fractions 16.Pythagorean Theorem 49.Trisection of an Angle 17.Pythagorean Triples 50.Nine-Point Circle 18.Finite Geometries 51.Complex-Number Operations by 19.Game Theory Geometry 20.Fourth Dimension 52.Snowflake Curve r3 21.Algebraic Structures J .A Mathematical System of 2 X 2 22.Traasfinite Numbers Matrices 23.Symbolic Logic 54.Mass-Point Geometry Non-Euclidean Geometry 55.Lattices 25.Probability 56.Egyptian Computational Methods 26.Boolean Algebra and Switching Net- 57.Greek Computational Methods works 58.History of Mathematical Symbols 27.Statistics and Statistical Inference 59.Gaussian Integers 28.Topology 60.Curves of Constant Breadth 29.Mathematics and Art 61.Women Mathematicians 30.Mathematics and Music 62.History of the Slide Rule 31.Linear Diophantine Equations 63.The Four-Color Problem 32.Duodecimal System (H.The Konigsberg Bridge Problem 33.Linear Programing 65.Development of Arabic Numerals 34.Congruences 66.Fibonacci Numbers 35.Inversive Geometry 67.Sundials NI ITI EMAT ICS PRO J ECTS. EN!! 1 Brrs AND !Al RS. GAMES. PUZZ LES. AND CONTES1S 385

I. ANCIENT CONIPUTATIONAL Glenn. William II_ and Donovan A. Johnson. Fun M ET HODS 1111h Malhemalics.Manchester, IVebster Pub. Raymond W.Romping ilnongh Mathe. lishing Co.. 1960. ouni(,).New Yolk: Alb ed A. Knopf, 19.17. Johnson, Donovan A.. and William II. Glenn.The Cajori. Hoban. /IIlisienv of Alathemalics.2d ed.. World of Meastnemeni.Nlanchester. Nlo.: Webster rev. and enl. New York: Nlacmillan Co.. 1919. Publishing Co.. 1961. Crook. Welton J.Abacus Arithmenc.Palo Alto. Kaufman. Getald.The Book of Time. NewYolk; Jit Calif.: Pacific Rooks. 1958. liar Messner. 1938. E% es.I 1m% arc!.An Inbadziclion lo the //is/my of Klaitchik, Maihemalical Reoealions.2d ed. Malhenzalics.3d ed.. rev. New York: Molt.olt. Rine. New Yolk: Dover Publications. 1953. halt Winston. 1969. Licks. II. Recrealimtsin Mathematics.Princeton. N. Freitag. Arthur. and Herta Freitag.The .Y zim bee J.: D. Van Nosnand Co., 1929. .Shay.Washington.1).C.:National Councilof National Council of Teachers of Nlathentatits. T cachets of Nlathentatics. 1960. Souie Book of Alathenzalical A pplicalions.Seven. Heath. Thomas little.ed. Works t:ncludingThe teenth Yearbook. New York: Bureau of Publica- Method of Archimedes].1897.1912. New York: thms. Teacheis College. Columbia University. 1912. Dover Publications. 1953. (neat y. V. "Calendar Plow ess." InSchool Science 1 logben.Lancelot T.Tl 11' onderf Irmldof and Alathenzalits. January1951. pp. 22-26. Alathenzalies. GardenCity. N.Y.: Doubleday C Co.. Schaaf. WilliamL. "The Measurement of Time" 1955. (bibliography).AlathenzalicsTeacher.February Johnson. Donovan A.. and William II. Glenn.Com 1953, pp. 115-17. puling Devices.Nlanchester. Nlo.: Webster Pub. Smith. David Eugene.IlisImy of Mathemalies.2 vols. lishing Co.. 1961. 1923. 1925. New Yolk: Dover Publications. 1958. Karpinski. Louis,Ilislmy of AriIhnielic. 1925. Re. 3. CODES AND CIPHERS print. New York:Russell k RussellPublishers. 1965. Bakst, Aaron. Aftilhemalic.s,Its Magic and illasiery. Kokomoor. Franklin IV.Mollie:moth's in Human /g- Princeton, N.J.: D. Van Nosttand Co.. 1952. lobs.New Yolk: Pientice.11all, 1912. Ball. Walter William Rouse. MaihemaiimiReerea, Eisen, Ilatold I)., and II. Glenn Ludlow.ilrithme- lions and Essays.11th ed.. revised by II.S. M. lieforColleges. NewYork: Nlacmillan Co., 1963, Coseter, 1939. Reprint. New York: Nfacmillan Co.. Newman. James Royed.'The World of Mollie. 1960. walks.4 vols. New York: Simon & Schuster, 1956; Encyclopedia Americana.1968 ed. S.. "(:ryptog. paper. 1962. Vol. I, pt. 2. mph)." Sanford. Vera.A ,Shot/ Ilisimy of Mathenzalics. Ms. Encyclopaedia Brilannica.1969. ed. S.v. "Cryptology.- ton:I loughton Nlifllin Co., 1930. Gamow, George.One, Two, TInee. . Infinity: Faces Swain. Robert L., and E. D. Nichols.Understanding and Specidalions of Science.Rev. ed. New York: Athhazelic.New You k;1 loll. It inehat t C NVinston, Viking Press, 1963. P. 206. 1965. Kraitchik, Nlaurice.Mathematical Recreations.2d ed. Turnbull. Ilerbct tW.The Gieal Alathenilicians. New Yolk: Dover Publications, 1953. New York: New York University Press. 1961: Simon Levine, J. "A Cryptographic Slide Rule."Mollie. k Schuster, 1962. malics Magazine,SeptembeOctober 1961. Willerding. Maigaet,Front Fingers to Compute's. Peck, LymanC. Semi Codes, Remainder Arithmelic, Pasadena, Calif.: Franklin Teaching Aids. 1968. and Mabices.Washington, D.C.: National Council of 'Teachers of Mathematics, 1961. 2. MEASUREMENT OF TIME Asimor, Isaac.The Clock We Live On. New York: 1. COMPUTERS AbekodSchuman, 1965. Adler, Irving.Thinking Machines.New York: John TheRealm of Aleaszire.Boston: Houghton Day Co., 1961. MilllinCo., 1960. Bakst, Aaron.Mathenzalics, Its Magic and Masleiy. Bolton. Lyndon.Time Measurement.Princeton, N.J.: Princeton, N.J.: D. Van Nostrand Co 1952. D. Van Nostrand Co., 1924. Berkeley, Edmund.Giant Mains, or Machines Thal Bradley. A. "The Day of the Week for Gregorian Think.New Yolk: John Wiley R Sons, 1949. Dates."Soipla Mathenizzlica,March 1955. "Simple Simon."Scienlific American,No Encyclopaedia Britannica,1969 ed. S.v. "Calendar." vember 1950. 386 ClIAVEER TEN

Bernstein, A. "Computers vs. Chess Players :' Ilogben, Lancelot '. Wonderful World of Malh- ilmeritan. June 1958. mettles. Garden City. N.Y.: Doubleday Co.. 1955. Diebold, ,John.zumnalion: The Advent of the Auto- Johnson, Donovan A.. and William II, Glenn, Un matic Factory. Princeton, N.J.: D. Van Nostrand detslanding Numeration Systems. Manchester. Nice.: Co., 1952. Webster Publishing Co.. 1960. Dindenur1. "1 :le Ride of the Computers," Scien- Jones, Button W. Eiemen/ary Coneepls of Mathe- tific American, September 1952. matics. 3d ed. New Yolk; Macmillan Co.. 1970. Gardner, M. "Logic Machines. Scientific American, Keedy, Mervin. Num ber Syslems. Reading,Nlass.: March 1952. Addison Wesley Publishing Co., 19o5. GIIICIlbelgCr,F."Use of Large-Scale Computers.- Kramer, Edna E. The Alain Stream of Mathematics. School Science and Mathematics, February 1954. Greenwich, Conn,: Fawcett Publications. 1961. Maga. E. "A History of Digital Computers." School Larsen, Hat old D.. and II. Glenn Ludlow. Aria:me. Science and ,Italhentalics, March 1962. lie for Colleges. New York: Macmillan Co., 1963. Newman, James Roy, ed. The World of Mathematics. Lieber, Lillian R. The Education of T, C. Mits. ,1vols. New York: Simon & Schuster. 1956; paper, Instrated by Hugh G. Lieber. New York: W. W. 1962. Vol. -1, pt. 19. Nome') g: Co.. 1944. Rees, s1. "Digital Computers." American Malhemati. Meyer, Jerome S.. and Senate !lantern. PIM will: OW cal Month ly, June-July 1955. AreW Math. New York: Fa \\Tett World Library, Schaaf. William L. "Modern Calculating Machines." 1961: Hawthorn Books, 1966. Mathematics Teacher, February 1952, pp. 110-12. Nioise, Edwin E. The Number Systems of Elemen- Tompkins, Charles B. "Computing NIachines and tary Mathematics. Reading. Mass.: AddisonWes- Automatic Decisions.-InInsight; inlo Modern ley Publishing Co., 1965. Mathematics. Twenty-third Yearbook of the Na- Newman, James Roy. ed. The World of Mathematics. eional Council of Teachers of Mathematics. Vash- 4 vols. New York: Simon C Schuster, 1956: paper. ington. D.C.: The Council, 1957. Pp. 372-403. 1962. Vol.1. pt. '3. Reid. Constance. From Zero lo infinity. Sd rev. ed. 5. CURVE STITCHING New York: Thomas Y. Cowell Co.. 1965. Cundy. Henry Mattyn, and A. P. Rotten. 11 lathe. Richardson. Moses. Fundamentals of Mathematics. medical 31 odds. 2d ed. New York: Oxford Uni- 3d ed, New York: Macmillan Co., 1966. versity Press. 1961. Sharp, Evelyn. A Parent's Guide to Ihe New ;Valhi!. McCamman. Cato! V. "Curve-Stitching in Geometry." mettles. New York:E.P.Dutton g:Co.,1964: In Muip'Sensory Aids in the Teaching of Afalize- Pocket Books, nal. Eighteenth Yeai book of the National Coun- Swain, Robert I.., and E. I). Nichols. Understanding cil of Teachers of Mathematics. New York: Bureau Arithmetic. New York: Holt, Rinehart g! Winston. of Publications, Teachers College, Columbia Uni- 1965. veisity, 1945. Pp. 82-85. Twaddei, Richard R. "A Look at the Base Nega- tive Ten." Mathematics Teacher, Februat y1963. 6. NUNIERATION SYSTEMS pp. 88-90. Altweiger, Samuel. Modern Malhemulirs. New York: Macmillan Co.. 1960. 7. NONIOGRAPHS Bakst. Aaron. Malhentaths, Ils Magic and Maslety. Adams. Douglas P. "The Preparation and Use of Princeton. N.J.: I). Van Nostrand Co.. 1952. Noniographic Climes in I ligh School NI:idle:in:tars." Crowder. Norman A. The Arithmetic of Computers. InAtultiSensoty Aids in Ilte Teaching of Mathe. Camden City, N.Y.: Doubleday g: Co., 1960. mettles. Eighteenth Vembook of the National Conn Davis. P. "Number." Scien/i/ir American. September cil of Teachers of AIathemati(s. New York: Bureau 1965. of Publications, "Teachers College, ColmnbiaUtni- Friend, John N. Numbent, Fun, and Fact. New York: \cn 1995. Pp. 164-81. Chitties Scribner's Sons, 1954. Brixley, John and R. V. Aintree. Fundamenlais of Fulkerson, Elbeit. "Writing a Number in Different College Mathematics. New York:I role. Rinehart Bases." Mathematics Teacher, May 1960, pp. 36.1- Winston, 1961. 66. Cajori, Florian. A Histbly of Mathematics. 2d ed., Glenn, William II_ and Donovan A. Johnson. Num. rev., and enl. New Yoi k: Macmillan Co., 1919. her Patterns. Manchester, Mo.: Webster Publishing EncyclopaediaBritattnica.1969 ed. S.. "Nomog- Co.. 1960. laphy." NIATIIENIATICS PIZOJEGTS, ENIIIIIITS AND FAIRS. G.011.:S.OUZZLES, AND CONTESTS 387

Meyer. Jerome S.. a MI Stu:utI Ion km. Fin, with the of Publications.eacheis Collegt:, Columbia Uni- New Math. New York. frawcet IWorldI .ibrary. vet sit .1915. Pp. 160-63. 1961: Hawthorn Books. 1966. Yates. Robert (:. Gem/re/ilia!Tools: A Malhematual Smith. David Eugene. llistov ofMalhe»talics.2 vols. Sketch and Model Book. St. Louis, Nlo.:Educt. 1923. 1925. New York: Dover Publications. 1958. tional Publishers. 1919. Vol. I. MAC1(: SQUARES Van assle. Lowell T. "Notes on a 'SpideC Nom° graph." MathematicsTeacher, November1959. pp. Irving. Magic House ofNumbers. New York: 557-59. John Day Co.. 1957. Wylie. C. R.. Jr. "Nomoglaphy."Mathematics Teach- Andrew.s, W. S. et al.Magic Squares and Cubes.1917. cr.November 1961. pp. 531-37. New York: Dover Publications, 1960. A nema. Amh ext.S. "Perfected Benjamin Franklin 8. NIEASURENIENTAND APPROXINIATION Magic Spun es."Malhe»talics Teacher. January Bakst. Amon. el pinoximaleComptuathm. Twelfth 1956.pp. 35-36. Yearbook of the National Council of Teachers of Ball. Walter \Yilliam Rouse. MathimudicalRower Nlathematic.s. New Yolk: Burean of Publications. lions and Essays.11th ed..eised by 11. S. M. Teachers College. Columbia University. 1937. (...()N etch. 1939. Reprint. New York: Nlacinillan Co.. Mathe»iaticsItsMagic andMostety. 1960. Princeton. N.J.: D. Van Nostrand Co.. 1952. Bernhard. II. "A Simple :Method of Cenemting Ilanks. J. Ilonston.Elements ofMathematics. 2d ed. Nlagic Square of Doubly Even Older."Scripla Boston: Allyn on. 1961. Malhematico.SeptemberDecember 1919. Bendk. Jeanne.lime Much and flow Many. New Friend. John N. Numb/Pis. Fun. and Fact. New Yolk: Vol k: Grawd lill Book Co.. 19.17, Charles Scribner's Sons. 195.1. Johnson. Donovan A.. and William II. Glepn.The Goldner. Martin. ed. The Scientific dmericanBobh Irmiti of Measurement. lancliester.mo.\ye!). of Mathematical Puz.zles and Diversions. NewVol k: ster Publishing Co.. 1961. Simon & Schuster, 1961. 1:okomoor Franklin \'.Mallumuttics iu Duman Guttman, S. "A Triply Magic Swim e." Scripta Mathe. fairs.New York: Pi enact:41:111, 10'12. manor, SeptemberDecember 1019. Larsen. Ilarold 1).. and II. Glenn Ludlow,Alithat Ileath. R. "A Doubly Magic Square," Scripts Mollie tic for Colleges.New York: Macmillan Co., 1963. matica, March 1955. Shuster, (:a1 N., and Fred L. Bedford.Field Work Horner. W. "Addition,Multiplication in Magic in Mathematics. Newfork: American Book Co.. Squares of Older $." Scripta Mathematica, March 1935. Distributed by Yoder Instruments, East Pales. 1955. tine. Ohio. !hinter. J. A. II. Hunter's Math Rutin Teasers. New York: Bantam Books, 1965. . MAP PROJECTIONS Kraitchik, Maurice..lfathematicai Recreations. 2d ed. Flexne, William W.. and Cordon L. Walker.Mill- New York: Dover Publications, 1953. Pp. 112-92. Lary and Naval Maps and Grids: Their Use and Licks,II.Recreationsin Mathematics.Princeton. Cmtsbnction.New Volk: Dryden Press, 19.12. N.J.: D. Van Nost and Co., 1929, Gieenhood, David.Mapping. Rev.ed, Chicago: Meyer. Jemine S., and Stnalt1 lanlon. Pun with the New versify of Chicago Press, 1961, Atoth.New York: Fawcett 'World Library. Creiticr. Samuel.Memento)), Topography and Map 1961; Hawthorn Books, 1966. Reading.New Yolk: McGrawllill Book Co. 19'11. Posey. I.. "A General Formula for Magic Squares of 1:ellaway, Gunge P. map majecthms. New York: Valimis Orders. Beginning with Numbers Different E. P. Dutton g: Co.. 19,19. fromUnity."School Science and Mathematics, Schaaf. William 1.. "Nlap Projections and Cartogra- Apia1910. phy" (bibliography).Mathematics Teacher, Octo 12. PAPER FOLDING her 1953, pp. 1013. Gardner, M. "Flexagons."Scientific American,1)e. 10. LINKAGES cember 1956. Johnson, DonovanA, Paper Folding for the Mathe- Goldberg. NI ithael. "1.inkages in Three Dimensions." matics Class.Washington, D.C.: National Council In MithiVenswy:lids in the Teaching of Mathe- of Teachers of Mathematics, 1957. matics.Eighteenth Yearbook of the National Coun- Joseph, Maigai et. "Ilexahexallexagons." Alai/rem/in cil of Teachers c)1 Nlathematics. New York: Bureau Teacher, April1951, pp. 2,17-8. 388 (:1 l. \ PTEI2 TEN

KIM. I. Geinne/i u ett tses 1n Pala, bold I Ilmittrt 01 Ilathrinatirs, No, lntk, nig. New Polk: Diner Publit at ions. 1905. Maintillail Co.. 1961. Satip. \ Paper Model for Solid Geometi." Gelb msl.\O. Twos, emlental A ma llalhrnlrtlitsTern lie,.huh 1916. pp. 185-86. be) 4. h alnialed In I Boum, New Yolk: Does higg. "Ccomem 01 Paper Folding."Iand Publit at ions. 1960, Salim! &rem.- and 3lathemoth J11111.195,1. De- I bictben. Loudon E. .11athtriatit% Ito the Milhous t miler 1931. New Yolk: ONloid lies.. 1911. Yates. Ruben C. "Paper Folding." In .11a/tiSensor1 Moir. Jane. Of Men and ,\ Sets. Ness York Dodd. ,lot, in the wreathing of Mathematics. Eighteenth Co. 1961: Dell Publishing Co.. 1962. Yearbook 01 the National Council of 'leathers of Newman. lames Ito,. (II, The 11 mbt at Moth: inatit lathemats. New York: Bureau 01 Publit at ions. 'Istd. Nes; Yk: Simon Si 1956: paper. Teadiet s( :ollege. Colombia University, 1915. Pp. 1962. Vol. I. pls. 2-8. 131-59. Oystein. .Vmaitet Metal and Its History. Ness 13. PARADOXES AND FALLACIES Volk: .ThCraw11111 Book Co.. 1915. Rakst. Amon. MothemantS. ItA .11ve and Masten, Shanks. I).. and J. Mem h. Jr. "Caktalation 01 r to I'inoenm. N.J.: I). Van Nostrand Co.. 1952. 1(10.000 Deanna.," liathinatoi Friena.Jiihn N. .Vuurbers. Pun. and Pail. New Yolk: January. 1962. Chalk. St Motel's Sons. 1951. Smith. David Eugene. I I Atm) of Mathematics. 2 vols. kasner. !Await!. and Jame, 12. Newman. Mathmatus 1923. 1925, New York: Dover Publitations. 1938. and the Intagruation. New York: Simon k Struik, Dirk J. ,1 Con: ire II:story of Mathentatio. rev. c I. New Yolk: Dos er Publications. 1967. 1953. LI mo. Mathentatirs in 11PeAte1 a Culture. Nes% 16. `1'11E P1TIIAC/01:EAN TIIP.ORENI Ymk: Oxford Unive.ity Pies.. 1953. Berger. Emil J.'.\ Model lotVisit:Ili/Mg the Ptha. Lait,h1k. ,,Ilathentatit al Heti ern 2d etl, gine:titThecn cm." ,llathmatos'/em her.April New N'tnk: Dover Publications. 1953. 1933. pp. 216-17. I.ieks.II./lc, reationsin Mat hemath prim et( in. Glenn. 11*illiaa II_ and Donovan A. Johnson. The N.J.: D. %%an Nostrand 1929. Pythagorean l'heitrent. Webster Meter. Jeonne S.. and Slum tI Ianlou. Pun with the Publishing Co.. 1960. S rte Sloth. New Vol 1::Fa svt cutWorld Libra) v. Loomis. Elikha Swit, The Pythagorean Propositunt. 1961:I lawthorn Books. 1966. 2d ed.. 1910. 1Vashington, D.C.: National Council Nto dm op. Eugene P. Riddles in Slathematics. Prime. of Teachers of NI:latent:ni., 1968. Intl. N.J.; Van Nostrand Co.. 191.1. Si taaf. "The Theorem ()IPthagoras." 11. OPTICAL ILLUSIONS Slather:nits 'readier. De: ember 1931. pp. 585-8S. Bakst. Amon. Mathematics. Its Magic I ,1lastrly. 17. PYTHAGOREAN TRIPLES Plimeton. N.J.: D. Van Nostrand 1952. Ihrd. pyamgo, can Ntimbets." School Si:etrie and Reeler, Nelson and rninklyilBranley. Expti. ,Ilathernatics. Fehromy 1959. went,' In Optiol Illusi ttttNew Polk: Thomas Datiig. lwobia.. Number. the Lang of Scietne. Y. Crowell Co.. 1951. Ith ed.. ter. and enl. New York: 1:icinillan Brandes, 1 G. "Optical Illusions." School Si e and 1951: Doubleday k Co., Anchor Books. 195(1. .11athernaties. Orober 1951. Dresden. mold. An Invitation to Mathematics. New 15. APPROXIIATION OE II Yolk:I loll. Rinehmt 5. %%Pittston. 1936. Baravelle. Herman von. "The Number 7:." Ilart. Philip J. "Pythagorean Numbers." ,Ilritheinato matics Teacher, Nlay 1952. pp. 310-48. Teacher, Jaitualy 1951. pp. 10-21. Cajoi, Florian. Ilistory of Madre:troths. 2d ed.. Jones. Button W. The Themy of Numbers. New rev. and mil. New fork: MaC11111bn Co1919. Volk: !lob, Rinehart k Winston, 1935. Cainalian. Walter II. "Pi and Probahilitr," Slathe- Klein. Felix. Elementary Mathematics front an Jr! macs T-acher, Fehrtaaty 195:1, pp. 65-66. yawed point of View, Tian slated from 3(1 Gelman Dant/ig. Tobias. Number, the Language of Science. ed. by E. It. Hedrick and C, A. Noble. 2 vols. Ness .0,th ed.; ce. and ea New York: Nlacmillan Co.. Dove, Publications, 1921. 1951: Doubleday 5 Co., Anchor Books, 1956. Long. Calvin T. Me:nen/m Introdru lion to A'unthet Eves. IIowani, iln lattoduction to the !thirsty of Themy, Boston: I). C. Heath Co., 1961. A/Mental/CS, 3d ed., rev. New York: Holt. Mitc- Merriman. Gaylord M. To Discover Mathematics. h:tit k Winston. 1969. New Yolk: John Wiley Sons. 1912. NIATHENIATK:s pRoJEcTs. EXHIBITS AND FAIRS. GAMES. PUZZLES. AND CONTESTS 389

Miksa. Ennui, L. "k Table ol Integral Solutions (11 1 tot,. Net% V. it Simon k Schuster. 1956: paper. bz = ); Trarher. April 1962. Vol, 2. pt. Ii. 1955. pp. 251-55. Rideodson. Night:N. Food lilac of Mathe tit %. Nloshan. Ben."Pi imitit e Ithagoi ea n 3d ed. New York: Nlatmillan (5o.. 1966. Malhe tits Tracho. November 1959. pp. 511-15. .john1). Thr compleat stratrgrd. Nett ()t c. Oystein. Number Thrmy and Its Ilistmy. New Yolk: NI(Graw.11ill Book Co.. 1965. York: NI(Glawdlill Book Co.. 1918. 20. FOURTH DINIENS1ON Rademacher. Ilan,. and Otto Toeplit/. The Knjoy. meld of Mather:laths. Plinceton. N.J.: Princeton Edwin A. Flathoul: Hourancr .1laul Univosity Press. 1957. -n1ions. 5th ed. New York: Barnes k Noble. Stewart. Bonnie NI. Themy of .V hers. New Yolk: Mac oti 1 la n Co.. 1952. Bakst. Aanni. Mollielies. Its Magic and Mastery. Wilson. john. "A Heuristic Approach to Pthagnrean Prin(eton. N.J.: I). Van Noorand Co.. 1952. Triples." Mail:maths rarher. Nlay 1969. pp. 357- Gamow. C./Inge. One. Tiro. Three (p0. Facts 1 S/iernlationc of Silence. Rev. ed. New Ytnk: Viking Press. 1963. IS. FIN FE GEONIEFRIES Kamer. Edwaid. and James R. Newman. Madre. Brixey. John. and R. V. Andree. Fundamentals of lies and thin I ination. New York: Simon k Colh'g Maths,lies. New York: llolt. Rinehart k Schuster. 1953. 1Yinston. 1961. Rline. NIonis. Mathematics in Western Gullme. New Ft emu!. John E. el Modein Introdurthm to Mathr. Yolk: Oxlind Unive:-sity Pies. 1953. unities. Englewood N.J.: PR:mice-11AL 1956. Kramer. Edna E. The Main Stream of .Ilathernalics. Lieber. Lillian R. The Education of T. C. Mils. Illus. Cleenssicli. Conn.: Fawcett Publication:. 1961. traced by Ilugh C. Lieber. New York: W. W. Nor- Lieber. Lillian R. The liduration of T. G ton k Co.. 1911. hated by !high C. Lieber. New Yolk: W. W. Nor. Iesei ve. Brut e E. Fundanirota/ concepts of Groin. ton k Co.. 19 H. etry. Reading. NI:is:4 Addison.WesIty Publishing Ilemy P. Gemm:by of Four Dimensions. C.o..1955. New Yolk: Dover Publication:. 1911. Norton. Monte S. Finite Mathe tical Systems. Ex- Maiming. Ilemy I'.. ed. The Fourth Dimension Sim. ploring Mathematics on Your Own sale:. New ply Explained. 1910. Reprint. New York: Dover York: NIc(raw4 fill Book Co.. 196:1. Publication:. 1960. Sawyer. W. \'. Prelude to Mathetics. Baltimme. Young. john We,ley. Lectures on Fundamental Coo. NId.: Penguin Book:. 1955. re/its of ellgebra I Geometry. New York: NI:u Stabler.E.R.AirIntroductiontoMatta' and millan Co.. 1911. Thought. Reading. Nla.s.: Addison.Wesley Publish. ing Co.. 1953. 21. ALGEBRAIC STRUCTURES Yarnelle. JohnE. Finite)lathe lit al Shut-lures. \dim Irving. The New Slathearatics. New Yolk: New Boston: I). C. Heath k Co.. 196.1. American Library. 1960. Allendmaler. Call B.. and C. 0. Oakley. Principles of 19. GAME THEORY Mathematics. 3d cd. New York: McGraw-11BI Rook Blaikwell. D.. and NI. A. Gershick. Throry of Games Co.. 1969. and Statistical Decisions. New York: John \l'iley Bush. George C.. and P. E. Obreanu. Bmir GourepIs Sons. 195,1. of Mathenurths. Nev York: I loll. Rinehart k Vin. Glicksman. AbrahamI. Lin /n/rodnetion to Linear sum. 1965. hog:intoning and the Theory of (; S. New York: Itt. Sic 'Nal. Introduction to Gontemporary Mathr. John \Viley k Sons. 1963. tics. San hatickto: Holden.Day. 1966. Kemeny. John G.. J. Lam ie Snell. and C. I.. Thomp. McCoy. NealII. Introduction to Modern MgrImr. son. 'Iii Introduction Io Finite Attain: Ins. 2d cd. Boston: Allen k Ba«m. 1960. Englewood Cliffs. N.J.: PrenticedIall. 1966. Nleserve. Bitice E.. and Max A. Sobel. Motile tics Luce. Robert 1).. and I I. Raiffa. Canirs and Decisions. for Secondary School Teachers. Englewood Cliffs. New York: John iley k Sons. 1957. N.J.: hulked fall. 1962. Nlosteller. nederick. "Optimal Length of P1:1for :t Nlooie. John T. Wen/rots of abstract :ilgebra. New Binomial Came." Alathernatirs Teacher. October Yolk: Nlacmillan Co.. 1962. 1961. pp. -111-12. Richardson.ft loses. rundaineineth. of Mathematics. Newman. Jame: Roy. ed. The World of Mathematics. 3d ed. New York: Nlacmillan Co.. 1966. 390 GlIAPTER TEN

