Reshaping College Mathematics

A project of the Committee on the Undergraduate Program in Mathematics

Edited by Lynn Arthur Steen MAA Notes Series

The MAA Notes Series, started in 1982, addresses a broad range of topics and themes of interest to all who are involved with undergraduate mathematics. The volumes in this series are readable, informative, and useful, and help the mathematical community keep up with developments of importance to mathematics. Editorial Board Warren Page, Chair Donald W. Bushaw Richard K. Guy Paul J. Campbell Frederick Hoffman Jane M. Day David A. Smith

1. Problem Solving in the Mathematics Curriculum, Committee on the Undergraduate Teaching of Mathematics, Alan Schoenfeld, Editor. 2. Recommendations on the Mathematical Preparation of Teachers, CUPM Panel on Teacher Training. 3. Undergraduate Mathematics Education in the People’s Republic of China, Lynn A. Steen, Editor. 4. Notes on Primality Testing and Factoring, by Carl Pomerance. 5. American Perspectives on the Fifth international Congress in Mathematical Education, Warren Page, Editor: 6. Toward a Lean and Lively Calculus, Ronald Douglas, Editor: 7. Undergraduate Programs in the Mathematical and Computer Sciences: 1985-86, D. J. Albers, R. D.Anderson, D. 0. Loftsgaarden, Editors. 8. Calculus for a New Century, Lynn A. Steen, Editor. 9. Computers and Mathematics: The Use of Computers in Undergraduate instruction, D. A. Smith, G. J. Porter, L. C. Leinbach, R. H. Wenger, Editors. 10. Guidelines for the Continuing Mathematical Education of Teachers, Committee on the Mathematical Education of Teachers. 1 1. Keys to Improved instruction by Teaching Assistants and Part-Time Instructors, Committee on Teaching Assistants and Part-Time Instructors, Bettye Anne Case, Editor. 12. The Use of Calculators in the Standardized Testing of Mathematics, John Kenel/% Editor: 13. Reshaping College Mathematics, Lynn A. Steen, Editor:

First Printing 0 1989 by the Mathematical Association of America Library of Congress Number 89-061338 ISBN-0-88385-062-1 Printed in the United States of America Reshaping College Mathematics

.. PREFACE...... vu MATHEMATICALSCIENCES ...... 1 Curriculum Background 0 Current Issues 0 Curricular Principles 0 Building Mathematical Maturity 0 Core Requirements 0 Teaching Mathematical Reasoning 0 How Much Theory? Sample Majors 0 Mathematical Sciences Minor 0 Examples of Successful Programs 0 IDe- partmental Self-study 0 Discrete Methods 0 Applied Algebra 0 Numerical Analysis. CALCULUS...... 19 Rationale 0 First Semester Calculus 0 Second Semester Calculus 0 Intermediate Mathematics Courses 0 Applied Linear Algebra 0 Multivariable Calculus 0 Differential Equations. COREMATHEMATICS ...... 25 New Roles for Core Mathematics 0 Four Questions 0 Abstract Algebra 0 Analysis. COMPUTERSCIENCE ...... 29 A Growing Discipline 0 Introductory Courses 0 Intermediate Courses 0 Concentrations and Minors 0 Faculty Training 0 Computer Facilities 0 ACM Curriculum 78. MODELINGAND OPERATIONS RESEARCH...... 41 Experience in Applications 0 Mathematical Modeling 0 An Undergraduate Modeling Course 0 Operations Research 0 Introductory Operations Research 0 Elementary Modeling 0 Intro- ductory Stochastic Processes 0 References. STATISTICS...... 55 Introductory Course 0 The Place of Probability 0 Alternative Arrangements 0 Instructor Preparation 0 Course Outline 0 Probability and Statistical Theory 0 Applied Statistics 0 Probability and Stochastic Processes 0 Preparation for Graduate Study. DISCRETEMATHEMATICS ...... 61 Introduction and History 0 Summary of Recommendations 0 General Discussion Needs of Computer Science 0 Syllabus 0 Preparation for Discrete Mathematics 0 Two Year Colleges and High Schools 0 The Impact on Calculus 0 Course Objectives 0 Sample Problems 0 Textbooks 0 References. CALCULUS TRANSITION:FROM HIGH SCHOOL TO COLLEGE ...... 85 Problem Areas 0 Accelerated Programs 0 High School Calculus 0 Successful Calculus Courses 0 Unsuccessful Calculus Courses 0 College Programs 0 Recommendations 0 References. CURRICULUM FOR GRADES11-13 ...... 91 Issues of Apparent Consensus 0 Recommendation 0 Issues Requiring Further Study 0 Gifted Students 0 Greater Integration 0 Statistics & Discrete Mathematics 0 Twelfth Grade Math- ematics 0 Calculus Review 0 Deductive Reasoning 0 Calculators 0 Alternatives to Remedial Courses 0 Use of Standardized Test Scores 0 Geometry 0 Collaborative Efforts. MINIMALMATHEMATICAL COMPETENCIES FOR COLLEGE GRADUATES...... lo3 Recommendations 0 Mathematics for Coping with Life 0 Mathematics Appreciation. MATHEMATICSAPPRECIATION COURSES ...... 109 Philosophy 0 Things to Stress 0 Things to Avoid 0 Course Organization 0 Examples of Topics 0 Two-Year Colleges 0 Films 0 Classroom Aids 0 Survey Monographs 0 Collections of Essays 0 Bibliography & References.

Preface

Calls for Change Background CUPM was established in 1953 to “modernize and For over 35 years the Committee on the Undergrad- upgrade” the mathematics curriculum arnd to halt what uate Program in Mathematics (CUPM) has helped pro- was even then decried as “the pessimislk retreat to re- vide coherence to the mathematics major by monitor- medial mathematics.” At that time total enrollment in ing practice, advocating goals, and suggesting model college mathematics courses in the United States was curricula. This volume brings together various curricu- approximately 800,000; each year about 4,000 students lum reports issued during the decade of the 1980’s. received a bachelor’s degree in mathematics, and about It provides a convenient reference for the mathe- 200 received Ph.D. degrees. matical community as it begins to reshape college In its early years CUPM concentrated on proposals mathematics in response to mounting demands for to strengthen undergraduate preparation for graduate change. study in mathematics. Spurred on by Sputnik and as- Many of the calls for reform have been expressed in sisted by significant support from the National Science published reports, for example: Foundation, the mathematical community in the United 1989 Curriculum and Evaluation Standards for School States matured rapidly from a servant discipline inden- Mathematics, published by the National Council tured to science and engineering to vigorous world lead- of Teachers of Mathematics. ership. By 1970 U.S. mathematics dlepartments produced 24,000 bachelor’s and 1,200 doctoral degrees-a 1989 Everybody Counts: A Report to the Nation on the Future of Mathematics Education, published six-fold increase in less than twenty years. by the National Research Council. Then the bubble burst. As student, interest shifted from personal goals to financial security, and as com- 1988 Changing America: The New Face of Science puter science began to attract increasing numbers of and Engineering, published by the White House students who in earlier years might have studied math- Task Force on Women, Minorities, and the ematics, the numbers of mathematics bachelor’s degrees Handicapped in Science and Technology. dropped by over 50% in ten years, as did the number 1987 The Underachieving Curriculum, published by of U.S. students who went on to a Ph.D. in mathemat- the Second International Mathematics Study. ics. In 1981, at the nadir of B.A. productivity, CUPM published a comprehensive report entitled Recommen- 1986 Towards a Lean and Lively Calculus, published dations for a General Mathematical Sciences Program. by the Mathematical Association of America. Significantly, this report advocated not a strengthened 1985 Integrity in the College Curriculum: A Report program in traditional (pre-doctoral) mathematics, but to the Academic Community, published by the a broad, innovative program in mathematical sciences. Association of American Colleges. Although CUPM did not create the movement towards mathematical sciences, its 1981 report helped le- 1984 Renewing US. Mathematics: Critical Resource gitimize a process that was well under way. As a con- for the Future, published by the National Re- sequence, mathematics programs in 1J.S. colleges and search Council. universities are now dominated by variations on two Other pressures for change are expressed in articles paradigms that reflect the two phases of CUPM activ- diffused throughout the literature on a wide range of is- ity. The first, a fading image of the CUPM recommen- sues, from remediation (too much) through Ph.D. pro- dations of the 1960’~~focuses on core mathematics as duction (too little), from students (greater diversity) to preparation for graduate study in mathematics. The mathematics (greater applicability), and from technol- second, reflecting the broader objectives of CUPM’s ogy (under-utilized) to pedagogy (too passive). What 1981 mathematical sciences report, focuses on math- all reports have in common is the case they make for sig- ematical tools needed for a “life-long series of different nificant change in undergraduate mathematics to serve jobs.” Most campuses support a mixed model repre- better the needs of students who will live and work in senting a locally devised compromise between these two the twenty-first century. standards. VIII RESHAPINGCOLLEGE MATHEMATICS

Issues of the 1980's volume-which were generally written three to ten years ago-align remarkably well with contemporary calls for During the 1980's several issues emerged that had change that one hears at every professional meeting. great bearing on the conduct of undergraduate mathe- Today's advice, by and large, is no different than yes- matics. Pressure from the computer science community terday's. It is just being said with greater urgency. created a demand for a freshman or sophomore course Here, for example, is a sample of recommendations in discrete mathematics. This posed issues of definition to be found among the reports in this volume: (what was to be included?), level (how much maturity was required?), and articulation (where could it fit into the ubiquitous calculus sequence?). No single answer ON GOALS: emerged, and experiments continue to determine locally A mathematical sciences major should develop a optimal strategies for meeting this important new need. student's capacity to undertake intellectually de- As American society moved towards greater concern manding mathematical tasks. with material well-being, pressure from many sources- A major in mathematical sciences should emphasize not the least being from parents and schools in affluent general mathematical reasoning as much as mastery districts-created enormous demand for high school cal- of various subject matter. culus. Suddenly large numbers of students came to col- The instructor's central goal should be to teach stu- lege with uneven preparation that partially overlapped dents how to learn mathematics, expecting that stu- standard introductory college courses. Problems of ar- dents will correctly retain only a tiny portion of ticulation between high school leaving and college enter- what was taught. ing became quite intense. The new Standards for school A mathematical sciences curriculum should be de- mathematics of NCTM promise to increase the diversity signed around the abilities and academic needs of of student preparation in years ahead, as some districts the average student, with supplementary work avail- adopt new programs and others retain old habits. able to attract and challenge more advanced stu- From a different source-mostly from parents and dents. public officials concerned with the quality of higher College students must understand the historical and education-came calls for assessment and evaluation to contemporary role of mathematics and be able to ensure that all students receive certain minimal skills place the discipline properly in the context of other from their college study. Quantitative literacy (or "nu- human intellectual achievement. meracy") joined the litany of demands generated by dis- Students should gain an ability to read and learn cussions of "cultural literacy" and "competitiveness." mathematics on their own. Such maturity is as Quantitative competence and mathematical apprecia- much a function of how mathematics is learned as tion of students who do not study mathematics for pro- what is learned. fessional reasons became-and still is-a major concern on college campuses. The Mathematical Association of America responded ON TEACHING: to each of these issues-discrete mathematics, school ar- Students should be led to discover mathematics for ticulation, quantitative literacy-by a variety of studies, themselves, rather than merely being presented with some under CUPM, some in cooperation with other or- the results of concise, polished theories. ganizations. Issued different times throughout the at The approach to most topics should involve an in- past decade, these studies supplement CUPM's com- terplay of applications, problem-solving, and theory. prehensive recommendations for an undergraduate pro- Applications should motivate theory so that theory gram with specific recommendations in areas of timely is seen by students as useful and enlightening. concern. They are all gathered in this volume, where Freshman courses in mathematics should be de- together they provide a thorough airing of issues perti- signed to appeal to as broad an audience as is aca- nent to reshaping college mathematics. demically reasonable. In the first two years, theorems should be uaed rather than proved. The place for theoretical rigor The More Things Change .. . is in later upper-level courses. Despite the many new issues that have arisen in re- The greatest challenge is that students enter college cent years (e.g., desk-top workstations, changing demo- with much less mathematics than they used to, but graphics, calculus reform), curriculum reports in this they expect to leave with much more. PREFACE IX

ON COMPUTING: ON CALCULUS: * Students should make full use of calculators and * Students should learn the content of the full four computers in all mathematics courses. years of high school mathematics before enrolling in All mathematical sciences students must be given calculus. an introduction to the basic concepts of computer Calculus in high school should be taught with the science. expectation that successful graduates would not re- * Computing assignments should be used in most peat calculus in college. mathematics courses. Colleges need to provide individualized placement for students who have studied calculus in high ON MODELING: school.

Applications and modeling should be included in a ON MINIMALEXPECTATIONS: nontrivial way in most college-level mathematical sciences courses. All college graduates should be expected to demon- All students majoring in mathematics should un- strate reasonable proficiency in the mathematical dertake some real-world mathematical modelling sciences. project. College remedial courses should not be a rehash- and certainly not an accelerated rehash-of tradi- ON WRITING: tional school courses. Students should find even remedial courses fresh, interesting, iand significant. * Explaining a mathematical result in terms of a real- world setting involves the need to communicate in a precise and lucid manner. This aspect of a math- A Context for Reform ematical scientist’s training should not be left to One must wonder, after reviewing all the arguments courses in other sciences or to on-the-job learning produced by the various committees whose reports are after graduation. contained in this volume, why so little has changed. Teachers of mathematics should employ strategies Why did it take seven years from the time that CUPM that encourage student reading, writing, and reflec- urged in 1981 that all beginning courses must be taught tion. in an effective and attractive manner for the community Students should be asked to make formal oral and to take a hard look at calculus? Why is it only now written presentations. A (non-original) paper serves rather than in 1981 that mathematicians are beginning the dual purpose of developing communication skills to realize the importance of writing assignments-both and introducing pedagogical flexibility. to learn to write and to write to learn? Momentum may be one reason. In the early 1980’s ON STATISTICS: there was very little support for educational change. The political agenda of the nation at that time was not New knowledge has rendered a course devoted solely supportive of issues in science and mathematics edu- to the theory of classical parametric procedures out cation. All educational activities at the National Sci- of date. ence Foundation were eliminated in 1980, only to be @ The traditional undergraduate course in statistical restored several years later. MAA and NCTM released theory has little contact with statistics as it is prac- separate reports (Prime 80, An Agenda for Action) into ticed and is not a suitable introduction to the sub- this thin atmosphere of uncertainty about science and ject. mathematics education. It should not be surprising to find, ten years later, that most of the problems identi- ON DISCRETEMATHEMATICS: fied in these reports are still evident today. 0 Discrete mathematics at the intellectual level of cal- Today, in contrast to 1980, many different organi- culus should be part of the standard mathematics zations are working together for the improvement of curriculum in the first two years. mathematics education. The Nationill Academy of Sci- Topics covered are less important than acquiring ences, the National Science Foundation, and many pri- mathematical maturity and skills in using abstrac- vate foundations have joined with the several mathe- tion and generalization. matical societies to work on a common plan for revi- Mathematics majors should be required to take at talizing mathematics education. NOW,after a decade of least one course in discrete mathematics. talking, everyone is finally moving in the same direction. X RESHAPINGCOLLEGE MATHEMATICS

A partial list of current activities that relate to col- in the first two years, on the mathematical sciences lege mathematics reveals clearly the breadth of current major, on service courses, and on the role of sym- support for reshaping college mathematics: bolic computer system. * The Undergraduate Curriculum Initiative at the MAA has recently published two volumes of papers National Science Foundation featured calculus re- dealing with calculus, one report on the role of com- form in its first wave of proposal solicitations. puters in undergraduate mathematics, and one re- * The Mathematical Sciences Education Board and port on the continuing mathematical education of the Board on Mathematical Sciences at the National teachers. A report on discrete mathematics is forth- Research Council have jointly established the Com- coming. mittee on Mathematical Sciences in the Year 2000 The present volume provides wisdom to support to analyze collegiate mathematics and make recom- these efforts. It reflects the best thinking of many ex- mendations for improvement. perienced mathematicians and teachers who have strug- The Division of Mathematical Sciences at the Na- gled with curricular questions facing college mathemat- tional Science Foundation is now supporting re- ics as part of their work for CUPM and other MAA search experiences for undergraduates, and is plan- committees. Although certain sections are obviously ning to add educational dimensions to many of its dated (e.g., discussions of computing, reference lists), new initiatives. the central message of this volume provides a philoso- * The MAA Committee on the Mathematical Educa- phy of instruction that is as valuable now as when it tion of Teachers is working with NCTM and with was written. the National Board for Professional Teaching Stan- We don’t need to look far for sound goals and objec- dards to revise the recommendations for the math- tives for college mathematics. Most of what we need can ematical education of teachers of mathematics in a be found in this volume. What remains to be done-as manner consistent with the new NCTM Standards. much now as ever-is to find effective means of turning Both AMS and SIAM now have committees deal- ideals into practice. ing with education, as well as liaison members on Lynn Arthur Steen, Chair CUPM. Committee on the Undergraduate The MAA Committee on the Undergraduate Pro- Program in Mathematics gram in Mathematics has subcommittees working St. Olaf College on recommendations for calculus and other courses March, 1989 Mathematical Sciences

In 1981 the Committee on the Undergraduate Pro- taught abstraction and application, emerging new prob- gram in Mathematics (CUPM) published a major report lem areas and time-tested old ones. The Panel sought entitled RECOMMENDATIONSFOR A GENERALMATH- to persuade mathematicians that the curriculum in the EMATICAL SCIENCES PROGRAM. This report comprises mathematics major should be shared among the various siz chapters that are reprinted here, with minor editing, intellectual and societal constituencies of mathematics. as the first siz chapters of the present volume. Alan The challenge was to be diverse without being superfi- Tucker, Chairman of the CUPM Panel that wrote the cial. 1981 report, has written a new Preface to introduce this The most concrete consequence of the Panel’s work reprinting. was its name, Panel on a General Mathematical Sci- ences Program. It asked that the mathematics major be renamed the mathematical sciences major-a change 1989 Preface explicitly adopted by hundreds of colleges and universities and implicitly adopted by the vast majority of In the eight years since the CUPM Recommendations institutions. The Panel recommended that first courses on a General Mathematical Science Program appeared, in most subjects should have a good dose of motivating issues in mathematics curriculum, such as calculus re- applications, particularly linear algebra and statistics, form and discrete mathematics, have become hot topics and that one advanced course should have a mathemat- in the mathematics community and have even received ical modeling project. This recommendation also seems extensive coverage in the popular press. The CUPM to have wide acceptance. There were several panel rec- Panel on a General Mathematical Sciences Program had ommendations that reflected trends already occurring the luxury of working in comparative anonymity, al- but being resisted by some mathematicians: requiring though ten panel discussions at national and regional an introductory course in computer science; not requir- mathematics meetings gave the panel some professional ing linear algebra as a prerequisite for inultivariable cal- visibility. The Panel’s basic goal was to give long-term, culus; encouraging weaker students to delay core ab- general objectives for undergraduate training in math- stract courses until the senior year; and not requiring emat ics. every mathematics major to take courses in real analysis The 1960’s and 1970’s had seen a variety of spe- and abstract algebra (i.e., other mathematics courses at cialized appeals made to college students interested in comparable levels of abstraction could1 be substituted). mathematics. For example, the discipline of computer Although it was unhappy with calcxlus, the Mathe- science emerged as an exciting career for mathematics matical Sciences Panel consciously avoided recommend- students. The earliest CUPM recommendations for the ing changes in calculus for fear that the inevitable con- mathematics major were aimed at preparing students troversy and the complexity of such an undertaking for doctoral work in mathematics. By the late 1970’~~ would undermine acceptance of its basic recommenda- there was a sense that the mathematics major had lost tions about the structure of a mathematics major. The its way, with upper-division enrollments in traditional Panel touched only lightly on the issue of discrete ver- core courses like analysis and number theory down by sus continuous mathematics, recommending exposure 60% from their levels five years earlier and with indus- to “more combinatorially-oriented mathematics associ- trial employers showing little interest in hiring mathe- ated with computer and decision sciences” (Tony Ral- matics majors. ston’s provocative essays about discrete mathematics To put these recent events in perspective, the Panel had not yet appeared). obtained a historical briefing from Bill Duren (the It was gratifying to the Mathematical Sciences Panel founding chairman of CUPM). He recounted over a cen- that its report was well-accepted: all two-thousand tury of swings of the pendulum between the theoretical copies printed have been sold (another two-thousand and the practical in American collegiate mathematics copies had been sent gratis to department heads). In re- education, and between training for careers of the fu- viewing the report for this reprinting, the only changes ture and training in classical, old-fashioned methods. have been to add a few additional references On the The Mathematical Sciences Panel sought to find other hand, there was one panel suggestion that has a common ground for the mathematics major which been ignored thus far and which merits consideration. 2 RESHAPINGCOLLEGE MATHEMATICS

It concerns the “modest” version of abstract algebra search are important subjects which should be incorpo- (in Section 111) in which time would be spent sensitiz- rated in undergraduate training in the general area of ing students to recognize how algebraic systems arise mathematics. naturally in many situations in other areas of mathe- Computer science has become such a large, multi- matics and outside mathematics (to keep algebra alive faceted field, with ties to engineering and decision sci- in their minds after they leave college). ences, that it no longer can be categorized as a math- ALANTUCKER ematical science (at the National Science Foundation, SUNY at Stony Brook computer science and mathematical sciences are dif- March, 1989 ferent research categories). A mathematical sciences major must involve coursework in computer science because of the usefulness of computing and because of computer science’s close ties to mathematics. Under- 1981 Preface graduate majors in mathematical sciences and in com- This report of the CUPM Panel on a General Mathe- puter science should complement each other. matical Sciences Program (MSP) presents recommenda- The new course recommendations presented in this tions for a mathematical sciences major. The panel has report do not, in most instances, replace past CUPM concentrated its efforts on general curricular themes and syllabi. They describe different approaches to courses; guiding pedagogical principles for a mathematical sci- for example, a one-semester combined probability and ences major. It has tried to frame its recommendations statistics course, or a multivariate calculus course within general terms that will permit a variety of implemen- out a linear algebra prerequisite. tations, tailored to the needs of individual institutions. The work of the CUPM Panel on a General Math- A prime objective of the original 1960’s CUPM cur- ematical Sciences Program was supported by a grant riculum recommendations for upper-level mathematics from the Sloan Foundation. The chairmen of CUPM courses was easing the trauma of a student’s first year of during this project, Donald Bushaw and William Lu- graduate study in mathematics. This report refocuses cas, deserve special thanks for their assistance. the upper-level courses on the traditional objectives of For information about other CUPM documents general training in mathematical reasoning and mas- and related MAA mathematics education publications, tery of mathematical tools needed for a life-long series write to: Director of Publications, The Mathematical of different jobs and continuing education. Association of America, 1529 Eighteenth Street, N.W., The MSP panel has tried to avoid highly innovative Washington, D.C. 20036. approaches to the mathematics curriculum. The em- ALANTUCKER phasis, instead, has been on using historically rooted SUNY at Stony Brook principles to organize and unify the mathematical sciences curriculum. The MSP panel believes that the primary goal of a mathematical sciences major should Panel Background be to develop rigorous mathematical reasoning. The word ‘rigorous’ is used here in the sense of ‘intellec- The CUPM Panel on a General Mathematical Sci- tually demanding’ and ‘in-depth.’ Such reasoning is ences Program (MSP) was constituted in June, 1977 at taught through a combination of problem solving and a CUPM conference in Berkeley. CUPM members de- abstract theory. Most topics should initially be devel- cided that a major re-examination of the mathematics oped with a problem-solving approach. When theory is major was needed. The CUPM model for the math- introduced, it usually should be theory for a purpose, ematics major contained in the 1965 CUPM reports theory to simplify, unify, and explain questions of inter- on Pregraduate Training in Mathematics and a Gen- est to the students. eral Curriculum in Mathematics in Colleges (revised in CUPM now believes that the undergraduate major 1972) was felt to be out of date. Following a six-month offered by a mathematics department at most Ameri- study, MSP reported to CUPM that the CUPM mathe- can colleges and universities should be called a Mathe- matics major curriculum should be substantially revised matical Sciences major. Enrollment data show that for and broadened to define a mathematical sciences major. several years less than half the courses, after calculus, in MSP was charged then with developing mathematical a typical mathematics major have been in pure math- sciences recommendations. ematics. Furthermore, applied mathematics, probabil- Five subpanels were created to develop course rec- ity and statistics, computer science, and operations re- ommendations in: MATHEMATICALSCIENCES 3