Sawer. W. W. helnde to Mal& tics. Baltimore. ing. Nlass.: ddison-Wesle Publishing Co.. 1969. NM.: Penguin Books. 1955. Exner. Rohm NI.. and Nlyrdm r. R,mkopf. Logic to Stabler.E.R. An Inhodurtionto Mollie lical /.../cutentetty .11othentati< s. Nen Vol k: N1cGraw I !ill Thonefil. Reading. Nlass.: Addison.Wesley Publish. Book Co.. 1959. ing Co.. 1953. Johnson. Donovan A. Logic and Ilearoning in Malin.. Weiss. Nlaric J.. and Roy Dubisch. Higher Algebra meths. New Yink: McGmw-IIIII !took Co.. 196:i. for the Undergraduate. 2d ed. New Vol k: John Jones. Ihnton W. Eirmentarr Concepts of Writhe, \vitey So3s, 1962. maths. :Id ed. New Yolk: Nlatmillan Co.. 1970. 22. TRANSFIN1TE NUMBERS Kemeny. John G.. J. Lamie Snell. and G. I.. Thomp. son. :121 Intiodnetion tel Finite Mathematics. 2d cd. IIiii ii clii Ii:, I. I.e(maid Modern riew of Cemn- Englewood Cliffs. N.J.: Prentice-11AI. 1966. eli y. San Fran( ism: W. II. Freeman k Co.. 1961. Itocton: n Cantor. Georg. Contributions to Founding of a Theory KenellY, John. info' ILogic. Bacon. 1967. of 7'. sfinite Numbers. New York: Dover Public:1- Kline. Nloiris. ,Ilathentalics in Irestrin Cullmr. New lions. 1915. York: Oxford University Press. 1953. Courant. Richard. and lierbett Robbins. What b Lieber. Lillian R. Mits. Intr. I Logic. 3d ed. Illus. Mathentatin? New York: Oxford University Press. crated by Hugh G. Lieber. New Vi ilk: W. V. Nor. 1911. ton & Co.. 1960. Dicsden. Arnold. An Invitation to Mathematics. New Pfeilfer.E. "Symbolic IA)gic.- Scientific American. Yolk: !loll. Rinelcut k Winston, 1936. December 1950. Gamow. George. One. Two. Three Infinity: Richardson. Moses. Fundamentals of .Ilathrtuntirs. Facts and Speculationc of Selena-. Rev. ed. New 3d ed. New York: Macmillan Co.. 1966. York: Viking Press. 1963. Stabler.E.R.:InInhodurtionto Mathematical Kasuer. Edward. and James R. Newman. Mathematics Thought. Reading. Nlass.: Addison-Wesley Publish. I the Inieigination. New York: Simon & Schuster. ing Co.. 1953. 1953. Stoll. Robert R. Sets, Logic. I Axiomatic Theories. Newman. James Roy. ed. The Ilforhl of Mathematics. San Francis( o: W. II. neeman k Co.. 1961. Vols. New Yolk: Simon k Schuster. 1936: paper. Suppes. Patrick. Introdurt. to Logic. Princeton. 1962. V('I. 3. pt. 10. N.J.: D. Van Nostrand, 1957. Reid. Constance. 11 Zero to Infinity. 3d lee. ed. New Yolk: Thomas Y. Clowell Co.. 1965. 24. NON-EUCLIDEAN GEOMETRY Intiodurtion to Higher Mathematics for the Geneml treader. New York: Apollo Editions. Bonob. Robot. Non-Eurlidean Orono:by. New York: 1962. Dover Publications. 1951. Vat n ci Ic. John E. Introduction toT,,,,,,fini;e Cooley. !laths R.. and II. E. \Vahlert. 11111°Inflien to fiahemaths. Boston: I). C. Ileath & Co.. 196.1. Mathentatirs. 3d ed. Boston: I hnightoll NIifflin Co.. Zippen. Leo. UR-A of Infinity. New York: Random 196S. !louse. 1962. Kasncr. Edwaid. and James R. Newman. Mathe molars and the Imagination. New York: Simon k 23. SYMBOLIC: LOGIC Schuster. 1953. 11Iendoerfer. Carl B. "Deductive Iio iiifathe. Klein. Felix. Eientenhor Mathematics fi o,,, an Ad. maths.-In Inlights into Modern Mathematics. mitred Point of View. Tianslated tom 3d Getman Twenty.third Yealbook of the Natitmal Council of ed. by E. R. Hedrick and C. A. Noble. 2 vols. New Teachers of NIathematics. Wasliiiigte iii. D.C.: The York: Dover Publications. 192.1. Vol. 2. (:ount il. 1957. lip. 65-99. Kline. Nhuris. "Connelly.- Scientific American, Sep. Alleudoerfer. Carl and C. 0. Oakley. Pat/rip/es of 'ember 1961. Mathematics. 3d ed. New York: NIcGrawilill Book ,,Ilathentatirs in II n Culture. New York: Co.. 1969. Oxford University Ness. 1953. Banks. J. !huhu'''. Fdements of Mathematics: 2d ed. Lieber. Lillian K. The Edmatimt of T. C. Slits. Illus. Bosun): Allyn k Bacon. 1961. trated by I high G. i.ieber. New York: W. W. Nor. Cluistian. Robert K. Logic and Sets. Boston: Ginn X: tern k Co.. 1911. Co.. 1965. Grometty. New York: Galois Dubisch. Roy. lattlices to Logic. New York: Holt. Institute of Alathematio and Att. 1966. Rinehart & Winston. 1964. Nlanning. Henry P. Geoutchy of Four Oimensionc. Durst. 1.11111411. The Olant ;;;;; r of 31 athe New York: Dover Publications, 1914. mATH EmATics PROJECTS. EN!!! BITS AND EA Rs. GA Ai puzzus. AND CONTESTS 391

NIEise. Edwin E. Elementary Geometry pont an Ad- rnold. BradfoldI!.Logicoul Boolean Algebra. vaneed Standpoint. Reading. Mass.: Addison-Wes Englewood aim. N.J.: Prentice-11AI. 1963. ley Publishing Co.. 1963. Bakst. Aaton. Mathematics. Its Magie and .11aIrry. Newman..james Roy. ed. The World of MathernalitA. Princeton. N.J.: D. Van Nostnnd Co.. 1952.

1 wits. New York: Simon Schuster, 1956: paper. Bell. Eric T. The Development of Mathentat'.s. 2d 1962. Vol 3. pts. 11 17. ed. New York: NIcGraw-Ilill Book Co.. 1915. Richaidson. Moses. Fundamentals of Mathematics. 3d Callowa). Jean NI. Fundamentals of Modern Mollie- cd. New York: Macmillan Co.. 1966. matirs. Reading. Nlass.: Addison-Wesley Publishing Sawyer. V. V. Preind to Mathematics. Baltimore. Co.. 1961. Nick: Penguin Books. 1955. Christian. Robert R. Logic and Sets. Boston: (lion Wolf. Ilatold. introduction to Non.Enclidan Geoni Co.. 1965. city. New Yolk:I lolt. Rinehart k Winston. 1915. Dubisch, Roy. Lattices to Logic. New Yolk:I lob. Rinehart k Winston. 1961. 25. PROBABILITY licksman. Abraham NI..and I y R udenuan. Bakst. Aaron. Mathematics. Its ,llagic and Mastery. Fundamentals for Advanced Mathematics. New Princeton. N.J.: D. Van Nostrand Co.. 1952. York: Ilolt, Rinehart Winston. 196I. Jacoby. Oswald. //ow to Figure the Odds. Garden Ilu. Sic Tsen. introduction to Contemporm Mathe- City. N.Y.: Doubleday C Co.. 1'917. matics. San Francisco: Ilolden-Day, 1966. (Dues. Button W. Elementary Concepts of Mt:the. Kemeny. John G.. J. Lamle Snell. and G. L. Thomp- rustics. 3d cal. New Yolk: Nlacmillan Co.. 1970. son. An Intiodurtion to Finite Mathematics. 2d cd. Kasper. Edward. and James R. Newman. Mathematio Englewood Cliffs. N.J.: Plenticollall. 1966. and the Imagination. New York: Simon & Schuster. Newman. James Roy. Ed. The World of Mathematics. 1953. 1 vols. New Yolk: Simon Shuster. 1956: paper. Kenteny. John G.. J. Laurie Snell, and G. 1.. Thomp- 1962. Vol. I. pt. 2. son. :In introduction to Finite Mathematics. 2d ed. Richardson.Nloses. Fundamentals of Mathematics. Englewood Prentice-Ilan, 1966. 351 ed. New Yolk: Macmillan Co.. 1966. Kline. Morris. Mathematics 1:: Western Culture. Neu Whitesitt. Join: E. Boolean Algebra and Its Applica- York: Oxford University Press, 1953. tions. Reading. Nlass.: AddisonWesley Publishing Kokomoor. Franklin W. Mathematics it: Human Al- Co.. 1961. /aim New York: Prentice-Ilan, 19.12. Klatner. Edna E. The Slain Sticam of Mathematics. 27. STATISTICS AND STATISTICAL Greenwich. Conn.: Fawcett Publications, 1961. INFERENCE Newman..james Roy. ed. The World of Mathematics. Allendoerfer. Carl B.. and C. 0. Oakley. Pei:triple IVols. New Yolk: Simon S: Schuster. 1956: paper. of Mathematics. 3d cd. New York: NIcGraw-Ilill 1962. Vol. 2. pt. 7. Book Co., 1969. Not duty. Eugene P. Riddles in Mathernatirt. Prince- Banks. J. 11ouston. Elements of Mathematics. 251 ed. ton. N.J.: D. Van Nosnand Co.. 194 -I. Bosom: Allyn Bacon. 1961. Richardson. Moses. Fundamentals of Mathematics. 3d Bross.I.D. Design for Decision. New York: Mac- cd. New York: Nlactnillan Co.. 1966. millan Co.. 1965. Robbins.I lerbet I. "The Them y of Probability." In Ilogben. Lam clot T. The Wonderful World of Insights into Modern Mathematics. Twenty.third ellathernalics. GardenCity.N.Y.:Doubleday Yearbook of the National Council of Teachers of Co.. 1955. NI athematics. Washington. D.C.: The Council, 1957. Dattell. blow to I.ie with Statistics. New Noll:: Weaver. W. "Probability." Scientific American, Oc- W. W. Not ton Co.. 1954. tober 1950. Johnson. Donovan A., :tad William II. Glenn. The 26. BOOLEAN ALGEBRA AND World of Statistics. Nlanchester, Mo.: Webster Pub- lishing Co., 1961. swiTcHING NETWORKS Morris. Mathematics in Western Culture. New Allendoerfer. Carl B.. aw! C. 0. Oakley. Principies of Von k: Oxford University Press, 1953. Mathernatir.,. 3d ed. New York: McGraw-Hill Book Moroney. \I. J. Facts from Figures. Baltimore. Co.. 1969. Penguin Books, Pelican Books. 1956. \ndree. Richard V. Seleetions from Modern Abstract Newman. James Roy. ed. The Irmlel of Mathematics. Algebra. New York:Ilolt. Rinehart S: Winston. 1 vols. New Yolk: Simon Schuster. 1956: paper. 195S. 1962. Vol. 3. pts. 5. 10. l 13. 392 CHAPTER TEN

Rich:111(km).Nloses. Fundamentals of Mathematics. Schaaf. William I... ed.Oin 3lathematual Iletitage: 3d ed. New Yolk: Macmillan Co.. 1966. liAays on the Natme and Cultmal Signifieamc of 28. TOPOLOGY .I;;(13I1 rematicA.Rev, ('d. . New York: Collier Books. Biadfo Id 11.Intuitive Concepts in Elmen- Schminger. Joseph. Madicounnwl Basisof the Arts. tary. Topology.Englewood Cliffs. N.J.:1)1.cm:ce- NewYolk: Johnson Reprint Corp,. 191$. 1 l:ill. 1962. Weyl. Hermann.Symmehr. Prin«qem. :;.J.: Ponce Bing. R. II. "Point Set Topology.- InInsights into tonUniveisit Press,1952. ,Modern Mathematics.Twenty-third Yearbook of the National Council of Teachers of Mathematics. 30. MATIIENIATICS AND MUSIC: Washington. D.C.: The Council. 1957. Pp. 306-35. Bushaw. Donahl Wayne,Elements of Ceneial Top- Cooley. Hollis R.. and It. E. Wahlert.Intuiduction ology. NewYolk: John Wiley $ Sons, 1963. to Mathematics. 3ded. Boston: HoughtonMifflin Courant. Richard. and Ileibert Robbins.What is Co.. 1968. Culver. Chalks Mathemati Ls?New Yolk: Oxford Univeisity ,Musical .4cousties. NewYork: NIc- 19.11. Graw-Hill Book Co.. 1956. !.tiler. L. "Kinligsbeig Bridges.-Scientific .4metuan. Kline. \Iurrii, .11athematies inII'estein Cultnre. New July 1953. York: Orford university Ness. 1953. Chap. 19. Newman. James Roy. ed. TheIrmld of illathematics. Gamow. George.one, Two. 7 / : ,...... / Facts and Speculations of Science. Rev. ed. New 01,. New Yolk: Simon g: Schuster. 1956: paper. Yolk: Viking Pte%S. 1963. 1962. Vol. 2. pt. 5. Ga hien ,Nial tin.New Alathematical pont Pinkerton. R. "fulmination Theory and NIelody.- Scientific .4metican.New York: Simon C Schuster. Seim: i fie m e dew n .Febrtary 1956. 1966. Rideont. Fheodore C. "Sebastian and the Wolf.' 3falhonalics Teacher, Februmy Johnson. Donovan and %Villiam II. Glenn.Top. 1955, pp. 84-86. °logy.,Manchester. Webster Publishing Co.. Steinhans. Hugo. illalhemalicedSnapshots. 3d Ameri 1960. can ed.. rev, and enl. New Yolk: Oxford Univeisity Ness. 1969. Jones. Milton W.Klementeny Concepts of ,paltry ?italics.3d ed. New Yolk: Nlacinillan Co., 1070. Kasper. Edwanl. and James R. Newman. ,Ilathentalw. 31. LINEAR. D1OPHANTINE EQUATIONS and the Imagination. NewYork: Simon kSchuster, Bell. Elk T. Mathematics,Queen and Servant of 1953. Science.New York: NIcGraw-Hill Book Co.. 1951. NIeseive. Bruce E. -Topology for Secondary Schools.- Bernhait. A. "Euclid's Ladder.- OklahomaUnivemity Newsletter. 1959. ,Ilathematics Teacher. November1953. pp. 465-71. NIciise. Edwin. and Floyd Downs.Gometty.Reading. Carmichael. Robert D.Diophantine :Inalysils. New Addison-Wesley Publishing Co.. 1966. York: Dover Publications. 1915. Newman. James Roy. ed.The World of Mathemafics, Theory of Numbers. NewYork: Dover Pub- 4 volt. New Yolk: Simon .3: Schuster, 1956: paper. lications. 1914. 1962. Vol.I. pt. 4. Coin an t.Richard. and Herbert Robbins. What Ls Northrop. Eugene P.Riddles in Illathematics.Prince- Mathentatics? New York: Oxford University Press. ton. N.J..: I). Van Nostrand Co.. 1944 1941. Steinhaus. Hugo.JIlathematical Snapshots.3d Ameri. Dickson. Leonard. ,110dernEkmenteny Themy of can ed.. rev. and enl. New York: Oxford U NumbeLc. Chicago:University of Chicago Press. city Press. 1969. 1939. Tucker. A. "Topology.-Scientific ilmetiean,January Gelfond. A. 0.The Solution of Equations in Inte 1950. gets.Translated by J. B. Roberts. San Francisco: \V. II, Freeman 3: Co., 1961. 29. MATHEMATICS AND ART Jones. Burton W.The Thmy of Numbers, New Kline. Morris. -Plojective Geometry.-Scientific ilniee. York: Holt. Rinehart 3: Winston. 1955. icon.January 1955. Meserve, Bruce E,Fundamental Concepts of Geoni Mathematics inWestern Culture. New eby. Reading.Alass.: Addison-Wesley Publishing Yolk: Oxford University Press, 1953. Co.. 1955. Newman. James Roy. ed.The World of Mathematics. Nile!). Ivan M.. and Herbert S. Zuckelman./In In. 4vols. New York; Simon fi Schuster, 1956: paper, tioduction 10 the Themy of Numbos.2d ed. New 1962. Vol. 3. York: John Wiley Sons. 1966, NIATHENIATICS PROJECTS. ENHIBITS AND FAIRS. GANIES, PUZZLES, AND CONTESTS 393

e. 0)steiu, Ntonber Them and Its History. New Jones. B. \V. Moduho .4tithmetn. Waltham. Mass.: York: IcGtaw11:11 Book Co.. 1948. Blaisdell Publishing Co.. 1961. Uspensky. jallICS s'.. and M. A.I leaslet. Elementary Law. C. "Arithmetical Congruences with Practical Ap Nimbi.; 'Army. New York: IcGraw-11111 'look plications.- Mathematics Magazine. Co.. 1939. 1958. Young. Ja«,I) \Villimn Albert. ed. 31onopaphs on :\leserl e, Bruce E. I rindamental Concepts of Grom. Topics of Minton Mathematics Relevant to the etty. Reading. Addison Wesley Publishing Elemen (my Field. 1911. New York: Dover Publica- Co.. 1955. t ions. 1955. RowP. D. "Algebraic Congruence." Ohlahonza 32. I-IE DUODEC:1MAL SYSTEM v idly Newsletter. December 1959. Schottllathe. mains Study Gt oup. Essays no Number Themy. 2 Andiews. F. "An Exclusion into Numbers." Atlantic Lois. Pasadena, Calif.: A. C. Yunnans. 19: Mouth h.. October 1931. \Pol.I. Nlarie J., and Roy Dubisch. Algebui for Banks. J. Houston. Elements of Al loth emalics. 2d ed. the Undergiadnate. 2d ed. New York: John Wiley Boston: Allyn k 1061. R: Sons. 1962. llstiont. John E. Basic Concepts in Modern illothe matics. Reading. NI:rs.: AddisonWesley Publishing 35. INVERSIVE GEOMETRY Co.. 1961. Adler. Claire Fisher. ,11011C111 Ge011leby: lntegrat. Lar.sen. 11. "An Application of Duodecimals." School ed First Course, New York: McGrawd till Book Co., Science and ,Ilathematics. June 1955. 1958. Larsen. 14:11old D.. and II. Glenn Ludlow. Aiithme. Com t. Nathan Altschiller. College Geornetry. New tic for Colleges. New York:Ittcluillan Co.. 1963. York: Barnes k Nobles, 1952. NIcl ann. Robe] t C..fir. "Decimals and Duodecimals, . "Notes on Inversion." Mathenzatics Teash .333 .. of .400." Mathematics Teacher, November e, December 1962. pp. 655-57. 1959. pp. 570-71. Davis. David R. Modern College Geometry. Reading. :\ forum. Robert L. "Decimal and Duodecimal Red- Addison.Wesley Publishing Co.. 1949. piocals." Mathematics 'readier, Niay 1963. pp. 333- Eves. Howard. A .Survey of Geomatly. New York: ,39. llolt. Rinehmt S: Winston. 1963. NI tidier. Fon( k J. Arithmetic: Its Sir-name and C011- Johnson, Roger A. Advanced Euclidean Geometry. (CPIS. 2d. ed. !Englewood Cliffs. N.J.: Pt entice-Hall, New York: Dover Publications, 1929. 1961. Spencer.Richard."Discovery ofBasic Inversion 33. LINEAR PROGRAMING "Theory by Construction." Mathematics Teacher, 1:k). 1964. pp. 303-6. Shmuel. "The Fundamental Theorem on Lin- Taylor. Edson 11.. and G. C. Baum>. .4n 1111)0(1110;0n ear Prop:num:lig." 11lothematies Teacher, janu- to College Geometty. Rev. ed. New Yotk: Macmil- al). 1961. pp. 25-26. lan Co., 19.19. Glicksman\brahana An Introduction to Linear Programming and the Theo?). of Gaines. New York: 36. CURVE FITTING John Wiley 5 Sons, 1963. Brixey, John. and R. V. Andree. Frindamentals of Lichtenberg. Donovan, and ,,Marilyn Zweng. "Linear College Mothenzeitics. New York: Holt, Rinehart hogramming ProblemsforFitstYear Algebra." S: Winston. 1961. illothernaticsroche?, Nlardt 1960, pp. 171-75. Cell, John W. Analytic Georneby. New York: John Panwalka, S."Linear Programming." Oklahoma Wiley C Sons, 1960. University Newsletter, Spring 1968. Duncan, H. Practical Curve Tracing. New York: Longmans, Green F: Co., 1940. 34. CONGRUENCES I lart. William. College Algebra. liostml: D. C. Heath Allendoerfer. Call B., and C. 0. Oakley. Thinciples of K: Co., 1966. illathematics. 3d ed. New York: McGrawill Book Landon, Melvin V. "Curve Fitting." Mathematics Co.. 1969. Teacher, May 1960, pp. 349-52. Andiee. Ridtaid V. Selections from Modern Abstract N'i'ne, William. Numerical Calculus. Princeton, N.J.: Algebra. New York: Holt, Rinehart g:\\Pittston, Princeton University Press, 1949. 1958. Morton, B. R. Numerical Approximations. New York: Goldsmith. N. "A General Test for Divisibility by a Dover Publications, 1964. Prime." School Science and Mathematics, April Richardson, Moses. College Algebra. 3d. ed. Engle- 1951. wood Cliffs, N.J.: Prenticediall, 1066. 394 (:11APTER TEN

Smatter, Joseph F. "A Note on Curve Fitting." Mathe- 4th ed.. ley. and cal. New York: Macmillan Co.. matics Teacher, May 1963,pp. 213-21. 1951: Doubleday k Co.. Anchor Books. 1956. "rhomas. Ce(wgeB.. Jr.. Calculus and Ana lyth Geom- Gamow. George. One. Two. There . In finily: eh. Reading. Mass.: ddison-Wesley Publishing Feu Ls and Speutlations of Science. Rev. ed. New Co.. 1967. Vol k: Viking Press.196:3. 37.THE TRACHTENBERG SYSTEM OF Gardner. Ni. "Niathematical Games." Scientific Anui. can. March 1961. ,June 196.1. RAPID CALCULATION Kasner. Edward. and James R. Newman. Alathe- Cutler. Ann. and Rudolph McShane. The Trachten- ;milks and the Imagination. New York: Simon berg System of Speed Mathematics. Garden City, Schuster. 1953. N.Y.: Doubleday k Co.. 1960. Newman. James Roy. ed. The lEorb I of Mathematics. Trachtenbetg. akow. "Any Boy Can Be a Computer: 4 %ols. New York: Simon k Schuster. 1956: paper, TrachtenInag Speed System." Life. 17 May 1963. 1962. Vol.I. pts. 2-3. 38. ALGORITIIISAND FLOW CHARTS Niven. Ivan NI.. and 11c:then S. Zuckerman. A» Mho. ditetion to the Theory of Nmnbers. 2d ed. New Additional Mathematics. EmebbeMathematics Series. Yolk: John Wiley Sons. 1966. Newton. Nfass.:Education Development Center. Ore. 0)stein. Number Theory and Rs nuttily. New 1967. k: McGraw-I fill Book Co.. 1918. School Nlathematics Study Group, A Igmithins, Coin. Rademacher.I fans. and Otto Toeplitz. The Enjoy. pu(ations. and Mathematics. Pasadena. Calif.: A. ment of Alathematies. Princeton. N.J.: Princeton C. Vic 1111a115. 1966. University Press. 1957. Ttakluenl)rot. Boris A. Algorithms and Automatic Sos. E. "Prime Numbels in the Bible." Sciipla Computing Machine's. Boston: 1). C. Heath k Co., matica. ,June 1919. 1963. Stein. Sherman K. Mathematics: The Man it Uni 39.THE ABACUS verse (An Introduction to the Spirit of JITathe Cajori. Florian. A Ilistoiy of Mathematics.2ded.. matins). 2d ed. San Francisco: W. 11. Freeman rev, and enl. New York: Macmillan Co., 1919. Co.. 1969. Talh»an. NI. "A Sequel Clook. Welton J.ilhacus 'bill:ludic.Palo Alto. toEratosthenes."Srrifrfa Calif.: Pacific Books, 1958. Mathenzatira, Match -June 1950. Glenn. William II.. and Donovan A. Johnson. Ex- ploring Alathematics on Yam Own. Ma» chest er. II. PERFECT NUMBERS Mo.: Webster Publishing Co., 1963. Anchee. Richard Selections from Modern Absbact lioghen, Lancelot T. Mathematics for the Millions. Algebra. New York: Holt, Rinehart F.: Winston. New Yolk: Oxford University Press, 1951. 1958. karpinski. Louis. Hislmy of Arithmetic. 1925. Re- Dantzig, Tobias. The Magic Numbers. New York: print. New York:Russell C Russell Publishers, Nlacmillan Co., 1962. 1965. Jones, Burton W. lemenloy Concepts of Mathe- Kojima, Takashi. The Japanese Abacus.Rutland, Vt.: matics. 3d ed. New York: Macmillan Co., 1970. CharlesE. Tuttle Co.. 1953. LeVeque. William J.Topics in Number Theory. Newman. James Roy. ed. The Muhl of Mathematics. Reading, Nlass.:Addison-Wesley Publishing Co.. vols. New York: Simon F.:. Schuster, 1956; paper. 1956. 1962. Vol. I. pt. 3. Niven, Ivan NI.. and Herbert S. Zuckerman. An litho. Saidld. Vera. A Short Vishay of blathentatics. I3os duction to (he Theory of Numbers. 2d ed. New ton:I tough ton Mifflin Co., 1930. York: John XViley Sons, 1966. NVillerding. Margaret. From Fingers to Computers. Nortlnop, Eugene P. Riddles in Mathematics. Prince- PasNlena. Calif.: Franklin Teaching Aids, 1968. ton, N.J.: I). Van Nosnand Co., 1944. Ore. Oystein. Number Theo)), and Its Ilistmy. New I0.PRIME NUMBERS York: NM:law-Hill Book Co.. 1948. Bell. Eric T. mathematics, Queen and Servant of Rademacher, Hans, and Otto Toeplitz. The Enjoy. Scion e. New York: McGraw-llill Book Co.. 1951. men( of Alathenzalies. Princeton, N.J.: Princeton Coolant, Richard, and Herbert Robbins. What is Univelsity Press, 1957. Mathematics? New York: Oxford Univetsity Press. Reid, C. "Perfect Numbers." Scientific American, 1911. March 1953. Danizig. 'Eobias. .anther, the Language of Science. School Mathematics Study Group. Essays on Number MATHENIATICS PROJECTS. ENHIBrrs AND FAIRS. GAMES. PUZZLES.AND CONTESTS 395

Themv. 2 vols. Pasadena, Calil.: A. C. Viomans. Jones. Burton W. The, 'Phony of Nurribeis. New 1960. Vol.I. You k: Holt. It inehm t Winston. 1955. Smith. David Eugene. Ili.stor of Mathematics. 2 vols. !tastier,Edward. :Ind JamesP..Nesrnt:ut. Alathe. 1923. 1925. New York: Dover Publications, 1958. ?natio and the imagination. New York: Simonk 1Vright. H. N. Mist Course in Them of Numbers. Schuster. 1953. New York: John Wiley Sons, 1939. Klein. Felix. Elementoy .11athenzatics ftorrian Ad. 2. FIGURATE NUMBERS vanced Point of View. Translated fun]] 3d Gelman Bowels. I !CHIT. and Joan BOMA'S. ;111M111CtiCal edition hr E. It. Hedrick and C. A. Noble. 2 vols. cuoiom. New Polk: Dover Publications. 1961. New Yot k:Dover Publications. 1924. Vol.1. Dodge, Winston E. "Letter to the Editor. ilrithme- May. Kenneth 0. Elements of 411 odern AI athe,natits. tie Teacher. October 1966. p. 473. Reading. Mass.: Addison-Wesley Publishing Co., Kraitchik, NIaurice. Mathernatical Roleations. 2d ed. 1959. New York: Dover Publications. 1953. NIetriman. Gaylold M. To Discover Mathematics. Rippey. Robert M. -"Some Challenging Series (or New York: John NViley C Sons. 19,12. Gifted Iligh School Students:* Mathematics Teach- the. Oystein. Number Them and Its Histmy.New er. January 1961. pp. 8-11. k: Malraw.1 1 ill Book Co.. 1918. Richardson. Moses. Fundarnerz0z/s of Mathenzatics. 3d 43. INTEGERS AS ORDERED PAIRS OF ed. New York: Macmillan Co.. 1966. NATURAL NUMBERS '16. THE GOLDEN SECTION Fraenkel, Abraham Adolph. Integers and Theory of Numbers. New York: Academic Press Inc.. 1955. Bell. Eric T. The Development of Mathematics. 2d Levi. Howard. Elements of Algebra. Mons, N.Y.: ed. New York: NI cGraw. 1 1 ill Book Co., 1915. Chelsea Publishing Co.. 1956. Coxeter. II. "The Golden Section. Phyllotoxis. and Waismann. Friedrch. Introduction to Alathematical Wythols Game:* Scripla Mathernatica. innSep- Thinking. hansb ted T. Reliae. New York: tenther 1953. 1:1 ederick Ungar Publish ing Co., 1951. Joseph.NIargat et."GoldenSectionCompasses.- ,Mathematics Teacher. May 1954,pp. 338-39, MARKOV CHAINS National Council of Teachets of NIathematics. Kemeny. John G.. and J. Laurie Snell. Finite Afar- Source Itooh of Afathenialical Applications. Seven- hull Chains. New Yolk: Litton Educational Pub. teenth Yearbook. New York: Bureau of Publica. lishers. Van Nostrand-Reinhold. 1959. lions.TeachersCollege,ColumbiaUniversity, Kemeny. John G.. J. Laurie Snell. and G. L. Thomp- 1912. son. An Introduction to Finite Mathematics. 2d ed. Newman. James Roy, ed. The World of Mathematics, Englewood Cliffs, N.J.: Prentice-Hall, 1966. .1 vols. New York: Simon C Schuster, 1956:paper. l5. REPEATING DECIMALS 1962. Vol. I. pt. 2. Adler, bring. The Neu, Mathematics. New York: Northrop. Eugene P. Riddles in Mathematics. Prince- New American Library, 1960. ton, N.J.: D. Van Nostrand Co., 1944. Allendoet fer, Carl B.. and C. 0. Oakley. Principles Raab. Joseph A. "Golden Rectangle and Fibonacci Sequence. as of Mallienzatics. 3d ed. New York: McGraw-Hill RelatedtothePascal Triangle.- Book Co.. 1969. Mathematics Teacher, November 1962,pp. 538-43. Andrews. F. "Dark Ages of Arithmetic." Atlantic Smith. David Eugene. History of Mathematics. 2 vols. Monthly, July 1935. 1923, 1925. New York: Dover Publications, 1958. Bakst. Amon. Mathematics. Its Magic and Mastery. Stephen. Sister NIatie. O.P. "Tile Mysterious Nmnber Princeton, N.J.: D. Van Nostrand Co., 1952. Phil." Mathematics Teacher, Mardi 1956,pp. 200 - 201. Banks, J. Houston. Elements of Mathematics. 2d ed. Boston: Allyn 8: Bacon. 1961. 47. HYPERBOLIC FUNCTIONS Courant, Richard, and Herbert Robbins. What is Brisey. John, and R. V. Andres. Fundamentals of Atathematics? New Yolk: Oxford University Press, College Mathematics. New York: Holt, Rinehart 1941. Winston, 1961. Dresden, Arnold. An Invitation to Mathematics. New Smith, David Eugene. Histmy of Mathematics. 2 vols. York: Holt. Rinehart 8: Vinston. 1936. 1923. 1925. New York: Dover Publications, 1958. !lardy. Godfrey II.. and E. M. NVright. An Intioduc- Yates, Robert C. A Handbook on Curves and Their lionto the Theory of Numbers. 4thed. New Properties. Anti Atbor, Mich.:J. W. Edwards, York: Oxford University Press, 1960. 1047. 396 CHAPTER 'TEN

18. CONTINUED FRACTIONS matits. New York: Bureau of Publications. Teach- Cajori. Florian. Ilistmy of Mathematics. 2d ed.. ls College. Columbia link ersity. 1915. Pp. 1-16-53. ley. and enl. New York: :Macmillan Co., 1919. The Trisection Mahlon, 1912. Reprint Courant. Richard. and Herbert Robbins. What Is (C:lassitsin Nlathetnatits Edt(ation soles. rol. 3). Alathentatus? New Yolk: Oxfo id University Piess. Washington. D.C.: National Count it of Teat hers 19.11. Of Nlathentatics. 1971. Dantrig. Tobias. Arionbet. the Language of Science. 50. THE N 1 NE-PO 1 NT CIRCLE 9th ed.. rev. and enl. New York: Macmillan Co.. 1954: Doubleday g7 Co.. Anchor Books. 1956. Dal is. David R. Modern College Geometry. Reading. Eves. Howard. An I ntioduclion to the Ilistoiy of Mass.: Addison.Wesley Publishing Co.. 1919. Mathematics. 3d ed.. re. New York: Holt. Rine- Nabb. MathematicsII.Skokie.III.: hart k xvitiston. 1969. Banks l'pshaw k Co.. 1959. hall.I leery S.. and Samuel Knight. Higher Algebia. Satterb. John. "Another \pploadi to the Niue-Point New Yolk: Nlacmillan Co., 1961. Chcle." Mathematics Teacher. January 1957. pp. Nloote. Charles G. "C:(quilluedFractions." ,Ilathe. 53-54. maims Teacher. April 1962. pp. 256-63. 51. COMPLEX-NUMBER OPERATIONS Olds. Carl I). Continued hactions. New York: Ran- BY GEONIETRY dom Ilouse. 1963. .4ddittonal Mathematics. Entebbe Nlathematics Series. 9. TRISECTION OF AN ANGLE Newton.Nlass.;Education Des clopment Center. 1967. Berger, Emil J.. ed. "A Trisection Device Based on Advanced Mathematics. Entebbe Nlathetnatics Series. the Instrument of Pascal." Mathematics Teacher. Newton. Mass.:Education Development Center. April 1952. pp. 287. 293. 1967. Butler. Charles Henry. and Frank Lynwood Vren. Somito. Juan E. "Vector Reptesentation of Nlithipli. Trigonometiy for Secondmy Schools. 3d eel. Bos tation and Division of Complex Numbers." Mathe- ton; D. C. !lead] 5 Co.. 1963. maths Teacher. May 1954. pp. 320-22. Nastier. Edward. and James R. Newman. Mathe- matics and the Imagination. New York: Simon g: 52. THE SNOWFLAKE CURVE Schuster. 1953. Fehr.Iiovvardh. Secondary B,150m, Koko nmr. Franklin \V. Mathematics in Human Af- D. C. Heath g: Co,. 1931. fairs. New York: Prentice-llall. 1942. King. Bruce W. "Snowflake Curves." Mathematics Leo. Mother. 0.5.1"Angle Trisection: An Example Teacher. April 1961. pp. 219-22. of 'Un deign ten ta bed' Nlath emats." Mathemat- Newman. James Ro), ed. The Inn Id of Mathematics.

ics Teacher, Nlay 1959, pp. 345-55. .1 vols. New York; Simon g: Schuster. 1956: paper. ,Nlerriman. Cayloid:NI. To Discover Mathematics. 1962. Vol.1, pt, 1. New York; John Wiley g: Sons, 1942. Jelome S.. and Stuart Hanlon. Pun with the 53. A MATIIENIATICAL SYSTEM OF New Math. New York; Fawcett World Library. 2 x 2 MATRICES 1961: Ilawthorn Books, 1966. Adler, Irving. The New Mathematics. New York: New Nlock, Alex J. "Trisecting Any Angle." Mathematics American Library. 1960. Teacher, April 1959. pp. 245-46. Allendoerler. Carl B.. and C. 0. Oakley. Principles Ore, Oystein. Number Theoiy and Its History. New of Mathematic.s. 3d ed. New Yot McGraw-Hill York: McCraw -11in Book Co., 1948, Book Co,. 1969. Pickett, Hale. " "Trisecting an Angle." Mathematics Bell,Eric T. illathematics, Queen and Servant of Teacher, January 1958, pp. 12-13. Science. New Yolk: M(Glaw.1 lin Book Co., 1951. Robusto, C. Carl. "Trisecting an Angle." Mathematics Blanton, Floyd. and J. E. Perry. Modern College Teacher, May 1959. pp. 358-60. .41geb, a. New York: McGraw-ill Book Co., 1963. Sackman, Bertram S. "The Thmalialvk."Maihematics Dokiani, Mar) l'.. et al. Motion Intioductmy Analy- Teacher, April 1956, pp. 280-81. sis. Boston: Houghton Mifflin Co., 1970. Todd. WernerS..Jr.,"Trisecting Any Angle." Lang. S. A. Linear Algebra. Reading. Mass.: Addison. Mathematics Teacher, October 195(1. pp. 278-79. Wesley Publishing Co., 1966. Yates. Robot C. "Trisection." In Altdti-Sensoo,ails Perlis, Sam. Themy of Matiices. Reading. Mass.: Ad- in the Teaching of Mathematics. Eighteenth Year- dison-Wesley Publishing- Co., 1952. book of the National Council of Teachers of Nlathe- Taylor. llowaid E., and Thomas L. Wade. Univc,szty \TI EMATICS PROJECTS, EXHIBITS AND FAIRS, GAMES. PUZZLES.AND CONTESTS 397

Fl esh an Mathemalics. New York: John Wiley Heath. Thomas Little, ed.!Folks [including The Sons. 1963. Method of Archimedes].1897.1912. New Yol k: Dover Publications, 1953. 5. NIASS.VOINT GEONIETRY Smith. Da id Eugene. History of Mallternalie.. 2 vols. Siumner. Harty. "Nlass Points," Mathe»ralics Siuden1 1923. 1925. New York: Dover Publications. 1958. Journal. November 1967. pp. I1. Unified Modern Mathematics, Course 2.pt, 1.Secon. SS. HISTORY OF MATHEMATICAL dal y School Mathematics Curriculum Improvement SYMBOLS St mh. New fork: 'I'eachets College l'ress. Colum. Bell. Eric T. The Developmeni of ,Ihilhe»ralres. 2d bia University. 1967. ed. New York: McGraw.Hill Book Co.. 1945. 55. LATTICES Cajori, Florian. A History of Slathemalics. 2(1 ed.. rev. and eta. New Yolk: Macmillan Co., 1910. Md, Gan ett. and Saunders MacLalle. A Survey Danttig, wnthias. ,Number, the Language of Science. of Modern Algebra. 3d ed. New York: Macmillan th ed., ley. and enl. New York; Nfacmillan Co., Co.. 1965. 195: Doubleday k Co.. Am hoe Books. 1956. Ore. Oystein. Number Theory and 11.5 History. New Ilooper. fed. Shakers of Slalbenzalics. New York: York: NI( Crawl fill Book Co.. 1948. Random House, Modern Library. 1958. Stabler.E.R.All Inboduclion10,Ilathemalical Karpinski. Louis. Ilislory of drithmelic. 1925. Re. Thought. Reading. Mass.: Addison.Wesley Publish. print. New York:Russell k RussellPublishers. ing Co_ 1953. 1965. 56. EGYPTIAN COMPUTATIONAL Kokomoor. Franklin \\'. 411zthenzalics in Human Af- METHODS fairs. New York: Prentice." hill. 1912. Sanfon1, Vera. A Short History of Shithrmatics. Caj(ni, Florian. A I I isior y of ,Ilathemalics. New York: ton: Houghton Mifflin Co., 1030. Macmillan Co., 1957. Smith. David Eugene. History of Mathe»zaths. 2 vols. Cordrey. William A. "Ancient Mathematics and the 1923. 1925. New York: Dover Publications. 1958. Develtylllent of Primitive Cultu're." Mathenzalics Vol. 2. Teat her. February 1939. pp. 51-60. Wilson. J. "Staking Out New Ground in lligh School DantAg. Tobias. Number, the Language of Science. Arithmetic." School Science and Mathenialics. April .1th ed ley. and cut. New York: Macmillan Co., 1951. 1951: Doubleday k Co.. Anchor Books. 1956. 1;11cylopaedia 13,111111 n len. 1969 ed. S.v. "Egypt" (Au. 59. GAUSSIAN INTEGERS cient ( :ivilitation and Culture: 7. Science, ,NIedi- One, Oystein. Number Theory and Its Ilislory. New eine. and Technological Processes). York: Mt Graw4lill Book Co., 1918. S v "Mit'lematics History ol" (Egypt). School Nlathematics Study Gimp. Essays on Number 11()gbett. Lancelot T. A/of/ter/macs for the Millions. Theory. 2 vols. Pasadena, Calif.: A. C. Vromans. New York: Oxford University Press, 1951. 1960. Hooper. Alfred. Maers of Mathenzaiies. New York: 60. CURVES OF CONSTANT BREADTH Random House. Modern Library. 1958. Ore, 0)stein. .Vngiber Theory and 11.5 Hislory. New Ilazaid. William J. "Curves of Constant Breadth." York: NkGrawdlill Book Co., 1948. 411a1hematies Teacher, Februm y 1955, pp. 89-90. Smith. David Eugene. Him»). of Mathe»ratics. 2 Rademacher, I lans. and Otto Toeplitt. The Enjoy. vols.1923. 1925. New York: Dover Publications. men, of Mathenzalics. Princeton, N.J.: Princeton 1958. Vol. 2. University Press, 1957. Struik, Dirk J. A Concise History of Mathematics, 3d 61. WOMEN MATHEMATICIANS rev. e i. New York: Dover Publications. 1967. Bell. Eric T. The Develolmicni of Mathematics. 2c1 ed. New York: McGraw-Hill Book Co.. 1915. 57. GREEK COMPUTATIONAL NIE.1.110DS Cajori, Florian. A History of Malbe»ratics. 2c1 ed., Bell. Elie T. The Development of Mathematics. rev. and en I. New York: Macmillan Co., 1010. ed. New York: NIcGraw.Hill Book Co.. 1045. Collide, J. "Six Female NIathematicians." Scripla Cajori. Florian. A Hislory of .1hilhematirs. 2d ed.. Malhematica, March.June 1051. Encyclopaedia Brit. rev. and MIL New York: Macmillan Co., 1919. arnica. 1969 ed. S.v. "Agnesi, Ninth Gaeta ii a." Heath. Thomas Little. .1Ilislory of Greek Slathc- Jones. Phalli) S. "Women iii Aniericall Mathematics moues. 2vols. London and New York: Oxford 20th Century." Mathematics Teacher, May 1957, University Ness. 1921. pp. 376-78. 398 clim,ER rEN

Smith. David Eugene. ni,,tor of Mathematio. 2 vols. 65. '1'111:1 DEVELOPNI ENT OF TI I E 1923. 1925. New York: Dover Publications, 195S. ARABIC: NUMERALS Yates. Robert C. A Handbook on Curves and Their magi( and umle?v. hopeities. At Arbor. J.\V.Ethvaids. Princeton. \.J.: 1). Van Nostrand Co.. 1952. 1917. Cajori. Florian. A History of Mathematic,. 2d ed.. re%. and eel. New York: Nlamillan Co.. 1919. 62. HISTORY OF THE SLIDE RULE Dan*. Tolqas. Nombey, the Language of Science. Cajori. Florian. A //i.,im of Alathematies. 2(1 ed.. 1i1i ed.. rel, and eel. New York: NIacmillan Co.. tee. and eel. New fork: Nlutillan Co.. 1919. 1951: Doubleday g: Co.. Anchor Books. 1956. Jones. Phillip S. "Tangible ArithmeticI: N: pier's De Wilt. C. "The Origin of Our Numeral Notation.- and Genaille's Rods." Mathematics Teacher. No. S(11001 Science and ,Ilathematiis. November 1917. vember 1951. pp. 182-87. limper, Alfred. Maheis of ,Ilathemaths. New York: Shuster. Call N. "(:onstrtuting (;lapIts with the Slide Random House. Modern Library. 1958. Rule." Mathematic.,Teacher. Jantray 195.1. pp. Karpinski. Louis. 111.11my of Arithmetic.1925. Re. 8-10. prim. New York:Russellg7 RussellPublishers. Note on the Use of the Slide Rulein 1965. Division." Mathematics Trachea. November 1952. Kokomoor. Franklin \V. .11athematics in !Inman p. 512. fobs. New York: Prenticollall. 1912. Ore. Oyste*n. Number Themy and /Eslistory. New 63. TI I E FOUR-COLOR PROBLENI York: NI(Graw.1 fill Book Co.. 1918. \Valter William Rouse. Mathematical Ilecea. Smith. David Etagene. History of Ma/he/oaths. 2 vo:s. lion., and Essays.1 1th ed.. revised by li.S. I. 1923, 1925. New Yolk. Dover Publications. 19;.8. Coxeter. 1939. Reprint. New York: Nfcmillan Co.. Vol. 2. 1960. Wilson. J. "Staking Out New Ground inI ligh School Bentham A, ""1"he FouColor Ploblem," OW/thou/a Arithmetic." School Science and Mathematics, April l'iaver.,it Newsletter. NIarch 1953. 195,1. Courant. Ric hard. and Herbert Robbins. What 66. FIBONACCI NUMBERS illatheinatics? New York: Oxford University Press. 19.11, Itother U. An /nhoduriton to Fibonacci Div. CoNeter, II. S. "NIap Coloring Problems." Seriph, weir StMay's College. Calif.: Fibonacci ASV). Ma Mona tit a 23 (1957):11-25. iation. 1965. Vadiman. Clifunt. ed. Mintasia Mathematica. New Eves. Howard. iln Intioduction to the !Usti,) of York: Simon g: Schnster. 1958. ,Ilathematics. :k1 ed., rev. New York:Ilolt. Rine- !Mho t. David. and 'Stephan Cohn.Vossen. Geometyy hart g: Winston, 1969, and the Imagination. Bronx. N.Y.: Chelsea Ptab Freund, John E. A Modern Introduction to lishiug Co.. 1952. motifs. Englewood Chills, N.J.. Prentice :fall 1956. Kasner. Edward, and James R. Newman. Alathe. Meyer, Jetome S.. and Stuall Hanlon. 11111 with the and the IlnatZinatiOn. New York: Simon g: New Math. New York:Fawcett ;World Library. Minster, 1953. 1961: Hawthorn Books. 1966. Rademacher, Ilans. and Otto Toepliti. The Enjoy 01 e. Oyst eh). Number Themy and Its I lictoiy. mon of .1Iathematics. Princeton, N.J.: Princeton Volk: Nlairaw4 WI Book Co., 1918. University Pies,. 1957. Rippe. Robert M. "Some Challenging for Cilted Iligh School Students." Afaibentalus Teach- er. funtialy 1961. pp. 8-1 I. 61. THE KONIGSBERG BRIDGE PROBLEM Walter William R4tise. ,Ifathemalical 67. SUNDIALS films and F,says. Ilth ed..I evised by II. S. M. Gould, R. E. Sundials. Bureau of Standards Ciuttlar Coxeter, 1939. Reprint. New Yotk: Nlacmillan Co., no. 102. Washington. D.C.: Govonment Printing 19(10. 011ke, 1933. Euler. L. "The Konigsberg Bridge." Scientific ihneii- Nlayall, R. Newton, and NlargaretI,, Nlayall Sun- can. July 1953. dials: How to Knorr, Use, and Mahe Them. New- Newman. James Roy ed. The World of Mathematics. ton Center, Mass.: Charles 'I'. Branford Co.. 1938. 1vols. No, Volk: Simon C Schuster. 1956: rape, Shuster. Carl N., and Fled L. Iledfold. Field Work 19(12. in Mathematics. New Yom k: American Book Co.. NI VI'l I EN1ATICS PROJEC'S. 1:N111111'1's AND FAIRS. 6:AND:S. PUZZLES, AND CONTESTS 399

19:;5.1 ribuiedbyYoderI list raments.East PIZ OD LIGEIZS AND DISTI:HICTOItS Palestine. Ohio. NI:V11.1ENIATIC:1. GAN1ES AN!) I\'.1111.NI. Stoessel. \\'c Nlad It and Itll'ott.sr: Classloom Construction of Sundials."..Irithnu, PUZZLES \ valon Hill Co. 11( Teach'''. Aiwa 1970. pp. 301-1. Clainage Emeriti ises cicallye Cuiseunite Co. of Amtlit:L Ntil 1.:1 listort«il Topics for the Mathematics Class- Ideal st11001 Supply Co. room.hirtfirst Yearbook of the National Council Educational Fun Games of Teacheis 1:011ematio, (Iva,hingion. D.C.: The 1111011t A1'111111104' Drill CallICN (:000(ii.1969) conesim .0 much material for the 1:11:111 GameCO. project topics listed; hove thatitisnot Ito luded 1:raeg Games. Inc,. under any of the individual headings. Consult the James 'W. Lang table of content, or the index 01 //k/ori«// Top/'(.c Pal ker llsothcss.I n c .(Soma Blocks) as a guide to subjet ts of site( i:tl hitt:lest. Itenewal Pimlints.