The calculus sequence, MAA meetings, at the PRIME 80 Conference, and indi- Computer science, vidually with dozens of mathematics department chair- Modeling and operations research, persons. The helpful criticisms received on these occa- Statistics, and sions played a vital role in shaping the panel’s thinking. Upper-level core mathematics. It should be noted that several people: warned that a The MSP project has had the cooperation of curriculum mathematical sciences major was unworkable because groups in the American Statistical Association, the As- of the diversity of techniques and modes of reasoning sociation for Computing Machinery, the Operations Re- in the mathematical sciences today. Others stated that search Society of America, and the Society for Industrial student course preferences had already “redefined” the and Applied Mathematics. Graduate programs in the mathematics major along the lines being proposed by subjects covered by those societies draw heavily on un- the MSP panel. dergraduate mathematics students, and except for computer science, undergraduate courses in these subjects are usually taught by mathematicians. Hence these cur- Curriculum Background riculum groups had a major interest in the design of a Many students today start mathematics in college at mathematical sciences major. a lower level and yet have specific (but uninformed) ca- The MSP panel coordinated its work with the Na- reer goals that require a broad scope of new topics of tional Research Council’s Panel on Training in Applied varying mathematical sophistication. Student changes Mathematics (chaired by P. Hilton, a member of MSP). are reflected in recent upper-level enrollment shifts and The Hilton panel had a much broader mandate than the explosion of new theory and applications in all the MSP panel. Its report addresses the unification the mathematical sciences. Uncertainties in curricu- of the mathematical sciences, the attitudes of math- lum produced by these developments have led the MSP ematicians, academic-industrial linkages, and society’s panel to look for guidance from past CUPM curricu- image of the mathematical sciences, as well as curric- lum development experiences and, farther back, from ula. The Hilton report presented a limited number the traditional goals of the mathematics major before of general curriculum principles with the expectation CUPM’s creation. No matter how great, the advances in that the MSP panel would develop fuller curriculum the past generation, the traditional intellectual objec- recommendations. The MSP panel recommendations tives of training in mathematics, defined over scores of have incorporated these principles (although the Hilton years, should be the basis of any mathematical sciences panel’s stress on differential equations has been dimin- program. ished). The MSP panel strongly endorses the Hilton Until the 19509, mathematics departments were pri- report’s emphasis on the importance within mathemat- marily service departments, teaching necessary skills to ics departments of proper attitudes towards the uses science and engineering students and teaching mathe- and users of mathematics and of a unified view that matics to most students solely for its liberal-arts role as respects the content and teaching of pure and applied a valuable intellectual training of the mind. The average mathematics equally. student majoring in mathematics at a better college in While CUPM and the Hilton panel have been rec- the 1930s took courses in trigonometry, analytic geome- ommending changes in the collegiate mathematics pro- try, and college algebra (including calciulus preparatory gram, the National Council of Teachers of Mathemat- work on series and limits) in the freshman year followed ics has been assessing priorities in school mathematics. by two years of calculus. While this program may today The 1980 NCTM booklet, An Agenda for Action, rec- seem to have unnecessarily delayed calcdus, and subse- ommends “that problem solving be the focus of school quent courses based on calculus, it did provide students mathematics in the 1980s . . .that basic skills in math- with a background that permitted calculus to be taught ematics be defined to encompass more than computa- in a more rigorous (i.e., more demanding) fashion than tional facility.” Recent nation-wide mathematics tests it is today. administered to students in several grades showed uni- The mathematics major was filled out with five or formly poor performance on questions of a problem six electives in subjects such as differential equations solving or application nature. Inevitably these mathe- (a second course), projective geometry, theory of equa- matical weaknesses will become more of a problem with tions, vector analysis, mathematics of finance, history of college students. mathematics, probability and statistics, complex anal- The tentative MSP ideas for curriculum revision were ysis, and advanced calculus. Most mathematics majors discussed by panel members at sectional and national also took a substantial amount of physics. Training of 4 RESHAPINGCOLLEGE MATHEMATICS secondary school mathematics teachers rarely included most, are trained in disciplined logical reasoning and, more than a year of calculus. In the early 19508, twenty secondarily, are versed in basic techniques and models of years later, the situation had changed only a little; top the mathematical sciences. In Warren Weaver’s words, schools did now offer modern algebra and abstract anal- these are problems of “organized complexity” as well as ysis. well-structured applied mathematics of the physical sci- In 1953, amid reports of widespread dissatisfac- ences. If mathematics departments do not train these tion with the undergraduate program, the Mathemati- quantitative problem-solvers, then departments in en- cal Association of America formed the Committee on gineering and decision sciences will. Undergraduate Program (CUP, later to be renamed The unprecedented growth of computer science as CUPM). CUPM concentrated initially on a unified in- a major new college subject parallels the theoretical troductory mathematics sequence Universal Mathemat- growth of the discipline and its ever-expanding im- ics, consisting of a first semester analysis/college alge- pact on business and day-to-day living. The number of bra course (finishing with some calculus) followed by a computer science majors now substantially exceeds the semester of “mathematics of sets” (discrete mathemat- number of mathematics majors at most schools offering ics). CUPM hoped its Universal Mathematics would programs in both subjects. However, computer science “halt the pessimistic retreat to remedial mathematics has not “taken” students from mathematics, any more . . . (and) . . . modernize and upgrade the curriculum.” than science and engineering take students from mathe- The first comprehensive curriculum report of CUPM, matics. Rather, computers have generated the need for entitled Pregraduate Training for Research Mathemati- more quantitative problem-solvers, as noted above. cians (1963), outlined a model program designed to pre- The shortage of secondary school mathematics teach- pare outstanding undergraduates for Ph.D. studies in ers nation wide has become worse than ever before. mathematics. Emphasis on Ph.D. preparation repre- This shortage appears to be due in large measure to sented a major departure from the traditional mathe- the greater attractiveness of computing careers to col- matics program and was the source of continuing con- lege mathematics students (indeed high-paying com- troversy. A more standard mathematics major curricu- puter jobs are currently luring many teachers out of the lum was published in 1965 (revised in 1972), but many classroom). Although the training of future teachers colleges also found it to be too ambitious for their stu- should include course work in computing and applica- dents. tions, such course work heightens the probability that For a fuller history of CUPM, see the article of W. these students will switch to careers in computing. Duren (founder of CUPM), “CUPM, The History of an On another front, pre-calculus enrollments have Idea,” Amer. Math. Monthly 74 (1967), pp. 22-35. soared as the mathematical skills of incoming freshmen have been declining (a problem that concerned CUP in its first year). The mathematics curriculum may soon Current Issues need to allow for majors who do not begin calculus until In 1970, 23,000 mathematics majors were graduated. their sophomore year, as was common a generation ago. The numbers of Bachelors, Masters, and Doctoral grad- At universities, the decline in graduate enrollments uates in mathematics had been doubling about every six has frequently over-shadowed the decline in undergrad- years since the late 1950s. The 1970 CBMS estimate for uate majors. Faced with heavy precalculus workloads, the number of Bachelors graduates in mathematics in shrinking graduate programs, and competition from 1975 was 50,000, but by the late 1970s only 12,000 were other mathematical sciences departments, university graduating annually. Enrollments in many upper-level mathematics departments appear less able to broaden pure mathematics courses declined even more dramat- and restructure the mathematics major than most ically in the 1970s as students turned to applied and liberal-arts college mathematics departments. Many computer-related courses. university mathematicians prefer to retain their current Yet while the number of mathematics majors is de- pure mathematics major for a small number of talented creasing, the demand for broadly-trained mathemat- students. ics graduates is increasing in government and indus- There are also several encouraging developments. A try. Mathematical problems inherent in projects to natural evolution in the mathematics major is occurring optimize the use of scarce resources and, more gener- at many schools. Students and faculty have developed ally, to make industry and government operations more an informal “contract” for a major that includes tradi- efficient guarantee a strong future demand for mathe- tional core courses in algebra and analysis along with maticians. These problems require people who, fore- electives weighted in computing and applied mathemat- MATHEMATICALSCIENCES 5 ics (a formal "contract" major at one school is discussed involve an interplay of applications, mathemati- below). cal problem-solving, and theory. Theory should Another important development is the emphasis on be seen as useful and enlightening for all math- system design, as opposed to mathematical computa- ematical sciences. tion, in current computer science curricula. The Associ- V. First courses in a subject should be designed to ation for Computing Machinery Curriculum 78 Report appeal to as broad an audience as is academ- delegates the responsibility for teaching numerical anal- ically reasonable. Many mat:hematics majors ysis, discrete structures, and computational modeling do not enter college planning to be mathemat- to mathematics departments. This ACM curriculum ics majors, but rather are attracted by begin- implicitly encourages students interested in computer- ning mathematics courses. Broad introductory based mathematical problem solving to be mathemat- courses are important for a mathematical sci- ical sciences majors. The MSP panel has been careful ences minor. to coordinate its work with computer science curriculum groups in order to minimize potential conflicts and Course Work maximize compatibility between computer science and mathematical sciences programs. VI. The first two years of the curriculum should be broadened to cover more than the traditional four semesters of calculus-linear algebra- Curricular Principles differential equations. Calculus courses should include more numerical methods and non- The goal of this panel was to produce a flexible set physical-sciences applications. Also, other of recommendations for a mathematical sciences ma- mathematical sciences courses, such as com- jor, a major with a broad, historically rooted founda- puter science and applied probarbility and statis- tion for dealing with current and future changes in the tics, should be an integral part of the first two mathematical sciences. The panel sought a unifying years of study. philosophy for diverse course work in analysis, algebra, VII. All mathematical sciences students should take computer science, applied mathematics, statistics, and a sequence of two upper-division courses leading operations research. to the study of some subject(s) in depth. Rigor- ous, proof-like arguments are used throughout Program Philosophy the mathematical sciences, and so all students I. The curriculum should have a primary goal should have some proof-orienled course work. of developing attitudes of mind and analyti- Real analysis or algebra are natural choices cal skills required for efficient use and under- but need not be the only possibilities. Proofs standing of mathematics. The development of and abstraction can equally well be developed rigorous mathematical reasoning and abstrac- through other courses such as applied algebra, tion from the particular to the general are two differential equations, probability, or combina- themes that should unify the curriculum. torics. 11. The mathematical sciences curriculum should VIII. Every mathematical sciences student should be designed around the abilities and academic have some course work in the less theoretically needs of the average mathematical sciences stu- structured, more combinatorially oriented math- dent (with supplementary work to attract and ematics associated with computer and decision challenge talented students). sciences. 111. A mathematical sciences program should use IX. Students should have an opportunity to un- interactive classroom teaching to involve stu- dertake "real-world" mathematical modeling dents actively in the development of new ma- projects, either as term projects in an opera- terial. Whenever possible, the teacher should tions research or modeling course, as indepen- guide students to discover new mathematics for dent study, or as an internship in industry. themselves rather than present students with X. Students should have a minor in a discipline us- concisely sculptured theories. ing mathematics, such as physics, computer sci- IV. Applications should be used to illustrate and ence, or economics. In addition, there should be motivate material in abstract and applied sensible breadth in physical and social sciences. courses. The development of most topics should For example, a student interested in statistics 6 RESHAPINGCOLLEGE MATHEMATICS

might minor in psychology but also take begin- might be: ning courses in, say, economics or engineering (heavy users of statistics). statilticsApplied /]‘cuyv ComputerTing

ProbabilityTheory Advancedfu\ ,Equ/ Differential Building Mathematical Maturity < NumericalAnalysis As noted in Principle I, a major in mathematical sciences should emphasize general mathematical reasoning REAL ANALYSIS as much as mastery of various subject matter. Implicit in this principle is that less material would be covered in many courses but that students would be expected to Core Requirements demonstrate a better understanding of what is taught, e.g., by solving problems that require careful mathe- The panel has found the question of whether to re- matical analysis. quire courses in algebra and analysis its most contro- This mathematical sciences curriculum would model versial problem. In light of the strongly differing opin- the historical evolution of mathematical subjects: some ions received on this subject, the MSP panel is making problems are introduced, formulas and techniques are only a minimal recommendation (Principle VII) that it developed for solving problems (usually with heuristic feels is reasonable for all students. Possible two course explanations), then common aspects of the problems sequences besides a year of analysis or of algebra are: are examined and abstracted with the purpose of bet- analysis and proof-oriented probability theory, analysis ter understanding “what is really going on.” The dif- and differential equations, abstract algebra and (proof- ference in this scheme between beginning calculus and oriented) combinatorics, applied algebra and theory of upper-division probability theory would be primarily a computation, or analysis and a topics-in-analysis semi- matter of the difficulty of the problems and techniques nar. While not a sequence, one course in analysis and and the speed with which the material is covered and one course in algebra also fulfill the spirit of this require- generalized, i.e., a matter of the mathematical matu- ment. Some departments will want to make stronger rity of the audience. In the course of two or three years requirements. The issue of theory requirements is dis- of such course work, there would be a steady increase cussed more fully below. in sophistication of the material and more importantly, Students should not be required to study a subject an increase in the student’s ability to learn and orga- with an approach whose rationale depends on material nize the ideas of a new mathematical subject. Students in later courses nor should they be required to memorize should be able to read and learn mathematics on their (blindly) proofs or formulas. Some upper-level elective own from texts. The MSP panel feels that such matu- courses should always be taught as mathematics-for-its- rity is a function of how a subject is learned as much as own-sake, but an instructor should be very careful not what is learned. to skip the historical motivation and application of a All courses should have some proofs in class and, as subject in order to delve further into its modern theory. the maturity of students increases, occasional proofs as The recommendation for interactive teaching (Prin- homework exercises. In particular, students should ac- ciple 111) seeks to encourage student participation in quire facility with induction arguments, a basic method developing new mathematical ideas. It constrains an of proof in the mathematical sciences. After review- instructor to teach at a level that students can reasoning performances of current students and programs of ably follow. Interactive teaching implicitly says that mathematics students 30 years ago, the MSP panel has mathematics is learned by actively doing mathemat- concluded that many able students do not now have, ics, not by passively studying lecture notes and mim- nor were they previously expected to have, the mathe- icking methods in a book. Without needlessly slowing matical maturity to take theoretical courses before their progress in class, an instructor should discuss how one senior year. On the other hand, by the senior year, can learn much from wrong approaches suggested by all students should be ready for some proof-oriented students. New mathematical theories are not divined courses that show the power of mathematical abstrac- with textbook-like compact proofs but rather involve a tion in analyzing concepts that underlie a variety of long train of trial-and-error creativity. concrete problems. For example, part of a flowchart Henry Pollak expressed this need in the Conference of courses leading to a senior-year real analysis course Board of Mathematical Sciences book, The Role of Az- MATHEMATICALSCIENCES 7 iomatics and Problem Solving in Mathematics (Ginn, came from traditions that go back to the roots of math- 1966): ematics. A carefully organized course in mathematics is While problem solving may traditionally be the pri- sometimes too muchlike a hiking trip in the mountains mary way of teaching mathematical reasoning to un- that never leaves the well-constructed trails. The tour dergraduates, the complexity and breadth of modern manages to visit a steady sequence of the high spots in the natural scenery. It carefully avoids all false mathematics and mathematical scienceis require theory starts, dead ends and impossible barriers and arrives to help organize and simplify learning. Rigorous prob- by five o’clock every afternoon at a well-stocked cabin. lem solving should lead students to appreciate theory . . .However, you miss the excitement of occasionally and formal proofs. In a mathematical sciences major, camping out or helping to find a trail and of making theory should be primarily theory for a purpose, theory your way cross-country with only a good intuition and born from necessity (of course, this is also the historical a compass as a guide. “Cross country” mathematics motivation of most theory). Students may find theory is a necessary ingredient of a good education. difficult, but they should never find it irrelevant. Further details about the course work recommendations in Principles VI, VIII, and IX appear in later Most courses in a mathematical sciences major chapters of this report. Discussion of courses in dis- should be case studies in the pedagogical paradigm crete methods, applied algebra, and numerical analysis of real world questions leading to matlhematical prob- appears in the last section of this chapter. lem solving of increasing difficulty that forces some abstraction and theory. As mentioned earlier, lower-level courses would concentrate on problem solving to build Teaching Mat hematical Reasoning technical skills with occasional statements of needed theorems, while typical upper-level courses would con- Because a mathematical sciences major must include centrate on problem solving to build technical skills a broader range of courses than a standard (pure) math- with occasional statements of needed ,theorems, while ematics major, many mathematicians have expressed typical upper-level courses would emphasize the transi- concern that it will be harder to teach the average tion from harder problem-solving to theory. mathematics student rigorous mathematical reasoning in a mathematical sciences major. They believe that Instructors should resist pressures to survey fully the major will develop problem-solving skills but that fields such as numerical analysis, probability, statistics, without more abstract pure mathematics, students will combinatorics, or operations research in the one cmrse never develop a true sense of rigorous mathematical a department may offer in the field. The instructor reasoning. The MSP panel thinks that a mathemati- of such a course should give students a sense of the cal sciences major with a strong emphasis on problem- problems and modes of reasoning in the field, but after solving is in keeping with time-tested ways of developing that, should be guided by the pedagogical model given “mathematical reasoning.” The question of whether to above. All syllabi produced by MSP siubpanels should require “core” pure mathematics courses, such as ab- be viewed in this light. Most instructors will cover most stract algebra and real analysis, in any mathematical of a suggested syllabus, but general pedagogical goals sciences major is discussed in the next section. should always take precedence over the demands of in- Historically (before 1940), the main thrust of the dividual course syllabi. mathematics major at most colleges was problem- The MSP panel believes that for generations math- solving. Most courses in the major could be classed ematics instructors have used the paraldigm mentioned as mathematics for the physical sciences: trigonometry, above to develop rigorous mathematical reasoning. Im- analytic geometry, calculus (first-year and advanced), plicit in this paradigm is a unity of purpose between differential equations, and vector analysis. Proofs in students and instructor. Most students like to start advanced calculus were symbolic computations. Proofs with concrete real-world examples as a basis for mathe- in number theory were, and still are, usually combi- matical problem solving. They expect the problems to natorial problems. The one abstract “pure” course in get harder and require more skill and insight. And they the curriculum was logic. A “rigorous” course did not certainly appreciate theory when it makes their work mean an abstract course, “mathematics done right.” A easier (although understanding formal proofs of useful rigorous course used to mean a demanding, more in- theory requires maturity). Interactive t,eaching also be- depth treatment tbat required more skill and ingenuity comes natural: students are interested in participating from the student. The past curriculum surely had some in a class that is developing a subject in a way that they faults, but its problem-solving and close ties to physics can appreciate. 8 RESHAPINGCOLLEGE MATHEMATICS

How Much Theory? Analysis and algebra are fine for some students but demand a mathematical maturity that many This section summarizes arguments for and against other undergraduates lack-these students memo- requiring upper-level analysis and algebra courses of all rize proofs blindly to pass examinations and never mathematical sciences majors, and why the MSP panel take the follow-on courses needed to appreciate the made its "compromise" decision. structure and elegance of these subjects; and Expecting controversy on several issues, the MSP Proofs and abstraction can equally well be devel- panel organized sessions at national and regional MAA oped through other courses such as applied algebra, meetings to get input from the mathematics community. probability, differential equations, or combinatorics. The main area of contention was how many courses to Mathematicians must face the reality of a general require in specific areas. The panel heard complaints change in the attitude of college students towards math- that some areas were being neglected or that only one ematics. The popularity of science and mathematics in course in a certain area would be so superficial as to be the 1960s drew more of the brightest students to mathe- worse than no course. However, most constituencies matics and also motivated all students to work harder at came to accept the need for compromise recommen- mathematics in high school. So the average mathemat- dations of limited exposure to several areas with stu- ics student was capable of handling a more theoretical dents left to choose for themselves an area to study in mathematics program. greater depth. On the other hand, one important issue Today, mathematics appears to be getting no more emerged on which a compromise position seemed to an- than its traditional (smaller) share of bright students, tagonize at least as many people as it pleased. This was and high school study habits are less good. However, al- the question of whether to require an analysis and/or most all of today's mathematics students still find a few an abstract algebra course and, more generally, how subjects, pure or applied, particularly interesting and much proof-oriented course work should be required in want to study this material in some depth. Also by the a mathematical sciences major. senior year, the MSP panel believes that mathematics In the early 1970'~~a majority of mathematics pro- majors do have the mathematical maturity to appre- grams required at least these two upper-level "core ciate, say, a moderately abstract real analysis course. mathematics" courses for all students. Recently, de- Examples of new approaches to teaching analysis and clining enrollments in these courses and student prefer- other core mathematics courses appear in subsequent ence for more applied or computing courses have forced chapters. many departments either to relax this requirement or Since there was agreement on the importance of some to introduce a new applied track which does not require theoretical depth, the MSP panel proposed the compro- these two courses. People favoring the requirement of mise of Principle VII, recommending "a sequence of two analysis and algebra argue that: upper-division courses leading to the study of some sub- 0 Not requiring them would speed an already dan- ject in depth." Because of the lack of consensus on the gerous deterioration in the intellectual basis of the analysis-algebra question, the MSP panel expects this mathematics major; issue to be debated and modified at individual institu- A major without at least analysis and algebra would tions. The faculty should not require courses that most be a superficial potpourri of courses-a major of no students strongly dislike, nor should faculty shy away real value to anyone, e.g., graduate study in statis- from any theory requirements for fear of losing majors. tics requires analysis and (linear) algebra; The faculty rather must motivate students to appreciate One cannot understand "what mathematics is the value of some theoretical course work. about" without these two courses-a major without these two courses simply should not be offered by a mathematics department. Sample Majors People in favor of not requiring analysis and algebra argue that: This section presents two 12 semester-course math- With a more applied emphasis the mathematical ematical sciences majors. Many other sample majors sciences major will attract more good students, could be given. The MSP panel believes that most ma- whereas requiring these courses would mean no jors should be a "convexcombination" of the two majors change (except for new applied electives) from the given here. Major A contains much of a standard math- 1960s type of mathematics major that today at- ematics major, while Major B is a broader program de- tracts only a marginal number of students; signed for students interested in problem solving. Both MATHEMATICALSCIENCES 9 majors should be accompanied by a minor in a related allow students to make statistics or operations research subject. a unifying theme. The common core of all majors would be three The MSP panel feels that a set of courses similar to semesters of calculus, one course in linear algebra, one either of the above two majors, or a mixture thereof, course in computer science plus either a second com- would be reasonable for most mathematical sciences puter course or extensive use of computing in several students. Some departments could offer several tracks other courses, one course in probability and statistics, for the mathematical sciences major. Special areas of the equivalent of a course in discrete methods, modeling faculty strength or student interest should obviously be experience, and two theoretical courses of continuing reflected in the curriculum. depth. Computing assignments should be used in most courses. When a liberal arts college mathematics de- Mathematical Sciences Major A partment teaches computer science, :such computing Three semesters of calculus course work must frequently be counted within the col- * Linear algebra lege limit of 12 or 13 courses permitted in one depart- Probability and statistics ment. This regulation is assumed in Major B. However, Discrete methods the MSP panel believes that counting computer courses Differential equations (with computing) this way unfairly restricts a mathematical sciences major. One alternative is to list computer courses through Abstract algebra (one-half linear algebra) the Computing Center. Two semesters of advanced calculus/real analysis The one fundamental new course in these sample * One course from the following set: abstract algebra majors is discrete methods. As mentioned in Princi- (second course), applied algebra, geometry, topol- ple VIII, the MSP panel feels that the central role of ogy, complex analysis, mathematical methods in physics combinatorial reasoning in computer and decision sciences requires that some combinatorial problem solving Mathematical modeling should be taught in light of the three semesters devoted Plus related course work: two semesters of computer to analysis-related problem solving in the calculus se- science and two semesters of physics, to be taken in quence. To this end, the modeling course should be the first two years. heavily combinatorial if students have not taken a formal discrete methods course. Mathematical Sciences Major B Major A would be good preparation for graduate Three semesters of calculus study in mathematics, applied mathematics, statistics, * Linear algebra or operations research as well as many industrial posi- Introduction to computer science tions as a mathematical analyst or programmer. Ma- Numerical analysis or second course in computer jor B would be good preparation for most industrial science positions and for graduate study in applied mathemat- * Probability and statistics ics, statistics, or operations research (for such graduate Advanced calculus or abstract algebra study, both advanced calculus and upper-level linear al- Discrete methods or differential equations gebra are usually needed). Representatives from many Mathematical modeling or operations research good mathematics graduate programs have stated that Two electives continuing a subject with theoretical they would accept strong students with Major B-type depth. training. Subsequent sections in this report contain recom- Many computer science graduate programs would ac- mendations for discrete methods, applied algebra, and cept Major B if the two electives were in computer sci- numerical analysis courses; for calculus, linear algebra, ence (although some other undergraduate computer sci- and differential equations courses; for upper-level core ence course deficiencies may still have to be made up in mathematics; for computer science; for modeling and the first year of graduate study). In a computer science operations research; and for probability and statistics. concentration within a mathematical sciences major, Major A is meant to be close to the spirit of the major modern algebra might be replaced by applied algebra suggested by the NRC Panel on naining in Applied (see below for more details). Major B with an elec- Mathematics. That panel viewed differential equations tive in the theory of interest and a second probability- as a unifying theme in the major. The proper mixture of statistics course would be excellent preparation for ac- Majors A and B (with appropriate electives) would also tuarial careers. Students interested in physical sciences- 10 RESHAPINGCOLLEGE MATHEMATICS related applied mathematics could modify either sam- one semester of introductory computer science, plus two ple major to get a good program. Both majors provide to four electives chosen from courses such as numerical preparation for secondary school mathematics teach- analysis, discrete methods, linear algebra, differential ing, when supplemented with teaching methodology equations, linear programming, mathematical model- and practicum courses (theory courses must include al- ing, and additional courses in calculus, probability or gebra and geometry). statistics, and computer science. Such a minor could Many smaller schools are being forced to offer a pro- easily be completed in three semesters. It has little gram in the spirit of Major B because almost all of prerequisite structure so that students can immediately B's courses have the needed enrollment base of students pick courses based on personal interests rather than ini- drawn from outside mathematics. tially "mark time" waiting to complete the calculus se- The courses involving numerical analysis, probabil- quence. ity and statistics, discrete methods, and modeling all Such a minor has several important points in its fa- can be designed as lower-level or upper-level courses. A vor. First of all, this minor is a collection of useful large amount of flexibility is possible in "repackaging" mathematical sciences courses which present concepts the mathematical sciences material. For example, a and techniques that arise frequently in the social and Computational Models course (see the 1971 CUPM Re- biological sciences. While this minor lacks the math- port on Computational Mathematics) could cover some ematical depth of the traditional type of mathematics numerical analysis along with a little applied probabil- minor, it nonetheless introduces students to important ity and statistics to be used in simulation modeling. modes of mathematical reasoning. Second, such a mi- A quarter system institution would have even greater nor will be attractive to students because it enhances flexibility in implementing this major. employment opportunities and prospects for admission to graduate or professional schools. Third, after the exposure to interesting mathematical sciences topics, Mathematical Sciences Minor some students will want to study these subjects further Just as a mathematical sciences major should be ac- in graduate school, either in a mathematical sciences companied by a minor in a related subject, so also do graduate program or as electives in other graduate pro- many other disciplines encourage their students to have grams. Fourth, this minor will bring more students into a minor, or double major, in mathematics. At some col- mathematical sciences courses, making it possible to of- leges, as many as half the mathematics majors have an- fer these courses more frequently. Conversely, offering other major. Unfortunately, while mathematical meth- more mathematical sciences courses each semester will ods are playing an increasingly critical role in social make a mathematical sciences minor, as well as the reg- and biological sciences and in business administration, ular mathematical sciences major, more attractive to students are generally ignorant or misinformed in high students. In addition, when more students are taking school and early college years about the importance of mathematical sciences courses and finding out how use- mathematics in these areas. ful mathematics is, the campus-wide student awareness The result is that many students either do not realize of the value of mathematics will increase. the value of further course work in the mathematical sciences until their junior or senior year, or their poor high Examples of Successful Programs school preparation forces them to take a year of remedial mathematics before they can begin to learn any of Proper curriculum is the heart of a mathematical the college mathematics they need. For such students, sciences program, but there are many non-academic as- a traditional six to eight course minor in mathematics, pects that also must be considered. A wide variety of starting with (at least) three semesters of calculus, is course offerings is not as important as the spirit with not feasible. When students in the social and biolog- which the general program is offered. This section dis- ical sciences come to realize the value of mathematics cusses salient features of some successful mathematics in the junior year, they have frequently had only one programs. "Successful" means attracting a large num- semester of calculus, or perhaps a year of calculus with ber of students into a program that develops rigorous probability. mathematical thinking and also offers a spectrum of The MSP panel believes that these students would be (well taught) courses in pure and applied mathemat- well served by a six to eight course mathematical sci- ics. Successful programs typically produce 5% to 8% of ences minor consisting of two semesters of calculus, one their college's graduates, although nation wide, mathe- semester of (calculus-based) probability and statistics, matics majors constitute only about 1%of college grad- MATHEMATICALSCIENCES 11 uates. Faculty and student morale is uniformly high in ence, Computer Science, and Operatioris Research. The these programs. As one would expect, teaching and re- course work in the mathematics graduate preparation lated student-oriented activities consume most of the track involves a problem seminar, Putnam team ses- faculty’s time in such successful programs, and there sions, and formal and informal topics courses (because is little faculty research. The professors’ pride in good of the limited demand in this area). All mathematical teaching and in the successes of their students leaves sciences majors must take a rigorous 25 semester-hour them with few regrets about not publishing. The set of core of calculus, differential equations, linear algebra, programs mentioned here is only a sampling of success- foundations, and computer science. Most courses are ful programs that have come to the attention of this peppered with applications and computing assignments. CUPM panel. More detailed information about these The mathematics faculty are heavily involved in re- mathematics programs is available from individual col- cruiting students by attending College Fairs and College leges. Nights and by visiting regional high schools to explain to students and counselors the many diverse and at- Saint Olaf College, a 2800-student liberal arts college tractive careers in the mathematical nciences, and the in Northfield, Minnesota, has a contract mathematics importance of mathematics in other professions. As a major. Each mathematics student presents a proposed result of this effort, 10% of the incoming Lebanon Val- contract to the Mathematics Department. The contract ley freshmen plan majors in the mathlematical sciences consists of at least nine courses (college regulations limit (the national average is 1%), and 7% of Lebanon Valley the maximum number of courses that can be taken in graduates are mathematical sciences majors. Many stu- one department to 14). The department normally will dents are initially attracted by the major in actuarial not accept a contract without at least one upper-level science (an historically established profession) and then applied and one upper-level pure mathematics course, move into other areas of applied and pure mathematics, a computing course or evidence of computing skills, but this pattern may change with the newly established and some sort of independent study (research program, computer science major. problem-solving proseminar, colloquium participation, Once the faculty have the “students’ attention,” they or work-study ). work the students hard. The students respond posi- Frequently a student and an advisor will negotiate a tively to the demands of the faculty for three reasons. proposed contract. For example, a faculty member will First, known rewards await those who do well in math- try to persuade a student interested in scientific com- ematics (besides the obvious long-term rewards, the de- puting and statistics that some real analysis and upper- partment awards outstanding students with member- level linear algebra should be included in the contract by ship in various professional societies in the mathemat- showing that this material is needed for graduate study ical sciences). Second, a personal sense of intellectual in applied areas, and in any case a liberal arts education achievement is carefully nurtured starting in the fresh- entails a more broadly based mathematics major. Con- man year with honors calculus for mathematics majors. versely, a student proposing a pure mathematics con- Finally, as at St. Olaf, a continuing (dialogue between tract would be confronted with arguments about not students and faculty allows students l,o help shape the being able to appreciate theory without knowledge of mathematics program. In fact, students interview can- its uses. In the end, the student and the faculty mem- didates for faculty positions and their irecommendations ber understand and respect each other’s point of view. carry great weight. The department keeps in close touch This understanding of each other’s interests natu- with alumni by sending each one a personal letter every rally carries into the classroom. Also, the contract ne- other year with news about the department and fellow gotiations “break the ice” and make students more at alumni. ease in talking to faculty (and encourage constructive Nearby Gettysburg College has a special vitality in criticism). The Mathematics Department offers minors its mathematics program that comes from an interdis- in computing and statistics, but the attractiveness of ciplinary emphasis. The department :has held joint de- a contract major in mathematics leads most students partmental faculty meetings with each natural and so- interested in these areas eventually to become mathe- cial science department at Gettysbyg to discuss com- matics majors. mon curriculum and research interests. Several inter- Lebanon Valley College, a small (1000-student) lib- disciplinary team-taught courses havc been developed, eral arts college in Pennsylvania, has only five math- such as a course on symmetry taught jointly by a mathematics faculty but its Department of Mathematical ematician and a chemist. An interdepartmental group Sciences offers majors in Mathematics, Actuarial Sci- organized two recent summer workshops in statistics 12 RESHAPINGCOLLEGE MATHEMATICS which drew faculty from eight departments. Mathe- had to limit severely their upper-level electives in or- matics faculty have audited a variety of basic and ad- der to keep class size down and preserve small group vanced courses in related sciences to learn to talk the seminars). language of mathematics users. Mathematics faculty As noted at the start of this section, the preced- bring this interdisciplinary point of view into every ing mathematics programs represent only a small sam- course they teach, giving interesting applications and pling of the excellent programs in this country. Sev- showing, say, how a physicist would approach a certain eral women’s colleges offer fine programs worth noting. problem. Needless to say, a large number of mathemat- For example, the Goucher College Mathematics Depart- ics majors at Gettysburg are double majors. ment has integrated computing in almost all courses and Frequently a separate computer science department has a broad curriculum in pure and applied mathemat- with its own major spells disaster for the mathematics ics; and the Mills College Mathematics Department has major at a college. But Potsdam State College (in the successfully promoted the critical role of mathematics economically depressed northeast corner of New York) for careers in science and engineering. The cornerstone has possibly the greatest percentage of mathematics of Ohio Wesleyan’s excellent mathematics program is graduates of any public institution in the country-close an innovative calculus sequence (with computing, prob- to lo%-despite competition from a popular computer ability, and diverse mathematical modeling). Georgia science major. The most striking feature to a visitor State University, an urban public institution with a to the Potsdam State Mathematics Department is the highly vocational orientation, has a Mathematics De- great enthusiasm among the students and the sense of partment that has broken out of the typical low-level pride students have in their ability to think mathemat- service function mode to offer a fine, well-populated ically. (While it is hard to measure objectively these mathematical sciences major. While research and grad- students’ mathematical development, leading techno- uate programs often dominate concerns about the un- logical companies, such as Bell Labs, IBM, and General dergraduate mathematics major at universities, math- Dynamics, annually hire several dozen Potsdam mathematics faculty at many universities work closely with ematics graduates.) undergraduate majors in excellent unified mathematical Classes have a limited amount of formal lectures. sciences programs. Three such institutions are Clemson Most time is spent discussing work of the students. The University, Lamar University (Texas), and Rensselaer emphasis on giving students a sense of achievement is Polytechnical Institute. due in large part to experiences of the Potsdam chair- Most universities today have separate departments man when he taught in a Black southern institution. in computing and mathematical sciences. To counter By instilling self confidence, he had helped able but ill- this division, the University of Iowa and Oregon State prepared students excel in calculus and even saw some University have developed unified inter-departmental go on to good mathematics graduate programs. The de- mathematical sciences majors. The MSP panel strongly partment has various awards for top students, a very ac- endorses such inter-departmental majors. At some uni- tive Pi Mu Epsilon chapter, publications about careers versities, most of the mathematical sciences, outside in mathematics and successes of former students, and of pure mathematics, have been housed in one depart- a large student-alumni newsletter. Upper-class mathe- ment. Although the MSP panel prefers a unified mathematics students are used to tutor (and encourage) be- matical sciences major (ideally in one department), sev- ginning students. They also communicate their enthu- eral of these non-pure mathematical sciences depart- siasm about mathematics to friends and teachers back ments have good undergraduate programs that may home. As a result, half the incoming Potsdam freshmen be of interest to other institutions: the Mathemati- sign up for calculus (although few departments require cal Sciences Department at Johns Hopkins University, it). the Mathematical Sciences Department at Rice Univer- The computer science major at Potsdam State is sity, and the Department of Applied Mathematics and viewed by the mathematics faculty as a great asset to Statistics at the State University of New York at Stony the Mathematics Department. The computer science Brook. major helps attract good students to Potsdam who often decide to switch to, or double major with, mathematics. Also the computer science program offers career Departmental Self-study and Publicity skills and needed mathematical breadth. Numerical analysis, operations research, and modeling are taught The MSP panel urges all mathematics departments in computer science (the Mathematics Department has to engage in serious self-study to identify one or more MATHEMATICALSCIENCES 13 major themes to emphasize in their mathematical sci- ematics and its study at their institution should seek the ences programs: an interdisciplinary focus in cooper- cooperation of local associations of the National Coun- ation with other departments; an innovative calculus cil of Teachers of Mathematics, which have long been sequence (integrating computing, applications, etc.); a working to promote interest in mathematics in the high work-study program or other individualized learning schools. experience; special strength in one area of the mathematical sciences (pure or applied); or a track directed towards employment in a regional industry (such as New Course Descriptions aerospace, automative, insurance). Some colleges have Finite structures are used throughout the mathemat- successfully developed a multi-option major, but usu- ical sciences today. Two new basic courses about finite ally such programs are the outgrowth of successful one- structures belong in the mathematical sciences curricu- theme programs that slowly added new options (for ex- lum, one addressing combinatorial aspects and one ad- ample, the multiple-major mathematical sciences pro- dressing algebraic aspects. Another topic, numerical gram at Lebanon Valley College, mentioned in the pre- analysis, has become more important with the growth ceding section, started with just an Actuarial Science of computer science. This section describes a numeri- option). The MSP panel’s advice is first to do one thing cal analysis course that is more applied and at a lower well. level than the previous CUPM numerical analysis rec- A departmental emphasis should be consistent with ommendations (Course 8 in the CUPM report A Gen- the general educational purposes of the whole institu- eral Curriculum for Mathematics in Colleges.) tion and the academic interests of the high school graduates who have historically gone to that institution. It is Discrete Methods Course very risky to design a mathematical sciences program This course introduces the basic techniques and about a theme that the mathematics faculty find at- modes of reasoning of combinatorial problem solving tractive and then to try to recruit a new group of high in the same spirit that calculus introduces continuous school students to come to the institution for this pro- problem solving. The growing importance of computer gram. Note that a thematic emphasis does not mean science and mathematical sciences such as operations that basic parts of the mathematical sciences program research that depend heavily on combinatorial methods discussed earlier in this chapter can be neglected. justifies at least one semester of combinatorial problem Following a departmental self-study and implemen- solving to balance calculus’ three semlesters of analysis tation of its recommendations for new courses or de- problem solving. velopment of industrial work-study contacts, etc., it is Unlike calculus, combinatorics is not largely re- next necessary to publicize the mathematics depart- ducible to a limited set of formulas and operations. ment’s program with brochures and visits to regional Combinatorial problems are solved primarily through high schools and College Fairs. Virtually all mathemat- a careful logical analysis of possibilities. Simple ad ics departments with large programs (where mathemat- hoc models, often unique to each different problem, are ical sciences majors constitute over 4% of the school’s needed to count or analyze the possiblle outcomes. This graduates) have extensive publicity programs. Such need to constantly invent original solutions, different publicity should emphasize the general usefulness of from class examples, is what makes the discrete meth- mathematics in the modern world, whether a student ods course so valuable for students. is a prospective mathematical sciences major or minor Like calculus, combinatorics is a subject which has a or an undecided liberal arts student. wide variety of applications. Many of them are related High school guidance counselors often do not realize to computers and to operations research, but others re- that there are other attractive mathematics-related ca- late to such diverse fields as genetics, organic chemistry, reers outside straight computing. Counselors tend to be electrical engineering, political science, transportation, afraid of mathematics because of their own personal dif- and health science. The basic discrete methods course ficulties with the subject. Some counselors have been should contain a variety of applicatiolns and use them known to discourage students from taking more than both to motivate topics and to illustrate techniques. the minimum required amount of high school mathe- The course has an enumeration part and a graph matics with the warning that students risk getting poor theory part. These parts can be covered in either or- grades in (hard) mathematics courses and thus hurting der. While texts traditionally do enumeration first, the their chances of college admission. graph material is more intuitive and hence it seems nat- College faculty trying to publicize the value of math- ural to do graph theory first (as suggested below). 14 RESHAPINGCOLLEGE MATHEMATICS

With the right point-of-view, many combinatorial C. Permutations and combinations. Definitions problems have quite simple solutions. However, the and simple counting; sets and subsets; binomial object of this course is not to show students simple coefficients; Pascal’s triangle; multinomial coef- answers. It is to teach students how to discover such ficients; elementary probability notions and ap- simple answers (as well as not so simple answers). The plications of counting. Optional: algorithms for means for achieving solutions are of more concern than enumerating arrangements and combinations; the ends. Learning how to solve problems requires an binomial identities; combinations with repeti- interactive teaching style. It requires extensive discus- tion and distributions; constrained repetition; sion of the logical faults in wrong analyses as much as equivalence of distribution problems, graph ap- presenting correct analyses. plications. Since the course should emphasize general combina- D. Inclusion/ezclusion principle. Modeling with torial reasoning rather than techniques, a large degree inclusion/exclusion; derangements; graph color- of flexibility is possible in the choice of topics. The ing. Optional: rook polynomials. course outline given below contains many optional top- E. Recurrence relations. Recurrence relation mod- ics. Some of the core topics, such as the inclusion- els; solution of homogeneous linear recurrence exclusion formula, might also be skipped to allow the relations; Fibonacci numbers and their applica- course to be tailored to the interests of students. tions. COURSEOUTLINE F. Optional topics: I. Graph Theory a. Generating functions A. Graphs as models. Stress many applications. b. Polya’s enumeration formula B. Basic properties of graphs and digraphs. Chains, c. Experimental design paths, and connectednesq isomorphism; pla- d. Coding narity. The preceding course outline is for either a one- C. Trees. Basic properties; applications in search- semester or a two-quarter course. A two-quarter course ing; breadth-first and depth-first search; span- has a natural structure, covering enumerative material ning trees and simple algorithms using spanning in one quarter and graph theory plus designs in another trees. Optional: branch and bound methods; quarter. There are several books available for part or tree-based analysis of sorting procedures. all of the discrete methods course. It is anticipated that D. Graph coloring. Chromatic number; coloring as this discrete methods course becomes more widely applications; map coloring. Optional: related taught, many more books will become available and the graphical parameters such as independent num- exact nature of the syllabus will evolve. bers. There are several obvious places where a computer E. Eulerian and Hamiltonian circuits. Euler cir- can be used in this course: ways of representing graphs cuit theorem and extensions; existence and non- in a computer and performing simple tests (e.g., connec- existence of Hamiltonian circuits; applications tivity); asymptotic calculations in enumeration prob- to scheduling, coding, and genetics. lems; network flow algorithm; and algorithms for enu- F. Optional topics: merating permutations and combinations. The peda- a. Tournaments gogical problem is that computer programming takes b. Network flows and matching time away from problem-solving exercises, possibly too c. Intersection graphs much time if a school’s computer operation runs in a d. Connectivity batch processing mode. e. Coverings A more advanced second course in combinatorics f. Graph-based games may also be considered. This course can treat core top- 11. Combinatorics ics in the discrete methods course in greater depth, and A. Motivating problems and applications. some of the optional topics. Other important topics are B. Elementary counting principles. Tree diagrams; Ramsey theory, matroids, and graph algorithms. The sum and product role; solving problems that course could concentrate on combinatorics or on graph must be decomposed into several subcases. Op- theory, or could be a topics course which varies from tional: applications to complexity of computa- year to year. Some of the texts listed below would be tion, coding, genetic codes. suitable for this second combinatorics course. MATHEMATICALSCIENCES 15

COMBINATORICS& GRAPH THEORY TEXTS later computer science courses. At thle same time, this 1. Bogart, Kenneth, Introductory Combinatorics, Pit- course can also be very rewarding to regular mathe- man, Boston, 1983. matics majors who should appreciate ithe new algebraic 2. Brualdi, Richard, Introductory Combinatorics, Else- structures such as formal languages and finite state ma- vier-North Holland, New York, 1977. chines that are so different from the structures in the 3. Cohen, Daniel, Basic Techniques of Combinatorial regular abstract algebra course. Substantial class time Theory, J. Wiley & Sons, New York, 1978. should be spent on proofs with special emphasis on in- 4. Liu, C.L., Introduction to Combinatorial Mathemat- duction arguments. This course is just as mathemati- ics, McGraw Hill, New York, 1968. cally sophisticated and capable of developing abstract 5. Roberts, Fred, Applied Combinatorics, Prentice- reasoning as abstract algebra, but the topics stress set- Hall, Englewood Cliffs, New Jers., 1984. relation systems rather than binary-operation systems. 6. Tucker, Alan, Applied Combinatorics, J. Wiley lz Indeed the abstract complexity of the basic structures Sons, New York, 1980. is much greater in applied algebra, but this complexity precludes the construction of logical pyramids built of GRAPHTHEORY TEXTS simple algebraic inferences common to many areas of 1. Bondy, J. and Murty, V.S.R., Graph Theory with abstract algebra. Applications, American Elsevier, New York, 1976. This course is an advanced version of the lower- 2. Chartrand, Gary, Graphs as Mathematical Models, division B3 Discrete Structures course in ACM Cur- Prindle, Weber, and Schmidt, Boston, 1977. riculum 68. The B3 course was the source of much 3. Ore, Oystein, Graphs and Their Uses, Math. Assoc. dissatisfaction because it contained ii huge amount of of America, Washington, D.C., 1963. material, and it required too great mathematical matu- 4. Roberts, Fred, Discrete Mathematical Models, Pren- rity for a lower-division course. The recent ACM Cur- tice-Hall, Englewood Cliffs, New Jersey, 1976. riculum 78 recommends that the B3 course be treated 5. Trudeau, Robert, Dots and Lines, Kent State Press, as a more advanced course and that it, should be taught Kent, Ohio, 1976. in mathematics departments rather than computer science departments. The B3 course was the subject of C o MBIN AT o RIC s TEXTS several papers at meetings of the ACM Special Inter- 1. Berman, Gerald and Fryer, Kenneth, Introduction est Group in Computer Science Education (SIGCSE); to Combinatorics, Academic Press, New York, 1969. see the February issues (Proceedings of SIGCSE annual 2. Eisen, Martin, Elementary Combinatorial Analysis, meeting) of the SIGCSE Bulletin in 1973, 1974, 1975, Gordon-Breach, New York, 1969. 1976. 3. Vilenkin, N., Combinatorics, Academic Press, New The B3 course contained both applied algebra and York, 1971. discrete methods. The MSP panel recommends that a 4. Street, A. and Wallis, W., Combinatorial Theory: separate full course be devoted to discrete methods (see An Introduction, Charles Babbage, 1975. the discrete methods course description earlier in this Section). Because some computer science courses may Applied Algebra Course devote a substantial amount of time introducing some (Editorial Note in 1989 reprinting: This course is of the topics in the above applied algebra syllabus, the now called Discrete Structures and is usually now exact content of this course will vary substantially from taught at the freshman level. The course discussed here college to college. For this reason the syllabus outline is more advanced and intended for the sophomore-junior was kept brief. At some colleges, applied algebra will level.) still have to be combined with discrete methods in one A traditional time for an applied algebra course is course (the computer science major may not have the in the junior year-when students would be ready for a time for two separate courses). The applied algebra modern algebra course. However, as noted above, many part of such a combined course would, in most cases, students will not be ready for algebraic abstraction un- concentrate on topics 1, 2, 3, 4, 6 in the syllabus. Many til senior year. The course builds on experiences in be- of the discrete structures texts listed below cover both ginning computer science courses that have implicitly applied algebra and discrete methods. imparted to students a sense of the underlying algebra of computer science structures, and formally presents COURSETOPICS topics like Boolean algebra, partial orders, finite-state A. Sets, binary relations, set functions, induction, basic machines, and formal languages that will be used in graph terminology. 16 RESHAPINGCOLLEGE MATHEMATICS

B. Partially ordered sets, order-preserving maps, weak students have a good understanding of how the art of orders. numerical analysis comes into play. C. Boolean algebra, relation to switching circuits. The course outline below presents a good selection of D. Finite state machines, state diagrams, machine ho- topics for a one-semester course. Error analysis should momorphism. be continuously discussed throughout the duration of E. Formal languages, context-free languages, recogni- the course so as to stress the effectiveness and efficiency tion by machine. of the methods. Alternative methods should be con- F. Groups, semigroups, monoids, permutations and trasted and compared from the standpoint of the com- sorting, representations by machines, group codes. putational effort required to attain desired accuracy. G. Modular arithmetic, Euclidean algorithm. An optional approach to this course would emphasize H. Optional topics: linear machines, Turing machines a full discussion (with computer usage) of one procedure and related automata; Polya’s enumeration theo- for each course topic (after the computer arithmetic in- rem; finite fields, Latin squares and block design; troduction). A sample of five such procedures is: computational complexity. 1. The Dekker-Brent algorithm (see UMAP module No. 264). APPLIED ALGEBRATEXTS 2. A good linear equation solver involving LU-de- 1. Dornhoff, Lawrence and Hohn, Frantz, Applied composition. Modern Algebra, Macmillan, New York, 1978. 3. Cubic spline interpolation. 2. Fisher, James, Application- Oriented Algebra, T. 4. An adaptive quadrature code. Crowell Publishers, New York, 1977. 5. The Runge-Kutta-Fehlberg code RKF4 with adap- 3. Johnsonbaugh, Richard, Discrete Mathematics, tive step determination. Macmillan, New York, 1984. Weekly assignments should include some computer 4. Korfhage, Robert , Discrete Computational Struc- usage; in total, four or five computer exercises for each tures, Academic Press, New York, 1974. major topic. Students should do computer work for 5. Liu, C.L., Elements of Discrete Mathematics, Mc- larger applied programs in small groups. However, Graw Hill, New York, 1977. the concept of utilizing %armed" programs with mi- 6. Preparata, Franco and Yeh, Robert, Introduction nor modifications should be stressed. Such an approach to Discrete Structures, Addison-Wesley, Reading, nicely brings out the strong interdependence between Mass., 1973. computers and numerical analysis yet does not over- 7. Prather, Robert, Discrete Mathematical Structures emphasize the efforts necessary to program a problem. for Computer Sciences, Houghton Mifflin, Boston, An interactive computer system using video terminals 1976. is ideal for this course. Microcomputers and even hand- 8. Stone, Harold, Discrete Mathematical Structures held calculators can also be used effectively. One or two and Their Applications, Science Research Asso- applied homework problems from each of the main top- ciates, Chicago, 1973. ics keep students aware of the balance that is necessary 9. Tremblay, J. and Manohar, R., Discrete Mathemat- between the art and the science of numerical analysis. ical Structures with Applications in Computer Sci- Prerequisites for this course should be a year of calculus ences, McGraw Hill, New York, 1975. including some basic elementary differential equations and a computer science course. For schools on a quarter system, two quarters should Numerical Analysis Course be a minimal requirement and the above material would In any elementary numerical analysis course a bal- be more than ample. One should spend the first quar- ance must be maintained between the theoretical and ter on numerical solutions of algebraic equations and the application portion of the subject. Normally, such systems of algebraic equations and the last quarter on a course is designed for sophomore and junior students the other topics. in engineering, mathematics, science, and computer science. Students should be introduced to a wide selection COURSEOUTLINE of numerical procedures. The emphasis should be more A. Computer arithmetic. Discretization and round-off on demonstrations than on rigorous proofs (however, error; nested multiplication. this is not meant to slight necessary theoretical aspects B. Solution of 4 single algebraic equation. Initial dis- of error analysis). At least one or two applied problems cussion of convergence problems with emphasis on from each of the major topics should be included so that meaning of convergence and order of convergence; Newton’s method, Bairstow’s method; interpola- ods; method of undetermined coefficients in differen- tion. tiation and integration; least squares approximations; C. Solution systems of equations. Elementary matrix parabolic (or elliptic or hyperbolic) partial differential algebra; Gaussian methods, LU decomposition, it- equations; numerical methods for multi-dimensional in- erative methods, matrix inversion; stability of algo- tegrals; multi-step predictor-corrector methods. rithms (examples of unstable algorithms), errors in NUMERICALANALYSIS TEXTS conditioned numbers. D. Interpolating polynomials. Lagrange interpolation 1. Cheney, Ward and Kincaid, David, Numerical to demonstrate existence and uniqueness of interpo- Mathematics and Computing, Brooks/Cole, Mon- lating polynomials and for calculation of truncation terey, Calif., 1980. error terms; splines, least squares, inverse interpo- 2. Conte, S. and DeBoor, C., Elemmtary Numerical lation; truncation, inherent errors and their propa- Analysis, McGraw Hill, New York, 1978. gation. 3. Gerald, Curtis F., Applied Numerical Analysis, 2nd E. Numerical integration. Gaussian quadrature, Edition, Addison-Wesley, Reading, Mass., 1978. method of undetermined coefficients, Romberg and 4. Forsythe, G.E. and Moler, C.B., Computer Solu- Richardson extrapolation (for both integration and tions of Linear Algebraic System,s, Prentice-Hall, differentiation), Newton-Cotes formulas, interpolat- Englewood Cliffs, New Jersey, 1967. ing polynomials, local and global error analysis. 5. Hamming, R.W., Numerical Methods for Scientists F. Numerical solution of ordinary differential equa- and Engineers, 2nd Edition, McGraw Hill, New tions. Both initial value and boundary value York, 1973. problems; Euler’s method, Taylor series method, 6. James, M.L.; Smith, G.M.; Wolford, J.C., Ap- Runge-Kutta, predictor-corrector methods, multi- plied Numerical Methods for Digital Computation, step methods; convergence and accuracy criteria; Harper & Row, New York, 1985. systems of equations and higher order equations. 7. Ralston, Anthony and Rabinowitz, Philip, First If this course has an enrollment of under 25 students, Course in Numerical Analysis, McGraw Hill, New non-standard testing can be considered, such as a take- York, 1978. home midterm. At the end of the term, instead of the traditional three hour examination, each student can write an expository paper exploring in greater depth Panel Members one of the topics introduced in class or investigating ALANTUCKER, CHAIR, SUNY, Stony Brook. a subject not included in the work of the course, ei- RICHARDALO, Lamar University. ther approach to include computational examples with analysis of errors. (Since most of the students will not WINIFREDASPREY , Vassar College. have had previous experience in writing a paper, topics PETERHILTON, Case Western Reserve/Battelle Insti- may be suggested by the instructor or must be approved tute. if student devised; scheduled conferences and prelimi- DONKREIDER, Dartmouth College. nary critical reading of papers guard against disastrous WILLIAMLUCAS, Cornell University. attempts or procrastination.) Some examples of final FREDROBERTS, Rutgers University. projects are: spline approximations; relaxation meth- GAILYOUNG, Case Western Reserve.