A SELECTED IIIBLIOGItAPIV ()I: GAMES ANI) PUZZLES

Allen. Layman E, "Thwaid .Autotelic Learning of Game." Slathemaths Teacher. December 1958, pp. \I: thelliatic al Logic." Mathonaths Teacher, .jantl. 61.1 -15. ary 1963. pp. 8-21. "Christmas in the Nlathematics Classroom." Ball.IV:titer \Valiant Itotist% mathenzaticai Re(rea .11a/henudis Teacher. November 1958, pp. 553-5.1. !ions and Essing.lIth ed.. revised by II.S. M. Johnson. Donovan :\.. and Clarence Olauder."Mathe- (:oxeter. 1939. Reprint. New York: Macmillan Co.. matital Games Build SkHk" Slalhonalics Teacher. 1960. April 1957. pp. 292-91. Brandes.Louis Grant. "(:ross.Number Pu//les:.1 Jones.PhillipS..ed."Algebraic Tit k:Fack:Foe." Teaching Aid for Classes of Genoa! NI:talent:tat s." Alathernatics Teacher, '-that') 1951. p. Mathenuaics Teacher, December 1957. pp. 567-69. Kraitchik. Nlaurit C. ,aalical Recreations. 2d ed. (:oxeter. II. S. M. /////oduction In Geometry. New New York: Dover Publications. 1953. l'ortz: John IN'ile K!. Sons. 1961. Loyd. Sam. Best Mathematical Puzzles of Sam Loyd. (.3111'1. nenlY Nialt)". and A. P. RIdlen.."aiheinath Edited by Nlartin Caulner. 2 pls. NM' fork: Dover cal Models. 2c1 ed. New York: Oxford Unieelsit Publications. 1959. 1960. Press. 1961. Helen A. allathematical Excursions. New Ducleney. Henry Ernest..hutisements in ,Ilathernatics. York; Dover Publications. 1958. 1917, New 'rock: Dover Publications. 1958. Niven. Ivan M.. and I lerbei t S. Zuckerman. rin Intro. Canic.,:ortyPuzzles and Other Cmions duction to the Theory of Min:hers, 2d ed. New Problems. 1tIs col., rev. and enl. New Yolk: Dover Yolk: John Wiley k Sons. 1966. Publications, 1958. O'Beirne. T. II. Puzzles and Paradoxes. New York: Duntait. Dewey C. "Ten Nlathematical eshinents." Oxford University Press, 1965. Slather:tals 'reacher. February 1965, pp, 102-8. Parker. Jean. "The Use of ''miles in Teaching Nlathe. Eves. Ilowaid, Introdntion to the !tinnily of mat it s."411athentaticsTeacher, April 1955.pp. Mathematics. Icol., sec. New York: Holt. Rine- 218-27. bast \ IN'inston. 1969. R king. Ociald R. "Some Continents on a Simple Gaidner. Nlartin. Alathrnatics, Slagle, and Alystery. ''mile." Mathematics Teacher, Apt a 1956. pp. 267- New York: Dover Publications. 1956. 69. Dallis. Patricia .11111. "Mathematical Bingo." Alathe- Scripture. Nicholas E. Fifty Mathematical Puzzles and mmics Teacher, November 1961. pp. 577-78. Oddities. Princeton. N.J.:I). Van Nostand Co.. Hillman. Thomas. P.. and Balbara Simis. "In the 1963. Name of Geometi y." Mathematics Teacher, Alm ch Vaughn. WilliamI). "Some Drill Techniques for 1968. pp. 261-65. 267. Arithinetie." Slathematics Teacher. October 1957. .Johnson. Donovan A. "Bridget. an Algebra Card pp. 136-37. OBJECTIVES NO ESTABLISH ESTABLISHED? OBJECTIVES

YES

MODES NO OF INSTRUCTION DETERMINE DETERMINED? MODES

.YES

DETERMINE STRATEGIES

SELECT METHODS

SELECT MEDIA

YES(

SELECT MATERIALS

A SYSTEMS APPROACH TO MATERIALS SELECTION 401

CHAPTER 11

A SYSTEMS APPROACH TO MATHEMATICS INSTRUCTION

by JACK E. FORBEs Purdue University. Calumet Campus, Hammond, Indiana JAM ES F. GRAY, S.M. NIarianist Provincialate, Marycliff, Glencoe, Missouri

Chapter 11 begins with a brief introduction to the general concept of a systems approach toproblem solving. Then the use of systems analysis in the solution of educational problems, particularly instructional problems in the. mathematics classroom, is discussed. Applications of "systems thinking" to both the analysis of materials and classroom management arc explored. Consideration is given to the various roles of the teacher in a clearly defined instructional system and to possible roles of an electronic computer in such a system. Finally, a detailed example of a systems analysis of a particular instructional problem is presented. 11.\ SYS H. IS .\ l'PROACI 1I () IIC'S1 \S 1 !WC HON

l'iesidnt John (..11 held one «. spoke of the ideal !rheaIAa SvAlettil instil IN tional s mem :is a Ito) On one end ()I' a log Leith NI:ukI lopkitts on the other. Tlic low srAtem seems licitiviiilto be used In the ((Hirst. (II time this simple s;stent has without an definition. tinder theassumption that it glom' complex. 11:1%e (lone mom with the lok is a pruttitie t °min that would hett translot ming it into the multiresotirted modern destood I) all. Where there :ire attemptsat s( heel and ( lasst (tom, IVe have learned less aboan definition. these tart (onsiderabl. In fact. there the hotalthough. after having dealt with him ale sevet al uses of the term system,NOIlle of which are50 elOSCI%related lot' a Yong time in large numbers. we are now 111:11. IIIthe s:IIIIC disc liss1011. again attempting 10 rea( It hint as an individual. Ille let III ,hilts 1.1'0111 011C IVICEVIII to :mother. \Ve shall gi% e one definition. then tentrit Nies' ()I us are not1:irkI lopkinses. but mod- to a distill sire approach to this topic. Tlw 1 merican ern tedinolog and research Inane done ttuu It 10 make :flailable to all ()I us the kinds of knowl- .1,s0( intion of School Administrators has defined edge .\ fat k I lopkins possessed about the boy. the :t sstent as an -arral. of resoiii( Cs designed and dedi(atd to achieve au ObjeCtIVC according learner. and about what i.. to be learned. to some plan ()Ia( lion. The simple 1,0 -Iog-Nlark I lopkins sstetn has It «mimics older. unit%. putp(xse. ariangement, and plan.-1 This (laud- be(mt moniamentall) «nnpli( :tiedilltoda's ( lassrooms. .11111014th we ina la( k the instill( Ike tion is inn lord' twithe: as the hest noras the genius of a Nlark I lopkitts. we. as tvachets. may most appropriate 1101 as the one we shall eventn- still 'cam mit( 11 about optinti/ing the learning all% adopt. Howe% cr.itseems a good matting point.. for itis stated in familiar language and. situation 01 our 4410141R It%fa( ing the fit( Ithat he asst of its sour( n.has considerable educa- our modem teaching on ironnwitt istomples tional a« eptabiliR. 11'e shall resist the instructional systemand learning to acton ibis teinpla- Lt neledge. lion to introduce the plethort 01 other definitions that appear. Their relationships 10 the given definitionvat.% \\ 1AT AppitoAci «msiderabl y. Some are mei el linguistic modiliotions: others (Wier ill11111)0E- TO INSTI:CTION? 1:1111 It'aS. Illsitild. we s11:111 (ICs(-1 ilk' :111(1 diller Tod:n eferem es It) ..1))1(1111...... 11StellIc :111:111A1." (1111:11c in Warn Itt'o ralliC 1:1111111:11'Ilse:, Of the and a "s%stenis apploach--«)11(epts 111.11 have be- 1.'01 (1 system. 'Then we shall explain what a sys- t:0111e NO (0:11111011 im the Ileitis of 1)114.'1110,s and gov- tems approach to classrooin instruction and Sys- ernmentate appearing in the ttl education. tents thinking on the partof the classroom Our goal in this ( Napier is 1101 the extension teacher mean to hic. of the dialogue going oil today aboutmstems. Commonplace usage kilts us to understand Ilse s)steills Will (1)1 11 we role ;he MC 01 11110 lerIlls Rather. itis to bring to the mathematicslass- .solar system, railway system. l)avis-firieht11sys- roomI earlier.IIIa (iSCIIINIVe:111t1descri ve rather than a t,ttical way. an explanation of tem of school board managemen 1, Bell Teiephone what is meant Icy .1 systems approach It) instruc- system. Chicago sche)! SYslem, htnnent sraent, tion. particularly altat a systems approachto his

Own classroom hist' uction might be like presently I. EDP and 11w Schant thinnnititaim.111.1.11ington. and in the immediate future. AmeticanAssociationof school ll l 1%7).p. 2. 404 Ely.\EN

Gwen system (of !midge bidding). system p» .wing policies to best.t( «mtplish the objet lives breaking the bank al Monte Carlo. Dewey de( i- of the system. They keep highly hammed on the mal system (I'm libral him Honing of the system. studk the vat ions Act entry bookkeeping system. native courses of action. and undenake the de. These systemsfallinto two quite distinct ( ision making thatproduces a mole (Mu-live categm ies. The word system is being used in two pursuit ()I the objecties. In lac I. the 11):11(, sysieni ielated, but quite dilletent. senses. is illat l'OnStalli Mate or redesign as management In the phrases solar system, railway system, attempts to put it into an optimal state In intro Bell Telephone system, Chi«igo school system, dm ing new things (bowl planes) and new inter- human system, the ,}stem is an existential system actions that ate mote poductive of the goal a complex. tlynami( situation in the teal world a( hievement (( tit bstone baggage( het king speeds imohing physit al objects. intent( lions among up the handling- of customers. makes mliednies these physical objet N. and basic laws that govern mole elisectise. and produces customer satisfac- these inter actions. Among these existential N}S. tion :01(1 hence heater use and gteater profit). (CMS, sonic (like the solar system) are natural. The basic, laws. operating policies. and rules of and some (like the Bell Telephone system) ate pmcedure ate often modified to the same cud not. Man-made system% Ihne two ( ham( teristits (stewattless-pilot coordination to a.oitl hijacking. not common to the natural systems. They an changed schultiling to avoid stacking- over busy designed to achieve certain goals. usually stated airports at busy limns). This example of an air- in tenni of behavioral outcomes. They also in- line system cleatiy fits the defiliftfoll of a system clude a management component whose function as stated by the . meri( Association or so11001 is to determine operating policies and rules Or procedure so as to achieve the objectives of the This discussion of existential systemswhether system in an optimal fashion. man-made or naturalcovers only certain of the An airline system provides an example of an system% mentioned in the introductory listing of existential system that is man-matle. Thcie ate commonplace uses of the word system. IVItat can goals: the transfer of people and freight at eco- we say of systems such as the Goren system of nomical costlevels,in adequate comfort and bridge bidding. the system for :neaking the hank with various services, such as meals, so as to de- at Monte Carlo. the 1)ewey decimal system for rive a profit. These goals ace often statedin library classification. the 1)avies-Brickell system highly specific behavioral terms. Very specific ar- of so hoof board management. of the double-entry kat times. for example. ale stated in timetables. bookkeeping system? Here the word system refers Often specific goals in terms of minimum per- not so much to the underlying reality the cow centage of plait arc established. plex dynamic situation that certainly is present The airline system. in addition to the specific. as to):I particula approach to) the handling of behaviorally stated objectives. imolves, a .ariety that dynamic situation. The system is the par- of things (equipment and people. in this case) ticular otgani/ation or strategy for handling the and .arious operations and interactions of these interacting entities that are plc:sent. This use of things (such as the operations of flying. ticket the word is illustrated by the following example: selling. landing. servicing). All these intent( tions Two people are faced with the same ate governed by laws and rules of procedure that reality or dynamic situation. One of them says. regulate the manner in which the various re "You go ahead and use your system:I am going sources are combined and the operations carried to stick to my on::: system." 1Vith the NallIC reality out for the achievement of the goals.Finally. there ate two systems, two means of approaching there is the management, whose function it is to the problems connected with this basic reality. set up these rules of procedure and these oper- two ways of handling or managing the situation. sysEms AupRoAcii NTAiENTAics rss.ntucTioN 405

Systems described by this sense of the word are because the context in which the term is used referred to ;tnumagenictil systems. should tell the careful listener or reader the sense Time is an obvious relationship between in which itis being used. sstems and managrinent systems. Fol Maur systems that exist in human allairs oper- the: °pet at ion of one pat titular existential system ate with their managers making a c ountless suc- there may be sevelal management systems. The cession of derision, without ever truly being dillcrence between these two uses of the %void aware of the unarm alternatives mailable to them syslon and the interrelationship of these two in their decision-making role or of the many types of systems can be seen in the example of tools far their use in the conscious pursuit of the game of bridge. The game itself is an example optimal decisions. A systems approach to man- of an existential system. There are entities agement of existential systems is a way of think- people and cardsinteracting under codified laws ing on the part of managers of such systems that that govern the play of the game. specify penal- makes them keenly :ovate of being involved in ties for their infringement. and :maid points for an existential system and alive to the fact that vations possible outcomes. (*.mainly bridge has they can do much to optimize the pursuit of the my specific behavioral objectives. front the over- objectives of the system by paying conscious all one of getting a higher score than the op. attention to the system they are managing and to ponents to the mole specific ones of taking enough its potential by analyzing its functions and 1), tricks to fulfill a contract (or to set one!). reach- entering into the decision-making process in a ing a slam contract if it is makable. attempting highly aware manner. ming whatever modern to make a particular finesse, and so forth. A tools are available. bridge game is a contest between two wattage- gronps. each attempting- to manage the SySieMSin Education existentia I system. the game itself, to its advantage. The methods used by each management group The establishment of a management system for illustrate the idea of a fainingentent One an all ead-functioning existential system is one of the two sets of partners may be using the thing. To start with a tasksuch as the education Goren system of bidding for best achieving the of the youngand to establish a management goals of the game. The other pair may use the system for both the design and the operation of Italian system, hoping for the same kind of re- an existential system to carry out this taskis, sults. Here the word Ayslem does not refer to the although not essentially (Inn ent. surely broader basic: game itself or to the interplay between the in scope. people and cards. It iefers to the management In the field of education, many apply a systems approach to the game. Thus it does make sense. approach to this broad task of the education of within one and the Sallie game of bridge, with its the young. Their concern is with the specification inflexible laws of procedure, to bring Iwo differ- of the general educational objectives of the cul- ent management systems to bear in older to try ture and with the roles to be played by various to optimize the achievement of the goals estab- structures in society in the attainment of these lished within the existential system. objectives by learners. They view not only the Because of the tight interplay of these two educational establishment but all other formal types of systems. one finds the word sy.slon being and informal institutions that contribute to the used in the Mine disc IINSinn, in almost succeeding edit( ational processfamily, peer group, church sentences. first to refer to the existential system group, communication media, and so forthas and then to icier to the management system. subsystems of their overall system. In short. their Though care should be exercised in such dual existential system contains whatever in the culture usage, it is rarely as confusing as it may sound contributes to the education of the young. Their 406

role as designets and managers of this system is GEN. INSTRUCTIONAL to Sla te in NI Wale terms what the culture means OBJECTIVES AS OETERMiNED BY b an -educated adult." to deteimine for various SOCIETY subs)stems the toles they are to play. and to construct a management system to facilitate the ient interaction of these subsystems to ensure DETERMINATION OF SOCIETAL THE AGENCY IN that all teguited goals are achieved by learners. SYSTEM SOCIETY TO WHICH EACH OBJECTIVE Others take the educational establishment it- IS ASSIGNED self as their existential :sMelli.IICItacit!) or explicitly assume that the overall goals for this system have been assigned. They construct a management system to determine which agencies within the establishment should be given the responsibility for inducing the achievement of

which goals. They also concern themselves with DETERMINATION OF PARTICULAR INSTITUTION the development of the financial and adminis WITHIN THE EDUCATIONAL trative structures necessary io the operation of ESTABLISHMENT TO WHICH OBJECTIVE IS ASSIGNED these agencies. EDUCATIONAL ESTABLISHMENT Still others concern themselves with the appli- SYSTEM cation of a systems approach to the management NO ACHIEVED BY of a single school district or building. finally. OTHER INSTITUTIONS others apply the techniques or NINIC111% analysis

at classroom level. YES Our concern in this chapter is with this last. ADAPTATION OF OW, BY admittedly limited. application or it systems ap LOCAL SCHOOL CONTROL: ASSIGNMENT OF SPECIFICA proach. In focusing on this, we do not mean to TION TO PROFESSIONALS ignot the importance of systems analyses 01 a LOCAL broader nature. It is merely that our interest and SOCIETAL SYSTEM I (we assume) that of our readers centers at the REFINEMENT OF DBJ. AND class100111 level. It is at this level that the class- ASSIGNMENT TO VARIOUS ADM. UNITS room teacher acts as designer, manager. and com- (ELEM., SEC., ETC.) ponent of an instructional system. Whenever a teacher accepts an assignnient to a I

class he accepts these design, management. and SPECIFICATION OF OE JECTIVES AND ASSIGN. component roles.1 fis choice is not to accept or OF GRADELEVEL !eject them but rather whether he will exercise OBJECTIVES (BY SUBJECT) them consciously and effectively to provide opti- SCHOOL DISTRICT I mum instruction or will let the existential system SYSTEM that is his responsibility operate somewhat "on SELECTION OF BASIC MATERIALS FOR its own" and hope for the best. INSTRUCTION

SCHOOL ESTABLISH ADM. BUILDING STRUCTURE FOR ACHIEVE. SYSTEM OF OBJECTIVES: PROVIDE Ficritr,1 1.1. Flow chart relatingvarions levels ADD. INSTR. MATERIALS ofthe edu«stion "system" \ SA S ELMS \ l'PRO \ CII 10 MA 1I IENI \ INs l'(. I-10\ 407

A ssstena, appioadi toI la,sio0111 Within the classroom itself the teat heras de. Will notClistlie ,t1(1 ess.Itflocs not pioVide a signer-managet,ut:tls /es leedbac k and modifies magical set 01 rules that will optintiie ()ppm tu. the original system design. The teacherout- nities but learning Im students. It is nothing mote poticnt assumes1 cspollsibilit lotittcliaditg the nor less than an otgani/ed approach to decision a( 11 ie% CHICO t of certain Obje( hi% es Is the learner. making. lisspelling out the stages in decision The experience in all of these roles is catalogued making it establishes a framework in which "best lot use ill his next assignment to cull huhu))in decisions tan be determined. uaaterialsselection «n»init tees. The teat her glass .1 role as sstellis desigiler at Your lasstoom is an instructional system.ott two hods. As part of cturicultim committees he ate iM01%Cd in the design and management of contributes to the assignment and statement of that system. These !acts ate not influenced objectives to be ac hieved beat nets in hislass- whether in not )ou opeta/c nitlt cons( ionslet (ig- loom s)stem. In this he applies the "first axiom" nition of the system's existence and operation. of systems- design: Sstents thinking is mewl) awarenos of the far! Objecti% es to be obtained within the ,ss- !ha! .srdent (old Itis tem must be stated specifically. in terms of recognition of the fact that you, as teacher.ate behaviors to he exhibited by learners as a pat ta managerial pattOf the systeill. Systems result of their presence in the system. thinking 1ecluires continual attentiveness to the specific objectives of the system. It includes an As part ofmaterials-selection conrinittees he awareness of applies two other -axioms- ol systems design: the wide variety of possible re sources and of their relationship to the objec- s)slem designed to achieve specific objet tives.Itis a habit of a mind that will think lives must he designed with these objectives openly and creatively about the possible alter- in mind. and the toles of components with- native pathways for pursuing objectivesa mind in the sst.eill Must he specifically stated. thatis open to feedback and modification of 'chat is, materials selection follows the es- plans to improve effectiveness.Itis :twat (mess tablishment of instructional objectives. Any that decision making is a constant process in material selected for use should contribute the classroom. one that often takes place inan directly to learner-achievement of some or inertial. drifting manner but that should bea all objectives. A s)stem cannot be designed conscious process that overlooks no good alterna- without both the !carnets and the objective.Itisan attitude that seeks toidentify tives being considered. (Good materials? problems; to obtain the data to make decisions To do what? For whom?) Furthermore, that are called for; and to !wing to the implemen- consideration not only of the isolated value tation ofthese decisions the most appropriate of materials but also of how they will fit of available resources in strategy. media. prow- with others as components of the overall (litre, and software. system of instructionis an important as- This is a hasty and informal description of a pect orSi ems thinking. ssfents approach to classroom instruction. Itis 3. No systems design can be considered final. deliberately soa generalopening sketchin .\ system is a dynamic thing. The design familiar termsto provide a cnn»onalit) of. must include ptovision for freqnnt moni- background. toring of its operation and for modification In what follows we shall add greater detail to of managet»ent or design to improve its this rough sketch. ht the next section we shall functioning. Thus. materials must be se- discuss Inlay various components available for lected and a plan established to provide use in instructional systems, and some of the this feedback. unique qualities these components possess. Then 408 c:11 \I' FER

we return to the (I )nainic situation of the teah- containedclassrooms.xvithother ompo- er-managed classtom» for a more detailed look nents of the sstein supphing instruction at the possibilities mailable today to a systems lot- the remainder of the class oriented teacher. Next. we considei computet- FAaltiatot, in a 11101e detailed 111:11111e1 th.tn managedinstruction.Finally. we illustratea in the past. s)stems apptoach to instruction.ttsittA apar- With a aliet ol modes ofinstitution ticular topic in mathematics as our example. mailableto induce the :uhievementof each objective and with imp oved tests and ANALYSIS OF SYSTEM CONIPONENTS devices for monitoring student performance and diagnosing student problems. the role The «mcept of an insnuctional sIstem ieguit es of the teacher .ts cnaltiator takes on added that the system designer assign the accomplish- dimensions and importance.Insteadof ment of specific instructional objectives to spe- simpl) determining who learned and xio cific materials. devices. procedures, and people. did not after ./. fact, a well-designed ms- Thus it is important that the designer have both iem will provide the teachei-ealnator \\hit a knowledge of what components are available the abilityto selectfor each individual for use and an understanding of the particular learner the strategy. methodolog), Imo«, capabilitiesofeach component.Inprevious dmes. and media of instruction that will chapters vat ions possible components ha\ e been a Mimi maximi/c his chances of learning. discussedindetailbutinrelativeisolation. It will make possible close monitcwing of Here we shall view each in terms of its role in Iris plop ess and precise diagnosis of learn- all overall system. ing problems itthese arise. In short. the tealieevaltiatorill a well-designedin- will have the oppor- Human Components of the System structional system utility Io prescribe the -best- instruction A major role of the teacher in a classroom in- lor ruck learnerand periodicalhto re- structional s)stem is, as it has always been. that assess and modify the original prescription of manager of the system. This role will be dis- to assure the achievement of the goals of cussed later. In addition, the teacher is a om- the instruction. po»enta maim teaching deviceof the s)stent. -Fedi» teacliing and open sI class organization and sched- piowss. We shall not repeat an) of that here. uling broadens his tole to include the following: not shall we engage in the unending debate be I.Lecturer, in team teaching situations tween "humanists" and others who claim this or I)iscussionleader,or tutor,withsmall that device will better perfon» essentially all of groupsorindividuals (supplementing the functions usually assigned to human teach- team teaching lecuties) during office hours ers. We shall old) commem that whatmer func- in open scheduling structures or. in a self- tions teachers ate to carry out can surely be \tittiI lis \PPROACII 101.\.1-11DIA ricsINSriwurioN 409

better dis(halge(II))(cachets ()ix:1.1611g Ivithin Have begun ond endedwith (omelets on the the hainework ()Ia carefully designed instruc- textbook assigned 10E use in the course. Today tionalsystemas describedinthe jneceding that is rack the case. This fact has been re(og- sect ion. niiedI))minpanies producing textbooks.for 'Teachers do not constitute the only human most now present -packages- (one') called .sys- component (II.111 instructional s)stetn. In Man) lents)including basic texts. together writ] man) St Ciii1 the 'Milian components include teacher other materialsinprinted and other media. aides assignedto aclass on a1 egular basis. Some of these ate designedlorteacher use; Fitch assistan« in the management 11111( lion as ()diets for use b%111(11V1(10:11 StIldelltS. We shall well as in the tutor...11 and mtluation hinoions use these ( lassili(ationsteac her use and student described abme inovides even mote opportuni- usein discussing printed (and other) mayo. ties for the teacher to make his best plofessional items of an instructional system. «mtribution to student lc:tilting. l'rinted Components fm. Teacher Use.The The the 01\ iNiting le( titiet5 110111vritIiimm the textbook and ac«nnpanying [cachet's mannal school or (minuttnit) as well as front mlleges comprise the primary printed component for and tutheisities adds still another human «)111, (cachet use in most instructional s)stents. How- pollen' to the s)stent. Far be%(>11(1 the entertain ever. the (cachet's inantuti or today is usually a went value of 511(11 appearances is (Itch educa- great departure hom the "answer book- of a few tional Naltle. 'These (an be planned and timed years ago. It is likely that the manual will in- tithe) to make 'col rmittibutions to the achieve- ( lude answers, statements of obje( tires, (011111H:1th ment ofspecificinstinctional objectives or to on methods and strategies. some backp,ronnd aid in ploviding .1meaningful emit-oilmen( in !nal heinals. suggestions lot mil( !intent activi- which specili()bjectives can be achieved. ties,lists of supplementary materials, and so the leatner himself is a human com- forth,It may. in fact. be essentially an outline ponent of the s)stent. Not only does he provide for a complete instructional s)stent ()I which the the inteinaliied changes that arc his learning. accompanying textis but one component. but also he contributes to his own learning and In any case, the text tisually defines the scope that of others as a questioner and discussant. on(1 sequence of the basic mitten( to be presented. ht (incstioning .111(1 discussing he participates in It serves as the basic -diction:1r)- of terms.It both the plesentation and maluation roles.In mav tend to limit or eettto determine the certain(lass-conduct pm( edures he also con- choices of hush tic-lion:II strategies that can be tributesto classroom management.Insmall used.It does not determine the objectives of a gimp work he may act as discussion leader. Ile coutseteachers often teach "tip. down, or out" may serve as ham-partner with a less able SM. limn a textbut itwill determine to what de- (lent. In the learner-centered (,lassloont he ma) gree other components (including the human select some of the materials he will the, decide ones) mustbeusedtoachievethe course which of several modes or io:orociioo he will objectives. use. choose the strategy or medium of instruc- Other teacher-use components in printed form tion he inefets, and make decisions as to which are alternate texts: background books; pictures, nonbasic objectives he will achieve or on which di:v:1111N, :111(1 chartsfor ClaS1100111 display; parts of basicinstruction he needs mote work. spiv 11.(1111/11( atm. masters On epared or blank): and diagnostic and achievement tests. The ever- l'rinte(1 illedium Components increasing mathematical sophistication of both elementary and secondary teachers has produced There was a 11111e Whet' a discussion of printed the eclectic attitude that makes alternate texts mate' ials used in a mathematics classroom would and backgiound books about mathematics and 410 c11.\P

its applications important components ol a ,s- ide the (ppm omits fog all to .0 hiese the spe- tem. The use oldisplasmaterials gives pre- cificobjectises ol the instruction. lewN of conning "T" lweNnIs "1"1""1 ol important fitcts, and illustrates the wide va- Audio and r stud Com pun en ts riety of connections of mathematics to the "real world. The availability ol welldesigned tests is Under this classification we include chalk and essential to the expanded evaluation thatis a chalkboards. overhead projectors and tranparen- basic part of a modern instructional ssstem. cies.slides, filmstrips, lihns (I(i-tinn and Smin This incicased sophistication of teachers. to "loops"). audiotapes. Ides ision. and combina nether with the increasingly professional role ol tions of these. Itis in the consideration of these the teacher made possible I)) the asailabilits of wmponents that the financial icsiraims on the other components of the system, has expanded system designer become very important. Thus it the tole of the spirit duplicator Iom its [Lich- is essential that the designer have au understand- tional one as a printer of tests to include the ing of what each «unponent can or cannot do to printing 01a wide variety of materials Inim aid in making decisions about the balance be- commercially available or teacher-prepared trots. tween instructional gain and cost. ter,.Williteststodiagnose studentneeds, Audio andI' isual Com lumen's f or Teacher knowledge of what is necessary to satisfy these Use. Chalk and chalkboard ale the traditional needs. and time to use this knowiedge. the sisual «imponems ol classroom instruction iu amount of supplementary materials for individ- mathematics. The extentofthetole olthe ual or class -wide use prepared and printed bs chalkboard in all phases of instruction could. teachers has increased until the spirit duplhatm we are sure,be establishedbsinters iews of rivals chalk as an adjunct to instruction. either mathematic's teachers or owners of dry. Pouted Components fo) Student Use. Again. cleaning establishments. Howes er. the osethead the plimats whited component is the textbook. pojectoi is becoming almost as common an ac liowevei, it is a(«impa»ied bs teacher - prepared cessors to instruction. Its tole pailiallv mei laps printed materials and bs workbooks for addi- that ol the (halkboaid but also extends beyond tional drill. Programed texts pros ide extra init.In large classes its use as a replacement for struction onbasictopics,reteachingfort e- the chalkboard pros ides imp' oed mediation, and presentation and teaching of The fact that the teacher can remain in one enrichment topics on an individual or small- place. facing the class, may be importantfor group basis. Alternate texts give students the acoustical reasons. In any class. the ability to oppm-tunits to use different strategies to achiese ruin back to any 'Not of a long presentation objectives. Books about mathematics, applica- (the part)ott'se alwass justerased in board tions of mathematics. and mathematicians con- work) may be important. tribute to an ens it omuent for learning by plac- Adsance prcpmation of transparencies con- ing. what is to be learned in a context that is taining diagrams. resultsto be used. and so relevant to the learner. forth. lor use in «mjunction with either chalk- Most pi inted (and other) materials for student boald or "continuous easel" demi) extends the use c designed to satisfy the needs of ind is Oita! tole ol the piojecun besond that of chalkboard students bs plosiding fog diffeiences in amount, replacement. The pi-epaiat ion of project tials «m- time, strategy, 01 medium of instruction.Itis wining well-«mstructed questions for poiodic Such student mato ials. regaidless of the medium use (luting piescntations as the basis fon "go on in which they are presented, that are central to go bac k" decisions is an important aspect of the concept of an instructional systemthat the the improsed esaluation function of awell- instruction must be sufficiently flexible to pro- designed system. \ ,1,..\L 'TR() \ to \I \till.\ Hu (.1 1():\ 411