Calculus

This chapter contains the report of the Subpanel on such as computer science (see “The Twilight of the Cal- Calculus of the CUPM Panel on a General Mathemat- culus,” which appeared under the title “Computer Sci- ical Sciences Progmm, reprinted with minor changes ence, Mathematics, and the Undergraduate Curricula from Chapter 11 of the 1981 CUPM report entitled in Both” in the American Mathematical Monthly, 88:7 RECOMMENDATIONSFOR A GENERALMATHEMATICAL (1981) 472-485). In his view, to ignore discrete meth- SCIENCESPROGRAM. ods, even in the first two years of college mathematics, would be absurd in this day. The subpanel does not disagree with the general Rationale sense of this position. On the other hand, the subpanel feels that the language, spirit, and methods of The Calculus Subpanel was charged with examining traditional calculus still permeate matlhematics and the the traditional calculus sequence of the first two years natural and social sciences. To quote Ralston himself, of college mathematics: two semesters of single-variable “The calculus is one of man’s great intlellectual achieve- calculus; one semester of linear algebra; one semester ments; no educated man or woman should be wholly of multivariable calculus. In approaching this task, the ignorant of its elements.” Perhaps the time is not far subpanel considered syllabi through which this sequence off when calculus will be displaced as the entry-level is implemented at various colleges and universities, the course, but it has not arrived yet. syllabus for the Advanced Placement Program in Calcu- The place for rigor. The subpanel believes lus, and alternatives to calculus as the entry-level course strongly that, in the first two years, theorems should in the mathematical sciences, for example, finite math- be used rather than proved. Certainly correct state- ematics or discrete methods. ments of theorems such as the Mean Value Theorem The subpanel eventually came to the conclusion that or 1’HGpital’s Rule should be given; but motivation, as the rationale for certain parts of the traditional calcu- long as it is recognized as such, and usage are more im- lus sequence remains valid, although some restructuring portant than proofs. The place for theoretical rigor is in and increased flexibility are warranted to reflect the dif- later upper-level courses. In this rega.rd, the subpanel fering mathematical requirements of the social and bi- agrees with the program philosophy outlined in the first ological sciences and, increasingly, of computer science. chapter, “Mathematical Sciences.” The general recommendations of the subpanel are thus: 1. To make no substantive changes in the first semester of calculus; First Semester Calculus 2. To restructure the second semester around model- The first semester of calculus, especially, contains a ing and computation, although leaving it basically consensus on essential ideas that are important for mod- a calculus course; eling dynamic events. This course has evolved through 3. To branch to three independent courses in the sec- considerable effort in the mathematical community to ond year: present a unified treatment of differential and integral a. Applied Linear Algebra, calculus, and it serves well both general education and b. Multivariable Calculus (in dimensions 2 and 3), professional needs. It is historically rich, is filled with c. Discrete Methods. significant mathematical ideas, is tempered through its Descriptions of the first and second semesters of calcu- demonstrably important applications, and is philosoph- lus, applied linear algebra, and multivariable calculus ically complete. Most syllabi for its teaching cover the are given below. The discrete methods course is dis- usual topics: cussed in the first chapter, “Mathematical Sciences.” A. Limits and continuity. The subpanel views its recommendations as conser- B. Differentiation rules. vative. Tony Ralston has argued, for example, that C. Meaning of the derivative. Applications to curve calculus need not be the entry-level course in the math- sketching, maximum-minimum problems, related ematical sciences and that a course in discrete methods rates, position-velocity-acceleration problems. is a reasonable alternative, better serving some areas D. Antidifferentiatwn. 20 RESHAPINGCOLLEGE MATHEMATICS

E. The definite integral and the Fundamental Theorem are still substantial reasons for keeping students in one of Calculus. "track" through the first two courses. In most Ameri- F. Trigonometric functions. can colleges, a "choice" in the second course would re- G. Jogarithmic and ezponential functions. Including a quire most students to be thinking seriously about ca- brief exposure to first-order, separable differential reer goals within a few weeks of arriving on campus as equations (with emphasis on y' = ky). freshmen. This does not strike us as realistic nor in the The first (and second) calculus courses should be 4- best interests of liberal education. Moreover, we con- or 5-credit hour courses. If less time is available, top- tinue to feel that many of the ideas and technical skills ics will have to be pushed later into the calculus se- arising in the second calculus course are reasonable to quence, with some multivariate calculus material left include at this point in the curriculum. Thus, the final for an analysis/advanced calculus course. Mathemat- conclusion is that a restructuring and change of empha- ics courses should not rush trying to cover unrealistic sis in the second semester calculus course is preferable syllabi. to its replacement. It might be desirable to add more non-physical sci- The Calculus Subpanel recommends the following ences examples to C (e.g., a discussion of the use of changes in the second calculus course: the word "marginal" in economics), although serious A. An early introduction of numerical methods. Imple- modeling examples should be postponed to the second mented through simple computer programs. Solv- semester. Integration as an averaging process can be in- ing one (or a system of two) first-order differential cluded in El but applications and techniques (numerical equation(s). or algebraic) of integration are better left to the second B. Techniques of integration. General methods such as semester. Exponential growth and decay are important integration by parts, use of tables, and techniques concepts that must be emphasized in G. that extend the use of tables such as substitutions and (simple) partial fraction expansions; less em- Second Semester Calculus phasis should be placed on the codification of special substitutions. There does not appear to be much slack or fat in the C. Numerical methods of integration. Examples where first semester of calculus. It is in the second semester, numerical and "formal" methods complement each therefore, when numerical techniques, models, and com- other, e.g., evaluating improper integrals where sub- puter applications can be introduced. Unlike the first stitutions or integration-by-parts make the integral semester of calculus, the second semester does not en- amenable to efficient numerical evaluation. joy the same consensus on either its central theme or its content. It tends to be a grab bag of "further calculus D. Applications of integration. Illustrate the "setting topics"-further techniques of integration, more appli- up" of integrals as Riemann sums. The emphasis cations of integration, some extension of techniques to should be on the modeling process rather than on the plane (parametric equations), sequences and infi- "visiting" all possible applications of the definite in- nite series, and more differential equations. Each of tegral. these topics is, in isolation, important at some stage in E. Sequences and series. These topics should have sub- the training of scientists and mathematicians. But it is stantially changed emphasis: less clear that packaging them in this way and having 1. Sequences should be elevated to independent them occupy this critical spot in the curriculum is justi- status, defined not only through %lased formu- fied today, given the pressing needs of computer science las" but also via recursion formulas and other and the non-physical sciences. iterative algorithms. Estimation of error and From time to time it has been urged that multi- analysis of the rate of convergence should ac- variable calculus should be started during the second company some of the examples. semester. But few institutions have implemented this 2. Series should appear as a further important ex- suggestion. And the subpanel believes that, in the ample of the idea of a sequence. Power series, as meantime, higher priorities for the second course have a bridge from polynomials to special functions, materialized in the form of applications and computing. should figure prominently. Specialized conver- The subpanel considered recommending branching in gence tests for series of constants can be de- the curriculum after the first semester of calculus, with emphasized. students advised to take courses more directly relevant 3. Approximation of functions via Taylor series, to their career goals. But it finally concluded that there and estimation of error, accompanied by implementation of such approximations on a com- be available from which students choose (with advis- puter. ing) their intermediate mathematics courses. Two of F. Differential equations. Should be treated with less these courses, whose descriptions follaow, are Applied (but not zero) emphasis on special methods for solv- Linear Algebra and Multivariable Calcdus. The third, ing first-order equations and constant coefficient Discrete Mathematics, is described in the first chapter, linear equations (especially the non-homogeneous ‘Mathematical Sciences.” case). More valuable would be: vector field interpretation for first-order equations, numerical meth- Applied Linear Algebra ods of solution, and power series methods for solv- For a large part of modern applied mathematics, lin- ing certain equations. Applications should arise in ear algebra is at least as fundamental as calculus. It is mathematical modeling contexts and both “closed the prerequisite for linear programming and operations form” and “numerical” solutions should be illus- research, for statistics, for mathematic,al economics and trated. Leontief theory, for systems theory, for eigenvalue prob- The new second course in calculus does not differ rad- lems and matrix methods in structures, and for all of ically in content from the traditional second semester numerical analysis, including the solution of differential course. It is a conservative restructuring that can be equations. The attractive aspect about these applica- taught from existing textbooks and based on modest tions is that they make direct use of wh,at can be taught modifications of many existing syllabi. But the intended in a semester of linear algebra. The course can have a change in “flavor” and emphasis should be more dra- sense of purpose, and the examples can reinforce this matic. About twelve lectures (of the usual 40 lectures) purpose while they illustrate the theory. must be modified substantially to achieve the desired A number of major texts have arrived at a reasonable computer emphasis. Numerical algorithms will thus consensus for a course outline. Their outlines are well figure prominently, along with the formal techniques matched with the needs of both theory and application. of calculus. Concepts not usually in a calculus course Applications can include such topics ars systems of lin- such as error estimation, truncation error, round-off er- ear differential equations, projections and least squares. ror, rate of convergence, and bisection algorithms will But the subpanel strongly recommends that more sub- be included. The theme for the course will be “calcu- stantial applications to linear models should be a cen- lus models.” Consideration of even a few UMAP-type tral part of the construction of the course. Many differ- models would be enough to change the nature of the ent applications of this kind are accessible and can be course significantly and to provide the intended “tying found in the texts mentioned. Thus, no rigid outline is together” of the traditional calculus topics that are in- required. The development of the suhject moves nat- cluded in the course. urally from dimension 2 to 3 to n, and although that A syllabus for the course could be constructed by is an easy and familiar step, it nevertheless represents starting with the second calculus course described in the mathematics at its best. The combination of impor- CUPM report, A General Curriculum for Mathematics tance and simplicity is almost unique ‘to linear algebra. in Colleges (revised 1972), or with the Advanced Place- Linear programming is an excellent final topic in the ment BC Calculus Syllabus. Topics to be diminished or course. It brings the theory and applications together. omitted include: emphasis on special substitutions in The changes in this course are ones of emphasis that integrals, l’H6pital’s rule except as it arises naturally in recognize that the course must be more than an in- connection with Taylor series, polar coordinates, vector troduction to abstract algebra. Abstraction remains a methods, complex numbers, non-homogeneous differen- valuable purpose, and linearity permits more success tial equations and the general treatment of constant- with proofs than the epsilon-delta arguments of calcu- coefficient homogeneous linear differential equations. lus. However, the main goal is to emphasize applica- Many of these topics will appear in examples but will tions and computational methods, opening the course not be emphasized in themselves. to the large group of students who nleed to use linear algebra. TEXTS Intermediate Mat hematics Courses 1. Hill, Richard, Elementary Lineo.r Algebra, Aca- Although the Calculus Subpanel recommends retain- demic Press, New York, 1986. ing a single track for students during their first year, it 2. Kolman, Bernard, Introductory Linear Algebra with just as strongly recommends that three different courses Applications, Macmillan, New York, 1979. 22 RESHAPINGCOLLEGE MATHEMATICS

3. Rorres, Chris and Anton, Howard, Applications of Applied Linear Algebra includes the solution of lin- Linear Algebra, John Wiley & Sons, New York, 1979 ear constant coefficient systems of differential equa- (paperback supplementary text). tions using eigenvalue methods. 4. Strang, Gil, Linear Algebra and Its -Applications, Although the Calculus Subpanel has not recom- 3rd Edition, Harcourt Brace Jovanovich, San Diego, mended a full course in differential equations in the cal- 1988. culus sequence of the first two years, it has suggestions 5. Tucker, Alan, A Unified Introduction to Linear Al- for a subsequent course. Such a course should not be gebra, Macmillan, New York, 1988. a compendium of techniques for solving in closed form 6. Williams, Gareth, Linear Algebra with Applications, various kinds of differential equations. Libraries are full Allyn and Bacon, Boston, 1984. of cookbooks; one hardly needs a course to use them. What is important is to develop carefully the models Multivariable Calculus from which differential equations spring. Modeling ob- This is the traditional multivariable calculus course viously means more than an application such as: at many colleges and universities. It is not a new course, According to physics, the displacement z(t)of a but for many schools it would represent a movement in weight attached to a spring satisfies 2”-bz’+kz = the direction of “concrete” treatment of multivariable 0. Solve for z(t) given that b = 2, k = 3, z(0) = calculus rather than the more recent elegant treatments 1, z’(0) = 0. making heavy use of linear transformations and couched For a more serious approach to applications, we refer to in general (high dimensional) terms. The course be- the art forgery problem at the beginning of Braun (see gins with an introduction to vectors and matrix alge- below) or indeed almost any of the models discussed in bra. Topics include Euclidean geometry, linear equa- the suggested texts. tions, and determinants. The remainder of the course The meaning of the word “solution” must be scru- is an introduction to multivariable calculus, including tinized. Different viewpoints must be introduced- the analytic geometry of functions of several variables, numerical, geometric, qualitative, linear algebraic and definitions of limits and partial derivatives, multiple and discrete. iterated integrals, non-rectangular coordinates, change A possible syllabus for a differential equations course of variables, line integrals, and Green’s theorem in the is: plane. A. First-order equations. Models; exact equations; existence and uniqueness and Picard iteration; numer- Differential Equations ical methods. B. Higher-order linear equations. Models; the linear al- The Calculus Subpanel has considered the place of gebra of the solution set; constant coefficient homo- differential equations in the curriculum. It recommends geneous and non-homogeneous; initial value prob- that the topic be treated at two levels: lems and the Laplace transform; series solutions. 1. Through methods and examples involving differ- C. Systems of equations and qualitative analysis. Mod- ential equations, spiraled through the calculus se- els; the linear algebra of linear systems and their so- quence, and lutions; existence and uniqueness; phase plane; non- 2. Through a substantial course in differential equa- linear systems; stability. tions, available to students upon completion of the Since some of these topics will have already been first-year calculus sequence and applied linear alge- introduced in courses from the calculus sequence, there bra. may be time for a brief discussion of partial differential We note here topics in differential equations that are equations and Fourier series. Existence and uniqueness part of the preceding courses: theorems are included here only because of the light Solutions of y‘ = ky occur in the first semester of they or their proofs might shed on methods of solution calculus. Exponential growth and decay are dis- (e.g., Picard iteration). cussed. TEXTS # Solution of second order linear differential equations are included in the second semester of calcu- The course can be taught using any of the many lus. Oscillating solutions occur as examples. In ad- reasonable differential equations texts with a modest dition, geometrical interpretations (direction field), amount of applications, supplemented by: numerical solutions and power series solutions are Braun, Martin, Differential Equations and Their included. Applications, Second Edition, Springer-Verlag, New York, 1978. multivariable calculus as prerequisites1 and could be Braun remains the only text to build extensively on taken a8 early as the second semester of the sophomore applications, but it has the serious drawback that it year if the two prerequisites were taken concurrently the is based on single-variable calculus and avoids linear previous semester. algebra. A somewhat radical alternative is a theoretical course Subpanel Members involving more qualitative or topological analysis emphasizing systems of equations. The subpanel does not DONKREIDER, CHAIR, Dartmouth Colllege. suggest a syllabus, but refers instead to V.I. Arnold, Or- ROSSFINNEY , Educational Development Center. dinary Differential Equations, MIT Press, Cambridge JOHNKENELLY, Clemson University. (paperback), 1978. GIL STRANG,MIT. This course would have applied linear algebra and TOMTUCKER, Colgate University.

Core Mathematics

This chapter contains the report of the Subpanel on both breadth and selective depth. (If size warrants, the Core Mathematics of the CUPM Panel on 0 General unified major can have several tracks, one for prepa- Mathematical Sciences Program, reprinted with minor ration for graduate study in mathemakics.) This sub- changes from Chapter 111 of the 1981 CUPM report en- panel is concerned with the role in a mathematical sci- titled RECOMMENDATIONSFOR A GENERALMATHE- ences major of upper-level core mathematics courses, MATICAL SCIENCES PROGRAM. and more generally with appreciation of the depth and power of mathematics. New Roles for Core Mathematics A prime attribute of a person educated in mathemat- In the 1960’s CUPM extensively examined curricu- ical sciences is his or her ability to respond when con- lum in core mathematics-upper division subjects that fronted with a mathematical problem, whether in pure comprise the trunk from which the other specialized mathematics, applied mathematics, 01’one which uses branches and applications of mathematics emerge. It mathematics that the person has not seen before. Our reviewed and revised its recommendations in 1972. students should be prepared to function as profession- See the Compendium of CUPM Recommendations pub- als in areas needing mathematics not by having learned lished by the Mathematical Association of America, es- stock routines for stock classes of probl.ems but by hav- pecially the 1972 revision of the General Curriculum ing developed their ability in problem solving, modeling for Mathematics in Colleges. With the current restruc- and creativity. This general pedagogical theme, that turing of the mathematics major into a mathematical was stressed throughout the first chapter “Mathemati- sciences major, new questions have been raised about cal Sciences,” guided the thinking of t,he Core Mathe- core mathematics curriculum. These questions concern matics Subpanel. the role of core mathematics in a mathematical sciences The report of the Core Mathematics Subpanel is major as much as syllabi of individual courses. This meant to be supportive rather than directive. What chapter focuses on four questions that were addressed an individual department does should reflect its con- to the Core Mathematics Subpanel by the parent Panel stituency of students, their needs, their numbers, and on a General Mathematical Sciences Program. the goals, character and size of the institution. The members of this subpanel represent a variety of institutions, public and private, liberal-arts colleges, Four Questions and research-oriented universities. All the members have seen at their institutions a divergence of the math- QUESTION1: Is there a minimal Jiet of upper-level ematics major from its form during their own under- core mathematics (algebra, analysis, topology, geom- graduate training, as career opportunities for mathe- etry) that every mathematical sciences major should matics majors have changed. In part, the members study? lament the passing of the mathematics program that ANSWER:No. There is no longer a common body nurtured their love of mathematics. At the same time of pure mathematical information that every student they acknowledge the challenge of the diversity of the should know. Rather, a department’s program must be present and future. They realize that it is not now tailored according to its perception of its role and the realistic for CUPM to recommend a core set of pure needs of its students. Whether pure mathematics is re- mathematics courses to be taken by all mathematical quired of all in some substantial way; whether it is used sciences majors in every institution. as an introduction to advanced work of applied nature While the mathematics major has generally broad- or as a completion to an applied program; or whether ened towards a mathematical sciences major, it is still pure mathematics is simply one track in a collection of possible for an institution, large or small, to elect to re- programs in a large department will be an institutional tain a traditional pure mathematics major, alone or in option. Departments must recognize this fact, establish conjunction with an applied mathematics major. But their programs with a clear understanding of objectives it is clearly more appropriate to work within current that are being met, and be prepared to share and ex- realities to fashion a unified mathematical sciences ma- plain these perceptions with their students. The limited jor with diminished pure content, a major incorporating resources of smaller departments must be exploited with 26 RESHAPINGCOLLEGE MATHEMA~E great efficiency and wisdom. Such departments may in the field. Syllabi and approaches in pure matheniat- face a difficult decision of whether to abandon certain ics courses must be adapted to changing constituencies traditional branches of mathematics entirely in order to with a careful balance of learning new concepts and offer courses and tracks best suited to their students. modes of reasoning and of using these constructs, a The underlying problem is that students enter college balance of “listening” and “doing.” Students should with much less mathematics than they used to, but they emerge from a course feeling that they have become expect to leave with more. There is a wide span of junior experts in some topics: they should know facts preparation among entering college students, they want and relationships, know some of the “whys” behind this an education that is specific to chosen career goals, and mathematics. the levels of mathematical and computational skills and It would be desirable for courses to be structured sophistication that accompany these goals have risen. with review stages that require reflection by students of Core courses such as abstract algebra and analysis are what analysis to use to solve a problem. The courses valuable for continuing study in many fields, but they need to contain assignments that ask for short proofs of are not essential for all careers. results and for application of concepts and techniques The Core Mathematics Subpanel and the parent from one problem to another (apparently unrelated) Mathematical Sciences Panel jointly recommend that problem. Proper judgment in the selection of a method all mathematical sciences students take a sequence of of analysis is the key both to constructing mathematical two courses leading to the study of some subject in proofs and to problem solving in applied mathematics; depth (see the first chapter, “Mathematical Sciences”). nurturing this ability is the critical challenge to instructors. Students should be required to present material QUESTION2: Should there be major changes in the both orally and in writing on a regular basis. Since content or mode of instruction of upper-level core math- students do not have to know a standard body of theo- ematics courses? rems for graduate study, the course content in algebra, ANSWER:While there will continue to be some stu- analysis, topology and geometry can vary according; to dents who plan to move toward a doctorate in pure, faculty interests and possible ties with stronger quanti- or applied, mathematics and an academic career, the tative areas of an institution (e.g., physics or biology). mathematical sciences major is seen by most students as The density of proofs in an upper-level course is al- preparation for immediate employment or for Masters- ways a controversial issue. It is traditional to feel tlhat level graduate training in areas outside of mathemat- one objective of such a course is to teach students how to ics (but where mathematical tools are needed). Thus construct proofs. However, this skill comes slowly and mathematics departments can no longer view their seldom arouses the same pleasure in students as it does upper-division courses as a collection of courses that in instructors. Some proofs are needed in any upper- faculty wish they had had prior to admission to gradu- level mathematics course to knit together the enlire ate school. Rather, departments must offer pure math- structure that is being presented, but one should priob- ematics courses that are compatible with the overall ably aim at piecewise rigor rather than a Landauesque goals of a mathematical sciences major, courses that totality. Students’ mathematical maturity will develop are intellectually and pedagogically complete in them- as much, and it will be far less painful. selves, courses that are both the beginning and the end The preceding pedagogical goals in core mathemiat- of most students’ study of the subject. The main objec- ics must accommodate the reality that courses such as tive in such courses now is developing a deeper sense of abstract algebra may only be offered in alternate years mathematical analysis and associated abstract problem- and that two-semester sequences or courses with core solving abilities. In these courses students learn how to mathematics prerequisites will be difficult to schedule. learn mathematics. With a broad mixture of students in infrequently-offered There is always a continuing need to re-examine the courses, instructors must be sensitive to the discourage- nature and content of any course. Some courses carry ment some students may feel in the presence of more baggage that may be there largely for historical reasons. sophisticated seniors. A frequent example of this is the traditional course in QUESTION3: How can the full scope of mathemaiics differential equations which is populated by isolated dis- be conveyed to students? Should this be done by one- coveries of the Bernoulli clan (and lacking in discussion semester survey courses that cover a range of fields? of numerical methods). Instructors are slow to discard topics that have a strong aesthetic appeal (for the in- ANSWER: Students pursuing specific career goals structors) but are no longer important building blocks in mathematical sciences and those taking upper-level COREMATHEMATICS 27 mathematical “servicen courses need to be made aware ways the power and usefulness of mathematical abstrac- of the depth and breadth of mathematics and the tion and generalization. greater mathematical maturity that their subsequent There are two important provisos about senior-year careers may demand. Mathematical survey courses do courses. First, when core courses cannot be offered ev- not appear to be the answer. They will not be able to ery year, they obviously must be accessible to most move beyond vocabulary and notation to give any sense juniors. Second, the mathematicalby gifted student of global structure in any of the fields covered. (whether a mathematics major or not) must be able Physicists seem to have been remarkably successful to take such senior core courses in the sophomore year in communicating some understanding about the “big without needing applied prerequisites that other stu- picture” to their students and laymen through exposi- dents naturally take before the core course. Such gifted tory articles that treat highly technical subjects by pre- students today are often directed towards popular ca- senting only a projection or shadow of the true struc- reers such as engineering or medicine and by their se- ture, but doing so in a way that does not seem to offend nior year would be too immersed in professional training their consciences. Similar approaches should be possi- to take the pure mathematics course that would reveal ble in mathematics using expository American Mathe- their mathematical research potential. matical Monthly, Mathematics Magazine, or Scientific It is worthwhile recalling that before 1950 few col- American articles. Following the reading of such an leges offered regular courses in abstract algebra, topol- article, a (once-a-week) class would discuss concepts, ogy, or up-to-date advanced calculus. The 1950’s and technicalities and applications in the article plus addi- 1960’s were memorable in mathematics education, but tional examples. Natural topic areas are complex analy- today’s students must be viewed as in the historical sis and two-dimensional hydrodynamics; number theory mainstream rather than as slow in learning to handle and public key cryptography; calculus of variations and abstractions. soap films; queueing theory and, say, toll road design. Individual institutions will differ greatly in the de- More traditional ways of projecting the wide-ranging sign of such senior courses. As noted in the discussion nature of mathematics are by rotation of courses and by of Question 2, these courses should require oral and providing seminars, extracurricular mathematical ac- written student presentations. The spirit of this recom- tivities, summer work opportunities, and by reference mendation could be achieved with a year-long course in and linkages to mathematics in courses in other depart- a subject such as differential equations or combinatorics ments. This breadth should also give a sense of the that begins with applications and lealds to abstraction rapidly changing nature of uses of mathematics and of or a course that begins with abstraction and leads to the need of learning how to learn mathematics. applications.

QUESTION4: Should pure mathematics courses be postponed for most students until the senior year to fol- Sample Course Outlines low and abstract from more applied courses earlier in In this section we discuss two approaches to the fun- the curriculum? damental upper-level core subjects of abstract algebra ANSWER:Many mathematical sciences students who and analysis. We suggest an ideal treatment and then prefer problem solving to theory appear to have con- a more modest version that is appropriate for most cur- siderable difficulty in their sophomore or junior years rent mathematical sciences students. The descriptions with abstract core mathematics. For these students, are stated in terms of student objectives. core mathematics may better wait until a senior year The philosophy behind each of the course descrip- “capstone” course(s) that builds on maturity developed tions is that the student needs a working understanding in earlier problem solving courses. This course (prefer- of the subject far more than a detailed intensive and ably year-long if only one such course is required) in critical knowledge. The instructor’s central goal is to a subject such as analysis or abstract algebra would teach the student how to learn mathematics, expecting build a student’s capacity (and appetite) for abstraction that students will correctly retain only a tiny portion and proof and for solving complex problems involving a of what was taught, but that when they need to refresh combination of analytical techniques. The course would their knowledge, they will be far better able to do so seek depth rather than breadth. The course should link than if they had never taken the course. Proofs are not abstract concepts with their concrete uses in previous of major importance, but in both approaches students courses, such as integration concepts used in limiting should be able to understand what the hypotheses of probability distributions. It should illustrate in several a theorem mean and how to check them. They should 28 RESHAPINGCOLLEGE MATHEMA= also be able to detect when seemingly plausible state- B. Study the class of continuous maps from a region in ments are false (and should be shown counterexamples Rn into Rm, and the special properties of maps in to such statements; e.g., integrals that should converge class C’ and C”. but do not). C. Study integration of continuous and piecewise ‘continuous functions over appropriately chosen sets, Abstract Algebra (Ideal) I bounded and unbounded, and then extend this to A. Give the student a guided tour through the algebraic integration with respect to set functions. “zoo,” so that he or she knows what it means to D. Extend to the theory of differential forms and de- be a group, a ring, a field, an associative algebra, velop a relationship between differentiation of forms etc. Include associated concepts such as category, and the boundary operator, via Stokes’ theorem. morphism, isomorphism, coset, ideal, etc. B. Show the student useful ways for generating one al- Analysis I1 (Modest) gebraic structure out of another, such as automor- A. Give the student a glossary of terms in point set phism groups, quotient groups, algebras of transfor- topology, appropriate also to a metric space and ap- mations, etc. plied to R”,and practice in their meanings. (Do not C. Give the student an understanding of the basic prove inter-relations, but state them clearly.) structure theorems for each of the algebraic sys- B. Introduce the class of C” maps from Rn into .Rm, tems discussed, as well as an understanding of their and discuss a few problems involving such functions, proofs. each motivated by a concrete “real” situation. Solve D. Give the student experience in using the preceding each of the problems by stating and illustrating the ideas and constructions and seeing how these ideas appropriate general theorems, and in a few cases, arise in other branches of mathematics (analysis, sketching part of the proofs. number theory, geometry, etc.). C. Discuss integration in terms of measurement ,and E. Show the student how algebra is used in fields out- averaging, extend this to Rn,and explain briefly side of mathematics, such as physics, genetics, in- techniques of numerical integration. At all stages formation theory, etc. give attention to improper integrals. Abstract Algebra I1 (Modest) D. Extend the notion of function to differential forms, A. Combine parts of A and B of Course I by show- illustrated with physical and geometric examples. ing students at least two different types of algebraic Motivate Stokes’ theorem as the analogue of the structures and several instances in which such an fundamental theorem of calculus, and arrive art a algebraic structure evolved or is constructed out of correct formulation of it without proof. Illustrate another mathematical structure. The goal is for a the theorem with examples, including some invdv- student to be able to recognize when a situation has ing the geometric topology of surfaces; if students’ aspects that lend themselves to an algebraic formu- background is appropriate, examples in physics (hy- lation; e.g., rings out of polynomials. drodynamics or electromagnetism) should be given. B. Describe part of the theory for one of the structures An analysis course can also be given an “advanced introduced in A and illustrate several of the deduc- calculus” emphasis including topics such as Fourier se- tive steps in the theory. Students should see the ries and transforms, special functions, and fixed-point nature of tight logical reasoning and the usefulness theorems, with applications of these topics to differen- of algebraic concepts, as well as come to appreciate tial equations. For further discussion of this approach, the cleverness of the theory’s discoverers. see versions one and three of Mathematics 5 in the C. Discuss at least one application of algebra outside CUPM recommendations for a Genera2 CurricuZum in of mathematics. Mathematics for Colleges (revised 1972). D. Assign students a variety of problems which require recognition of algebraic structures in unfamiliar forms, proof of small deductive steps, and use of Subpanel Members theory in B. PAULCAMPBELL, CHAIR, Beloit College. Analysis I (Ideal) LIDA BARRETT, Northern Illinois University. A. Give the student a working knowledge of point set R. CREIGHTONBUCK, University of Wisconsin. topology in Rn and analogous concepts for a metric MARGARETHUTCHINSON , University of St. Thomas. space. Computer Science

This chapter contains the report of the Subpanel on ences major. A mathematical sciences undergraduate Computer Science of the CUPM Panel on a General program and a computer science undergraduate pro- Mathematical Sciences Program, reprinted with minor gram should complement one another to the advantage changes from Chapter IV of the 1981 CUPM report en- of both departments and their students (for example, titled RECOMMENDATIONSFOR A GENERALMATHE- see the description of the interaction at Potsdam State MATICAL SCIENCESPROGRAM. in Chapter I, "A General Mathematical Science Pro- gram"). A Growing Discipline Computer Science is a new and rapidly growing sci- Introductory Courses entific discipline. It is distinct from Mathematics and Electrical Engineering. The subject was once closely The foundation for a computer science component in identified in mathematicians' minds with writing com- a mathematics department is a one-year introductory puter programs. In the beginning, however, computer sequence. Courses CS1 and CS2, prolposed in the As- scientists concentrated on the discipline's mathemati- sociation of Computing Machinery Curriculum 78 (see cal theories of numerical analysis, automata, and re- last section of this chapter), are excellent models for cursive functions, as well as on programming. In the this year sequence. The Subpanel on Computer Science past decade, theories developed to understand problems endorses the objectives of these two courses, and rec- in software design (compilers, operation systems, struc- ommends that all mathematical sciences majors should tured programs, etc.) have blossomed. These theories be required to take the first course and strongly encour- involve the analysis of complex finite structures, and in aged to take the second course in this sequence. If the this sense have a strong mathematical bond with the second course is not required, substantial use of com- finite structures common in operations research and di- puters should be an integral part of other mathematical verse areas of applied mathematics. sciences courses. More importantly, these computer science theories The primary emphasis in the first 'course should be are needed by analysts who design algorithms for com- on: plex problems in the mathematical sciences. For this Problem solving methods and algorithmic design reason, all mathematical sciences students must be and analysis, given an introduction to the basic concepts of computer Implementing problem solutions in a widely used science. Further, facility in computer programming is higher-level programming language, required of all mathematical sciences students so that * Techniques of good programming i~tyle,and they can perform practical computations in mathemat- Proper documentation. ical sciences courses and in subsequent mathematical Lectures should include brief surveys of the history of sciences careers. computing, hardware and architecture, and operating Although only one-third of the country's colleges systems. and universities now have computer science depart- The second course should include at least one major ments, the number of students currently majoring in project. The course should cover topics such as recur- computer science taught in a computer science depart- sive programming, pointers, stacks, queues, linked lists, ment (approximately 50,000 students) is greater than string processing, searching and sorting techniques. the number of all majors in mathematics, mathemati- The concepts of data abstraction and algorithmic com- cal sciences, and applied mathematics. The computer plexity should be introduced. Proofs of correctness may science recommendations in this chapter are designed also be discussed. for institutions where computer science is taught in a Good design and style in programming should be em- mathematical sciences department or in a mathemat- phasized throughout both courses: the use of identi- ics department. When a separate computer science fiers to indicate scope, modularity, appropriate choice department exists, that department's diversity of com- of identifiers, good error recovery procedures, checks puter science offerings will enhance a mathematical sci- for integrity of input, and appropriate commentary and 30 RESHAPINGCOLLEGE MATHEMAE documentation. Of course, efficient algorithms and cod- cepts in programming languages, data structures, and ing should also be stressed. There is a strong tendency computer hardware. among students to worry only about whether their pro- The intermediate-level courses should be taught by grams run correctly. Through class lectures and care- a computer scientist, that is, by an individual who has ful grading of programming assignments, the instructor significant graduate-level training in computer science must teach the students the importance of good design, (see below). style, and efficiency in programming. The Subpanel on Computer Science, in concurrence A source of useful commentary about introductory with ACM curriculum groups, strongly rejects the idea computer science courses is the SIGCSE (Special Inter- of a set of courses that each address a specific pro- est Group on Computer Science Education) Bulletin. gramming language, e.g., a sequence of advanced FOR- The bulletin is published quarterly, and issue #1 each TRAN, COBOL, RPG, and APL. The argument for year, which contains papers presented at the SIGCSE such a sequence is usually based on the employability annual meeting, is especially valuable. of students completing it. If indeed this argument is Most introductory texts have many sample projects. valid, and there is some question about that, it is a short In addition, the following three texts are good general range benefit. Students completing such a sequence will sources of computer projects. soon find that the lack of underlying concepts will put 1. Bennett, William R., Scientific and Engineering them at a severe disadvantage. However, it may be ac- Problem-Solving with the Computer, Prentice-Hall, ceptable, resources permitting, to have one “vocational” Englewood Cliffs, New Jersey, 1976. elective course that studies a second higher-level lan- 2. Gruenberger, Fred and Jaffray, G., Problems for guage such as COBOL. Of course, it is also natural to Computer Solution, John Wiley & Sons, New York, discuss new programming languages in several interme- 1965. diate (and advanced) computer science courses. How- 3. Wetherall, Charles, Etudes for Programmers, Pren- ever, the new language would not be the focus of the tice-Hall, Englewood Cliffs, New Jersey, 1978. course, but rather a tool used in learning and illust,rat- Mathematicians teaching introductory computer sci- ing fundamental concepts. ence often emphasize numerical computation in pro- The role of numerical and computational mathernat- gramming assignments. At the introductory level, the ics in computer science has diminished in recent years. computer science issues involved in numerical computa- While the ACM Curriculum 68 treated numerical anal- tion are quite simple, Assignments requiring symbolic ysis as part of core computer science, today numeirical manipulation and data organization present more sub- mathematics is considered by most computer scientists stantive programming problems and, in general, require to be simply another mathematical sciences field that more thought. The following is a sample assignment has overlap with computer science. Numerical math- that could be given late in the first course: ematics is very important in a mathematical sciences Write a program which obtains a five-card major, but it is not a part of the computer science com- poker hand from some source (terminal, input ponent . deck, or file), prints the hand in a reasonably Following the CS1 and CS2 courses, the ACM Cur- well-formatted style, and determines whether or riculum 78 specifies six additional courses in core com- not the hand contains a pair, three of a kind, a puter science. straight, a full house, etc. CS3 Introduction to Computer Systems CS4 Introduction to Computer Organization CS5 Introduction to File Processing Intermediate Courses CS6 Operating Systems and Computer Architecture CS7 Data Structures and Algorithm Analysis Intermediate-level computer science courses building CS8 Organization of Programming Languages on CS1 and CS2 should address basic underlying issues The syllabi of these courses are given at the end of in computer science. In describing computer science this chapter. Ideally, all six of these courses would be in the first two years, the ACM Curriculum 78 report offered. A concentration or a minor in computer sci- states that the student should be given “a thorough ence would commonly consist of CS1 and CS2, followed grounding in the implementation of algorithms in pro- by two of CS3, CS4, and CS5, and two of CS6, CS7, gramming languages which operate on data structures and CS8. For the purposes of a mathematical scieiices in the environment of hardware.” Thus these courses program, it may be justified to place more emphasis on should develop general topics about algorithms, con- the software oriented areas. This would imply, if there COMPUTERSCIENCE 31 was difficulty in offering all six courses, that CS3, CS5, 4. Memory management (15%) CS7, and CS8 would be most useful. Then CS3, CS5, 5. Process management (15%) CS7, and CS8 would be offered once a year, and CS4 6. Systems design (10%) and CS6 offered as topics courses every other year. 7. Exams (5%) At many schools, it may not be feasible to offer at It is important to note that an individual wishing to least four of these intermediate courses in computer sci- go on from these courses to advanced .work in computer ence on a regular basis. Then one can combine parts of science may have to make up, as deficiencies, areas in these intermediate courses to provide a significant offer- core computer science that are not represented in these ing in two courses above CS1 and CS2. In this case, only condensed pairs of courses. two computer science courses, one elementary and one intermediate, would be offered each semester. One approach would be to combine topics from CS5 and CS7 Concentrations and Minors; into one course, and topics from CS3, CS4, and CS6 into the other. This would yield two courses with the A computer science concentration in a college mathe- following sort of syllabi (for more details about these matics department can be defined as an option within a topics, see the ACM Curriculum 78 syllabi at the end mathematical sciences major or as a “stand-alone” mi- of this chapter): nor. A computer science minor should consist of about Al. Algorithms for Data Manipulation six courses, ACM Curriculum 78 couirses CS1 and CS2 1. Algorithm design and development illustrated in plus four intermediate courses. areas of sorting and research (25%) A computer science concentration within a mathe- 2. Data structure implementation (30%) matical sciences major has three components: 3. Access methods (25%) A. Mathematics: 5-plus courses; 4. Systems design (15%) B. Computer Science: 4-6 courses; 5. Exams(5%) C. Applied Mathematics: 3-plus courses. A2. Computer Structures A. The mathematics component would include 1. Basic logic design (15%) the three semester freshman-sophomore “calculus se- 2. Number representation and arithmetic (10%) quence” plus linear algebra. As recommended in Chap- 3. Assembly systems (35%) ter I, “A General Mathematical Sciences Program,” any 4. Program segmentation and linkage (15%) mathematical sciences major should contain upper-level 5. Memory management (10%) course work of a theoretical nature, typically algebra or 6. Computer systems structure (10%) advanced calculus. In a major with a computer science 7. Exams (5%) concentration, algebra is the natural area. Specifically, This approach focuses on data structures and soft- the applied algebra course given in Chaper I would be ware issues that relate to operating systems. An al- excellent for the computer science concentration. The ternative approach could concentrate on programming course’s syllabus incorporates most of the topics of the languages and algorithms involved in computer systems ACM 78 discrete mathematics course (required of com- performance. This theme could be realized by combin- puter science majors). A small department could offer ing topics in CS3, CS5, and CS8 into one course, and applied algebra and standard abstract algebra courses topics in CS4, CS6, and CS7 into the other course. This in alternate years. Logic and automata theory are at- would yield two courses with the following syllabi: tractive electives in the mathematics component if a Bl. Language Types and Structures mathematics department wishes to focus on more the- 1. Assembly systems (25%) oretical aspects of computer science. 2. Program segmentation and linkage (15%) It should be noted that several computer science ed- 3. Language definition structure (10%) ucators have questioned the reliance on calculus as the 4. Data types and structures (15%) basic mathematics for future computer scientists; ACM 5. Control structures and data flow (20%) Curriculum 78, for instance, requires a (freshman) year 6. Access methods (10%) of calculus. They advocate a mathematics component 7. Exams (5%) based on discrete mathematics with only one semester B2. Algorithms for Computer Systems of calculus (taught, say, in the junior year). See A. Ral- 1. Basic logic design (20%) ston and M. Shaw, “Curriculum 78--Is Computer Sci- 2. Algorithm design and analysis (20%) ence Really that Unmathematical?”, Commzlnications 3. Procedure activation algorithms (15%) ACM23 (1980), pp. 67-70. 32 RESHAPINGCOLLEGE MATHEMAT?

B. The computer science component would include since so many smaller colleges are located away firom ACM Curriculum 78 courses CS1 and CS2 plus two to the metropolitan areas where most technical and scien- four intermediate courses, as described in the preceding tific employers of such adjunct faculty are found, this section. The syllabi of ACM Curriculum 78 core courses solution will not be useful to most smaller institutions. are given at the end of this chapter. A crucial point that must be emphasized when using C. The applied mathematics component should in- existing non-computer science faculty (i.e., mathemati- clude a course in numerical analysis and a course in cians) to teach computer science courses is that com- probability and statistics. The third applied mathe- puter science cannot be treated like most other new matics course would be discrete methods, which would mathematics course topics which mathematicians will cover the combinatorial material in the ACM Curricu- (quickly) learn as they teach it. Mathematicians un- lum 78 discrete mathematics course in greater depth, in- trained in computer science are very likely to teach cluding operations-research-relatedgraph modeling (see computer science badly, hurting both the students and Chapter I for a full description of this course). The the mathematics department’s reputation. Therefore, CUPM Mathematical Sciences Program panel recom- if a current mathematics faculty member is to be used mends that all mathematics departments should offer to teach computer science, especially beyond the first a discrete methods course. Other good courses for the course, he or she must first acquire some formal educa- applied mathematics component are ordinarily differen- tion in computer science. tial equations, mathematical modeling, and operations The most plausible approach to such computer sci- research. The 1971 CUPM Report on Computational ence training is through some program of released time. Mathematics describes courses in computational mod- The pertinent questions about the training are: how els, in combinatorial computation, and in differential long? where? and how financed? equations with numerical methods; these courses com- Assuming that the mathematician who is to be bine topics from a variety of mathematical sciences and trained is, at most, familiar with programming in a computer science courses and hence are particularly at- high-level language, then full-time study for one year is tractive to small departments. the minimum period needed to acquire the background, In either the computer science concentration or mi- knowledge, and experience necessary to teach several nor, all six computer science courses are needed for of the intermediate-level core computer science courses. future graduate study in computer science. Incoming Since one year is also the maximum period which would graduate students with less preparation are commonly be administratively or financially feasible, this shoiild required to make up undergraduate course deficiencies. be viewed as the canonical period for faculty training in computer science. Part-time study over a longer period or a succession of summers can also be considered. Faculty Training However, both because the needs to train faculty in For the foreseeable future, the dominant factor af- computer science are pressing and because intermittent fecting computer science instruction at all institutions, study is almost always less effective than continuous but particularly at smaller colleges and universities, will study, at least one faculty member in a mathematics be the extreme shortage of qualified computer scientists department should have completed a one-year program in academe. At smaller colleges and universities it may of full-time study in computer science. therefore be effectively impossible to hire a computer The most logical place at which to study computer scientist to teach core computer science courses. Among science for the purpose of becoming able to teach it is at the possible solutions to this problem are: a university with undergraduate and graduate (prefer- 1. Using adjunct faculty to teach computer science ably Ph.D.) programs in computer science. Although courses. there are exceptions, the current level of computer sci- 2. Using existing (non-computer science) faculty to ence instruction in American colleges and universities is teach computer science courses. so uneven that only at such institutions can one be rea- The first solution is acceptable for some courses. Al- sonably assured of an atmosphere in which there will be though one cannot build a program with adjunct fac- the necessary broad understanding of the principles of ulty and although staffing courses with adjunct faculty computer science. Such an atmosphere is particularly is never as desirable as using full-time faculty (e.g., stu- important for an academic mathematician preparing to dent advising is a particular problem), this is a feasi- teach the subject. ble way to get computer science courses taught when Another possibility which should be mentioned is for such faculty exist in the local community. However, the faculty member to spend one year at one of those COMPUTERSCIENCE 33

(relatively few) major industrial firms with good in- great speed or memory size, such as “number crunch- house training programs in computer science. An addi- ing,” and smaller computers for student programs and tional attraction to this idea is that it might be possible other instructional purposes. to arrange an exchange in which a member of the firm taught at the college for a year. CENTRALIZED FACILITIES Methods of financing such a program of faculty train- Historically, so-called “economies of scale” encour- ing in computer science are fairly obvious: aged the development of increasingly larger computers; Through released time at full pay from the mathe- and of increasingly larger organizations to administer matician’s home institution. them. Such computer systems are caplable of providing Through grants from current, and hopefully new, a great variety of services with a low cost for each ser- federal programs; officials of both the MAA and vice. In addition, the organizations .which administer ACM are currently pressing NSF to provide more these systems can play an important, role in develop- funds for this purpose. ing and supporting instructional uses of computing on Through grants from private foundations; individ- campus. ual institutions and departments may be more ef- On the other hand, the very size of such facilities fective than professional associations in obtaining and the organizations that administer them create cer- such private funds. tain problems. First, large systems have a high unit Through corporate sponsorship of participation in cost, in the range of half a million to several million in-house training programs or academic-corporate dollars; replacing or enhancing such i3 system involves exchanges. a major administrative decision. Second, instructional users of such systems must often compete with other powerful and better-financed constituencies; either separate facilities are needed to reduce competition among Computer Facilities instructional, research, and administrative uses of the computer, or policies are needed to allocate the services Facilities to support computing in mathematical sci- provided by a single facility. And third, large organiza- ences instruction can be provided in a variety ways, of tions can be bureaucratic and inflexible. ranging from one large centrally administered system to many small personal computing devices. The suitabil- DEPARTMENTALCOMPUTERS ity of a particular means depends not only upon its in- For the last ten years minicomputers have provided tended applications, but also upon factors such as cost, an alternative to a large centralized facility. Lower unit ease of use, and local politics. At present, computing costs (around $100,000 or less) and the possibility of services in most colleges and universities are provided local control have made it attractive for academic and by a large centralized facility, the Computing Center. administrative departments to acquire facilities of their Growing numbers of institutions, however, are begin- own. Such facilities can be tailored t.0 a department’s ning to decentralize computing on campus. Three cur- needs and can provide almost as many services as a rent modes of providing service are discussed below: large centralized system. Centralized facilities Minicomputers, however, are not necessarily the an- * Departmental computers swer to every department’s computiing needs. First, Personal computers. there is the question of which services they will provide. There is a fourth mode that is primarily a form of access Second, there are hidden costs associated with admin- to centralized or departmental computers: istering any computer facility: personnel are needed to Terminals operate and maintain the facility and to provide tech- The second half of this section discusses the cost and nical assistance to users. Small departments run the ease of implementation of various applications with dif- risk of diverting attention from their primary task of ferent types of computing facilities. teaching mathematics to the subsidiary task of manag- It should be noted that it is possible for an institu- ing such an enterprise. One way to deal with such hid- tion to form a consortium with nearby schools to op- den costs is for departments to contract with a central erate a common central computing facility or to buy campus organization to manage their facilities. Third, time (and services) from commercial computing centers. there are inconveniences for students faced with using, This option allows an institution to have a mix of com- and first learning to use, several different departmental puting, using large computers for problems requiring systems. Of course, this difficulty can be overcome by 34 RESHAPINGCOLLEGE MATH EM AT^

requiring departments to purchase compatible systems INTRODUCTORYPROGRAMMING and by interconnecting all systems. Any of the three types of facilities can serve as a vehi- Many academic computing specialists expect inter- cle for teaching beginners to program and for introduc- connected departmental computers to become the dom- ing computational examples into elementary mathemat- inant means of academic computing in the next decade. ics courses. Such uses typically involve large numbers of PERSONALCOMPUTERS students writing relatively simple programs. Larger facilities tend to provide a greater choice of programming The recent development of personal microcomputers languages, although modern languages such as PAS- provides another alternative for instructional comput- CAL and PL/I are becoming increasingly available even ing. Very low unit costs (one or two thousand dollars) on microcomputers. Larger machines tend to be faster make computing possible for departments otherwise un- also; even though use of such machines is shared, stu- able to afford or gain authorization for large facilities. dents will find that they process simple programs much Microcomputer facilities suffer from many of the same faster than microcomputers. Costs, however, tend to be problems as minicomputer facilities. In addition, mi- roughly equal for simple interactive computing on the crocomputers are limited in the services they provide, three types of facilities-around $2.00 per hour. These are slower than their large competitors, and may not costs can be reduced significantly by using larger ma- be designed for rugged use by large groups of students. chines in a noninteractive, batch-processing mode. This Still they can prove quite adequate for elementary ap- mode of use, while predominant in the past, is becom- plications. Further, by being less intimidating and more ing less popular as minicomputers and microcomputers exciting than larger computers, they can play a role in make a more responsive computing environment avail- overcoming a student’s ‘‘computer anxiety.” able and affordable.