The (A (Them! projector has (ended to limit \ ite%% dimension has been added to hint, the role ()I slides and filinsit ip,. Seldom ate these !Mitsui!) use 1)%the .1% ailabilit of of used to pi csent (halo ants 01 'mimed inlot mat ion. them. incdia, Ninglc film tii. (0111.1in motion I lowe%er. the)tentain the pl hum y devices lot picutic sequences with petiodic automatic-stop the presentation ofpi(tot ial Ieinesentations of sequel» es lotfilinsuip lc% ieuing of impot tam stati(situations. quitthe%atec lass) o0111- flames of the 1111n. eXtension de% i« .s.Ininging into the classt (tom \tuliotape call be used 1)) (cachets to listen in lait.t, experiences not alread) pesent there. on then own leaching and on class dis( ussions. fret (all1)10%1(1(4 adequate substitutes lor ob- to aid in hoinmentent of !unite presentations. servations of teal world situations. In compat 1- Small group disc tissions can be taped lor use ill son with 1)111111(1 displa) picunes. both slides and monitoring student !canting. 1)1 ill tapes(an he filinsuips 11.1%ethe ;id% :images of easietview. plo(111«.(1 to 111 ide (nal (hill without clue( of detail and easier storage. Slides pm- tem her imokentent. Tapes of present:clic»), and tid flesibilit%01 sequem ing. %%idiom distia(- discussions (an save limns of tea( her (oncluct of tions.111111,161)s ate generall%easier touse. inal,..1,1) or remedial %vork. Either can be used ill conjunction with audio- Television has a tole as a in esentation device tape continentalies. for centrally located films or filmstrips. One film \\lien the (1)namics of (hanging situations librat) in a large s( hoof s)stent with multi(han- lather than the details of static situations are to tie! tele% ision ttansmission lacilities Pt 0% ides im- he illustrated. motion picture films replace film- mediate availabilit) of this floral') to many (lass stripsOrslides. Sixteen-millimeter -lullreel" rooms.In addition. television can be used to films inovide overt ice's and re% ice's and con- provid the ackantages of large group instruc- tibute to lootivaticm lor learning.In man) tion without the necessit) lot physical together- cases they are used by teachers for reviews of ness of the group. Special interest topics can be content or methodology to aid them itr prepar- presented to a few students in each of many ing their presentations. The) often contain «on- classrooms.Illlarge or regular group instruc- plete cleveloi»»ents of enrichment topics for use tion. TV call give each student a "front row seat.' by itsubset of the class while the teacheron- to view details of visual presentations. In some (limes basic instruction %%ith whets. computer-managed systems, television is a major Single.conceptfilms (8.1nin"loops-) provide presentation and response.ac cepiing component., the initial confrontation with a problem impor- 0di0and Cmisponents for Sistdent tantto all instruction. The simplicity of their L' e. Several of the uses of such components by use makes them both a teacher-use and student- sintlents to 'Amide i»tliitIttaliiatiott of instruc- use component. tion were mentioned above. Student work at the A number of mathematical concepts have an clialkboatd is a major day.toclay meats of evalu. essentially dynamic characteristic:. Films make it ating student progressinInallyclasses. The possible to show the change from thinking of a%ailability of %ideotapes, films, filmstrips, audio, individual objects to thinking of a set of objects tapes. and so forth containing key palls of pre- Or from thinking of two sets to thinking of the sentations is a source of additional or makeup one set that is their union. They make possible instruction for individuals or small groups of the visuali7ation ofthe limiting procedure in- students. Remedial or enrichment packages con- trinsic to the determination of the area or cir- taining audio, tistial, and printed components cumference of a circle. Other «)mponents of a are valuable adjuncts to basic instruction. Au. system call show discrete steps ill these processes. diolape can be used by students to provide Only film can show the changes that produce needed oral - stimulus drill.treat -the -tape drills these steps. introduce time (.0111101 when thisis desirable. 41") (.1 1.1z.

stinnuatt. oust tuted subsyYiettis (on- do sow( "teal- mathematics.Ellis. ill unit. mat taining isual. and other components can inotit thcit!raining 01 basi( 181 Is and be used to piocitu lo one or a lets students apploximate teplications 01 %vital has occulted. .\ late) suction 01 this (11.11)11 (ontains dis- 01 %%mild o« In. in tea( Iteli (nicht( ted hist' tu ti011. (ti.sion 01 (onlinntis as 111811.1gui otl111,1111111(111 al s)steills or subssleills.. \s a (onlponent oI .110(Iels, Devices, sstelil a (ompliter is, lust 01 all. an efficient cal- Game,c (is (;oinlionents ( ulating machine*.It can also be used to shun. late experimentationIt 1)10(111(ing.on com- lte use of models and manipulative devices to mand. the qualuitatitaspects 01 experimental intiochu C con( elms and to 10rin the basis lor iestills.Its steed of1 01111)111811011 aIlotyS shun - absttactions is «innon. They lore) 1)81't or both latiug ino«.sses by pioducing tea( het demoo,taioos :111(1 student experhoco- 00(111 ling cli tete results.(1:01. example. a Nolion..lcachei.mildth led games involving teams tomptiter plogramedto a ten.pla(e of students are a major method ml bringing time i)tintout ol tallies 01 «nitiol into le:tilling. (, :cue, that simulate teal- tvorld situations :Ile be«niiing impoilant «mr / \ n pollen'sofinstItictioninsoda! studies and (1 + otherarras.Little ofthis has liven cloneill lotin( casing posit itintegral %aloes ()I n Bites mat heinati(s. I lotet Cr. enrichment games .111 impressit e example 01 the limit 1)101ess4 01 ("E(Iuations.- "Sets.- "\VI:F .N PROOF." etc.) «nnse.learningcompute'.progiamingand can play.111 impoitant pa« in the mathematics «nnputtiniented inail,ematics mat be one of (las%room. the objectives for some 10111seS ill111:1111C111:111( S. IIIthe Daditional dass100111. knowledge of student plogiess was provided both the teacher 21/achines as Cora / onenis and the student inic».inallv. on a clay-t041 ba- peliodi1 the hum. \la( hines hate an ;n11)011.1111 role as presentation sis: With (let iceslot man) of the components discussed duction of boll, :11clit 411181i/cc! and latge gimp picviouslt.1 !etc. refetence isto machines that institution mote e feedback com ale iheinse/ves the instructional «nliponent. In pollen's hat e beeine ne(essav. part 1( 111ar. we shall discuss calculating machines. The earl%tea( hing ma( lines.101 use with «impute's. and ill tchines designed specifically to inch% ichiali/edinstill( lion. %tetedesignedto plot-41e 01 10111101 feedbac k to tea( her or st ticlent. putt ide immediate «mliolled feedback to the l'se of calculating machines can he both a learner andItle«nd ml!earner progress that means and .1 goal of hist' tiction. lII SOIIIC C011seS could he moult(); ed by the teacher as desired. 011C Of the cottise objectives is learning to use a These tathei C. usuallt manually oper desk (alculator.In some. desk (alculatots are ated devices were the foernimels 01 (intently tools lot use in plobleins Icquiling length) Loin- atailable sophisticated. clectli(all) °limited ma- imitations. The availability of calculators may chines using- filmstrips and audiotapesillthe allott the innodtulion ol applied ploblems that 0(.51111,1110u of inatelial and ',emitting %%linen ate Closer approximations of reality (only prob- or 01811 espouses. lems in mathematics books hat e "nice" answers; Feedback (onnponents lot laige group insuuc- teal world problems seldom (to!). lion are also (pike sophisticated. At the mini- Some research indicates that the use of cal. mum. they allowlecturersto determine im- culatois may allow tveakel students the oppor- mediate!) the percent of .tclass choosing oath tunity to citcnint cut Ago) Wind( "blocks- and tesponse to .1 multiple-choice question displayed NYS TENIS NIIIIROACI I1'0 NI11 IF NIA FR'S IV I:RUC TION 4'13

INS I IZI'GrIONAI. SYSTENIS

Nine we 'chilli to the main theme ol this (hap. ter. With the list within as a gettetal outline ()I systems thinking and the se«mdas a catalog -'3.&iaggferr44. of systems components. we Ionsideras our exis. tential s)stem a single mathonatiisc lass (at an) gradele\ phsi( co\ honment.the learners. the content. and the teacher:a teacher who k intent on using a mstems :wpm:tilt to maximiie the ellicienc of learning by hisstu- dents. What is the classroom teacher's mole in the design and management of the classroomin- structional N)Sielliio which he normally finds himself? 'Imo answer this we mustset the scene. We picture a typical school )ear anda teacher who has been assigned to teach mathematics to a set of students lor that sear. The organim- tionalstrut( !me isthe familiar selfcontained class ol approimatel) thittk studenN basic test has been selected lor use. Enough money is mailable for some additional tit:umiak (lams. filmstrips. models. en.). This institutional s)stem is teally a sults)stem of a larger system. Hence we shall assume that the students wine into this particular course (:otiliesyulIttivihcon Learning Sy3leinsCynthia) with apasthistory of mathematics in which certain behm lot al objecik es. consideml asre. FicatREI1.2.Combi»ation automated and (wiredin«uningbehaviorsforthepresent manual-operated lectern used to control mul- course. have been achieved. I:miller. the teacher timedia presentations andto accumulate has below him the goals ol this particular sub- student-response data On a magnetic tape mstetna particular set of terminal behaviors or recorder outcomes to be achieved b) the learners. While these me often described e» glob() as "teaching thecontentoffoutth.gradearithmetic-or h om a tanspatenc) or slide.lore elaborate de- -teaching the d'i'd semester of algebra.-we vi

( ()Vet ed.' butI))sa lel )acclui,itiort of the DETERMINE SEQUENCE helms iota! ()bin of this pat ti( tilar segment AND MODIFICATIONS cuniculuttt. (Jul assumptions. them are or COURSE of the run OBJECTIVES that the students in tin, (muse !lase been :tde quately prepared. with. or «turse. some satia- tion in the entering behaviors: that the teacher DETERMINE DIAGNOSTIC ARE A has a careltd specificationin behavioral toms AND REMEDIAL PROLE VARIETY OF YES DUNES TO BE USED: ()I' the objectives of the present segment ()f the ENTERING BEHAVIORS EXPECTED>, OBTAIN NECESSARY curriculum: and that these are reasonable out- MATERIALS comes to expect in the time allotted lor the NO particular class in question. Faced %%id) this new class. the classroom teach- DETERMINE GENERAL er begins to function as :ts)steins manager On INSTRUCTIONAL STRATEGIES TO BE USED. MAKE TIME two levels. In the large. he must conceive of and ASSIGN, TO BEHAVIORS plan the entire year's work. This is principally a design task. Then. at any particular point within the ;ear. he must engage in the anise mill- agement (ilthe system at a particular point in DETERMINE IN. STRUCTIONAL MODES TO time and space. Ile has to be engaged in huhu- BE USED (SMALL GRP , ing- learning of a specific bit of mathematics. INDIVID ETC.)

The Teache -Managed Instructional System in the Large MODIFY CLASSROOM DESIGN, SELECT ADD. MATERIALS, DEVICES, ETC. The classroom teacher has a special design and TO IMPLEMENT STRATEGIES management role to play with respect to his initial phoodog or the entire year's work.Itis here that he must view the behasioral objectives 1:u.t.up. 11.3. Flow (haft of a clawmun system for the entire course, plan the sequence in which "in the large" these are to be attained. and make some rough time allotments tospecificobjectives (Figure ensure their aival for the at the appropriate I 1.3). 1-k must pay particular attention to the time. Ile should carefully analyze his classroom sequence in which the attainment of objectives the arrangement and design of the room itself is plumed because sonic objectives are not ter- and the possible permanent acquisition and minal objectives but are. rather, prerequisite be- placement of an overhead projector, togethel haviors for the attainment or others. Sequen- with other auxiliary materials such as those cing will depend On both the content itself and describedinprevious chapters ofthisyear- the particular teaching strategy he intends to book. Does he unconscious!) tend to arrange his use. Ile must make at least a rough estimate of classroom for this particular class just as he did satisfactorylevels of performance for various when he was teaching ail eutilel) different sub- goals. These levels ma) differ, depending on the ject at an entire!) different level to an entire!) particular objective. different type of student? If so, this is decision This first approximation of his detailed plan by inertia. Ile should seriously consider whether for the oldie year must include decisions con- there is anything in the nature of the new course cerning resource materials so that those that are that would suggest changes inclassroom ar- not immediately available can be ordered to rangement. Is there anything about the ability \1*,`1"1NIS APPROACH TOI.V1'111;NIA`IICS 415

levels or si/e ()I this class that vould make the Mani( sstt mime again and doer mires which use of dillet tilt patterns of ( lassloom oigani/a past behaviors ate ilccessal y (Altering behaviors lion mole leasible. that would make easier the for learning the new material. tt,e ()I cm i( Innen! maul ials. and so forth? D/61.0tosis. hating rctiewed these fac tolsex pe( led out( owes. the content to be taught. need- Gla.ssoom Sr.slems 'Ill inking in the Small edinitialbeim% iorshe thinks ofthese with regardtohis own particular class.Illsfirst Impolunn as the teacher's role is in planning managerial decision is diagnostic. II he is about this course in the laige. a major task of teaching to engage in sok ing quadratic equations by lies in enteling the classroom day by day to assist factoring. he will «.rtainly need to know the the students and manage the (Iasi as theyat- «unpetence or this patirtdar class in the lactor tempt 10 fear II some specific111:1111e111:16( .s. This ing of quadratics and their understanding of is the pla« where the teacher is «instantly en- the real import of the following fart:lot real gaged ill a decisionmaking tole that vitally af- 11umbets a and b.ab= 0 if and 0111)ifa 0 fects the degree to which his class achieves the orb desired objectives. If he has taught these topics to his class earlier Now 0111 selling be«)nies even more delinite. in the year he I11:1)rely on his Own judgment The (lass is somewhere in die middle of the and recollection of the past to decide that they year's work. The teacher has induced adequate are Nttllicientl well gasped by the class 10 more achieententofthe behavioral goalsofthe ahead into the new material. with only passing c nurse Op to this1101111:111(1ISabout tO tackle reference to the fact above and with no real a new bit of materiala new set of behavioral need to review factoring.II' he isless sure of objectives to be achieved by learners. their skill in I:tooting, he aright present a few As the NyNI erns Olallager, the teachers atone problems at the Wald. asking students to work thinItrsttillns to 111(. objectives of the new them and respond immediately. This is an ex- unit. These objectives are sufficiently spelled out ample of all essential component of a systems ape it behavioral terms. Still.itis important for proacha process of feedback whereby through the teacher to be acutely (ear of the behavioral out the instructional process information from specifications Of this new unit.II. having com- the ongoing situation is read as frequentlyas pleted work on facto! ing it elementary algebra. possible and decisions ale made and plansare for example, students are about to embark on changed in the light of this information, this the solution or quadratic equation,, the teacher feedback. IOUs( he sharply :Ind specifically aware of the Several choices are possible at this point(see types of problems he will expect the students to rigtueI 1..1).If the teacher reads the classroom be able to work on the completion of this unit, response to these few problems in fiuloring as the various conceptual questions he will expect highly positive, he may proceed with the intro- the students to be able to answer, the various duction of the new material. If the response is procedures that he will expect the students to somewhat disappointing, he immediately makes have moaned, and the various distinctions he the decision, well within his scope, to stop and will expect the students to he able to make. reteach or reexplain the points of difficulty with Besides making himself aware of the specific the factoring process, confilming success by some behavioral objectives to be achieved in the new additional problems. unit, the teacher must review the mathematics If the results of this informal diagnosis arc involved and be sine that he is thoroughly fa- truly bad, the new topic should be postponed milia with it and the way it is presented by the 1111tilremedial materials have been used. Al- materials to be used. Next he looks at the inathe- though the teacher should be ready for this el IAIII- ER LUNEN

10. ENT NO IS BEHAV. KNOWN INFORMAL DIAG PROVIDE 1011MAE 0 BE PRESENT, ADE OUAlt ) DIAGNOSIS

PROVIDE INFORMAL DIAGNOSIS

YES pllovtoi Avolop 4EASsw101 III`.11

pROVIDI APPIIOP INDIVID REM(

SPECIFIC ADJ, TERM Ow TERMINAL OBJ. CLASSWIDt OR APPROPRIATE I OR ALL) INDIVIDUALLY

SPECIFY STRATEGIES TO (IC USED Fuand: 1...1chwromn AyAlent"inIhe Atnall"--4 churl for leaching a p«ithalar bit of nutihentaiies within the overall ( mo,se SPECIFY INTERMEDIATE OBJECTIVES ..$eq item e

SPECIFY MODES Of INST.( , SELECT SUP MAT CLASS, PROCEDURES, ETC,

SPECIFY TEMPORAL SCO OF INSTR STEPS

PROVIDE APPROP, INSTIL FOR STEP (STARTING WITH 0

IEVALUATE

YES SEO ACV, TO NEXT COMPLETED, STEP

YES

REVIEW AND EVAL, ACHIEVEMENT OF TERMINAL OBJ. I14. 1111 I\11 III() \I N.1111 NI I1( 1 INN! (,1 417 eentitalio, itis to be hoped that it %vottld not the past that thole ate some belia%imal obje( 0( (111 111:11 the le:1(11C1 would not misjudge the Inc. not piesent in the blneprin, he %,as gken piepaiation ()I the ,Nindenh to this extent. II he that ale ncessai .It.1later point in the murse 11:1(1IKT11 (1111/10111 :11/011111111' factoring Nkills. lothis paiticulatway()Itea( !ling ()diet new he should haw made .111 ea) lie) decision to ( 011- matt:ILI!.Phis is .111 espedally (1 tidal kin41 of .1

(111(1 :1 tonna! diagnosis with .111iti.depth test decision .tteaclwrhat things. p)operly 011 IW)Ol:MI(111:11 1.1(uning types that intist emphasiied and adlieved now. will make the be used Wei in the new matetial. This diagnosis going match easiel Wei ouc 1V11.1t things. iI not would local the title state of the !miming skills opta ly .1((11111t(1 11011. 11111 (au.r g) eat (I Oh( tilt% ()I his students and would lead him to additional de( isions ast4)what kind ofremedial %vork rewiring, Ara, (:outent. !Ening made a (ale. would 11.ne 10 be (1011e :II11111 11011 Mil( 11 lid iinento)of ma cssalcntet ing belia iois and W0111(1 1/ 11Celk(1, :111(11))W110111. 011e del kiln) haing plovided any remedial institution that might lead to lcMedial work the entire class. was ne(ssar%. the teacher is now )cad) to begin .thother decision might he thatitis indicated the new ()nit()Iinstitution.I fere he is laced ior mil% a few students and (0111(1 he a(( o)u. with some ofhis major dedsions. Ile ahead% !dished b palling them with mole (.Viable stu- 11.1s deter mined the lumina! beim% iffis he will dents. In di)eoling them into piogiained mate- (.xpect horn this particular class. Now he must ials to be done .tt.1 separate time. and so forth. sele(t his (nei all :11111 11011:11 stiatel.gv for inchu The teacher. as a farsighted manager. might ing the attainment of these objectives. 11:oe arianged to hme wady merhead pro. The teaching of a new topican be viewed as jector and seveial transpal elides With example. pimeeding in duce stages. or phases. and problems on factoring that (mid be used First thereis the introductory. or overview. easily and efficientlyin the classroom for both phase.IIdone successfully.this produces in diagnostit imposes and. if the remediati.)11 indi- %01%ement ofthe !camels with what isto be catedistelativelylight.for iemediaticol with learned.Itis the most dramatic. partof the the entire group. teaching ptocess, designed to provicie impact. So far we have seen that II": teacher as systemt, motivation. interest. and effect. manager must make :t number of decision: 1:111) alternate strategies are possible het e. may In 10 beata valiet% of materials and l'se of some kind ()I discovery method is quite media in the assessment and reanimation of the «mmion. P0r some topics a formal.almost beli.niois that ale prerequisite to leant. lecture type of presentation, skillfully paced so ing the new material. The teacher also has that students can anticipate each step

Nil lilt ion (lull gt oup, Nina!! gi (nip, hulk idnaliied) loph tomes plum.. Oiler. that of acquisitional that best fits the strategy and iihniutional mode. activities.%vitt' each lealner engaging in thos Of course. general decisions on materials have activities that will produce hi, achievement of heel made as part of the overall «nirse plan; bin theobjectivesexpectedofhim. 1 ferethe these must be 'dined. and often modified. when icache.managel's chief ded,ion, c in icgaid the actual instruction of a unit begins. to apinopriate %cork material, and activities .Vier the rather diamati( entry into ihe unit ploblems. exercises. pioje(ts. tcadiug,. and No provided bythe introdiu lot comes the forth. These. in tutu. must be paired %vial ap. second, or developmental. phase of the instruc- piopriate instructional modes. tion. The hist phase was a lough sketch. -This 1)e( ision, concerning- a...... igninents ale perhaps one is a carefully constructed design. lleie the the most impol tam the leacher makes. Of (misc. es,ential points of the topic ale presented in as,igninents must be reasonableinterms of an orderly sequence:results made leasonable number of exe« i,es and expectedlength of by the introduction are established. The learn- %yolk time. However. this is not N111[16(111. The er heroines aware of the full scope of the behav- easy decision. "I)() the firmfifteen problems. iors he is to acqvire from the instruction. is seldom the best one. The work to be done The teacher is again confronted with decisions must be specific:di;chosento contributeto regarding stiategLs, clas-aoom Olgaiiiiation, and ,Indent achievement of the lull tank of objec- matel ia IS. As a systems manager he will tives with iespect to the material in the assign- 1.111 10 01/1.1111adequate feedback so that initial ment. Surely the assignment must be tailored to decisions (an be modified. Ile will make choices the needs of the class and. itis hoped, to the among available materials. sometimes choosing needs of individuals in the( lass. Again, only ales,.than-pei feet medium of presentation to knowledge gainedfloutasensitivefeedback take advantage of a particularly good piece of structure makes this possible. software (a good filmstrip may he better than Not only may it be necessary to make dillerent

.1 mediocrefilm. althoughhis knowledge of assignments for diffetent stlents. perhaps using media tell, him film is the superior medium for different suategies and media. in (tido to obtain plc:sent:aim' of a particular topic). achievement of basi( obje( Byes. bin it may also 'ithin this phase the teacher makes another be necessary to modify terminal objectives for key decision, of a somewhat different type. No some students. a( ( eluting achievement of a mini- one but the teacher (an decide the degree of mal set of goals by sonic. adding emichinent ob. precision of language and the level of nmthe. jcctives (in breadth or in depth) for others. Only matical formalism (rigor)that are appropriate (arefnl planning %%ll either reveal these needs for his class. Of course. correct use of certain or provide tesonrces to satisfy them. A systems language and ability to reproduce or construct approach to instruction is designed to produce certainformal arguments traybe behavioral in the most efficient manuet, maximum possible objectives of the course. Beyond this, however. goal achievement br each student. Lack of pro is the general level of formalism to be used iu vision of the necessary flexibility in all aspe(ts the development itself. Between the tote lean- of instruction near the end of the acquisitional ing of facts and techniques and the logical de. phase can nullify the effects of a presiosly well- velopment of completely rigorous mathematics exe( uted plan. Pros iding this Ilexibilit) will be is a broad spectrum of mathematically and ped- a stern test for the best systems manager. Itis agogically defensible positions. No one but the here that materials selection is most vital. for teacher himself can decide the optimuni posi- itis here that individual differences must be tion for his particular class. taken thoroughly into account. The teacher who Following this onlerly development of [In: believes students learn only what. aml when. In: .\ SYSDIS APPROAC:11 TO .\IXIIIENIATICS INSTRUM'ION 419

Is lea( hillg till 1111(1 his task at this stage .\ 11101C reliable Ino(cdntc is to ill( Illde (111 it110t impossible. The teache -manager btiel(11:1g110 11«IlleSt 1.011Sorexeit isesof a who understands the full lange of possibilities singlecumept imitate lewd."' 1.throughout the 101- instru(lioub.other minponnts ofthe teat hing P10° e51.111101111:11 Incscntalion of these s.stem, has taken the steps to dequile the neces on the boaid. .1 minute or two 101 thought. and salt vain. 01 materials. tan operate successful!. .1 discussion of the mesnita ale all that is macs- inseveraltypesofelassiouniorgani/ational Nay. For example, alter the topi(01 absolute sit tat tut espet haps sininhanemisl.and has de- %aim has been introduced. the definition ex- veloped the feedback structure thattells him plained. and some bash impel ties discussed. the what is needed. by whom and when. will receive emit e class might be asked to write an :111sWer at this time the int.off for all of his haul work. 10 ilIC (IOC:41011 111 x is less than zero. which 01

The payoff is knowing each student has been the following is the absolute value of x: 01 gken all optimum learning- opportunity. The distribution 01 Iesponses and a ,IN- Ilioughotit the above discussion emphasis ClINNI011 of the titieNtioll col be very inloiniative has been plated oil the many decisions involved to both teacher and students. in ail) instructional 1)101 C5s. These decisions (10 ()I (misc.. One 01 the immediate benefits of