TERMINALS ADVANCEDPROGRAMMING Terminals are used for remote, interactive access to Advanced programming is more distinguished from large computers. Some have small memories and prim- introductory programming in its requirements for more itive editing capabilities. Departments often have a sophisticated languages and for facilities to handle large greater choice in selecting terminals to connect to com- programs. Microcomputers at present do not meet puter systems than they do in selecting the systems these requirements; the languages they provide are themselves. Cost, speed, and durability are primary quite restrictive, and large programs exceed their ca- factors influencing the selection of a terminal. By these pacity. Execution times and costs for large programs criteria, video terminals are preferable. The availability tend to be lowest on large machines under batch proof graphical output and local editing features are other cessing, but minicomputers are becoming competitive factors to consider when choosing terminals. Hard-copy both in price and speed. (printing) terminals are more expensive and tend to be PROGRAM DEVELOPMENT AND MAINTENANCE slower than video terminals, but they do provide users with a permanent record of their work, and so some Program development is influenced heavily by the printing terminals are necessary (medium or high speed computing environment in which it occurs. Corive- printers can be used in conjunction with video termi- nient interactive editing capabilities accelerate the task nals to provide this record). Video terminals may also of writing and correcting a program; microcomputers, be used in conjunction with television monitors to pro- with almost instantaneous response, do a particularly vide classroom displays of computer output. For such good job of editing. Facilities for file storage enable output to be visible in a large classroom, either many program development to be spread over several ses- monitors must be provided or the video terminals em- sions. Large machines provide less expensive storage ployed must use larger, and hence fewer, characters in and much faster retrieval of information; they also fa- their display. cilitate sharing programs among users and provide centralized backup. Microcomputer facilities can distribute Applications the costs of file storage by requiring users to purchase individual floppy disks, but unless a centralized store The suitability of a particular computing facility de- is provided through a network, sharing information can pends most upon its intended applications. The rest of be difficult. this section discusses the most common academic uses of computers and how well different types of computing GRAPHICs facilities serve these uses. One of the primary attractions of personal microcom- COMPUTERSCIENCE 35 puters is their ability to generate graphic displays and To teach how to design, code, debug, and docu- to enable users to interact with these displays. Larger ment programs using techniques of good program- systems, unless specifically tailored to graphic applica- ming style. tions, tend to have primitive graphic facilities at best. COURSEOUTLINE: APPLICATIONPACKAGES The material on a high-level programming language Application packages available for various machines and on algorithm development can be taught best as an provide aids for numerical and symbolic computations. integrated whole. Thus the topics should not be cov- Typical areas of application include statistics, linear ered sequentially. The emphasis of the 'course is on the programming, numerical solution of differential equa- techniques of algorithm development and programming tions, and algebraic formula manipulation. Such pack- with style. Neither esoteric features of a programming ages are more widely available on larger machines. language nor other aspects of computers should be al- Large computations often require an unacceptably long lowed to interfere with that purpose. time on microcomputers (several hours) and may exceed the memory size of small computers. TOPICS: A. Computer Organization. An overview identifying MISCELLANEOUSAPPLICATIONS components and their functions, machine and as- Word processing systems facilitate production of sembly languages. (5%) course notes, research papers, and term papers. If good B. Programming Language and Progrcrmming. Repre- word processing facilities are available, they are likely sentation of integers, real, characters, instructions. to quickly generate heavy faculty use. Simple word Data types, constants, variables. Arithmetic ex- processing software is available for personal computers, pression. Assignment statement. Logical expres- but a minicomputer (or powerful $5,OOO-plus microcom- sion. Sequencing, alternation, and iteration. Ar- puter) is needed for good mathematically-oriented word rays. Subprograms and parameters. Simple I/O. processing software, such as the UNIX system. Large Programming projects utilizing concepts and em- computers often have poor word processing capabilities. phasizing good programming style. (45%) Data base systems are of more use in the social C. Algorithm Development. Techniques of problem sciences than in the mathematical sciences, but can solving. Flowcharting. Stepwise refiinement. Simple be used to provide real data for analysis in statistics numerical examples. Algorithms foi: searching (e.g., courses. Such systems require a centralized file store on linear, binary), sorting (e.g., exchitnge, insertion), a larger computer. merging of ordered lists. Examples taken from such Real-time data acquisition is of interest in the natural areas as business applications involving data manip- sciences. They can also be used to provide real data for ulation, and simulations involving games. (45%) mathematical analysis. Dedicated microcomputers are D. Ezaminations. (5%) better suited to laboratory instrumentation than are shared machines. CS2. Computer Programming I1

ACM Curriculum 78 OBJECTIVES: * To continue the development of discipline in pro- The following computer science course syllabi are re- gram design, in style and expression, in debugging produced from the ACM Curriculum 78 Report in Com- and testing, especially for larger programs; munications of ACM, March 1979, pp. 147-166. (Copy- To introduce algorithmic analysis; and right 1979, Association for Computing Machinery, Inc.) To introduce basic aspects of string processing, re- a computer science They provide eight core courses for cursion, internal search/sort methods and simple major. data structures. CS1. Computer Programming I PREREQUISITE:CS 1. OBJECTIVES: COURSEOUTLINE: To introduce problem solving methods and algo- The topics in this outline should be introduced as rithm development; needed in the context of one or more projects involv- To teach a high-level programming language that is ing larger programs. The instructor may choose to be- widely used; and gin with the statement of a sizable project, then utilize 36 RESHAPINGCOLLEGE MATHEMATICS structured programming techniques to develop a num- and the nature and orientation of CS4 described below. ber of small projects each of which involves string pro- Enough assembly language details should be covered cessing, recursion, searching and sorting, or data struc- and projects assigned so that the student gains expe- tures. The emphasis on good programming style, ex- rience in programming a specific computer. However, pression, and documentation, begun in CS1, should be concepts and techniques that apply to a broad range of continued. In order to do this effectively, it may be computers should be emphasized. Programming meth- necessary to introduce a second language (especially if ods that are developed in CS1 and CS2 should also be a language like Fortran is used in CS1). In that case, utilized in this course. details of the language should be included in the outline. TOPICS: Analysis of algorithms should be introduced, but at this level such analysis should be given by the instructor to A. Computer Structure and Machine Language. Mem- the student. ory, control, processing and 1/0 units. Registers, Consideration should be given to the implementa- principal machine instruction types and their for- tion of programming projects by organizing students mats. Character representation. Program con- into programming teams. This technique is essential trol. Fetch-execute cycle. Timing. 1/0 Operations. in advanced level courses and should be attempted as (15%) early as possible in the curriculum. If large class size B. Assembly Language. Mnemonic operations. S,ym- makes such an approach impractical, every effort should bolic addresses. Assembler concepts and instruction be made to have each student's programs read and cri- format. Data-word definition. Literals. Location tiqued by another student. counter. Error flags and messages. Implementation of high-level language constructs. (30%) TOPICS: C. Addressing Techniques. Indexing. Indirect Address- A. Review. Principles of good programming style, ex- ing. Absolute and relative addressing. (5%) pression, and documentation. Details of a second D. Macros. Definition. Call. Parameters. Expansion. language if appropriate. (15%) Nesting. Conditional assembly. (10%) B. Structured Programming Concepts. Control flow. E. File I/O. Basic physical characteristics of 1/0 and Invariant relation of a loop. Stepwise refinement of auxiliary storage devices. File control system. K/O both statements and data structures, or topdown specification statements and device handlers. Data programming. (40%) handling, including buffering and blocking. (5%) C. Debugging and Testing. (10%) F. Program Segmentation and Linkage. Subroutines. D. String Processing. Concatenation. Substrings. Coroutines. Recursive and re-entrant routines. Matching. (5%) (20%) E. Internal Searching and Sorting. Methods such as G. Assembler Construction. One-pass and two-pass as- binary, radix, Shell, quicksort, merge sort. Hash semblers. Relocation. Relocatable loaders. (5%)1 coding. (10%) H. Interpretive Routines. Simulators. Trace. (5%) F. Data Structures. Linear allocation (e.g., stacks, I. Ezaminations. (5%) queues, deques) and linked allocation (e.g., simple linked lists). (10%) CS4. Introduction to Computer Organization G. Recursion. (5%) H. Ezaminations. (5%) OBJECTIVES: To introduce the organization and structuring of the CS3. Introduction to Computer Systems major hardware components of computers; To understand the mechanics of information trans- OBJECTIVES: fer and control within a digital computer system; To provide basic concepts of computer systems; and a To introduce computer architecture; and To provide the fundamentals of logic design. To teach an assembly language. PREREQUISITE:CS 2. PREREQUISITE:CS 2. COURSEOUTLINE: COURSEOUTLINE: The three main categories in the outline, namely The extent to which each topic is discussed and the computer architecture, arithmetic, and basic logic 'de- ordering of topics depends on the facilities available sign, should be interwoven throughout the course rather COMPUTERSCIENCE 37 than taught sequentially. The first two of these areas F. Ezaminations. (5%) may be covered, at least in part, in CS3 and the amount of material included in this course will depend on how CS5. Introduction to File Processing the topics are divided between the two courses. The OBJECTIVES: logic design part of the outline is specific and essential to this course. The functional, logic design level is em- To introduce concepts and techniques of structuring phasized rather than circuit details which are more ap- data on bulk storage devices; propriate in engineering curricula. The functional level To provide experience in the use of bulk storage de- provides the student with an understanding of the me- vices; and chanics of information transfer and control within the To provide the foundation for appllications of data computer system. Although much of the course mate- structures and file processing techniques. rial can and should be presented in a form that is inde- PREREQUISITE:CS 2. pendent of any particular technology, it is recommended COURSEOUTLINE: that an actual simple minicomputer or microcomputer system be studied. A supplemental laboratory is ap- The emphasis given to topics in this outline will vary propriate for that purpose. depending on the computer facilities available to students. Programming projects should be assigned to give TOPICS: students experience in file processing. Characteristics A. Basic Logic Design. Representation of both data and utilization of a variety of storage devices should be and control information by digital (binary) signals. covered even though some of the devices are not part of Logic properties of elemental devices for processing the computer system that is used. Algorithmic analysis (gates) and storing (flipflops) information. Descrip- and programming techniques developed in CS2 should tion by truth tables, Boolean functions and timing be utilized. diagrams. Analysis and synthesis of combinatorial TOPICS: networks of commonly used gate types. Parallel and A. File Processing Environment. Definitions of record, serial registers. Analysis and synthesis of simple file, blocking, compaction, database. Overview of synchronous control mechanisms; data and address database management system. (5%;) buses; addressing and accessing methods; memory B. Sequential Access. Physical characteristics of se- segmentation. Practical methods of timing pulse quential media (tape, cards, elk.). External generation. (25%) sort/merge algorithms. File manipulation tech- B. Commonly used codes (e.g., BCD, ASCII). Coding. niques for updating, deleting and inserting records Parity generation and detection. Encoders, de- in sequential files. (30%) coders, code converters. (5%) C. Data Structures. Algorithms for manipulating C. Number Representation and Arithmetic. Binary linked lists. Binary, B-trees, B*-trees, and AVL number representation, unsigned addition and sub- trees. Algorithms for transversing and balancing traction. One’s and two’s complement, signed mag- trees. Basic concepts of networks (plex structures). nitude and excess radix number representations and (20%) their pros and cons for implementing elementary D. Random Access. Physical characteristics of disk, arithmetic for BCD and excess-3 representations. drum, and other bulk storage devices. Algorithms (10%) and techniques for implementing inverted lists, mul- D. Computer Architecture. Functions of, and commu- tilist, indexed sequential, and hierarchical struc- nication between, large-scale components of a com- tures. (35%) puter system. Hardware implementation and se- E. File I/O. File control systems and utility routines, quencing of instruction fetch, address construction, 1/0 specification statements for allocating space and instruction execution. Data flow and control and cataloging files. (5%) block diagrams of a simple processor. Concept of F. Ezaminations. (5%) microprogram and analogy with software. Prop- erties of simple 1/0 devices and their controllers, CS6. Operating Systems & Corn],. Architecture synchronous control, interrupts. Modes of communications with processors. (35%) OBJECTIVES: E. Ezample. Study of an actual, simple minicomputer To develop an understanding of the organiza- or microcomwter svstem. (20%) tion and architecture of comr>ute:r svstems at the 38 RESHAPINGCOLLEGE MATHEMATE

register-transfer and programming levels of system D. Evaluation. Elementary queueing, network models description; of systems, bottlenecks, program behavior, and sta- To introduce the major concept areas of operating tistical analysis. (15%) systems principles; E. Memory Management. Characteristics of the hier- To teach the inter-relationships between the oper- archy of storage media, virtual memory, paging, :peg- ating system and the architecture of computer sys- mentation. Policies and mechanisms for efficiency of tems. mapping operations and storage utilization. Mem- ory protection. Multiprogramming. Problem of PREREQUISITES:CS3 AND CS4. auxiliary memory. (20%) F. Process Management. Asynchronous processes. Us- COURSEOUTLINE: ing interrupt hardware to trigger software procedure This course should emphasize concepts rather than calls. Process stateword and automatic SWITCH case studies. Subtleties do exist, however, in operating instructions. Semaphores. Ready lists. Implement- systems that do not readily follow from concepts alone. ing a simple scheduler. Examples of process con- It is recommended that a laboratory requiring hands-on trol problems such aa deadlock, product/consumers, experience be included with this course. readers/writers. (20%) The laboratory for the course would ideally use a G. Recovery Procedures. Techniques of automatic and small computer where students could actually imple- manual recovery in the event of system failures. ment sections of operating systems and have them fail (5%) without serious consequences to other users. This sys- H. Ezaminations. (5%) tem should have, at a minimum, a CPU, memory, disk or tape, and some terminal device such as a teletype of CS7. Data Structures and Algorithm Analysis CRT. The second best choice for the laboratory experience would be a simulated system running on a larger OBJECTIVES: machine. To apply analysis and design techniques to non- The course material should be liberally sprinkled numeric algorithms which act on data structures; with examples of operating system segments imple- To utilize algorithmic analysis and design criteria mented on particular computer system architectures. in the selection of methods for data manipulation in The interdependence of operating systems and archi- the environment of a database management system. tecture should be clearly delineated. Integrating these PREREQUSITES : C S 5. subjects at an early stage in the curriculum is particularly important because the effects of computer archi- COURSEOUTLINE: tecture on systems software has long been recognized. The material in this outline could be covered sequen- Also, modern systems combine the design of operating tially in a course. It is designed to build on the founda- systems and the architecture. tion established in the elementary material, particularly on that material which involves algorithm development TOPICS: and data structures and file processing. The practical A. Review. Instruction sets. 1/0 and interrupt struc- approach in the earlier material should be made more ture. Addressing schemes. Microprogramming. rigorous in this course through the use of techniques (10%) for the analysis and design of efficient algorithms. The B. Dynamic Procedure Activation. Procedure activa- results of this more formal study should then be in- tion and deactivation on a stack, including dynamic corporated into data management system design deci- storage allocation, passing value and reference pa- sions. This involves differentiating between theoreti- rameters, establishing new local environments, ad- cal or experimental results for individual methods and dressing mechanics for accessing parameters (e.g., the results which might actually be achieved in systems displays, relative addressing in the stack). Imple- which integrate a variety of methods and data struc- menting non-local references. Re-entrant programs. tures. Thus, database management systems provide Implementation on register machines. (15%) the applications environment for topics discussed in ithe C. System Structure. Design methodologies such as course. level, abstract data types, monitors, kernels, nuclei, Projects and assignments should involve implemen- networks of operating system modules. Proving cor- tation of theoretical results. This suggests an alterna- rectness. (10%) tive way of covering the material in the course; namely, COMPUTERSCIENCE 39 to treat concepts, algorithms, and analysis in class and could be specified and analyzed one at a time in terms deal with their impact on system design in assignments. of their features and limitations based on their run- Of course, some in-class discussions of this impact would time environments. Alternatively, desirable specifica- occur, but at various times throughout the course rather tion of programming languages could bc: discussed and than concentrated at the end. then exemplified by citing their implementations in various languages. In either case, programming exercises TOPICS: in each language should be assigned to emphasize the A. Review. Basic data structures such as stacks, implementations of language features. queues, lists, trees. Algorithms for their implementation. (10%) TOPICS: B. Graphs. Definition, terminology, and property (e.g., A. Language Definition Structure. Formal language connectivity). Algorithms for finding paths and concepts including syntax and basic characteristics spanning trees. (15%) of grammars, especially finite state, context-free, C. Algorithms Design and Analysis. Basic techniques and ambiguous. Backus-Naur Form. A language of design and analysis of efficient algorithms for in- such as Algol as an example. (15%) ternal and external sorting/merging/searching. In- B. Data Types and Structures. Review of basic data tuitive notions of complexity (e.g., NP-hard prob- types, including lists and trees. Constructs for lems). (30%) specifying and manipulating data types. Language D. Memory Management. Hashing. Algorithms for features affecting static and dynamic data storage dynamic storage allocation (e.g., buddy system, management. (10%) boundary-tag), garbage collection and compaction. C. Control Structures and Data Flow. Programming (15%) language constructs for specifying program con- E. System Design. Integration of data structures, trol and data transfer, including DO . . . FOR, DO sort/merge/search methods (internal and external) . . .WHILE, REPEAT . . .UNTIL, BREAK, subrou- and memory media into a simple database manage- tines, procedures, block structures, and interrupts. ment system. Accessing methods. Effects on run Decision tables, recursion. Relationship with good time, costs, efficiency. (25%) programming style should be emphasized. (15%) F. Ezaminations. (5%) D. Run-time Consideration. The effects of run-time environment and binding time on various features C S8. Organization of Programming Languages of programming languages. (25%) OBJECTIVES: E. Interpretative Languages. Compilartion vs. inter- To develop an understanding of the organization of pretation. String processing with language features programming languages, especially the run-time be- such as those available in SNOBOL, 4. Vector pro- havior of programs; cessing with language features such as those avail- To introduce the formal study of programming lan- able in SPL. (20%) guage specification and analysis; F. Lezical Analysis and Parsing. An introduction to To continue the development of problem solution lexical analysis including scanning, finite state ac- and programming skills introduced in the elemen- ceptors and symbol tables. An introduction to pars- tary level material. ing and compilers including push-down acceptors, top-down and bottom-up parsing. (10%) PREREQUSITES:CS2; RECOMMENDED:CS3, CS5. G. Ezaminations. (5%) COURSEOUTLINE: This is an applied course in programming language constructs emphasizing the run-time behavior of pro- Subpanel Members grams. It should provide appropriate background for ALANTUCKER, CHAIR, SUNY-Stony Brook. advanced level courses involving formal and theoretical aspects of programming languages and/or the compila- GERALDENGEL, Christopher Newport College. tion process. STEPHENGARLAND, Dartmouth College. The material in this outline is not intended to be BERT MENDELSON,Smith College. covered sequentially. Instead, programming languages ANTHONYRALSTON, SUNY-Buffalo.

Modeling and Operations Research

This chapter contains the report of the Subpanel on ing approaches using materials from pure and applied Modeling and Operations Research of the CUPM Panel mathematics, such as the methods of G. P6lya and R.L. on a General Mathematical Sciences Program, reprinted Moore. Problem solving teams for ccimpetitions such with minor changes from Chapter V of the 1981 CUPM as the Putnam contest and special departmental prac- report entitled RECOMMENDATIONSFOR A GENERAL tica exist in many colleges. Special courses or seminars MATHEMATICALSCIENCES PROGRAM. on modeling, case studies, and project-oriented activity are becoming more common, as are mathematics clinics and consulting bureaus. Co-op and work-study Experience in Applications programs, summer internships, and various other stu- This chapter is concerned with mathematical model- dent exchanges have been successfully implemented at ing and associated interactive and experienceoriented some inst it utions. approaches to teaching mathematical sciences. Math- The Modeling Subpanel believes that applications ematical modeling attempts to involve students in the and modeling should be included in a nontrivial way in more creative and early design aspects of problem for- most college-level mathematical sciences courses. Con- mulation, as well as provide them with a more complete cern with applications has been an important historical exposure to how mathematics interfaces with other ac- force and a major cultural ingredient in the develop- tivities in solving problems arising outside of mathemat- ment of all mathematics. Further, thie Modeling Sub- ics itself. Model building is a major ingredient of opera- panel strongly recommends that all mathematical sci- tions research and the contemporary uses of mathemat- ences students should obtain first-hand, experience with ics in the social, life and decision sciences. In addition realistic applications of mathematics from the initial to being important in their own right, these newer uses stage of model formulation through interpretation of of mathematics provide a rich source of suitable materi- solutions. This can be done in a project-oriented mod- als for interaction and modeling which complement the eling course in one of the alternate out-of-class modes many modern and classical applications of mathematics mentioned above. Such an experience yields insight into in the physical sciences and engineering. the place of mathematics in the larger realm of science. This chapter is intended to assist mathematics fac- It provides an appreciation for the need for interdisci- ulty in implementing the main panel's recommendation plinary interaction and the limits of specialization. It that mathematical sciences majors should have sub- offers a chance for individuals to make use of their own stantial experience with mathematical modeling. Sub- intuition and creative abilities, to sense the great joy of sequent sections discuss the modeling process in some personal accomplishment, and to develop the confidence detail; provide specific suggestions for conducting stu- to confront similar problems after graiduation. Finally, dent projects, applications-experience-related courses such experience may assist students in choosing careers and other such programs, along with general recommen- and fields for future study. dations concerning modeling courses at different levels; explain the field of operations research and the requirements for graduate study. The final two sections Mathematical Modeling present outlines for four courses in operations research and modeling, and a compendium of resources and ref- Modeling is a fundamental part of' the general sci- erences for modeling courses. entific method and is of primary importance in ap- Learning and doing mathematics is a rather individ- plied mathematics. A model is a simpler realization ualized and personal activity. The typical classroom or an idealization of some more complex reality cre- lecture in which students are passive spectators has ob- ated for the purpose of gaining new knowledge about vious limitations. Students need supervised hands-on a real situation by investigating properties and impli- experience in problem solving and constructing rigor- cations of the model. Models may take many differ- ous proofs. A large variety of alternate teaching tech- ent forms, from physical miniatures to pure intellectual niques and special programs have been developed in at- substitutes. Study of a model will hopefully provide un- tempts to meet this need. These include problem solv- derstanding and new information about real phenomena 42 RESHAPINGCOLLEGE MATHEMATE which are too complex, excessively expensive, or impos- right side from Urnathematicalmodel" to "mathemati- sible to analyze in their original setting. cal solution." This is the deductive activity of finding We tend to take the amazing effectiveness of models solutions to well-formulated mathematical problem. It for granted today. The reader should give a moment's is usually the most logical, well-defined and straight- thought to the following examples. One can learn a forward part of modeling, although not necessarily the great deal about a proposed aircraft from wind tun- easiest. It is often the most immediately pleasing, el- nel experiments before building a costly prototype, and egant, and intellectual part. This "side" of the "mod- one can learn much about flying an existing airplane eling square" is the one covered best in standard ap from a computer-aided cockpit simulator. Simple com- plied mathematics courses. Unfortunately, most teach- puter simulations can provide insights into the complex ing of applied mathematics is confined to discussing just flow or queueing behavior of traffic in a transportation model-solving mathematical techniques, with superfi- system. Theoretical studies about elementary particles cial treatment of the other three sides of the square, have provided new insight into fundamental physical whereas these other sides often involve much more ccre- laws and have guided subatomic experimentation. ativity, interaction with other disciplines, and communication skills.

Real World Mathematical END BEGIN Problem Model t INTERPRET RESULTS PROBLEM OF ANALYSIS PRESENTATION Interpretation MathematicalI I in Original Settin; Solution IN0 A Simple Model of Mathematical Modeling SPECIALIZATION Figure 1 The process of mathematical modeling can be simply represented with the diagram in Figure 1. One begins with a problem which arises more or less directly out ANALYSIS OF PREPARATION FOR of the "real world." One builds an abstract model for purposes of analysis, and this frequently takes a mathematical form. The model is solved in this abstract setting. The solution is then interpreted back into its original context. Finally, the analytical conclusions are I MODIFICATIONOF MODEL REp!~t?%?~l!oN I I I I compared with reality. If they fall short of matching the real situation, then modifications of the model may . be called for, and one proceeds around this cycle again. A Refined Model of Mathematical Modeling One often proceeds back and forth within a cycle and Figure 2 makes successive iterations about this figure many times before arriving at a satisfactory representation of the The bottom arrow in Figure 1 is concerned with real world. translating or explaining a purely mathematical result The creation of new knowledge via this modeling in terms of the original real world setting. This involves route is at the heart of theoretical science and applied the need to communicate in a precise and lucid man- mathematics. We will use the word "modeling" to de- ner. (Inexperience in this skill, according to many em- scribe the complete progress illustrated in Figure 1. Fre- ployers, is a serious shortcoming in mathematics gradu- quently this term is used only for the model formulation ates). This aspect of a mathematical scientist's training step (the top arrow in the figure). A full discussion of should not be left to courses in other sciences or to on- the four steps in this modeling paradigm follow. Addi- the-job learning after graduation. tional steps refining the modeling process are sometimes In describing the meaning of a mathematical solution inserted; for example, see Figure 2. one must take great care to be complete and honest. It First consider the downward pointing arrow on the is dangerous to discard quickly some mathematical so- MODELING& OPERATIONSRESEARCH 43 lutions to a physical problem as extraneous or having no translating into mathematics, maintaining the essential physical meaning; there have been too many historical ingredients while filtering out a great amount of excess incidences where “extraneous“ solutions were of fun- baggage, and arriving at realistic and manageable intel- damental importance. Likewise, one should not select lectual limitations. The three basic elements of a model out just the one preconceived answer which the “boss” are: is looking for to support his or her position. A deci- 1. A logical mathematical structure such as calculus, sion maker frequently does not want just one optimal probability, or game theory; solution, but desires to know a variety of “good” solu- 2. An appropriate interpretation of the variables in tions and the range of reasonable options available from that structure in terms of the given problem; and which to select. 3. A characterization with the structure of all laws and There is an old adage to the effect that bosses do constraints pertinent to the problem. not act on quantitative recommendations unless they To build such a mental construct, one must concep- are communicated in a manner which makes them un- tualize, idealize and identify propert,ies precisely. A derstandable to such decision makers. This communi- model builder must carefully balance the tradeoffs be- cation can often be a difficult task because of the tech- tween coarse simplifications and unnecessary details- nical nature of the formulation and solution, and also often the effects of such tradeoffs are n.ot apparent until because large quantities of data and extensive computa- subsequent validation (three steps later in the modeling tion may need to be compressed to a manageable size for process). the layman to understand in a relatively limited time. This initial part of modeling is clearly the most es- If mathematical education gave more attention to this sential and valuable part of the whole ]process. It is usu- aspect of mathematical modeling, there might be wider ally the most difficult part. Eddington said “I regard recognition and visibility of mathematicians in society the introductory part of a theory as the most difficult, beyond the academic world! because we have to use our brains all of the time. Af- A major step in real world modeling is to validate terwards we can use mathematics.” ]Model building is models critically and to check out solutions against the an art, and must be taught as such. original phenomena and known results. This step, represented by the left upward arrow in Figure 1, may involve experimentation, verifying, and evaluating. Two An Undergraduate Modeling Course major criteria for evaluating a model are simplicity and This section discusses various approaches to design- accuracy of prediction. Questions about the range of ing mathematical sciences courses concentrating on the validity, sensitivity of parameters errors resulting from modeling process. The resources list,ed at the end of approximations, and such should be investigated. In this chapter contain a wealth of additional information many cases, a modeling project will simply confirm from on models, the modeling process and specific modeling another perspective properties that are already believed courses as well as references to supplementary materials to be true. The real gain from modeling activity occurs which the reader may find useful in c’ourse design. when the modeling leads to discovery of new knowledge Practitioners in the physical or social sciences or en- (which subsequently is confirmed by other methods). gineering have an instinctive feeling of what the mod- Modern mathematics education rarely involves itself eling process is all about, even if they are not able to with this left hand side of the modeling process, except articulate it well. Modeling is an important part of perhaps for an occasional “eyeballing” of an answer or their work-a-day activity. For the most part, however, in projects undertaken by a mathematics or statistics they prefer to leave the analysis and structure of the consulting clinic. By omitting this activity, mathemati- modeling process itself to workers in other disciplines, cal education misses an opportunity to become involved like mathematics, or to philosophers of science who are with real-world decision making, judgmental inputs, the trying to understand the abstract thieories underlying limitations of its mathematical tools, and other more these results and how scientists get their results. human aspects of science, as well as the reward of wit- How does one go about acquiring experience in real- nessing the acceptance of a new theory. world modeling? The wrong place to start is looking at Finally, consider the top arrow in Figure 1 which big models in the scientific literature which are broad represents the heart of the modeling activity. The con- in scope and the epitome of their kind. Indeed, one struction of an abstract model from a real situation is could probably learn more about sculpting by looking the really creative activity and an important compo- at the pieces that Michelangelo discarded than by look- nent of all theoretical science. Building models involves ing at the Pieta. The mathematical techniques with 44 RESHAPINGCOLLEGE MATH EM AT^ which one is familiar will be a primary limiting factor program by J. Truxal, et al., McGraw-Hill publisher). in understanding models. Another factor is that real- A range of courses emphasizing the modeling process world problem areas have their own peculiar “empirical is clearly possible between the case study approach and laws” and “principles” which are commonly known to the experiential approach. specialists in an area but are not easily accessible to the It is important to note that the scope of the engineer- casual reader. ing approach to modeling is much broader than just, the Apprentice modelers need some help and guidance technical aspects of the problem at hand. In designing in selecting model areas for study which will build their a solution to a problem, engineers must take into ac- modeling skill without discouraging progress. The ideal count time constraints and build into their models pre- way to do this within the college curriculum is to begin scribed economic and other technical constraints as well the modeling process as early as possible in the stu- as consideration of the impact of their design on soci- dent’s career and reinforce modeling over the entire pe- ety. Engineers do not build elaborate models to explain riod of study. That is, the modeling process should be the fundamental workings of nature nor do they seek an integral part of the curriculum. Most mathematics the best possible solution to a problem in the absence departments, for a variety of reasons, are not prepared of the proposed application of that solution. In spite to give modeling such a major emphasis. For them, a of these differences, there is obviously a large overlap more reasonable approach is to design a course specifi- between the engineering and mathematical approaches cally around the modeling process. to modeling. Efforts to emphasize the modeling process in under- We now characterize the components of a modeling graduate courses on a broad scale began in the 1960’s course in a way that readers should find useful in de- and were promoted mainly by engineers, operations re- signing a course to fit their own local needs. The Table searchers and social scientists. Extensive discussions of on pp. 46-47 organizes much of this information for easy modeling in mathematics courses developed later. The reference. There are six basic aspects of teaching mod- modeling process has been brought into the classroom eling that must be considered: in many ways but two particular approaches are worth 1. Prerequisites. For whom is the course intended? describing in some detail. 2. Effort level. How long-a few weeks, a semester, a First there is the case study approach in which the year? modeling process is described in a series of examples 3. Course format. Experiential or case study ap- that are more-or-less self-contained. The examples se- proach? Team or individual work? Instructor’s role. lected by the instructor are designed to bring out the Communication skills used. basic features of the modeling process as well as to in- 4. Resources available. Computer system, remote ac- form the students about basic models within a disci- cess, good software packages (students should be- pline. An excellent early example is You and Technol- come familiar with using some major software pack- ogy: A High School Case Study Tezt developed by the age). Access to expertise in fields considered. Ap engineering departments of the PCM Colleges (Chester, propriate handouts to keep students progressing. PA), edited by N. Damaskos and M. Smyth. 5. Source of problems. Real-world or contrived? The second approach applies “hands-on” experience Open-ended or can student answer all questions by to problems that may only be vaguely described. This looking them up in the literature? approach is sometimes called “open-ended” or “experi- 6. Technical thrust. What technical areas should the ential,” because it is not clear at the outset what kind course emphasize, or avoid? Continuous or discrete of a model will be successful in analyzing a problem, models? Deterministic or stochastic? Role of com- or indeed whether a particular problem is well-posed in puter programming. any sense. An interesting sidelight on this approach to We now expand a little on two of these compo- teaching the modeling process is that the models pro- nents, effort level and course format. The level of effort posed by students for a particular problem depend not devoted to a modeling course can range from “mini- only on the students’ breadth of knowledge but, as much projects,” using a team approach to short projects as anything else, on time constraints and computer (and within an established course, to major projects which other) resources available. Engineers popularized the last an entire year. The mini-project format requires experiential approach in the early sixties with the high a great deal of organization and preparation to make school program Man Made World, mostly as a means of it work. See Borrelli and Busenburg ‘‘Undergradu- exposing students at an early stage to engineering as a ate Classroom Experiences in Applied Mathematics” profession (a text of the same name was written for this (UMAP Journal, Volume 1, 1980) for one approach to MODELING& OPERATIONSRESEARCH 45 structuring a mini-project program, together with its students know their work will get exposure, they are pro’s and con’s. The one-semester case study course, motivated to write good reports. judging from its popularity, is the best understood and It is valuable to integrate the modeling process into trusted of modeling courses. There are good textbooks the curriculum as widely as possible and not just as and a great many modules written for use in such a an add-on special course with no connection to any course (see list at end of chapter). other mathematical sciences course. While most case studies texts on mathematical mod- A problem with most modeling courses is that the eling are designed for upper-level courses, the text You material in them quickly becomes dated. When stu- and Technology (mentioned above), supplemented with dents discover that they are working on the same modules, can easily be adapted for use in a freshman projects or models as their classmates did last year, case studies course. Such a course might also present they lose enthusiasm. What is needed is a format an opportunity for students to see the fundamental for automatically updating the matlerial. A constant differences between engineering and mathematical ap flow of real-world problems, as come into a mathe- proaches to modeling (this issue is treated nicely in matics consulting clinic, is a great advantage. You and Technology). An extensive outline is provided below for a special custom-made, lower-level modeling course. Operations Research Experiential modeling courses are not used as often Operations research is a mathematic a1 science closely as case study courses. Since the experiential approach is connected to mathematical modeling. Although some typically used on open-ended problems where the out- notable contributions were made prior to 1940, oper- come is difficult to predict in advance, this approach ations research grew out of World War 11. The analy- is especially risky for a mathematics instructor who is sis of military logistics, supply and operational prob- teaching a modeling course for the first time. Neverthe- lems by scientists from many different disciplines gen- less, experiences of various colleges over the last several erated the techniques and approaches that evolved into years show that the experiential approach is feasible and modern operations research. This sub-ject studies com- that, whatever happens, students and instructors find plex systems, structures and institutions with a view it a rewarding experience. Several successful formats towards operating such multiparameter systems more for experiential modeling courses have emerged. All efficiently within various constraints, such as scarce re- seem to use the team approach with occasional guid- sources. Operations research analyse:! are used to op- ance by consultants, as needed. It should be noted that timize current activities and predict ifuture feasibility. many industrial employers treat such experiential mod- The complexity of its problems has made operations eling as job-related experience in assessing a student’s research heavily dependent on high-speed digital com- job qualifications. References at the end of this chap- puters. It is now used in fields in which decisions were ter contain descriptions of the well-known Mathematics traditionally made on the basis of less quantitative ap- Clinics in Claremont and other experiential modeling proaches, such as “experience” or merle hunches. There courses (interested readers can write directly to Harvey is frequently a major concern with “people” as well as Mudd College for first-hand advice). “things,” and the man-system interface in a complex We close this Section with some important general social activity. Major national concerns such as produc- points to keep in mind when designing any modeling tivity, environmental impact and energy supply have a course. large operations research component. To encourage initiative and independent work, stu- The approach in operations research is multidis- dents should have access to, and be responsible for ciplinary in nature, and uses common sense, data, using, support resources such as documentation of and substantial empiricism (heuristics) combined with software and previous student projects. new, as well as repackaged traditional, mathematical If high standards are imposed on writing of re- methodologies. The principal mathematical theories of ports, then these reports deserve some exposure; operations research are mathematical programming and they should not just be shoved in filing cabinets and stochastic processes. Major topics in these theories are forgotten. Instructors should encourage students to mentioned in the operations research course contents in seek publication of a paper based on their reports, if the next section. Operations research lnas major overlap warranted, or an article in the campus newspaper. with the fields of industrial engineering, management Abstracts of recent reports should be made avail- science, mathematical economics, econometrics and de- able to students early in a modeling course. When cision theory. 46 RESHAPINGCOLLEGE MATBEMAE