1101.11 11,M bet 11//Ar of .1,..,:,(ems approach. They 511t11 .111 M110;16011 01 diagnostic question k that ale inherent!.present..\:,SICIIIN:111;t1NISof the students begin to know what they do not institution is intendet1 mil. to point out their know. Then. instead of listening complacent!) existence and to provide a haulm-01k within Ion tiller 01 10111 11101C (1:1,N or development on!) which oulerl. decision making can take place. to find ow :it the end that they die tompletely It seems that three alternatives exist. One is to lost when the) thought the. %vele doing well, ignote the net essit.for mull de( sion making. the) immediate!. ale forced to recognize their thatk. to allow decisions 10 occur 1).default deficiencies and focus clearly on the basic defini- (conscious!.firtittionsdously (leciding- notto tions berme too much complex structure has decide is a decision when the tesponsibilitv 01 been built on top of them. They ate much more making decisions is you's!). .\ swiml isto be alert to the subtle difficulties and the types of :male of the needto make derision, but to responses they inns; be able to give. negle(I to obtain the lamination, acquile the All through the introduction and develop. materials. 01 generate a plan that trill prodme ment stages of the material, the teacher should -best approximations of best decisions.- The use a variety of data for evaluationfrom facial thind k the s.stematit approach 10 decision mak- explessions. to the questions of the students, to ing encomixtssed by the teun spiettm apprina h. their responses to such diagnostic questions as I.:radar:lion. The major evaluation of student just illustrated. Each bit of information causes perform:nit e at the end 01 the insult( (ion on a 111111to slightly modify his approach: to repeat unit is only' a small 1):11of the overall evaltiaa or dwell longer on certain concepts Or types lion procedure. Front the beginning of the intro. of exercises, and so hnth. The more effective (Intuit.. phase of the insult( (ion the teat her, as this ongoing diagnosis and remediationis,the classroom manager. must be acutely aware of less of a problem there will be with the laiger what he intends to achieve With each evaluation at the end of the unit or topic. 1 1011:11 activity and of the degree Of achieve- The evaluation at the end of the unit must ment he attains. Many teachers rely entirely be ealefully designed to measure the degiee of illinformal observation of student reactions attainment by the students of all objectives ex- for this evaluation. However, student reactions pected of diem. These tests should be designed are not always good indicators. (Students ale not only to rate the students and provide grades. sometimes not aware that they have misconcep They should also be diagnosticillnature. If 420 (:I 1AI' ELEVEN

SOU' students ha% e not .11 hie% Cd ObjeCt I% es that let oglIt/C not onb d es among students .ire necessan lot their !wog' esl itt the next unit. butdifferences among the same stmlents on these deficiendes must be teniu before going clas, and his abilityto gke highl on. The outcome of the evaluation should indi- specific emoutagement to the pool student Ito (ate the need, degree. and scope of necessary does stuprisingly well or to teasingly elicita Ittrthr instruction on the unit being tested. better tesponse hots a good student who has (,00(I cla),I)day evaluation should Inc:chide a goofedarepricelessadvantages. The atol classwide disasteratthis postteststage. How- leaching extends fat beyond the science of teach- ever.itis possible that some students have ing, anditis hoe that the !Inman teacheris responded well to bits of the unit but exhibit it t epla«,a hie. on!) «mlusion when the bits ate put together. In the :tea of motivation. one of the ,neatest As a ,),tents manager. a teacher must be we- elements is inninsi(:itlie, in .1 Student', sens lt:tied lor a wide range of results. Some or all of of adlieVCIllent. his feeling that he has su(cess- thetechniquesdescribedforpreinstruction full) accomplished things up to this point and remediation of deficiencies in entering behavior iswell prepared and capable ofa«-ontplish- ate apptopriate here. Again, many decisions are 'items to come.(l'o the extent that the com- necessary. Surely "They haven't achieved the pute'helpstoinclivi(htali/einstruction and objectives, btu well go on anyway- is among helps diagnose and bring specific temedies to the least acceptable. the leaning needs of each ofthe students,it The Teacher-Manager and Ihe also helps to motivate the student to «mtintie Motivational Components of the System on in an effective plogram.) As the next section willillustrate,there are Nevertheless. the human condition being what ways inwhicha compute-managed instuc- itis.the extrinsicinteractions ofthe human tional s)stent can be superior to a teacher-man- teacher withthe huntan personalities ofhis aged one. For example, the computer's ability students oiler many opportunities for the human to keep complete, instantaneously available t cc- teacher to optimi/e the clitantic situation that ords of the progress of each studenthis ac- is the classroom instructional system. The teach- complishments. his difficultiesan(to present er can make various decisions about his inlet- rapidly l0 hill] a selection of we..,,,,en special actions with the (lass that will definitely allect items that might temedy particular deficiencies the learning processdecisions based on instinct cannot be matched in a teachetmanaged class- and insight. the gestalt of the given moment, a loom. feelingfor aestheticprinciples, and leactions Ilowcver. the human teacher has many unique to indefinable elements of the cia,,romn mmo- abilities. One in particular deserves (omment. ,phe,e. Thew me the essential human element, This is his ability to .nteract with the student of an otherwise highly structural systems ap on a nonmathematical plane but in relationship proach to elasstoom instruction. to the !canting situation. In other words, the A Résumé teacher hastnotivationalresourcesthatthe Inbrief,a classtoom teacher who desiresto computer lacks.Motivation being what.itis actconsciously and effectivelyinthe rolein (the generator or starlet lot the sell-actkit) that which he is inevitably castthat of the manage' is the essence of learning). this is a nemendous of a classtoont instructional systemmust know, advantage. do, and be many things. Such nonmathematical things as the teacher's i.Ile must know well the mathematics he is personality. his hums' in the students and re- to teach. spect for them, his sense, of humor, his willingness 2.I-le IlltiNt be well aware, of the specific be- to help. his sensitivity to nuances. his ability to havioral objectives for the instruction. sYSTEIS APPRoACII TO mATIIEMATICS INS l'IWCI`loN 491

3.lie must be well a«putinted with the par- and resonrc es. lor both are limited: he must be ticulat group of stndents he has inthis sensitive and open so as to be receptive to the c lass. feedback provided him front the learning situa I.lie must decide On teaching strategies lor non. the innodution of the new material, its The role of a teacher as manager of a class- development. and its acquisition. room instructional systemis one of the most 5.Ile must decide on the most appropriate challenging and most human of roles. A teacher pat ternof classroom mg:tin/anonc on e- who learns to use a systems approach as he sponding to these strategies. views his teaching assignment can make a great Ile must select and plan the use of the many decisions that will improve the learning Narious media that will prove most effec- situationfor students and by that very fact tive for the topic in question and for this increase the pleasures of teaching and the joys particular class. of accomplishment for the teacher himself. The 7.Ile must make afurther set of difficult world of work and occupation can become an decisions tep,anlingthespecificsoftware art lot and each day a series of episodes of loteach medium that best promotes stu- challenge and excitementepisodes not without dent learning. their difficulties and adventmes but far removed S. I lemust make immediate decisions of from the feelings of aimless drifting or mechani- diagnosis and es ahlaii011. Ieqcling. and cal pacing that so frequently tend to creep into engaging in temedial work to assure the our teaching if we are not alert to the potential achievement of the objectives pursued. He at our disposalnot used to systems thinking. must choose the diagnostic materials and the temedial materials. 9.lie must select complementary and enrichment materials for the individualiia- tion of instruction. If).lie must make decisions much less easily arrived at. much mote elusive. about lac- torsofmotivation and classroom atmosphere that tend ultimately to promote a psychological climate in which attention to and pmsnit ofthe &sited objectives will be eagerly engaged in by the students. II.Ile must evaluate the extent to which the objectives have been attained and make a value judgment astothe need for reteaching and the extent to whichthis should be done. The qualities that these requitements demand of the teacher include the following:(1)an adequate knowledge of mathematics,(2)an "I NAVE A Mu t,* 1140.45 AlECOM) -ro se understanding of and rapport with his stuents, DIFFEREur -ruts YEAR.." (3)a great deal of flexibility and adaptability, Reprinted p orn Educational Technology and (I)an ability to come to a decision with some degree of rapidity as opposed to vacilla- tion. He must be efficient in the use of time 422 (:11,\PITI:EI.l\I

COMPUTER-MANAGED mole. his lank in( lass, the class aet age. ,tlist INSTRE'GTIoNAI. SYSTEMS of questions answered imoth.e(tl together with \Viten one begins to mite on the uses 01 com- (ortect answe's lor them, his cumulatise course puters in the management of instructional sys- aserage. and so forth.In addition. extenske tems. he is immediate!) aware of the timeliness anal)ses of test data may be printed out lor the oI comments in the field. A complete, detailed teacher. The performance of each student may be«nne part 01 his course record. sun ed in the description 01the state()Ithings atthe time of writing would be merely a historical account omputer's memory lor later use. The use ()I at the time 01 publication. The held is develop- telet)pewritets allows the procedure above to be adaptedto ingtoo lapidl)lor titnel)teporting in book constucted-responseexamina- lorm. However. an account of what has been tions ()Ia short-answer twe. Electronic pencils and is being done indicates directions the field allow the compute' to determine what position is taking with respect to both the role of the on a television screenis being indicated and computer in the management of instruction and hence to record 'espouses to a wide variety of the plans of manttlacttners for making com- visually presentedquestions. At present,the puter-managed systems or subsystems available evaluation of neither oral responses nor lengthy, consuucted responses to schools. Thus it is possible to present a com- ispossible. The accept, posite picture of what was, is, and most probably aul« of handwritten. rather than merely typed. is to come. construction seems desirable. Computer people It should be emphasi/ed that our discussion say that these possibilities present difficult. but deals only with the computer as manager of the not insoluble. problems. actual instruction of students. Its classroom use just as a satiety of response modes is possible. as a sophisticated calculating machine. as well so is a wide variety or modes Of presentation of as its other uses as a component 01 a system, questions in addition to the traditional printed tests and the use of teacher-controlled oral, trans- was discussed earlierillthis chapter.Its other uses in schoolsthe usual business-machine him-- !harem). or slide ph esemat ions of questions. These lions. record storage, construction of schedides. include the use of andiotape, alone or in con- and so lorthare relevant to our discussion only junction with visual presentations by means of as they increase the probability oftile television, and the use of filmed or videotaped of computerfacilitiesfor use inthe presentations of questions that include a dynamic management of instruction. element. Itis with the use of the computer to present &alma M a nage m en questions as well as to record responses that the The use of computers Ior test scoring is a com- shift from its use as auevaluation tomponent ponent role that has become widely accepted, to at least sonic degree of evaluation manage- evenexpected. Now thisalmosttraditional ment takes place. Its simplest management func- evaluation role is being extended in a number tionistime control of stimulus presentation of ways. Several Nail:11.1011S in the mode of re- when not only correctness but also response time sponseexist.Multiple-choicetestingcanbe constitute the behavior being measured. used with test consoles. With thew, the student, Far more «mnplicated evaluationmanage- gun having "sighted on" lour identification ptn- ment functions can be carried out. By using the poses, communicates his answer ditectl) to the decision-making capacity of the computer the computer by pressing appropriate buttons of sequence of questions to be presented to the inset ling a stylus in appropriate holes. He may student can be determined as he takes the lest, receive the results practically as soon as he com- with his response to a single quest, on or his pletes the test. The printout may include his 'espouse pattern on a block of questions used \ SYSTILS \PPROACII 1 () \TICS INS1`1W(XION 423

to determine what he is asked to do next. This F.% en mole sophisticated computer rolesin (an pimide much more cnalliatkcinfotth at ion cnalt)atimi and malnation-assignIllent manage- than simpl)the number of answers that ate ment appear certain. right or wrong. For example. it0 11 show that a student knows his addition facts when the% are illanagementof1)rill plesentedinthe normal (additiontable)se- Coniptiterinailagellientofdrill and practice quence butnot When they are presentedin includes all 01the management functions de- random oider.It makes possible the presenta- scribed lot evaluation. together with se% etal other tion of a "hard- addition problem: horizontal important features. The means of communica- form. with Girrying.If this is done correctly, tion between student and computer are those the studentis good at addition. Ifitis done described above. incorrect!). the same !mildew (or a comparable Ill a computer progiam for drill management one) ma) he plc:scatted in vertical form. then the computer ma) make (lain:160n decisions one involving other additionfacts.then one to determine both the type of problem to be with no carrying, and so forth. The Iesponsc presented and the appropriate 'CVO of difficulty pattern on such a sequence gives a clear picture withinittype. Some such decisions are: of exactly what the student knows and does not know about addition. This adds all important "He is perlorming above grade levelill climetion to both achin emelt( and diagnostic subtraction but poorly inmultiplication. Give hint More Intlitiplicati011, less subtrac- testing-. tion drill." The decision-making function of the «mu- puler (an be used to make individualized assign- "lie misses multiplication problems in which nine appears as a factor. Give him ments for students 011 the basis of their response (It ill on multiplication facts involving nine." patterns 011tests. 011 multiple-choice tests not "I-le does addition problems of this level only the identification of questions answered of difficulty well when they are pesentedin incortectly but the particular incorrect answers verticalIon». poorlyinhorizontalI'm in. selected call be used to branch students to vari- Lower the level of difficulty and bring him back through the lower levels using hori- ous reviewor remedialtopics.Patternsof zontal rot tn." responses to a series of constructed-response questions (an lorn1 the basis for comparable assign- Such in ograms stole a le«nd of each student's ment decisions. In them%. even bet eel decisions pet 101111am e so that tittle is continuity to sttc- muld be made b) haing the computer compare essk edrillsessions, althoughthey may be the sequence or mina' collstrlittedresponses several cla)s apart. The) make decisions about with man) such possible sequences storedillits st,t)ing- 01 the dilhuilt)le% el mentor). The piacticalit) of this is limited by on the basis of student performance. They ma) the large numbet of I easonable responses. which allow the student two or more attempts at the leads to a telatkely long Iesponse time by the same problem. They may refuse to accept incor- cotnptitel. When the number of acceptable re- lett allsWerS by locking all but the correct type- sponses is limited so that the comparison prob- writer key. They may accept "Help" or "Show lem is manageable, decision making is based on me" requests Irom students. Such requests may, little more information than that available from leadthe computer to do the next step of a rightwrong patterns or frommultiplechoice ptoblem or to work indetail One or more questions.(Six acceptable answers to each of wmparable problems. They may provide all- Went) questions gives WI' possible lespollse pat- sweis to mole complicated questions from stu- terns. Tilele are "0111)- possible right-wrong dents. If answers to students' questions arc not patterns.) illthe computer's memory, it may record the 424 c a l \r rilt N

questions :and, *-1.74.; ill Wil. ask the

Ott I for certain kinds of drill and may (nen author suchdrill Im inserting dillerent numbeis in

preprogiamed drill patterns. The) man pota time control On lesponses by refusing to accept / the answer ,alleya certain time OF b\ giving WaM then. The response time oI the student, as well as his iesponse. Ina\ bc«one a part of the student's lewid I'm use ill tonne decision making, 111 «iniptitei-stmlent interaction may be printed out for useIt\[cachets and 1)\computer programets to improve the system. liefoie continuing\\ idI des( riptions of \\ hit «tumult:Is can do. itis 1111)01 Lint that wc stop to ashen that they (an do nothing other than what the) ale told to do b) the program the) are Theycanpiesentno material e(e1)1 that "hidl the)Ita\v been ptogiamed to plesent. When they present .111 CaSICI1)101)- 1CM.itis problem the pro tam writer determined to be easier. When the\ "make decisions'. based on student performance. they merelyact on the decision made a priori by the programer. Itis not the purpose ofthis interjection to comfort the leader by "putting the computer in its plate.- Itis rather to emphasi/c the extreme importance of the software in any computer-managed system. This software includes both the computer software (the computer pro.

Courtesy of News and PublicaIion5 Service, Stanford University grain that determines all of the computer's decisions and actions) and the instructional software (the actual instructional material presented FIcului: 11.3. The setting and the student in a compnl- to the student). IVeaknesses in either of these er-managed or computer-assisted instructional system soltwale «nnponents willproduce weaknesses (taken al Waller 1-lays High School, Palo Alto,one of in instruction. To quote a saying of computer a number of ,schools linked by teletype to the computer people: "Garbage in, garbage mu!" The quality facilities at Stanford inproject of (7m/inter-assisted instruction generated by Patrick Suppes) of instruction from a computer-managed system cannot exceed the quality of the system's soft- wareno matter how large. impressive, or expensive the computer.

A compiele Compnler-Managed Syslani

Itispossible to extend the computer'sman- \ SYS l'ENIS WNW \CH FO \TIII.:,MATICS INS IltIVI-ION 425

agemem function to include not onl) etalua- torequestrepeats and reviews,to ask ques doll. assignment, and drill but also most other lions. and to select some of his own topics of aspects of the instructional process. Such a «mt- instruction.Ile willinstructthe computer to pntel-manage(I system might opetate as f011(): give hint certain information ofto do certain 1\11(11 a student enters the system he is exten- continuations lot him. Ile will use the compute' si el) tested. This testing trill establish .1 com- to simulate experiments and real world situa- plete profile of the student. This will certainly tions from which dismt ct ies can he made. include a plecise analysis of his cuter lug mathe- Not all of a student's learning will be (lirectl) matical !what iot. This mat be done b) establish- monitored b) the computer. He will be instruct- ing his position on several patallel K-12 strands ed to do certain things with traditional compo tepresenting the "big ideas- of school Iliadic nents or lush actionohmipolative (lea ices, book,. walks.Itwill include both concept (levelop- teachers, fellow studentsand then to return to ment and skill development lot each strand. the computer for a "dim ussion- of these act it it ics. Part of this prelimina) evaluation will pro- Ile will, no doubt, be scolded, «unplimented, tide data lot the computer to use in making its coaxed, encouraged perhaps even "loved"by initial decision as to the optimum instructional his complicated electronic tutor. He will cer- strateg)(dis(tver), tell-re«mstru(t. etc.)to be tainly, computer people claim, be taught much used with that student. Perhaps data will be more, with much greater efficiency than has ever collected as the basis for deciding how his learn- before been possible. They will, they say, have ing isbest mediated (erbally. pictorially. ab- built both a better log and a better Mark flop. stractly.«mcretel). en.). No doubt. informa- kips and will have made both available to all tion trill be collected for use in decisions on learners. trim' media should he usedinpresentations. Some measure of mathematical ability will be Plans for Making GMT Available made as the basis for deciding on step site, pacing, and reasonable terminal objectives for the Three problems of the availability to schools of computer-managedinstruction(CNII)require student. This studentprofilewillhe storedinthe consideration. These are: (nnputer's mentor). .\s he proceeds through the 1. Tile availability of the hadnune (C01111)tli initial sequence selected by the computer, it will (MN.I espouse stations.etc.)to individual must:wily monitor his progress and use the in- schools formation it obtains to modify his record in its The availability of good computer soft- mentoty.Thesemodificationsmaylead to ware (computer programs that manage the changes inits initial decisions on how bestto instruct ion) instruct him. 3. The availability of good instructional soft- Presentation to the learner will be by means ware (theinstructionalmaterialtobe oftypewriter,television. and audiotape, and presented to the student). possibly by compiler-constructed oral language, The firstof theseis an economic problem. using elec tioni«it «tits to simulate human vocal Thee possible solutions ate often proposed. sounds. The learner will communicate with the One is that of shated time systems. In such a (unpile' by typewriter and electronic pencil, system a school or school district tents response using the pencil to point to symbols on a tele- stations and buys time on alarge computer vision sct (Tit of to mote them. Etentttall) lie will owned and operated by the computer manufac- probabl)use oulinal) handwriting and voice. tut er of other organiiation separate from the Not all decisions concerning instruction will be school itself. Such a computer installation can made b) the «mtputer. The lea net will be able manage a wide v aricty of instruction for man) 426 (Ai \PEtt 11,

studentsill many geogr..phicallocations. The ( ()tubule to be skin until enough ofitis «ist to ea( 11 school depends on the actual amount able to n.11 lant signih0.1111111-st 11001 utilisation.

of computer time ituses for instructional and Then the increasing.1\ ailal)ility 0)1 «mintier ulheh putposcs (bookkeeping, scheduling. et(.). Utilities and the inol einem nit!' compute'.

.\nother possibility is the rental or purchase of managed systems 01 .111 inc teasing number 0)1 a «unpiler of moderate slit by a large school peopl 111111C C(111(.111011.11 «M1111111111 I% ill1)10- distlict or sevet al small ones for their exclusive duce .1 steady. adequate supply of MA1NV:11'e. At

tise..\ !so. some manufacturers produce 1)111 Inst. both compincr sOf1 1%.11('.111(1111.41 11( 11011.11 sophislit .11ed belic C hush ( OM- sollIV.Ile will he rented or bought with the cont. InIt ers and the accompanying response systems plum EA-entually, no doubt. institutional soft. to he priced Ivithin the budget of an individual \%ate tespec %yin be de\ eloped by individual schoolbuilding. Such system, generallyeel) teachers or groups c)Iteachers Ion u,c 111!hell heavilyonoutof«)111pitterstorageofboth 01%11classes01101 for1)111111Call(111 by computer computer and instructional softare. using the manufacturers and softivate publishing compa internal «minuet. memory mainly lot student !tie,..1, textbook, and odic' naditional 01111110

profiles and general management ptograms. Ilellls 01 111.4111(11011 ale1101V deelOpC(1 :111(1 The prochwtion of (impute!. software (com 1)111)11,shed. inner pr(;4ratus to manage instruction) has pre- . \s with .111\ re's 111s1 1 11( li011.11 1001. the quality sented an interesting ploblein. Ithas requited ()I matetial is likel% to be tine\ en lot some time.

the development within the wmputer industry Itis impot tam that potential «mstunets (II I and other research establishments of people with material be a\vae of this. They must look be- knowledge of both what computers can do and yond the impiessi e 11.11(11y:tie. They should ask how students learn. question, such as the following: Similarly.the development 01instructional "Is the content dint is presented collect and software (Mater sal, lor student use) has requited a«.itrate?" the involvement ()I people with knowledge of What are the objectives of this instruction? computers. theories of learning, and the content Ate these adequate for our students?" to be presented. ".\re data available to show that this in. Furthelmore. bothNorm:it e problems Imve stroction does prochice achievement of these been confronted in an almost paradoxical situa- objectives with students such as ours? tion. Computer facilities will not he genetalh \Vhar isthe per-stnclent cost of this available until software exists, but their general stuction? Is this cost justified by improved availability is necessary in order for large num le:tiltingorinc leasedchic fence or other hers of people to become involved ill software factors:- developmcblit. Computer-managedinstruction is nothing The necessity for the availability or a «im- more than :t way of managing instruction. The puter facility for software development means quality and value of the instruction can be deter- that most of this material is being- produced "in mined only by the :111:Ave's to questions such as house" by computer manufacttners and !elated these. companies or by tescarch projects associated with large school systems or universities. Some had- A SYSTEMS ANALYSIS wae manufactutels have purchased traditional OF AN ITEM OF INSTRUCTION textbook publishing companies, intendingto "put their books on the computer." Thisap In this section we shall illustrate the various proach has not. in general, been successful.It aspects of a systems apploach to instruction by seems certain that development of software will applyingthistechnique toaspecific content \ sys\\ \ vo m \ Em\Tics c,:s rizu(; o:\ 427

item. intetsection of sets. We shall give what 5.Construct sentences including theterms amounts to the initial s)stent-plaiting phase of el luent of and membel <4.3111eN oC oll the institution 'elated to this topic. including: jeds. and Verbal deSdp11011S of IIlesets I. Spc(III(ation of entering behaviors to indicate that the named objects ale or 2.Specification of terminal behavims ate not members ol the described sets. hhitial s)stent design. including determina- .Read s1111bOIS h as tion of instructional strategies and assign. . . as -the setwhose elements ale mem Orloociioos hospecific (oolpoocov, . _ - ... and as "the set of the s\ stem. of re«)gni/e the use of l'hat is. we shall show ghat might be either the braces enclosing sbols to indicatea set first draft of an outline prepared by editorsor whose elements are lepresented 1)) theen- authors for a «mipany producingsystems mate- closed bols. tials or the lirst step in elalsr00111SySle11 plan- 7.Construct a bruesnotation mbol for a ning by a teacher whose general goal isto teach set. giveu, a verbal (lescription deterndo. intersection of sets to his class. It isnot our pur- ing pose to delend the specific behaviors We asSlIllle S. Remgniie sbols end sec!I))a simple 01' eXpCd. I101'(I() We el:11111 0111'lleelsi011S closed curse as a II:1111e for theset whose strategy or media ate tlonlyor even the best elements are rept esented by the enclosed ones. NVe intend only to illustrate bya specili( smbols. Gise averbal description or xample the ideasdiscussedearlierinthis braces notation for the setso leiesented. chapter. and conversely. 9.Read :1-7- ;1.2)as"Set.\ Assumed Eniering Behaviors isthe set whose elements are I. 2." That is.recog- Students entering this instructionmust be able niie the use of capital lettersas names for to do the lollowing- things: sets and the use of the equality symbol to I.F:s:hibit the :Wilkto think of a collection canned names for the same set. of objects as a single entity by using «)I- 10. Respond -The et ,par set"or "The null lective nouns and singular verbsto de- set" to questions suchas this: What set scribe «Meico's. (he team is_ has as its elements all peoplemore than The (lass .) thirty feet tall? Construct verbal descrip Y. Construct sentences including :hewon lions of the empty set. set and a sufficient qualifying description \vibe 0 or I I I. jItt lesponse to a requestto to indkate that certain given objects ale write a symbol that represents theempty. to be thought of as parts of a single col- set. Read A 0 as "Set A is the empty lection. (The set of boys in this100111. set." The set containing Mars.your pencil. and 19.Given two sets. identify eachas a subset my dog.) or not a subset of the other. !(lentil)the Rams element of :Ind mem- Ct.Given a set. determine someor all sub- ber Of by esponding correctlyto ques- sets of it. including- identification of both lions such as these:Is John a member the empty set and the entire setas subsets. (element) of the set of boys in this class? 14.Given a set. construct one or more sets of Is France a member (element) of theset which itis a subset. of «ninnies in North America? 15.Else the symbols C and with either . Given objects and,a verbal description of braces notation or letter names fortwo a set. an rectly sort the objects to separate sets to «rectly identify one as a subset mentbets of the set front nonmembers. of the other. 428 (Al \Ill IIII\