Anatomy of a Modeling Course

Ingredients Background and Source Material Remarks PREREQUISITES: Lower Division. Single variable cal- Case study approach most likely. See, If the team approach is selected t,hen culus, a science course with lab, some e.g., "You and Technology" or suit- there can be some flexibility in these computing. able UMAP modules. prerequisites. Upper Division. Multivariable calcu- For experiential approach and case If modeling course is not required., lus, linear algebra, computation and study approach consult appropriately then some thought must be given as some computer programming, basic noted reference. to how students can be attracted to prob/stat., some diff. eqns., a science such a course: descriptions in reg- course with lab. istration packets, posters, note to advisor, etc. EFFORTLEVEL: Partial Course. Recommended min- Mini-projects are a possibility here. Important that mini-project work not imum of 2 weeks out of a 3 hour See Borelli and Busenberg. Format of be simply added to standard load of course preceded by a tooling up pe- mini-projects can be effectively struc- the host course-it should replace: riod. tured. See Becker, et al., "Handbook some required work; e.g., an exam. for Projects." Full Course. May be designed to fit Many possibilities exist for modeling Format of instruction can seriously into special options, either to give courses for a full semester-see items affect the student's interest as well as job-related training or introduction below. For a discussion of pros and his capacity for effective work-see to modeling process with important cons, see Borelli and Busenberg. "Format" section below for possibili- models in a discipline. ties. COURSEFORMAT: Case Study. The modeling process Material selected from modules, text- Advanced students can be asked presented via examples that are books, conference proceedings, or to lecture on material that is well more-or-less self-contained. journals. enough organized. Ezperiential. Hands-on approach to Internships, work-study programs not open-ended projects incorporating the appropriate for inclusion here. modeling process. Some possibilities are: 1. Problem-centered Course. Class Needs highly experienced instructor Oral presentation and written reports divided into teams to work on a se- to select and present the projects and are emphasized. Most demanding of quence of projects and share experi- watch over progress of the teams. instructor's time. ence. Class size limited by instructor's energy. See Borelli and Busenberg for more details. 2. Mathematics Clinic, Consulting Composition of team is critical. See Team communication skills highly Group. Intensive, industry-supported Claremont Clinic Articles for details. emphasized in Clinic program and is team effort on a single project, usu- Because of time constraints, able sup- crucial to success. Team has main ally for one year. port staff must be readily available. responsibility for work, instructor advises. Student handbook at Cla.re- mont Clinic (by Handa) available on request. 9. Research Assistance. Students aid MIT has a highly organized program A danger here is that the success of faculty in research work. which does this. Mostly, however, it's the faculty member's research may catch-as-catch-can. The Institute of take precedence over the impact on Decision Science, Claremont Men's the students' education. Students" College, has developed a classroom needs could get lost in the shuffle. approach to such work. 4. Mini-projects. Team approach on See Borelli and Busenberg. Emphasizes writing skills, highly short projects within an established structured activity; see "Handbook course. for Projects" by Becker, et al. MODELING& OPERATIONSRESEARCH 47

Anatomy of a Modeling Course

Ingredients Background and Source Material Remarks

RESOURCESAVAILABLE: Computer. Good access to a high Computer graphics capabilities and level computer (preferably with time- knowledgeable (and accessible) con- sharing capability) having good soft- sultants at the computer center add ware packages is very important for not only a professional touch but also the success of most modeling courses. help teams live within their time constraints. Ezperienced Consultants. Access to A successful, long-term program de- Be sure that consultants help is ac- knowledgeable colleagues, experts in pends to a large extent on the Direc- knowledged by the students in all local industrial firms, and talented tor’s ability to secure willing assis- written reports, even if it is only of a computer center personnel are all tance from able consultants. casual nature. helpful in keeping a team’s progress from faltering. Supplemental Materials. Handouts on For project work, see the Handbooks how to work in a team on projects, by Becker, et al., Handa, Seven and or where to go for help, etc., lessen Zagar, and for computer graphics, the student’s feeling of abandonment Saunders, et al. (all were developed when working on projects. at Harvey Mudd College and are available on request).

SOURCEOF PROBLEMS: Real World. Open-ended problems See Borelli and Spanier for a descrip- Used only in experiential type model- submitted by local industrial firms tion of one effective method of re- ing courses. or government agencies which are of cruiting sponsored projects from in- current interest to them, or problems dustry. MIT has a highly organized from current research of colleagues. way of advertising current research of its faculty and laboratories and whether undergraduates can play a role or not. Contrived. Open-ended problems The modeling books in the references Used mostly in experiential type pulled from a variety of sources: from are good sources of problems. modeling course. technical journals, suggestions from colleagues, books, etc. Case Studies. Reasonably well self- Good sources in modules, proceedings Used only for case study type of contained descriptions of completed of conferences on case studies and modeling course. projects or problems. books. TECHNICALTHRUST: Discrete-OR. Problems whose models Deterministic and istochastic methods involve discrete structures, program- are both possibditi’es here. ming, or optimization within discrete settings. Also interpolation with finite structures in continuous settings. Continuous. Problems whose models involve differential or integral equations, continuous probabilities, or optimization within continuous setting. Computer. Problems with main goal the production of software either at the systems level or solvers for a class of equations in continuous settings, along with error analysis of same. For DEC users, the IMSL package is a good all-around one to have available on the system. 48 RESHAPINGCOLLEGE MATHEMA‘=

There are many opportunities for mathematical sci- by most college mathematics professors with appropri- ences majors to pursue graduate studies or find employ- ate attitudes if they are willing to undertake some self ment in operations research and related fields. Indus- study. Indeed, faculty without formal operations re- trial mathematicians in all fields find themselves faced search training who are going to teach such a course with operations research problems from time-to-time. should be strongly encouraged to learn about the iield Thus it is important for mathematical sciences students by attendance at short courses, participation in a de- to have some exposure to operations research and its partment seminar on the subject, or by sabbatical leave applications, and also knowledge of its career possibil- (or other released time) at universities or industrial lab- ities. This classroom exposure to operations research oratories with operations research activities. can occur in conjunction with undergraduate modeling experience or in a specific course on operations research. The current relevance and naturalness of this Course Descriptions subject are immediately clear to students, and realistic Four sample courses on operations research and mod- projects at various levels of difficulty are readily avail- eling are described below. Only more general remarks able. An interesting article by D. Wagner about opera- are given for the courses in operations research and tions research appeared in the American Mathematical stochastic processes since these have become fairly stan- Monthly (82, p. 895). Students should also be referred dardized in recent years. More specific details are pro- to the booklet Careers in Operations Research, avail- vided for an elementary-level modeling course using idis- able from the Operations Research Society of America, Crete mathematics and for a more advanced modeling 428 Preston Street, Baltimore, MD 21202. course using continuous methods. These are merely il- A student interested in graduate work in operations lustrations of the wide variety of different sorts of mod- research should have a solid preparation in undergrad- eling courses which can be taught. The 1972 CU:PM uate core mathematics: calculus, linear algebra, real Recommendations on Applied Mathematics contain a analysis, plus courses in probability, introductory com- detailed description of a physical-sciences oriented mod- puter science and modeling. A course in operations re- eling course. Such a modeling course continues to be search itself is more important as a way to learn if one very valuable and in no way should be considered dat,ed. likes the field than as a prerequisite for graduate study. Many basic intermediate-level courses in the physical A substantial minor in a relevant area outside mathe- sciences are also excellent modeling courses, from the matics (as recommended for all mathematical sciences point of view of a mathematical sciences major. majors in the first chapter, “Mathematical Sciences”) is important. This outside work should include a sampling Introductory Operations Research of quantitative courses in the social sciences, business, Much of the material in an introductory operati’ons or engineering (if available). Experience solving some research course for undergraduates has become fairly problems involving substantial computer computation standard. The course covers primarily deterministic and an exposure to nontrivial algorithms are also desir- methods. Most publishing companies have good intro- able. ductory operations research texts (the text title may At some institutions, mathematics departments are be Linear Programming, the course’s main topic). The now preparing to offer an operations research course for level of this course can vary depending on the prerequi- the first time, while other institutions may have many sites and student maturity. It is normally an upper-level operations research courses offered in mathematics, eco- offering with a prerequisite or corequisite of linear al- nomics, business, industrial engineering and computer gebra. Calculus and probability should be required if science. In either extreme and situations in between, stochastic models are also included. mathematical sciences students are best served by some An operations research course can be a “pure math- form of interdepartmental cooperation, or at least co- ematics” course which stresses the fundamental prop- ordination of offerings. If a mathematics department erties of systems of linear inequalities, basic geometry is planning to offer an operations research course when of polyhedra and cones, discrete optimization and com- none previously existed at the institution, mathematics plexity of algorithms. Most operations research courses, should work closely with other interested departments. however, emphasize the many applications which can be In planning this first course, mathematicians could solved by linear programming and related techniques of seek contacts with local industry to obtain practition- combinatorial optimization. Such courses usually de- ers as visiting lecturers. On the other hand, an in- vote some time to efficient algorithms and practical mu- troductory operations research course can be taught merical methods (to avoid roundoff errors), as well as MODELING& OPERATIONSRESEARCH 49 basic notions of computational complexity. While prob- mathematical developments and indicate the current lem solving and modeling are important, a first oper- relevance of mathematics to contemporary science and ations research course should cover some topic in rea- policy making. sonable depth and not be merely a collection of simple The course should maintain an open-minded and techniques and routine applications. questioning approach to problem solving. Much of the class time should be devoted to student discussions of COURSECONTENT their models and how to improve them. Students should The course should start with a brief discussion of the be asked to make formal oral and written expositions. general nature, history and philosophy of operations re- Many of the topics coveredare also suit,able, with proper search. Some of the older texts such as Introduction to adjustments, for upper-level courses or for lower-level Operations Research by C. Churchman, R. Ackoff and “mathematics appreciation” courses. (Readers inter- E. Arnoff, Wiley, 1957, and Methods of Operations Re- ested in the latter courses should coneult the 1981 Re- search by P. Morse and G. Kimball, Wiley, 1951, de- port of the CUPM Panel on Mathematics Apprecia- vote extensive space to history. The instructor should tion, reprinted later in this volume.) Nlot all of the top- not spend much time on history at the beginning of ics mentioned below can be covered in any one course, a course but instead should weave it into discussions and frequent changes in course content are necessary to throughout a course. maintain the originality of problems. The first half of the course in usually devoted to lin- No one current textbook appears appropriate for this ear programming: its theory, the simplex algorithm, course, although a simpler “prepackage:d” version of this and applications. The course then continues on to a se- course could use the high-school-oriented text You and ries of special linear programming problems, such as op- Technology with supplementary modules. The course timal assignment, transportation, trans-shipment, net- described below is an example of how various sources work flow, minimal spanning tree, shortest path, PERT can be assembled (as handouts or on library reserve) to methods and traveling salesperson, each with its own form a modeling course, in this instarnce emphasizing algorithms and associated theory. Basic concepts of modeling in the social sciences. graph theory are normally introduced in conjunction with some of the preceding problems. If time per- COURSECONTENT mits, elementary aspects from decomposition theory, Overview and Patterns of Problem Solving. Intro- dynamic programming, integer programming, or non- duction to the nature of modeling and problem solv- linear programming may be included. ing. The role of science, engineering and social sci- It is difficult to find space in an introductory opera- ences in making and implementing new discoveries. tions research course for even a small sampling of prob- The nature of applied mathematics ,and the interdis- ability or stochastic models. If possible, it is better to ciplinary approach to problems. Illustrations of prob- include this material in a second course. Similarly, there lems solved by quick insight rather )than by involved is usually little time available to discuss game theory, analysis. Many books have chapters on modeling and except possibly for showing that two-person, zero-sum problem solving; also see Patterns of’ Problem Solving games are equivalent to a dual pair of linear programs. by M. Rubinstein, Prentice-Hall, 1975, or “Foresight- Game theory is probably best treated in a separate Insight-Hindsight” by J. Frauenthal and T. Saaty, in “topics” course. Modules in Applied Mathematics, vol. 3 (W. Lucas, editor), Springer-Verlag. Elementary Modeling Course Graph and Network Problems. A large variety of The following course on mathematical modeling and problems related to undirected and directed graphs and problem solving is intended for freshmen and sopho- network flows can be assigned and discussed at the out- mores with a solid preparation in high school math- set with no hint of any theory or technical terms. At ematics. The primary objective is to provide lower- a later stage, a lecture can be devoted to theory to de- level students with a first-hand experience in forming velop a common vocabulary. The language and general their own mathematical models and discovering their approach of systems analysis can be developed. The own solution techniques. A secondary goal is to intro- four-color theorem can be discussed. References are duce some of the concepts from modern finite math- Applied Combinatorics, by F. Roberts, Prentice-Hall, ematics and illustrate their applications in the social 1984, Graphs as Mathematical Models by G. Chartrand, sciences. The instructor might supplement these main Prindle, Weber and Schmidt, 1977, and Applied Com- themes with brief discussions of some important recent binatorics by A. Tucker, Wiley, 1980. 50 RESHAPINGCOLLEGE MATHEMA=

Some lecture time can be spent illustrating how graded exercises and class discussions. Some of the top graphs are applied: to simplify a complex problem, ics listed above can be treated in this mode. Other suit- such as Instant Insanity (Chartrand, p. 125 or Tucker, able topics are: the Apportionment Problem (Fair Rep p. 355), or the more difficult Rubik’s Cube (Scientific resentation by M. Balinski and H. Young, Yale Press); American, March, 1981); for purely mathematical pur- measuring power in Weighted Voting situations (W. poses, such as to prove Euler’s formula V - E + F = 2 Lucas in Case Studies in Applied Mathematics MAA, and use it in turn to prove the existence of exactly five 1976); Cost Analysis (C. Clark in same Case Studies regular polyhedra; or to examine R. Connelly’s flex- on harvesting fish or forests); some simple topics from ing (nonconvex) polyhedra (Mathematical Intelligencer, statistics such as Asking Sensitive Questions, module by Vol. 1, No. 3, 1979). The analogy between transporta- J. Maceli (in Modules in Applied Mathematics, vol. 2, tion, fluid flow, electric and hydraulic networks can be W. Lucas, editor, Springer-Verlag); and Social Choice illustrated (see G. Minty’s article in Discrete Mathemat- Theory and Voting (Theory of Voting by R. Farquhar- ics and Its Applications Proceedings of a Conference at son, Yale, 1960). Indiana University, ed. M. Thompson, 1977). In addition to the class project, teams of two or Enumeration Problems. (Tucker, 2nd ed., Chapter three students can spend a few weeks on a mini-project. 5 or Roberts, Chapter 2.) Some practical uses can be Many of the topics above can be applied to a local prac- covered briefly, e.g., to probability problems or the Pi- tical problem. Scheduling, inventory and optimal al- geonhole Principle. Computational complexity and its locations are good topics, as are gaming experiments, application to hard-to-break codes can be discussed. simulations and elementary statistical studies. More theoretical topics, ranging from walking versus running Value and Utility Theory. Expected utility versus in the rain to designing the inside mechanism of the expected value; St. Petersburg paradox; construction of Rubik’s Cube are also possible. Some attempt at dis- a money versus utility curve: axioms for utility; assess- cussing possible implementation of a mini-project re- ing Coalitional Values (see module by W. Lucas and L. sult, e.g., with a campus administrator, is encouraged Billera in Modules in Applied Mathematics, vol. 2, W. in order to show the practical difficulties of implement- Lucas, editor, Springer-Verlag). ing mathematically optimal procedures. Conflict Resolution. Some three-person cooperative game experiments and analysis; the Prisoner’s Dilemma Introductory Stochastic Processes for two or more persons (H. Hamburger in Journal of The purpose of this course is to introduce the stu- Math. Sociology 3, 1973); illustrations of equilibrium dent to the basic mathematical aspects of the theory of concepts; two-person zero-sum games, e.g., batter ver- stochastic processes and its applications. This course sus pitcher (Economics and the Competitive Process by can equally well be offered under such alternate titles as J. Case, NYU Press, 1979, p. 3; also see The Game of Applied Probability or Operations Research: Stochas- Business by John McDonald, Doubleday, 1975, Anchor tic Models. Stochastic processes is a large and growing paperback, 1977, and Game Theory: A Nontechnical field. This course will lay background for further learn- Introduction by M. Davis, Basic Books, 1970). ing on the job or in graduate school. A Discrete Optimization Problem and an Algorithm. The prerequisite for this course is at least the equiv- Possible topics are the complete and optimal assign- alent of a full course of post-calculus probability incllud- ment problems (UMAP module 317 by D. Gale), or the ing the following topics: random variables, common marriage problem (D. Gale and L. Shapley, American univariate and multivariate distributions, moments, Mathematical Monthly 69, 1962, p. 9). conditional probability, stochastic independence, c:on- ditional distributions and means, generating functions, Simulation. See chapters on simulation in many and limit theorems. Such a course is fairly traditional books and “Four-Way Stop or Traffic Light? An Illus- now, but if most students have had just the integrated tration of the Modeling Process” by E. Packel (in Mod- statistics and probability course suggested by the Statis- ules in Applied Mathematics, vol. 3, W. Lucas, editor, tics Subpanel, then the beginning of the stochastic pro- Springer-Verlag). Additional ideas from Inventory The- cesses course would have to be devoted to completing ory, Scheduling Theory, Dynamic Programming, and the needed probability background. It is also desirable Control Theory, e.g., lunar landing, can be included. for students to have some experience with basic matrix Projects and Mini-projects. At least one significant algebra and with using computer terminals. project type activity should be pursued over several The course should slight neither mathematical the- weeks by the whole class by means of a sequence of ory nor its applications. It is better to cover few topics MODELING&I OPERATIONSRESEARCH 51 with a full discussion of both theory and applications waves and calculus of variations. The course should give to survey theory alone or to cover only applications. a solid motivation for more advanced courses in these The course emphasizes problem solving and develops an topics. A (non-original) paper on a topic of interest acquaintance with a variety of models that are widely to the students serves the dual purpose of developing used. Stochastic modeling and problem formulation are communication skills and introducing pledagogical flex- different activities that should be treated in a modeling ibility. course. If many students do not subsequently take a A course on continuous modeling usually has as a modeling course, then the instructor should consider prerequisite a course in differential equations, although allocating some time (assuming course time did not the modeling can be taught concurrently or integrated also have to be devoted to probability) to a module in one course, using a book such as Martin Braun’s on stochastic modeling in business or government (see Differential Equations and Their Appkations (second list of modules below) or to a real problem at the local edition), Springer-Verlag, 1978. Continuous modeling college, e.g., modeling the demand for textbooks in the problems frequently involve concepts from natural sci- bookstore or utilization of campus parking spaces. ences. In this case, it is important that either an appro- Computers should be used in this course in two ways: priate background is required of students or the techni-

0 As a computational aid to perform, for example, cal essentials are adequately introduced in the course. matrix calculations needed in Markov chain theory; The texts by Lin and Segal and by Haberman (see and below) are well suited for this course. Selections from As a simulation device to exhibit the behavior of the two-volume Lin and Segal text can be used to pro- random processes. vide a solid basis for physics and engineering modeling Understanding randomness is difficult for undergradu- using both classical subjects, such as fluids, solids and ates and discussion of data accumulated in simulation heat transfer, and modern subjects, such as fields of bi- studies can help overcome students’ deterministic biology. The text’s broad coverage probably includes an ases. introduction to an area of expertise of the instructor to which he or she can bring personal research insights. COURSECONTENT A course which requires a little less sophistication Bernoulli process; Markov chains (random walks, can be designed around Haberman’s book. This text’s classification of states, limiting distributions); Poisson topics in population dynamics, oscillations, and traffic process (as limit of binomial process and as derived theory require less scientific background than topics in via axioms); Markov processes (transition functions mechanics and mathematical biology, lbut still provide and state probabilities, Kolmogorov equations, limiting an excellent basis for modeling discussions. For exam- probabilities, birth-death processes). ple, population dynamics provide a good introduction These basic topics have numerous applications that to dynamical systems. Topics in regular and singular should be an essential feature of the course. In addition, perturbation theory can be presented in the context of some applied topics can be covered such as quality con- oscillations. Traffic theory provides a vehicle for intro- trol, social and occupational mobility, Markovian deci- ducing continuum mechanical modeling in which the sion processes, road traffic, reliability, queueing prob- processes are readily appreciated by students. Here the lems, population dynamics or inventory models. In- “microscopic” processes involve cars amd drivers, and structors can find these and other applications in the interesting models are obtained by car-following theory. many good texts on stochastic processes. Also see the Traffic flows also involve partial differential equations modules and modeling texts listed at the end of this and shock waves. chapter.