Ili.Id ell 111 ) the lrse ()I a 11 1v it IIII t inlet .1 Is 6.Identil% this dual use 01 sunbols ao.114). s in (i 5 I) as c\lncssiuns lot the Bolts to the uses of and ill at ithinetic. stint and piodu(oI the numbers tepte- 7.Respond "1 he set of all elements in both seined. IZespond colic( 11% 10 theb use as set A and set B when asked to define the institutions to (lett:indite simpler symbols tom the intern'( 110)1 N1 01 .1 arid 0 I B t Ior the sum and product(6 3 = ? tot A (1 1:). 5-1 4 ? Ans%veis: 9. 2(I), 17.(:onsti Ito real number instati«s of each 8.Respond (on te( 10 examples of any of ()It he held pi ()pert ies («minimale 1n op. the spec ial cases :1 n B A n B B. cities of addition and multiplication. as, 1 n B with the gken setstepie so( intive!novelties.et(.).Stateelticli set (cellby bra«'s or endosmc notation. piopert is illustrated Ir a gi'.elI instance. 9.Identity A (I B B oink dent Io B C A and A B. Identify A n B = /I as equiv- Expected Termimd Behaviors alent 10 :1 C B and B 7 A. Correctly re- As a testi!' of the instill( tion. students are CN spond 10 :Ind (olsiro(t examples 01 these. VeLled to be able to do the following: 10.identii "1 and B ac disjoint WIN" as I.Gken two sets lepiesented b enclosure equkalent to A n B Correctly lea notation. ideiktheset ()Iconnnon si)ond to :Ind «nistruct examples ()I' this. members as the intensection or the sets. Describe the intersection of the two sets II.'.'ranslateI %erbal des( tiptions ()I the inter. as -the set \ehose members are section of Iwo sets that b a h ide die term /at/Tu./lion into a description not con _ . .. . mining this term and vice versa. as in the 2. Recogniie and «insult( I eal.woildex- amples to just ifthe use 01 the (elm inter. following, examples: The intersection of .section in mathematics. the set of all girls and the set of all redheaded people is the set of all redheaded 3.Given two sets tepresemed by enclosure girls. The set of all men over forty years notation, use braces notation 10 rep esem of age is the intersection of the set of all their intelsection. men and the set of all people over forty .1. Given two sets iepresented by braces nota- years of age. tion. construct enclosurnotation tepre- sentations of them. showing appropriate 12.Distinguish veibally between A n B and demons intheir intersection. Construct B n A. Give a hem ;500 aigument to show that for all sets A and B, A It B = t1ItA. abracesnotation representation of this hit ersect ion. Identify this as the (finnimative property intersection. Recogni/e and con- 5.Recogniie A n B as a symbol for the in- of set tersection or set A and set B. When the struct instances of lids piopertv. elements of A and B are given, recogni.ie 13.Identiktheuse of parenthesesI))dis- A fl B =? as an instruction to construct tinguishing (serbally) between a representation for this intersection for (el n B) nand A n (ll n c). example. A =I.2.3).11 = (2.3.5), Give a heuristic aignmem to show that :\ n B = (2,3). Similar ly. re«igniie lor all sets A, B. and C. fa, bl (I fa, e,(l)asit symbol for the (el It B) (I; n c). intersection of the represented .sets and Idemilv this as the associative property (a, b) (1(a, r, =? as an instruction or setintersection. Recogniie and con- to produce fa). stuct instances of this property. NYsTI.MS \ PPR° 1(3 II O M tics INs ['RU(: 1"ION 429

The I Migoof aSrAl below loimal institution begins. Theii menntal Itthe desito, that 5%e it;, listedin, use ill lolinal institution is des( lilted later. sututi0na1 .1(thilies in the older 01!licitp1 o .\CTIVIT1 T\VO: I)IAGNOSIS posed use. Illthin emit :1(1i% it% we lime listed Pupme the «mnibution (tI 111.11 :11.1 15 il)to Objellke I o obtain .IIIinventors of Ihe abilities of the .1(11Ie1C111C111. 111.11(1 ills to he used in the uti5 ii%. students with iespe( tto the lists of expc( led en-

and .1 plan 101the institutionitself.1)etailed ing behaviois and expected let beliaioiS des( iptious ale gi%en 01 sonic materials to pro- 3 fat erials

side.1 1.11i011:11('101 111(11selll 1 1011. 110 I)iagoostitIcst Inc.isto ing .ill expe(ted enteiing .111C111)111.11 been 111.1de 10 select eNOlit (01111/0- beliai(ots and .tt least the key tut mina! luhavims 11(111%, fiddler has the Merall (0s1 Of the SyS1e111 expected been (011sidel(11.In Wel le1iS1011S of this pre For (Pleqi0115 "ncring liminaly design. tradolfs of(ost againstin- key enteihig behaviors may be presented on stitutional elli(ien(% would be (onsidered. natispaiemies for use in class dis(ussions, The institution pimidcdisstItictuledac- Cm, ((tiding to the three phase procedwe des( ri bed If this is :111 is01.11Cd 101)il 011 sets. 101111:11 CV:1111:1- in the se( Lion headedI eat her:NI:waged hist rill li011of entering studentbeim% iorsis recom- 1;01011 S1s1IIIN," \VC 11:1%e adopted a guided dis. mended. If itis part of a unit on sets, inlormal (0erY sta(rgY '61 h Inininlal S('11):11111(1H:160H 0:1111:11 1011 may be sufficient. of learning.Itis assumed that this system sill ii AD jusTmENT F014 ,\ he used WI111111:1 grouprogrtss administrative structure. Purpose, .\CTIVITV ONE: To ensure that all students begin instructionon PltEINSTRUCTIC)N ORIENTATION the topic with all necessary entering behaviors Purpose To preplan inodifu ations in the instructional To pros ide aninlornialintrodiutiontothe sequeme for those students(if any) who have topi( and :t motivation for the inathennitkaluse :dread%a«piired some expelled terminalbe- of the term 1'H/enter/ion havior:, Alaieriais Mater/Ws Prints (poste's) showing teal wolld situations One -reelIliinnifilm (Meting developmental illustrating the everyday use of the term inter- aspects of all entering behaviors seriion: intersections of streets. roads, wallsat The film medium is particularly appropriate the corner of a room: the "area of possible con. for illustrating the dynamics of the change from flirt" of two dogs lied to dillerent posts so that thinking about individual objects to thinking their circular ranges overlap; and SO forth of a set with these objects as IllellIberS. 1'41111es of intersections of sets of pointstwo Printed work sheets for student activityill the intervals on a line. two rays on thesame line, a(quisitionalphase of allentering behaviors two lines illitplane. and s0 10111MR1 of sets Traitspaiencies containing the major points of of objects represented by enclosure notation the film to be used In allpion' es the intersections should be Use shown ill the Aallie distinguishing color to focus If the testilts of the diagnostic test are:tclass- attention on the sameness in otherwise different wide disaster, as might be the case if this is the first instruction Oil sets ill a course, the unitoil Use intersection should be deferred until a complete These pictures should be posted several days unit on basic set concepts has been taught. If 430 ( II \VII p,[I \

%nil It(11.,:i.1 I On. Ir.4111., of ( in for a small subset 01 rw 4)1!HMI'. nolnlion loI eincm'Ill the( lass.possible (le( isions ate the 1(111(i ving: intelw( Of IVo l'se the film and voik sheets as sell.in- stilt( tional maul ial. (1)0.4(1 ,). \tlisir( ()in: lot%volt, in and outside ()I( lass. l'ossibly I.ohl),,nen(je%. Willi Mole examples like those limittheset 01 trimilml()hie( Ii%es101 01) the In hos. (..1)« ia1.1% %with sets I eiriescolcil 1)% 'hese iticlents. losme notation l'se the film aml %yolk sheets with these stitdents 011 a small gioup ;Intl tO non basis .0 het -led. ossioll. thing outside 01(lass or ( lass While others the allead!1:iiiiiliar int.: %%Nil k outside this unit 01 institution. -1'llat AI of 'hew? l'w the !dm ot natipat en( iVN !Or le might %'.d (all 'hew '(oilinion vans': \Illy? Is I et die k sheets for iiiclass ill 1)(01 ,1 and set 11? Is it in the inter hot' ti( [ion. 601 mole able students .ser%- Archon of them. Nev.:" Student. ( whim( Imii- ing gum!) !cadet s for small gimip %yolk. tem es .11(11 a. -The intermItion of set Itis to be hoped that such cli,,,asters trill not and set is the set whose menthe/. ale

own. I Imvevei. the diagnostntest ma% feral . . . and ;incise( tion ser( e. but isolated. deli( ienf ie. among some (P of these sets wmains . _ ." the students. i'%( 01 the !din 01 Ilan Talendes. Transpalm ies .tre toed for feedbm k and lor 1:1 dim 11.1011. and %VIII,. on apinOin 1310 %%'0i k additional nisi tu don. 35 needed. sheets indiidtialf (0 in small gimps ( om. STEP 2 OF patable vaktiesses should prmide stillident it l'IVITN mediation. \limit deli( ien( ies into be removed In time 01 Purpose unit the film 01 tianspat elides and class (limit. To indu« achioement of the met 011 the list sion. or the lemma! ()I delidem ies wet be - ()I timinal objectkes: 1)tioubleshooting at appiopi I.Given 1,18«.. notation lot two sets. (in. places In the 111511t1( 1101131 5etitlent e. shim en( lostne notation for them and Ina( es notation lor their intersection. 1 1 diagnostif testing tercels prior a«ptisition ()1 some tel obje( lives Ir some or all St tl- Alateriais dents. modiluation ol the sequence that follows Thice-inintite SInin silent film trill he lie( essary lor them. Such modituation hn Transpai em ids individual students (all take the font either 01 sheets adding blithe' 0.111 i( !intent) teiminal obje(thes i'l(c:frk with huh% idualiied insult( Initial activities to in- The lilm. tie,1 for Inc introduction. shows the

c dick attaillimmt or of %%.01 1, outside the following: unit of non ti('tion. Two sets represented by braces notation ()vet lapping closed curves .\(:11VITY 14)t_tlt: S'Is1.1) I 01: tisnibolsfor elements that"slideinto- Purpose closed ,tirves. with dente-lb of the inter- To india«..0 hie\ement of the lollo%%ing se( tion sliding into superposition in the object ives: med. ;)ping part I.Iclentilif ation of the intersection of given Rraces notation for the intersection sets 'Hie film runtimes. with several examples with 2.Rationale for the use of the term inter- nonempty inteisections and With neither set a seethm subset of the other. Transparencies with Cool- sI ms molt( )(,1 II ) \I 1( s \ s I RI ( 1( ),\ 431

p.1 I a blc 10111(111 (a be used to eplaw the film minim! able itiox tote the 1(01 (11s. anti lootlec(11).)( k and additional iihntutioo. it ( 11.0.1011 .111(1le( dim( needed. 1 1 ((a sheet.( ()ain (ill: "St.opose we don't Itae a list of ele111011% "(.iell 11.1,e, ne )1,;1 jun101 IN%() 01 NONI and B. Ilex et tildes.. teplesent sit II( elk 10.11iC n01.10011 101IIIe11n bid( es theirintelse( Bout" Studentsanswer. "re., UM:111011 101 01(41 11111c1 W( 11011Mid. fl It." The the, ()I n all instttu tion 1101:n1011 lur 11110 c( 11011s %VII WillI he and as a s)11114,1for a tooth ate dim thsed. inlet illediale (C11( ioNttl( 1101;111011) Ime )(on ...CCU..11( 11(1011111e usage 1)(010," MIT\ SIX: NTFI' 3 OF lead.to analogy withti + 1and5 6ill ItASI(: INS' FR rc-ri(),N at ithinti(. "Can you des( Own it without Purf)we knowing the elements of and /IF' leads to To indtue :Ode% emelt' of the next (liceOn the the statementoftheformal deltnitionof list of tetutinal obje(Iie.: A C as the set of all elements in both set 5, Recogni/e '1 11:IN110111 :1 %%1111101 101 .4 and let IL the 1111e1 11'111011 u1 :1 :11111 Is :111t1 :111 111,41111 11"011, 'sheets lequiring )(prated CX11111111011of 11011 10( 011.11111( 1 .1 '',.11111)1C"ICI)rCNC111:10 allbehaviors above plovide thea( quisitional 111111 101'1111' NC( 11011 GI fl :).1:11,( phase Of this act it icy. similat the ol (t with Ina(es notation loot sets. Ac.TiVIT) EVALCATioN Pinpon. 6.Identil% this dual the a. analogousto the the. Of andill aithineth. To detetmine a( hies einem of terminal objec- 7.(;he a homal definition of the interm ti% esI through 7 tton of set and ,W1 /I. Maier:a/A ,Material, 1(Itieeutent test on tho.e ()hie( the!. Use Threenninute S111111 silent film 'Fr:mit:new ie.. objectkes eniomixiss the bash ideas and 11'orksheets notation related to the interse( lion ofset.. IZe Use maining obje( in\ olse spe( ial (:i.e. and hi- The lilm itarodu( es the (Oncepts in the follow !cresting con.quences of this «m«.pt. Vitltsome lug sequen«.: (1a..es or some students in a class. the tem:tin- ."' example such as ing obje(tise. (especially I I. 12. and 1.1)ma) be c. it = e, g, It) «nhideted etni( !intent. In an)case. esalttation appears. and 11e( Cbhary additional instruction forall) sin- The following sentence appears: deot. needing it should pre( (ale (or possiblyle- The intersection of A and B is pla( etfurther new instruction. 'lime is given for the students to respond AC,TIVIT1' STF.1) 01: orally. then the symbols appear. BASIC: INSRUCTION Immediately below this the followingap- Purpose pears: To induce achievement of the following objec- A n tives: This sequence of sentences is tepeated :um- S. Respond correctly to examples of: thee example, with time for student'espouse n . . , , . A 1111 =B, A 1111 =A. allowed for il thild,t oscine as well as the 9.Identity A fl ItB asequivalentto second sottnce. 'nen examples are given with 11 C and A B. and identify the second sentewe emitted. AnII =A Tiansp:trewies (or dialkboard pictin es)of as equivalent to A C B and B A. 432 (.11 1'11.1; 1 1 1.V1-N

10.Identil "A and 1; are disjoint sets- as hal kboa I (11%el bal dem I iption,01Iwo m, and captivalent to A fl B O. .1,k, for a %erbal des( Option01[heir inirsertion tvluit IIdoe, not( omain the tem: intersection. ianlhoncies with (nerlat, .kltet this description has been given. the inter- Work Sheets section 1)1this inters(tion and a Atilt! given Nei Use (rinsered. the ate', the The three cases in objective 8 :Ile Mitsui:tid idea. thing Itatisparencies with mrlas. For example. Gi% eii Iwo Nets: rl fl1; 0 isillustrated by showing setA, Tile Net 01all.uulcnt. in ihiN 11001 represented by enclosure notation. with a setB The .et 01allredheaded people such that Afl B=shown on an overlay. Pie Inii:weii0n 01 the given sets: overlay contains the question "11 flB =?" The set 1)1 .t11 tdlteaded studems in this The cases/1 flB=11 and /1 flB = Bare school minparably illustrated.:11ter these have been Given a ['did set: Niihii Within it cuntntcnt and the questions an- The set ()I all football piaets swetedby students.ateache -leddiscussion centers on these questions: "How 'big' can the Itnetse( lion 1)1 the intoset tion ()I the lostIwo intosection of two sets be? How 'small' can it sets and the thild set: be:- These lead to the obseications ill objective 'lite set of all tdheaded lootball plaetsin 9 and the obsenation and terminolog% in objec- this school tive 10. The saute liteistepeated by starting with Work sheets illitstrace all special cases, as well the second and third sets and then intodming as nonspecial Lases for discrimination. Sets are the lust set. and linalh 1) smiting with the first tepresented by enclosure notation and braces and third sets Mid then introducing the second notation and ate determined by verbal descrip- set. Students are asked if the last set obtainedis tions. S. 'dents (determine intersections. lespond the smut. in The .m,we,Stari,.uu ink), D i., "13 D A," or "A andBare disjoint omi hoodoctioo to the final two objectives. sets" Ivhen sets and intersections are given. and Now the process is ieversed. .A set whose ele- construct sets A and B, given that A fl B = A, ments are cles(libed bt sevetal characteristics is /3. or 0. given. and the students construct set:whoa ACTIVITY NINE: STEP 5 OF intersection is the given set. Discussion of the BASIC INSTRUCTION range or possible answers for each example is ill. Purpose solo This game can be ex- To induce achievement of the following ter- tended by having one student or a team of stu- minal objective: dents construct descriptions for ()diets to trans- late. At:dim:11)es with time delays for student 1 1.Translation or verbal descriptions containing the term intersection into other responses. or work sheets. can provide challeng- verbal descriptions. and conversely. ing- examples for all students and easier drill for 41/a ferials 'hose less able. ranpaenies AcTivrry -ITN: STEP I; OF BASIC A 'idiom pes INSTRUC;FION Work sheets Purpose Use "1-o induce the achievement or the final terminal A "restatement game- provides the basic instruc- objectives: tion. The teacher presents on transparencies (or 12.Distinguish between A flBand B fl A. \ITNIS ,\I'1.1:0 \CIIII) NI \ \ 11C s INS 11:1"C rIoN 433

.1igtte that ft B B A. ()I)Nel ye simi- .\1:11IT1' ELEVEN: EVALl'.\TION larity with at b = b -i a and a = b a rpose lor ntutthrtN. Name this the /omit:is/alive To determine student perlormance with ',rapt-) ly. ilicakated by analogy with num respect to all terminal objectives bee properties. lo deal mine %chat reteaching. it identily use of parentheses and argue that am. is lictes,ary n it) c). 31alerials Name this property by analogy with .\ chievement test la a + ) Use :cal(at b) r a(b1-) for numbers. ttStis(INC(' both as a measure of aehiete- Material., inem and as an indicator of possible need for Four- to sixminute lihn ientedial insult( tion. Whether suchinstruction ranspalencies. with ot erla.s is done at this time or later is determinedby the 1Vork sheets position and role of this unitin the overall Use course Nu tic The film provides the intioduction: Enclostne notation for a set (labeled set /I) Co)Ilinuing Toward a "Bess Design.' appear:, Enclosure notation for a tiondisjoint set it It should be leempliasi/cd thatthisis a pre. lades in. Labels for It and n It appear. liminary design for a system. Thenext phase %could be plodution (or acquisition) Pinions ligm e moves to left of the picamC. of indi- Same set B appears at the right. Then set vidual components. testing of these inisolation fades in and label C n A appears. Final- and in sequence. and revision of either halivid ly :1 ft B = B n ..I appears. mil components or or the entire designuntil the requited behaviors are huhu ed withthe in- Next a comparable sequence lor (.1 n 1;) n tended audience. hroughout thisprocess both aod n (it n C) i, :shown. Teacileried (ii,ett.,- instructional and economic laiorsare consid- sion, thing iioNparcories with overlays,recom elect. .\ surest, approach to instruction includes ttels the sequence from the film and provides both the preplanning- illustrated byour prelimi- other examples for discussion. "Transparencies nary design and the determination toaccept and or chalkboald le% iew ol properties or operations act upon feedbackuntil a "best design"is On numbers leads to compatison and naming of achieved that will. width. given restraints,pro. these properties of set intersection. llort: sheet duce optimum instruction for all studentswho drill provide, the accptisitional phase. rile to learn within the system. C11 \ I' 1 IA:El\ 1.N

liII;LIOGIZAPl11'

1. Ile% non. Robot "The1 ot.tl Sstenis Com ept 1:11iis. Geiald I.. Ihe Ssteins\ ppioath." Reseal% h Implitations." 4/...D5.11»zih». Seinen]. .11onzio). janual 1968. pp. 18-20. ber 1966. pp. 8-10. 9. David. "Components of a CAbernetit 2.Bright. R. Lottk. "Cool Judgment, for New Direc- Insult( 64nial Sstent." Edioatzonai Teihno/ogy, tions," lournal of Pain(aln)noi Data note$%ing, 15 April 1965. pp. 5-10. Spiing 1968. pp. 8940. (..upon. Elliot. "Games in the Classioom.- Saim- 10. /o New (:onzpzziti:ed \ special isqle de- din lleyzete. 15 April 1967. p. 62. %met! to the potential ol the al1101111:1111111 ICU/111- 1. Exton. Elaine. "1 low Will 'Systems Appio.uh' (1011 and its implitations 14) our sot wty. ,Saizodav \Heti, Role of School lloalds:" ..Imezican School nevzew. 23 July 1966. llood journal. September V)67. pp. 13-11. II.()ettinger.\ ittliony(.. M ths of I:(luuttional 5.Itoihes. jack E. "A S)stenis App14)ach to Institu- 1 et Ittiology."Sat today Revule. IS Nla 1968. p. 76. tion: Bat kgiound-and Futtne." Canadian Audio l'zzzal lirozeze. May 1966. pp. 22-25. 12.Silveri). Leonaid C. "A Cbernetic Ssteni Model Hills. R. jean. The Concept of System. Eugene: for ()cc-tip:164mA 1Am:16mi." Educational Tech- Ititheisity of Oiegon. Center for the Advanced nology, 30 janualy 1968. pp. 3-9. Study of Ethic:164)1ml Administration. 1967. 13.11rilber. Thomas P. "A Rationale for the Employ. 7.kapler. Philip G. "An Instructionallanagentent mcnt ol S.stenis \nal)sis as an Aid in the Nlan- Strategy for Indiidualind Learning." Phi Della agentent of Public StItools.- lonznal of Edzzca- Kappan. janmo 1968. pp. 260-63. 'zonal Data l'tme$szng. \\tinter 1967-68. pp. 5-12. APPENDIX 436

APPENDIX

pRoDucRs AND DisTRIBuToRs OF INsTRucT1oN.\1. Alps

Assoc iation Eihns, Nihon Hiadle (a). \ school Supply, Inc. 600 Madison . \se. ,Nlass. 01101 New Yotk. N.Y. 10022 Blentnet ,\Itthipluation 137 \11»our N.E. Atulio laster Corp, \llama. Ga. 30321 IZec olds. Inc. 17 15th St. .\(adenticIndustries Inc. 161 Gwen B.is lid. New Yolk. N.Y.. 10017 \Vilmne. 111. 60091 1751 \\walnut ,\\e. \udio.isual S)stems Bums. N.Y. 10.153 Rtooks ,Nlanufacturing Co. 1219 V. 4th . \se. P.( ). Bo, II 195 \ddIson-Wesley Demer, Colo. 80218 Publishing Co.. Inc. C111(11111:111, Ohl° 15211 \salon 11111 Co. RIIIIIIIICItter CO.. Inc. School IleacIpl:01(bis Sand Hill Rd. 1517 Ilarlold Rd. 19151 touttnan St. Baltimore. Md. 21211 Menlo Palk. Calif. 91025 gtooklut. N.Y. 11237 As is Eihns Buhl Optical Co. Ihe Achaticc.ment Pla«.ment 1000 Beech\s c. Institute Milbank. Calif. 91506 Pinslungh. Ira. 15233 169 N. 9th St. Bus( It Film and plipment Co. lbookhn. .11211 r) 21 IS. Hamilton .Si. \eto Educational Ptoducts Ilailm Films, Inc. Saginaw. NlicL 48602 St. Charles. III. 6017 1 6509 De lamgple \ye. \estheoinett. I lolls wood, Calil. 90028 1903 Clayton \\e. C. Ballard Co. C \Itsia. 88210 2$ S. P (:ada«). Inc. \IN Industries. 1.kermot v. Calif. 9 1550 310 \\,..poll, St. 253 State St. Bausch and Lomb Optical Co. Chicago, III. 60607 St. Paul. Mitt. 51107 635 St. Paul St. Cadd holler Inc. \ itecptipt. Rochester. N.Y. 11602 Box 5097 20 Jones St. Beckle.Canly Co. Inglewood. Calif. 90310 New Rochelle. N.Y. 10802 190)) N. Narragansett Ave. rnivetsity of California American Book Co. Chicago. III. 60639 Fsiension Media Center 300 Pike St. Bell and Howell Co. FilmDist Tann ion Cincinnati. Ohio 4 5202 7100Ic(-:ortnick Rd. 2223 Fulton St. \merit act (:rayon Co. Chic ago. III. 60615 Rerkelev. Calif. 94720 Div. of Joseph Dixon Crucible Co. Ilen.G.Pnalucts. Catr Plastics. Inc. P.O. P.os 2067 162 Sagamme Ave. 129 W. Stye' ion' St. Sandusky. Ohio 4870 Williston Park. N.Y. 11596 Chicago'I. 00610 \O Instiument Co. Better Sc ientific C.C.N1: Can: .os«), Inc. ScientificI tistrument Div. 37 Williams St. 312 Western Ave. Buffalo. N.1'. 14215 Boston. ,lass. 02119 Boston. 02135 Applied Sciences. Inc. Pubikele Enterprises. Cen«) Educational Aids 123.15 hit lid ,\ve. SI5 Washington St. See Central Scientific Co. Cleveland. Ohio H106 Newtonville.lass, 02160 Central Sc ientific Co. \ ithmetic Clinic Charles Beseler Co. Cemo Center 4502 Stanford St. 219 S. 18th St. 2600 S. Kostner Ave. Chase. 21027 East Orange. N.J. 07018 ( :hie art). Ill. 60623 \ rithmet it al Principles. \ ssn. Chanting,; 1.. Bete Co.. Inc,. Champion Publishing Co. 5848 N.E. 42d :\ve. Fedetal St. 612 N. 2d St. Pot tland. (le. 97213 Greenfield.lass. 01301 St. Louis. Mow 63102 \any International 1)iv. Hook-1.:0) Inc. Chan oisCatsen Corp. 2372 Linden Pdvd. 144 9 37th St. 5 Daniel Rd. Brooklyn. N.Y. 11208 lhooklyn. N.Y. 11218 Fairfield. N.J. 07006 Associated School Distributors. Inc. Stanley Bowmar Co.. Inc. Cheintronis Co. 220 W.I:Iclison Ave. 4 Ric)adway 175 Vega Dr. (:hie ago. 111. 60606 Valhalla. N.Y. 10595 Goleta. Calif. 93017 437

(11111(1(1.th Equipment Co.. Inc. 'I he Curia Co. Ealto ational id Publishers 155 F. 2:1(1 St. P.O. Box 3.111 371 E. 55th St. Nett fork. N.V. 10010 Van Nits. (:alif.91.105 ookl) it.N.Y. 11203 Clam agenteipises. lad. Curtis\ tolio-Visnal Materials Educational \ 1109 S. \Vooster St. Indep0101011«. Square 29 Nlarblc 90035 Philadelphia. Pa.19105 Pleasamxille. N.1. 10570 idge Pioducts and Equipment Custom I:abricators Inc. Educational Dexelopment P.O. Box 910 8702 Bessemer .\ve. 1:1boratories Ilarrison. At k. 7260! Cleveland. Ohio. 14127 I Itintington. N.Y. 117.14 John Colburn .\ssociates, Educational hut Games 1215 \Vashington .\v. P.O. Boy 56 \\Film(0. III. 60091 \Vinnetka. Ill.60093 Daintee Tos, I ihus. Ito. Echo at ional Plathings Colonial 65 Toledo St. 752 Spring St.. N\\' 1706 Ilaes \xe. E. Farmingdale. N.. 11735 .\ t Lntta. Ga. 30308 Sandusky. Ohio -11870 Dana and Co. Inc. Coral ass Insittnnent Filticational Piojections. Box 201 and Optical Co.. inc. 527 S. (:onttnetce St. Barrington. R.I. 02806 101 1. 25th St. Jackson. Miss. 39201 The John Day Co.. Inc. New Volk. N.1'. 10010 Educational lteading .\ ids Corp. 62 W. 511t St, Conispa« p. Glen Cute and \"oice lids. New Volk. N.Y. 10036 Carle Pla«.. N.1'.11514 350 (:t eat Neck Rd. Donn Pless Farmingdale. I.,I.. N.1'. 11735 Educational Supply Co.. Ito. 3026 S.W. 10th St. 2823-25 E. Gage Axe. Cooper Biotite), Co. Des Moines. Iowa 50321 29 New Volk Ave, I luntington Park. Calif. 90255 Dem )yer-Geppet tCo. Educational Teaching .\ ids \Vestbitt). N.Y. 11590 5235 Raxenswood Ave. (:oopetatixe I:cue:idol, Service .\. Daigger and Co. Chicago. III.60610 159 E. Kituie St. Itadnor ltd. Design Center. Inc. Dvlattaie. Ohio 13015 Chic ago. III.60610 Pla \all. Inc. Edultaid of Ridgewood Corbett Blac kboaid Stein ils Eugene Dieugen Co. 1250 E. Ridgewood .\ ve. r) IS :lid\ve. 2,125 N. Sheffield Ridgewood, N.J. 07.150 N. Pelham. N.Y. Chicago, III,6061i El«) Cure. Colonel Institutional Films Italph D. Dotter NIontgomtyville. Pa.18936 65 K, South Water St. Mathematical Iletbert M. Elkins Co, Chic ago. III.(10601 2 :0)5 Gill .St.. SE 10031 (:ontinerce Ave. I1 untsville. Ala.35801 Gouge E. (:t am Co, Tujunga. Calif.910.12 301 S. 1.a Salle St. Doubleday and Co., Inc. Encc lopaedia Britannic a Indianapolis, Incl. 16206 School and Lihnny Div. Educational Cot porn ion Cittloid Engineering Equipment ;at den City. 1..1.. N.Y. 11530 125 N. .\lichigan .\ve. 1200 Columbia Rd. Dover Publications. Inc. Chicago, III.60611 Joanna. S.C. 29351 180 Varick St. Engineering Instruments, Inc. New York. N.Y.1001.1 Creative Playthings 16,1 laiigatis ltd. Edinburg ltd. DuKane Corp. Peru. Ind. CranIntiv. N.J. 08512 E.S.1t.I tic aeative Publications 103 N.I hit Ave. 350 St. P.O. Box 328 St. Charles. III. 60174 Otange, N.J. 07050 Palo ,\ Ito. Calif. 9 1302 Ut naSlide Co. Exchishe Plat ing Card Co. Thontas V. Clowell Co. See 5(ieme Related \Iatetials 711 S. Dealluon St. 201 Palk Ave. South (31 is ago. III.60605 New Vol k. N.1'.10003 Exton.,\ids The (:.Thrti Ruler ( :o. Eastman Kodak Co. Dover Plains. N.Y. 12522 6 Britton1)1., 3 13 State St. Exe Gate House. Inc, Bloomfield. Conn. 06002 ROChester, N.Y. 14 650 1.16.01 Archer .Ave. (:ttisenaire Co, of \merica. Inc. Fa kel and Ballind Jamaica.N.Y.11.135 12 (:hutch St. 25 Elmwood Ave. New Rochelle. N.Y. 10805 1,a Grange Palk. III. 60525 F (:tit tic ultunlaterials Corp. Edmund Scientific Co. E:dui:Eck Inc. 1319 Vine St. 100 Edswrp Bldg. 5901 S, County Rd. 18 Philadelphia. Pa.19107 Barrington. NJ, 08007 Minneapolis, Minn. 55436 438

hild C.( hilts StippIN Co. 1 It 111) 1g 1'111(1 pl Is) s and histitiment (:orp. Nlilmatik, \\Is. 33200 8 1271 am wide 75 Mall Dr. Geodesli\ \kokw. III,1)0076 ( k. N.Y.1 1725 P.O. Bo \ 3179 I it11).nli Itadetnati. 111(. 1..ininhar ianspalent Globes spokan. Wash. 99201 1011. hic \(. 1007 \\',u ringlon\t e. Geer Instructional \ids Co. Philadclphia, l'a,P1131 Philadelphia. 19113 1229 Ma\ine Dr, Ilcid( I and 1 !rider Feaion Publishets \ \'.n tie. Ind.16807 232 \ladison \. 216) Paik Blvd. Ginn and (.o. \lp lotk, N1, 10016 Palo \Ito. Calif. 91306 450 \\'.\Igon(piin ltd. (gint(.11 1(1 1 111.11111 \ssociates of (:alilotnia \rlington Heights, Ill,(i0001 I 101e,( ismar 11)59 Santa NIoni( a Blvd. Gould Scienlifn \Vcg I os Migeles. (:alil. 90025 Bo \ 671;3 Gelman\ 111110.11'4) 'louse Washington, 1),C,201)2(1 11( stet and \..o(

132 Paik \N. South 11 122 Ilan tI lines Blvd. New YInk, N.Y.10016 3750 \loitio ANe, P.( ). Bo\ 20812 Fold .\lotor Co. It( )( hcsteiN.Y. 11603 Dallas.Iles:.7522(1 Fitgineering Stall GlaphiCiaft lloh, Rinehait .111(1 \\Ingot]. 111). l'he \inerhan It(1. P.O. Bo\ 509 383 Madison \se. Uc.ti hot u, Ali(1]. 18121 Westpm I. Conn. 06880 Nett lock. N.Y,10017 rot marli] (sgegoIN Magnetic Indusiries, 111(. Iloughlon Mifflin Co. lice Jolt Plasti(s (:o. 317 SAY, 5th St. 13(1 St. Fortune Games Pompano Beach. Fla.33(160 Nett Yolk. N.Y. 11(036 1517 Leer St. Whi- j.( :tide. 31(1 !bones well Photographic Tex. 75007 815 Hudley St. 5301 S. 1;141:111w.l anklin Publications. Inc. Lakewood. Colo. 80215 I illleton. (:olo. 80120 o Vioniatt's G. W. Sc pool Supph Specialists Iltibbaid Scientific Co. 367 S. Pasadena Ave. P.O. Bo\ I.1 P.O, lioN 103 Pasadena. Calif.91 105 hesno. Calif. 93707 \ollhblook. III.60062 \V.I I. rieeinan and Co. 1 indson Ihodul Is 660 Mat ket St. Pinehmst Sta. Sail hancisco, Cali',I,) 1101 hetet!. \\*ash.982(11 I Tans heyer. Inc. P.(). Box 2,15 I hill and 1( Co,. \Vestwood. lass.02090 (:hit ago. III. 60600 Friden Div. j.I.. Hammett Co. Ideal St pool Supply Co. 'Ilse Singer Co. Ilahhhett Pl. 11000 S. Laveigne AVe. 2350 \Vashington Ave. Brainnee. Mass. 02151 San Leancho. Calif. 94577 ();11.1.1W11. III.(10'133 Jam Handy Oiganiiation Knout l'ullettonl Sales Co. 2821 E. Gland Blvd. 706 \\Illiamson Bldg. 811 St, Detioit. Mid]. .18211 ClcN eland. ()hi() ,1.111.1 Glendale. (:alif.91203 Ilanotnt Bra(e Jovanovich. km. IndianaI Ilk et sits' 757 3(1 ANT, Midi() Visual (:enter New York. N.Y. 1(1017 Bloomington. Ind. 7,101 G.\ I: corp. I hillier and Row. Publishers Industrial and Itesem(11\ssociates 5Ist 51 E. 33d St. I lempstead New York, N.Y.10020 New Ymk. N.Y. 10016 1,eitlown. N.1'. oO Industries. Inc. Miles C.1 lartley Institutional\ids. Inc. Box 310 666 Geneva l'1. P.O. Box 191 Rig Spring. Tex. 7972(1 Tampa. Fla. 33606 Mankato, Minn, 56001 The GanglerGefin y Co. I hiwald histru(tional ,Nlatrials Co. 113 Bil(hood Rd. 12.15 Chi(ago Ave. 1.103.1 l'inetoc k Baltimow. Md, 21228 Evanston. III.60202 Ilousion. Tex. 7702.1 (*.amid Press Ilaves St hoot Publishing Co.. Inc. The I ',Still( to Col P. 123 \V, Park Ave. 321 Penwood ANT. Cedar flollom. and NIaltliews Rd. Champaign. III.61820 WilkinsbIng, Pa,15221 Paoli. Pa.19301 GelSten Supply Co.. Inc. .Karl Heil/. Inc. The Instructor Publications, Inc. 911-13 S. 11111 St. 979 3(1 Ave. 7 Bank Si. Los Angeles, Calif. 90015 New York. N.Y.10022 1)aosille.. N.1'. 14.137 P 1: 0 DC (11t \ \I) D1511:1111I Olt 1\,1 RC(I ION \I\II), 439

1111(110.d lte)entic ice Rohm.' Poodic.1., \ lank o ,c 110(41 Suppl% Co. C.s.I le.a.itt% I 1%11(' (0.1111('111%. ( .0000. \ 02021 I caching1.1\es( 0/whit:nor P.O. 110\ 2_9I \1.011 \ ide 1).(.20229 I.asi Paulson. N.J. 0711)7 Si nhlidge 0111)( enici Intent.' iional him But eau. Inc. 1\iact; sq in In idgc. \ 01566 332 \lichigati \e. 59SS\ I anc 11.let.\ve. Compam (.hicago, III, 60001 St.I (mi.. Mu. 51117 336 Kit \e. lo)%a State Citiveisity 1.1%1)144 Coq). s)%ractIse. N.V. 13205 Butcati of \nclioVi.tial Instruction 2 Pine st. \lath:m(01i(of Pentis%hania Inc, lcma (:ii%.14owa 122 10 San Fiatici.(0. Calil. 9 1111 211 Cle.«.tit St. \Valdiant. \Ia.:, 1)2151 La( I.011 i'11)(illl Is. Inc. \ 1.101(111.0i( al PieI ea( luing\ 31 CI111.11 Aye. Iactoncla Manufacturing Co. and Conoluction 1\its Ne(%.111.. N.J. 07102 Huffier 7C0 man !toad I .111(41 Shirker. Philadelphia. Pa. 19131 \ Vat :tic k.hite 10\ $01 England (ell%\ rithmelic Games CICA. Calif. 95685 2900 71st\ve. \lalei Lab. lame. X. Lang P.O. Boy 1911 Fla. 33155 P.O. Rox 221 Ion Plastics Co. Bit; Spring. rex. 79720 Mound. Minn. 55361 N1.011 Media Ito\ 3006 Lino (.o. 101 1.1111e. Cali!. 90500 II K.\I\smuiatcs 17 11\V. Liberly ,Nlaigatel Joseph P.O. lion, 1107 \tin \limn -18103 1)alibiti. Conn. 06810 1501 N. hope(' Ave. LaPine Scientific Co. Nlik%aukee. \i.. 53200 ,Nlaili.C-Nlatic. 0001 S. Litox\ ye. Co. 31)17 N. Stile The Jml 60629 ( :hie ago,I II. Oklahoma Citv, Okla. 73105 310 N. 2d St. lw Learning Center \litmeapolis. 55,101 Flententai Dept. .\ ( :1 e ojectot s.Inc. 1215 \a.hingion ,\ ye. l't in) eion. N.J.08510 . Lei /. Inc. \Vilmene. III.60091 Ic(.raw.1 lill Book Co. Kalali Game CO. Ito( kleigh. N.. 07617 V. Lieu Co. Educational Games and Aids 27 Maple \ 330 \V..120 St. 11441)10A. Mass. 023.13 1615 Del Ain) Blvd. New York. N.V.10036 The 1\alait Co.. In(. Cat son. Calif. 907.16 Link Red School !louse lediaGraphics, 11 tiltenith St. 119 IS Iiiiiietotika Blvd. (toint. 06062 38 Main .St. linnetonka.linn.55313 The Kenchey Nlantilacturing Co. .\latiasquan. N.J. 08736 Meredith (:o1p. 207 2d Ave. S. Lowe Co..Inc. 27 \V. 25111 St. Education Div. P.O. Box 629 Ho park ye. San \15044, (:alif. 9.1101 New Yolk. N.Y. 1001(1 1:eutter Plochuts Co. Lufkin New Volk. N.Y. 10016 Ohio 15202 Ilk. of Cooper Industries ,Meridian Co. 1:etnuortIn Educational Seri«. P.O. Boy 728 1976 limes P.O. Rox 3031 \ pex. N.C. 27502 San Fiamisco. Calif. 9112,1 Buffalo. N.V.11205 I.-\V Plm). Inc. Midwest Publications, Inc. kettflel and Es.er Co. 15.153 Cabrito ltd. P.(). Box 3(17 500 (:ent)al Ave. \'an Nus, Calif.91.106 Birmingham. Mich. 98012 Northfield. 111.60093 NI litniesota and 1:evsi111e View Co. NI3( ale.ter Bicknell Co. latittlacittring Company ,Nleadville. Pa.16335 181 Henry St, 3NI Center Miles Kimball Co, New Ilaven. Own. 06507 Si. Paul. ,Minn. 55101 111 \. 5th \ve. ,Nlacalester Scientific Co. Models of Indust' Inc. Oshkosh, xi.4.51901 Div. of Itatheon Education CO. 2100 5111.Si. Kindle Matualacturing Co. Route 11 1and Everett Tpk. Berkeley. Calif.9 1710 P.O. Box 11606 Nashua. N.11,03060 Modern Lcat nitu Aids Palo\ Ito. Calif, 91306 The Macmillan Co. Div. of ,Modet it Talking Heinle Knowledge Builders S( pool Dept. Service, Inc. 31 1)(4 144(4 Sq,\rest 866 3d Ave. 1212 .\ venue of the Americas NeW Yolk, N.Y. 10003 New Volk. N.V. 10022 New fork. N.V. 10036 440 \ Pl'EN DIN

\Ione\ 1.111agettient Institute 11.1),1. ptibithittv., (44. I 14 outwhold Fittatue Corp. Opt is1)111.1N (>1)11). ktulio isnal Prudential Plaia \ i11111g01111'INdie Industrial 355 la.xingunt\Ne. (.11i(ago,III.(i11601 Center NeVIII L. N.Y. 10017 \loin:towel) \Vald Co. NIontgomet 1S9311 tedetilk Post (o. 1100 l'itiNet sit \Ne. 0"" I'nblishin14 (o. 333 SibleN St. St. patti. mititt. :1510I \ ilk. N.Y. 11137 St. Paid. Nli1111.11101 .10(1I \1(0.'e 0\lool Piers Pottet's Photographi( 112 \\. Lake Rd. Ili 1th\Nr. \ppluattons (.o. Penn Van. N.'.1.1527 >.(V l'ok. N.Y. 10016 150 Iletti(ks Rd. ,NloN i(7N1 it r Cot p. Oialid Mineola. N.Y. 11,101 1001 I. iellehou\ ye. Genetal\ nilinemo hih11 (:01p. 1 he (Jim les T. Powner Co. Dettoit. Mi( It.18207 1111111s1)11 (.11N N.Y. 13790 Roy 796 .\loNer (:hi(ago. III.60(190 IndlisttkN Plena« 1 lall 25 NlilNan Dr, him album' Book I)iv. \ston. (Mt. Pa( ill(Coast Publishers Fttglewood Clills. N.J. 07632 Canada 1085 Campbell \e. Ptindle. \Veber .111(1 Sdtmidt. NItindtts Co. lettlo Palk. Calif. 91025 13 State St. P.O. Roy 3716 Paillatd. Roston. Nlass 02109 Ithetdale. NI11, '20810 1900I of er ltd. Iho(hut, ol the BeltaNioral I inden. N.J. 07036 S( blues PallteNs hitt( atbmal 1110 I)(.11\N(.. N- 7715 E. Cal VCV .\ve. (:ampbell. Calib 95008 Itosentead. Calif. 91770 hoje( don ()tubs Co.. Pantheon Books. Ili)'. 271 Ildt\Ne. Nam° I. (bange. N.J. 07018 901 lanes\ ills Ave. 22 F. 51st St. New York, N.Y. 10022 1'sN1 boll ogi( at Sri \I' Ft..\tkinson. 1Vis. 53538 Paxton Equipment and Supply 1102 S1.11110111 St. National of "feathers of 7.101 S. Pulaski Rd. Chase, Mat land 21027 Nlathentaths Chi( ago. III. 60629 120116th St.. NA. Iles Corp. Pentagon ,111:11t tun Cot p. IVashingt(1.1).(:. 20036 21 Harriet Dr. National Film Bo:n(1 of Catta(h) 1885 lats11:111\Ne. SNOsset. N.Y. 11791 St. Paul. Minn. 5511)1 P.O. Roy 6100 Philogiaph Publications lad. Montt eal 3. Que. Fulham I ligh Sued 69-79 ---R-- (anada London. S.W. 6 RaNtheon Learning S)stents Co. National tionalhiltus England r.S. I lighwav 12 Fast 99 S. Nlain St. Mi(higan City. 111(1..16360 Ph\ si( sReseal( 11Laboratories Sluing Valley. N.1'. 10977 P.O. gox 555 R.C. \. National S(hool StippIN and Commet( ia 1Ely( ttoni( I lempstead. L.L. N.1'.11551 1'9(6'1)111Na S s11'11111)i,. Pi( kenI ndustriei 79 \. Monroe St. 111(Ig, 15.5 136 E. (:tuierrez (;Iii( ago. III. 60603 Flout and Cooper Sts. Santa Batbata. Calif. 93102 Camden. N.J. 08102 NiltN 1)iv. 1 he Plawa) Gaines St. Regis Paper Co. C. N. Mclae I:ene( al Ptodiuts. I Ile. 330 Pinson Valley Pk(v) 17nadilla, N.1'. 13171 )1(1 (:otintry Rd. Itioningliant.\la. 35217 Plaxall. ineola. New Volk 11501 Notthern Signal Co. it 26 16th \Ne. Responsive Enviumments Corp. Sauk it le. \ 5:080 1.011g Island City. N.1'. Learning Materials I)iv. NON alum. Inc. Playskool Nlantibuturing Co. 20(1 Syk a ,\ve. 253 State Si. 3720 N. 1:ediie ANC. Englewood (Aills. N.J. 07632 Itolatape. St. Paull. Nlinn. 55107 (:hi(ago. III.6061$ 1301 ()lymph' RIvd. Number Films Plenum Publishing Col p. Santa lotti(a. California 90100 17350 Gresham St. 227 \\. 17th St. Northridge. (:alif. 91324 New York. N.Y.10011 \. NNstiont and Co. The Phinouth Press St. Paul Rook and Stationery Co. 3333 Elston .\ 1232 \\'.79111 St. 1233 \kr. County Rd, "F.," (ilk:Igo. III. 60618 (:hicago. III.60620 St. Paul. Minn. 55112 1.101)L'Cl.1:S \NI) Dls num. roR, (II. s 1 RI (.110N \I\ID> 441

SI Ik( Taper (-""111)."" solo I ti'rtliu.ttional Equipment stmulaid l'oplitti mid di%1)1%. ts1.1 1111. I quipmni 0. 3.300 Pinson Valle% Pkm. Midge sq. 1911 Piamitl, \ Ia. 31217 Ncm ton. \lass. 02195 (.It it' it m. III.60025 1111 I s. ',Linen Co. s,alittclu'\\ uldt 'u)(11011( s/ II1 cm Bing s 7300 N. Linder \%e. 1102 '. filth St. 1 161 ( Intent street skokit.III.60076 I int olit. Nebr. 68106 \thol. \Liss. 01331 s I\s Inst:attion.11 \Lncli.il.. 1111 lit\ Lite.,III(. st ix I«. and 'moles 122. 8111 I 1)6111 St. Distiibuons Ltd. .1111. 91710 Blom_ N.V.10171 9 \lal.tir iNlews sic( Is-Vaughn Co. 1 monto 159.( ',th,nht Rubber P.O. p.0\ 2025 (..nisida \Ltnul,ututil g Co. \milli.I es. 75767 238 Polk st. I.I L slieldon hquipimin Stlling Plastits Co. Ilitinington. Ind.16750 .Nlitskrgt on. NI i(11,19113 Slit Ilield S/hool (.o. sift,( 0111p,im \lotint.tinside. N. I.07092 15(11 s, \ li(higan\ 531 N 7th St. Sipes Publishing Co. Chi( ago.III.60616 \ hin. 51111 lo ( liester St. St hoof Poulin is Co. Int. sigma I'utrlltt ises, Ch.iiiipign. III.61820 12I. 2:41 St. Bos 15185 summit Indusiiies Neat k. N.V. 10010 Bo\ 115 Dem er.( olo.50211 Palk. III.(1003(1 St hold Sriit r Co, shalom and St limier. Int. 017 S. La Itira \%e. liomald swim (Amiyany. I V. 3911i St. Lot,\ngeles. Cal a. 90036 17 XV.iiten St, Nem loll,. N.Y. 1001 S I hr hu Co. Neat lolls. N.Y. 10007 lie I.. W. Singer Co.. 111t. I Eineison PI, Sstenis lor Fdiu.ition. Inc. Bosion. Mass. 02111 201 P. 50th St. 612 N. Nli(higati \% Nm York. N.Y. 10022 Chit ago. III,60611 ien«. Poulin( tions skool, \ids Coip. Ros 113 5702 Bessemer ( ) ontomom II Vie. 5 :1066 (;Ire Claud. ()Iiio 1.1127 NIalerials I.d.C.q) P.O. Bo\ 1009 81111111 (s.m4m.1 .N111(11.1111 531 N. 7111 410 Palk \%e. hanston. III.602(11 55111 Nmv yolk. N.Y. 10022 sdpe IZA.se.11,11\ ssm iates. "readier\ \ 259 F. Erie St. Smith Ssiciti Co. 168:1 South 7(1(1 Vest \'ood, Ptah 81087 Chit ago,III.(10611 56 S.F. Futrrald mini). 55111 heat lei. Publishing Cm potation Stien«. Seininais. Ile, (Publisheis of Giade Teathei P.( ). hoe 1019 Sot iet% for Visual Education, Int. 1315 I)i%eisey Pot land. (beg. 97213 2 :1 Lerm A% e. (3ii(ago. III.60611 Stien«.s NI:m.6:11s (:enter Daiien. Conn. 06820 59 4th\ ve. Soma Blot ks Teat hing \ ids Nem. York. N.Y.10003 Parker Poo. 1m. 159 W. I:iniir St. St imili( Edutatioual P.O. 1105 900 Chit ago. III.60610 Piodiutsoip. Salem. (1197(1 hrth11itolor Int. :10 E. 12d St. Spee(1.1'1) Geometly Ruler Co. (.onittiei( i.tl and 1.dinational 1)1%. N'em- Vol k. N.Y. 10017 1()shortie \ve. 299 1:alintis 1)r. Stoll. l'oiesinati and Co. Nlatyland 21228 Costa Mesa. (:alit.92627 190(1 E. Lake Ave. Spindler and Sauppe Te(iiilas p. Gleminv. III.6(1025 1329 Grand Central \%e. 195\ ppleton St. Ilol%oke. Nlass,(11010 S(iitddield iNlatinfacturing Co. (dendale. (:,ilif.91201 31) 1:ignaCiapli Corp: 1783 1 oga(i BoN 261 San I)iego. ('alif. 92113 P.O. Box 20158 Park Ridge. III. 60068 Seals. loebiuk and Co. 108071 laily !lines BI d. Tomb. Int. Consumer lamination Service:. 1)allas. "I"t'. 75220 P.(). lion 1711 1)ept. 703 Stand:n(1 EMI( :Ivo.' Society, Inc :Many, N.Y. 12201 7.101 Skokie Blvd. 130 N. Wells St. Fein Tripp Skokie. III.6(107(1 Chit ago. III.60606 I)imtha. Calif. 93618 442 \I IA 01

1 IZLTIN he ViN.X co. Wesicin Pnblishing 1.(Imation Di(. of 1:(1(11 Coq). 1'.0 Ilox 11107 122(1 \hound e. 6111 W. Mequon 1(1, \ tigcles. Cali!. 90011 It.o Inc, Wisconsin 13101 Mequon. Wis. 53092 WE *N I'ROOF I IT See \.11on Ilill ( o.1 lloi\ 71 Charles E. Tuttle Co.. Inc. NV- New 1:.1%(11. C111111. 06501 28 S. Main Si. IZtitland. Vt. 05701 Wabash Instrument Coip, P.O. 110N 2012 I weedy Transparencies 80)1 Manchester Ave. ,d010. City. low.) 51101 W.11).1.11. Ind.16992 20$ Hollywood .\ve. 10111) .111(1 Sons. 116. I. Orange. N.J. 0701$ I. Weston Walch Publisher 605 3d ..\. Box 1(175. Nlain Post ()flue New Yotk. N.1'.10016 Pottland. Nlain 01101 II, Wilson Cow. r \\*Aker Educational Book Cot p. 555 \V. Taft Dr. rnited Chemical and School Suppb Co. 720 5th \ve. S, I loll:m(1. III. 60173 2115 Como \,.. SE New York. N.Y. 10019 \VwhI \Vide Gaines Nlinnapolis. Minn. 55.111 Walker PtOducts Ohio 3015 rtikisal Education 1530 (:aniptis Dr. I,. NI. Might Co. and Visual Sits Berkelev. (:alit. 91708 686 E. Nlaiiposa St. 221 Park Ave. South Wang Laboratories. Inc. Sliadena. Calif. 9101)1 New York, N.Y., 1000:1 $36 Nolth St. Wyk I .aboratol ics Tewksbury. NliiSl. 01876 128 NIalyland St. Weber Costello Co. El Segundo. Calif, 90215 \Pi( tot -\l'in"grkPll 1900 N. Nairaganseti 1)k of Kalart \9( tor (:(n p. Chic ago. III. 60639 I Itilienins St. Webster Div. Plainville, Conn. 06062 NI(Graw.Hill Book conip Xerox Corporation vit,,dex. 110. Curriculum hogranis Nlan( hester lid. IBroadway See. 600 Madison , \ Ye. Nlanchester. Nlo. 63011 I lolbrook. N.Y., 11711 New York. N.Y. 10022 l'iking Co. Webster Paper and Supply Co. Slbany. N.Y.12201 113 S. Edgeniont St, -Y- Los .5ngeles. Calif. 90001 Weems. Platte. Vision 111(0E1)ov:tied 18 Maryland .\ve. Yoder Instruments 1501 N. Ditimin Ave. \tinapolis. \I(1. 21.101 E. Palestine. Ohio 1.1.113 1.0s \ilgeIes.C.alil. 90063 Welch Scientific Co. \ Ipiodtios See SaigentWelch Scientific Co. -z- Nlinnesota .'lining and Western Publishing Co. Inc. Manta:Rim ing Co, S(160)I 1.ibraly Dept. 'Lima Educational Nlatelials 3N1 (:enter 850 3d Ave. 1715 (11e.stnut St. Si. P:1111. NIinn.55101 New York. N.Y. )022 Philadelphia. Pa.19103