Continuous Modeling References on Modeling A primary goal of a continuous modeling course is to present the mathematical analysis involved in sci- Modules entific modeling, as for example, the derivation of the heat equation. The course should also give an introduc- A. MODULEWRITING PROJECTS tion to important applied mathematics topics, such as Claremont Graduate School (Department of Mathemat- Fourier series, regular and singular perturbations, sta- ics): bility theory and tensor analysis. A few advanced top A Fractional Calculus Approach to a Simplified Air ics can be chosen from boundary layer theory, nonlinear Pollution Model for Perturbation Analysis. 52 RESHAPINGCOLLEGE MATFIEMATE

Continuous-system Simulation Languages for DEC- Stochastic Models for the Allocation of Fire Com- 10. panies. * Free Vibrations in the Inner Ear. B. MODULESDEVELOPED BY INDIVIDUALS * Modeling of Stellar Interiors. Subsurface Areal Flow Through Porous Media. Undergraduate Mathematics Application Project (UMAP): UMAP has several hundred modules cov- Variance Reduction for Monto Carlo Applications Involving Deep Penetration. ering all areas of application. Selected modules ap- Voting Games and Power Indices. pear in the UMAP Journal (four issues a year), published by Birkhauser-Boston. UMAP catalogue Mathematical Association of America’s Committee on available by writing to: UMAP, Educational De- the Undergraduate Program in Mathematics Project, velopment Center, 55 Chapel Street, Newton, MA Case Studies in Applied Mathematics (designed espe- 02160. cially for open-ended experiential teaching). Measuring Power in Weighted Voting Systems. c. PROCEEDINGS OF MODELINGCONFERENCES A Model for Municipal Street Sweeping Operations. 1. Discrete Mathematics and Its Applications, Pro- * A Mathematical Model of Renewable Resource Con- ceedings of a Conference at Indiana University, ed. servation. M. Thompson, 1976. * Dynamics of Several-species Ecosystems. 2. Mathematical Models in the Undergraduate Cur- * Population Mathematics. riculum, Proceedings of Conference at Indiana TJni- MacDonald’s Work on Helminth Infections. versity, ed. D. Maki and M. Thompson, 1975. Modeling Linear Systems by Frequency Response 3. Proceedings of Summer Seminar on Applied Mathe- Methods. matics, ed. M. Thompson, Indiana University, 1!)78. Network Analysis of Steam Generator Flow. 4. Mathematical Models for Environmental Problems, * Heat Transfer in Frozen Soil. Proceedings of the International Conference at the University of Southampton, 1976. Mathematical Association of America Summer 1976 5. Proceedings of Conference on Environmental Mod- Module-writing Conference (at Cornell University De- eling and Simulation, Environmental Proteciion partment of Operations Research): Agency, 1976. * About sixty modules covering virtually all areas of 6. Proceedings of a Conference on the Application application, such as biology, ecology, economics, en- of Undergraduate Mathematics in the Engineer- ergy, population dynamics, traffic flow, vibrating ing, Life, Managerial and Social Sciences, ed. P. strings, and voting. Knopp and G. Meyer, Georgia Institute of Technol- Selected modules from this conference along with ogy, 1973. MAA applied mathematics case studies (ii) above 7. Proceedings of the Pittsburgh Conferences on Mod- were published by Springer-Verlag (New York, eling and Simulations, Vols. 1-9 (1969-78), Instru- 1983) in four volumes, edited by William Lucas. ment Society of America. Rensselaer Polytechnic Institute (Department of Math- 8. Proceedings of the Summer Conference for College ematical Sciences), published in Case Studies in Mathe- Teachers on Applied Mathematics, University of matical Modeling, by W. Boyce, Pitman, Boston, 1981: Missouri-Rolla, 1971. Herbicide Resistance. 9. Information Linkage Between Applied Mathematics * Elevator Systems. and Industry, ed. P. Wang, Academic Press, 19’76. * Traffic Flow. Shortest Paths in Networks. Articles on Teaching Modeling Computer Data Communication and Security. 1. J. Agnew and M. Keener, A Case-study Course Semiconductor Crystal Growth. in Applied Mathematics Using Regional Industries, State University of New York at Stony Brook (Depart- American Mathematical Monthly 87 (1980). ment of Applied Mathematics and Statistics): 2. R. Barnes, Applied Mathematics: An Introduction A Model for Land Development. Via Models, American Mathematical Monthly 84 A Model for Waste Water Disposal, I and 11. (1977). A Water Resource Planning Model. 3. C. Beaumont and R. Wieser, Co-operative F’ro- Man in Competition with the Spruce Budworm. grammes in Mathematical Sciences at the Univer- * Smallpox: When Should Routine Vaccination be sity of Waterloo, Journal of Co-operative Education Discontinued. 11 (1975). MODELING& OPERATIONSRESEARCH 53

4. J. Becker, R. Borrelli, and C. Coleman, Models for 6. C. Coffman and G. Fix, ed., Constructive Ap- Applied Analysis, Harvey Mudd College, 1976 and proaches to Mathematical Models, Academic Press, revised annually. 1980. 5. R. Borrelli and J. Spanier, The Mathematics Clinic: 7. R. DiPrima, ed., Modern Modeling of Continuous A Review of Its First Seven Years, UMAP Journal Phenomena, American Mathematical Society, 1977. 2 (1981). 8. C. Dym and E. Ivey, Principles of Mathematical 6. R. Borton, Mathematical Clinic Handbook, Clare- Modeling, Academic Press, 1980. mont Graduate School, 1979. 9. B. Friedman, Lectures on Applications-oriented 7. J. Brookshear, A Modeling Problem for the Class- Mathematics, Holden-Day, 1969. room, American Mathematical Monthly 85 (1978). 10. F. Giordano and M. Weir, A First Course in Math- 8. E. Clark, How To Select a Clinic Project, Harvey ematical Modeling, Brooks/Cole, 1!985. Mudd College, 1975. 11. P. Lancaster, Mathematics Models of the Real 9. C. Hall, Industrial Mathematics: A Course in Real- World, Prentice Hall, 1976. ism, American Mathematical Monthly 82 (1975). 12. D. Maki and M. Thompson, MathLematical Models 10. L. Handa, Mathematics Clinic Student Handbook: and Applications, Prentice Hall, 1976. A Primer for Project Work, Harvey Mudd College, 13. F. Roberts, Discrete Mathematical Models, Prentice 1979. Hall, 1976. 11. J. Hachigian, Applied Mathematics in a Liberal 14. T. Saaty, Thinking with Models, AAAS Study Arts Context, American Mathematical Monthly 85 Guides on Contemporary Problems No. 9, 1974. (1978). 12. E. Rodin, Modular Applied Mathematics for Begin- B. MODELINGIN VARIOUSDISCIPLINES ning Students, American Mathematical Monthly 84 Mathematical modeling is such an integral part of (1977). physics and engineering that any text in these fields 13. R. Rubin, Model Formulation Using Intermedi- is implicitly a mathematical modeling book. ate Systems, American Mathematical Monthly 86 1. P. Abell, Model Building in Sociology, Shocken, (1979). 1971. 14. M. Seven and T. Zagar, The Engineering Clinic 2. R. Aggarwal and I. Khera, Management Science Guidebook, Harvey Mudd College, 1975. Cases and Applications, Holden-Day, 1979. 15. D. Smith, A Seminar in Mathematical Model- 3. R. Atkinson, et al., Introduction to Mathematical building, American Mathematical Monthly 86 Learning Theory, Krieger Publishing, 1965. (1979). 4. D. Bartholomew, Stochastic Models for Social 16. J. Spanier, The Mathematics Clinic: An Innovative Pro- cesses, Wiley, 1973. Approach to Realism Within an Academic Environ- ment, American Mathematical Monthly 83 (1976). 5. M. Bartlett, Stochastic Population Models, Methuen, 1960. Books on Mathematical Modeling 6. R. Barton, A Primer on Simulation and Gaming, Prentice Hall, 1970. For further references, see Applications section of A Ba- 7. S. Brams, Game Theory and Pollitics, The Free sic Library List, Mathematical Association of America, Press, 1975. 1976. 8. C. Clark, Mathematical Bioeconom,ics, Wiley, 1976. A. GENERALMODELING 9. J. Coleman, Introduction to Mathematical Sociol- 1. J. Andrew and R. McLone, ed., Mathematical Mod- ogy, Free Press, 1964. eling, Butterworth, 1976. 10. P. Fishburn, The Theory of Social Choice, Princeton 2. R. Aris, Mathematical Modeling Techniques, Pit- University Press, 1973. man, 1978. 11. J. Frauenthal, Introduction to Population Modeling, 3. E. Beltrami, Mathematics for Dynamic Modeling, UMAP Monograph, 1979. Academic Press, 1987. 12. H. Gold, Mathematical Modeling of Biological Sys- 4. E. Bender, An Introduction to Mathematical Mod- tems, Wiley, 1977. eling, Wiley, 1978. 13. S. Goldberg, Some Illustrative Ezamples of the 5. G. Carrier, Topics in Applied Mathematics, Vol. I Use of Undergraduate Mathematics in the Social and 11, MAA summer seminar lecture notes, Math- Sciences, Mathematical Associatiion of America, ematical Association of America, 1966. CUPM Report, 1977. 54 RESHAPINGCo LLEGE MATHEM AT-

14. M. Gross, Mathematical Models in Linguistics, Mechanics, Mathematical Association of America, Prentice Hall, 1972. 1977. 15. R. Haberman, Mathematical Models, Mechanical 26. J. Pollard, Mathematical Models for the Growth of Vibrations, Population Dynamics and %fit Flow, Human Populations, Cambridge University Press, Prentice Hall, 1977. 1973. 16. F. Hoppensteadt, Mathematical Theories of Popu- 27. D. Riggs, The Mathematical Approach to Physiolog- lations: Demographics and Epidemics, SIAM, 1976. ical Problems, Macmillan, 1979. 17. J. Kemeny and L. Snell, Mathematical Models in the 28. T. Saaty, Topics in Behavioral Mathematics, MAA Social Sciences, MIT Press, 1973. summer seminar lecture notes, Mathematical Asso- 18. C. Lave and J. March, An Introduction to Models ciation of America, 1973. in the Social Sciences, Harper and Row, 1975. 29. H. Scarf, et al., Notes on Lectures on Mathematics 19. C. Lin and L. Segal, Mathematics Applied to Deter- in the Behavioral Sciences, MAA summer seminar lecture notes, Mathematical Association of Amer- ministic Problems in the Natural Sciences, Macmil- ica, 1973. Ian, 1974. 30. C. von Lanzenauer, Cases in Operations Reseawh, 20. D. Ludwig, Stochastic Population Theories, Holden Day, 1975. Springer, 1974. 31. H. Williams, Model Building in Mathematical Pro- 21. J. Maynard-Smith, Models in Ecology, Cambridge gramming, Wiley, 1978. University Press, 1974. 22. B. Noble, Applications of Undergraduate Mathe- matics to Engineering, Mathematical Association of Subpanel Members America, 1976. 23. M. Olinik, An Introduction to Mathematical Mod- WILLIAMLUCAS , CHAIR, Cornell University. els in the Social and Life Sciences, Addison Wesley, ROBERTBORRELLI, Harvey Mudd College. 1978. RALPHDISNEY, VPI & SU. 24. E. Pielou, Mathematical Ecology, Wiley, 1977. DONALDDREW, RPI. 25. H. Pollard, Mathematical Introduction to Celestial SAMGOLDBERG, Oberlin College. Statistics

This chapter contains the report of the Subpanel on gle course in this area. The course proposed below gives Statistics of the CUPM Panel on a General Mathemat- students a representative introduction to both the data- ical Sciences Progmm, reprinted with minor changes oriented nature of statistics and the makhematical con- from Chapter VI of the 1981 CUPM report entitled cepts underlying statistics. These broad objectives raise RECOMMENDATIONSFOR A GENERALMATHEMATICAL several issues that require preliminary comment. One SCIENCESPROGRAM. year of calculus is assumed for this course. The course should use Minitab or a similar interactive statistical package. Introductory Course Statistics is the methodological field of science that The Place of Probability deals with collecting data, organizing and summariz- Probability is an essential tool in several areas of the ing data, and drawing conclusions from data. Although mathematical sciences. It is not possib1.e to compress a statistics makes essential use of mathematical tools, es- responsible introduction to probability and coverage of pecially probability theory, it is a misrepresentation of statistics into a single course. The Statistics Subpanel statistics to present it as essentially a subfield of math- therefore recommends that probability topics be divided ematics. between the courses on probability anti statistics, dis- The Statistics Subpanel believes that an introduc- crete methods, and modeling/operations research as fol- tory course in probability and statistics should con- lows: centrate on data and on skills and mathematical tools * Probability and statistics course: Axioms and basic motivated by the problems of collecting and analyzing properties; random variables; univariate probability data. The traditional undergraduate course in statisti- functions and density functions; moments; standard cal theory has little contact with statistics as it is prac- distributions; Laws of Large Numb’ers and Central ticed and is not a suitable introduction to the subject. Limit Theorem. Such a course gives little attention to data collection, Discrete methods course: Combinzrtorial enumera- to analysis of data by simple graphical techniques, and tion problems in discrete probability. to checking assumptions such as normality. Modeling/operations research cour,Se: Conditional The field of statistics has grown rapidly in applied probability and several-stage models; stochastic areas such as robustness, exploratory data analysis, processes. and use of computers. Some of this new knowledge This division is natural in the sense that the respective should appear in a first course. It is now inexcusable to parts of probability are motivated by and applied to the present the two-sample t-test for means and the F-test primary concerns of these courses. for variances as equally legitimate when a large literature demonstrates that the latter is so sensitive to non- Alternative Arrangements normality as to be of little practical value, while the The subpanel is convinced that two semesters are former (at least for equal sample sizes) is very robust required for a firm introduction to both probability (e.g., see Pearson and Please, Biometrika 62 (1975),pp. and statistics. Many institutions now offer such a two- 223-241,for an effective demonstration). However, the semester sequence in which probabilit,y is followed by Statistics Subpanel does not believe that a course in statistics. The subpanel prefers this structure. In this “exploratory data analysis” is a suitable introduction sequence the statistics course should be revised to incor- to statistics, nor does it advocate replacing (say) least porate the topics and flavor of the data analysis section squares regression by a more robust procedure in a first of the proposed unified course. With probability first, course. But it does think that new knowledge renders added material in statistics can also be covered, such as a course devoted solely to the theory of classical para- Neyman-Pearson theory, distribution-free tests, robust metric procedures out of date. procedures, and linear models. While the Statistics Subpanel prefers a two-semester Institutions will vary considerably in their choice of introductory sequence in probability and statistics, en- material for this statistics course, but the subpanel reit- rollment data shows that most students take only a sin- erates its conviction that the traditional “theory-only” statistics course is not a wise choice. If experience Course Outline shows that many students drop out in the middle of a two-course sequence, the unified course outlined be- I. Data (about 2 weeks) low should be adopted, followed by one of the elective Random sampling. Using a table of random digits; courses suggested in Section 3 of this chapter. simple random samples, experience with sampling variability of sample proportions and means; strat- Instructor Preparation ified samples as a means of reducing variability. Since the recommended outline is motivated by data Ezperimental design. Why experiment; motivation and shaped by the modern practice of statistics, many for statistical design when field conditions for liv- mathematically trained instructors will be less prepared ing subjects are present; the basic ideas of control to teach this course than a traditional statistical theory and randomization (matching, blocking) to red.uce course. Growing interest in “applied” statistics has, of variability. course, led many instructors to broaden their knowl- COMMENTS:Data collection is an important part of edge. Some background reading is provided for others statistics. It meets practical needs (see Moore) andjus- who wish to do so. The publications listed here contain tifies the assumptions made in analyzing data (see Box, material that can be incorporated in the recommended Hunter and Hunter). Experience with variability helps course, but none is suitable as a course text. In order motivate probability and the difficult idea of a sam- of ascending level: pling distribution. Students should see for themselves 1. Tanur, Judith, et al., eds., Statistics: A Guide to the results of repeated random sampling from the same the Unknown, Second Edition, Holden-Day, 1978. population and the variability of data in simple experi- An elementary volume describing important ap- ments such as comparing 3-minute performance of egg plications of statistics and probability in many timers (see W.G. Hunter, American Statistician, 1977, fields of endeavor. pp. 12-17, for suggestions). 2. Moore, David, Statistics: Concepts and Controver- 11. Organizing and Describing Data sies, W.H. Freeman, 1979. A paperback with good material on data collec- (about 2 weeks) tion, statistical common sense, appealing exam- Tables and graphs. Frequency tables and his- ples, and the logic of inference. tograms; bivariate frequency tables and the mislead- ing effects of too much aggregation; standard line 3. Freedman, David; Pisani, Robert; Purves, Roger, and bar graphs and their abuses; box plots; spot- Statistics, W.W. Norton, 1978. ting outliers in data. A careful introduction to elementary statistics Univariate descriptive statistics. Mean, median isnd written with conceptual richness, attention to percentiles; variance and standard deviation; a few the real world, and awareness of the treachery more robust statistics such as the trimmed mean. of data. Bivariate descriptive statistics. Correlation; fitting 4. Mosteller, Frederick and Tukey, John, Data Analy- lines by least squares. If computer resources permit, sis and Regression, Addison-Wesley, 1977. least-square fitting need not be restricted to lines. Good ideas on exploratory data analysis, robust- COMMENTS:In addition to simple skills, students ness and regression. must be trained to look at data and be aware of pit- 5. Box, George; Hunter, William; Hunter, J. Stuart, falls. Freedman, Pisani and Purves have much good Statistics for Ezperimenters, Wiley, 1978. material on this subject, such as the perils of aggrega- Applied statistics explained by experienced tion (pp. 12-15). The impressive effect on a correlation practical statisticians. Some specialized mate- of keypunching 7.314 as 731.4 should be pointed out. rial, but much of the book will repay careful Simple plots are a powerful tool and should be stressed reading. throughout the course as part of good practice. 6. Efron, Bradley, “Computers and the Theory of Statistics: Thinking the Unthinkable,” SIAM Re- 111. Probability (about 4 weeks) view, October, 1979. General probability. Motivation; axioms and bisic A superb article on some new directions in rules, independence. statistics, written for mathematicians who are Random variables. Univariate density and proba- not statisticians. bility functions; moments; Law of Large Numbers. STATISTICS 57

0 Standard distributions. Binomial, Poisson, expo- truth. Moreover, most instructors will find it easy to nential, normal; Central Limit Theorem (without supplement a methods text with mathematical mate- proof). rial and problems familiar from previous teaching. It is More ezperience with randomness. Use in computer much harder to supply motivation and realistic prob- simulation to illustrate Law of Large Numbers and lems, and it is psychologically difficult for both the Central Limit Theorem. teacher and student to skip much of the probability in COMMENTS:Probability must unavoidably be a mathematical statistics text. pressed in a unified course that includes data analysis. The following books are possible texts or reference Instructors should repeatedly ask “What probability do material for the course described above. All of these I need for basic statistics?” and “What can the students have essentially the same shortcoming of being too ‘un- learn within about four weeks?” It is certainly the case mathematical.” The appropriate combination of level of that combinatorics, moment generating functions, and sophistication and content is not now a.vailable under a continuous joint distributions must be omitted. Some single cover. The class of books below fall in the “inter- instructors may be able to cover conditional probability mediate” level between an elementary statistics course and Bayes’ theorem in addition to the outline material. and a first course in mathematical statistics. Box, George; Hunter, William; Hunter, J. Stuart, IV. Statistical Inference (about 6 weeks) Statistics for Ezperimenters: An Introduction to Statistics us. probability. The idea of a sampling Design, Data Analysis, and Modei! Building, John distribution; properties of a random sample, e.g., it Wiley & Sons, New York, 1978. is normal for normal populations; the Central Limit Moore, David and McCabe, George, Introduction to Theorem. the Practice of Statistics, Freeman, San Francisco, Tests of significance. Reasoning involved in alpha- 1989. level testing and use of P-values to assess evi- Ott, Lyman, An Introduction to Statistical Meth- dence against a null hypothesis; cover one- and ods and Data Analysis, Second Edition, Duxbury, two-sample normal theory tests and (optional) chi- Boston, 1984. square tests for categorical data. Comment on ro- Neter, John; Wasserman, William; Whitmore, bustness, checking assumptions, and the role of de- G.A., Applied Statistics, Allyn and Bacon, Boston, sign (Part I) in justifying assumptions. 1978. Point estimation methods. Method of moments; maximum likelihood; least squares; unbiasedness and consistency. Confidence intervals. Importance of error estimate Additional Courses with point estimator; measure of size of effect in a test of significance. Probability and Statistical Theory Inference in simple linear regression. COMMENTS:A firm grasp of statistical reasoning is CONTENT: Distribution functions; moment and more important than coverage of a few additional spe- probability generating functions; joint, marginal and cific procedures. For much useful material on statis- conditional distributions; correlations; distributions of tical reasoning such as use of the “empirical rule” to functions of random variables; Chebyschev’s inequal- assess normality, see Box, Hunter and Hunter. Don’t ity; convergence in probability; limiting distributions; just say, “We assume the sample consists of iid normal power test and likelihood ratio tests; introduction to random variables.” Applied statisticians favor P-values Bayesian and nonparametric statistic!a; additional re- over fixed alpha tests; a comparative discussion of this gression topics. issue appears in Moore. COMMENT:This course is designed to complete the traditional probability-then-statistics sequence. Since RECOMMENDEDTEXTS the students have already completed a semester of The Subpanel is not aware of a text at the post- study, they should be capable of tackling a good text calculus level that fits the recommended outline closely. on mathematical statistics such as the one by DeGroot Instructors should seriously consider adopting a good or by Hogg and Craig. The book by Bickel and Dok- post-calculus statistical methods text rather than a the- sum is a little more difficult than the other two, and the oretical statistics text. A methods text is more likely teacher would have to supplement it with the topics in to have examples and problems which have the ring of probability. 58 RESHAPINGCOLLEGE MATHEMATE

TEXTS: classic but covers only discrete probability. The book 1. Mendenhall, William; Schaeffer, Richard; Wackerly, by Olkin, Gleser and Derman is at a slightly lower level Dennis, Mathematical Statistics with Applications, and is more "applied" but will require the instructor Second Edition, Duxbury, Boston, 1981. to provide some supplementary materials. The book 2. Larsen, Richard and Marx, Morris, An Introduction by Chung is excellent but must be read with a "grain to Probability and its Applications, Prentice-Hall, of salt." The book by Breiman is also excellent but Englewood Cliffs, N. Jers., 1985. expects much of its reader. A new book by Johnson and 3. DeGroot, Morris M., Probability and Statistics, Kotz also looks interesting but is restricted to discrete Addison-Wesley, Reading, Mass., 1975. probability. The books by Chung, Feller and Breirnan 4. Hogg, Robert and Craig, Allen, Introduction to are difficult for the average student. Mathematical Statistics, Macmillan, New York, TEXTS: 1978. 1. Olkin, Ingram; Gleser, Leon J.; Derman, Cyrus, Probability Models and Applications, Macmillan, Applied Statistics New York, 1980. CONTENT:This course uses statistical packages to 2. Larsen, Richard and Marx, Morris, An Introduction analyze data sets. Topics include linear and multiple re- to Probability and its Applications, Prentice-Hall, gression; nonlinear regression; analysis of variance; ran- Englewood Cliffs, N.J., 1985. dom, fixed and mixed models; expected mean squares; 3. ROSS,Sheldon, A First Course in Probability, Sec- pooling, modifications under relaxed assumptions; mul- ond Edition, Macmillan, New York, 1984. tiple comparisons; variance of estimators; analysis of 4. Chung, Kai Lai, Elementary Probability Theory with covariance. Stochastic Processes, Springer-Verlag, New York, COMMENT:The new introductory course will proba- 1974. bly attract more students from other fields than the tra- 5. Feller, William, An Introduction to Probability The- ditional probability-then-statistics course. This course ory and Its Applications, Volume I, John Wiley & is an excellent follow-up for such non-mathematical sci- Sons, New York, 1950. ences students. Its topics are among the more widely used statistical tools. Students should be expected to Preparation for Graduate Study use a statistical computing package such as Minitab of SPSS for many of the analyses. The book by Miller and There are a large number of career opportunities for Wichern is a possible text for this course. statisticians in industry, government and teaching. For example as of 1977, the Federal Government employed TEXTS: over 3500 statisticians. plus 3500 statistical assistants I- 1. Miller, Robert and Wichern, Dean, Intermediate and numerous other employees performing statistical Business Statistics, Holt, Rinehart and Winston, duties but classified in different job series. A recent New York, 1977. report by the U.S. Labor Department, reprinted in the 2. Neter, John; Wasserman, William; Kutner, Michael, New York Times National Recruitment Survey, predicts Applied Linear Statistical Models, Second Edition, an increase of 35% in the demand for statisticians dur- Irwin, 1985. ing the 1980's. This compares to a predicted increase 3. Morrison, Donald, Applied Linear Statistical Meth- of 9% for mathematicians and 30% for computer s:pe- ods, Prentice-Hall, Englewood Cliffs, N. Jers., 1983. cialists. Preparation for a career in statistics usually involves Probability and Stochastic Processes graduate study. An undergraduate major in statistics, CONTENT:Combinatorics; conditional probability computer science, or mathematical sciences is the rec- and independence; Bayes theorem; joint, marginal and ommended preparation for graduate study in statistics. conditional distributions; distribution functions; dis- It is desirable for such a major to include solid courses in tributions of functions of random variables; probabil- matrix theory and real analysis, in addition to courses ity and moment generating functions; Chebyschev's in- in probability and statistics. Most statistics graduate equality; convergence in probability; convergence in dis- programs require matrix algebra and real analysis for tribution; random walks; Markov chains; introduction fully matriculated admission. Either one of the sample to continuous-time stochastic processes. programs in the report of the General Mathematical COMMENT:This is a fairly standard course and a Sciences Panel in the first chapter would be adequate number of texts are available. The book by Feller is a preparation for graduate study in statistics. However, STATISTICS 59 major A is preferable to major B, and both should in- [2], discusses preparation for a career i.n government. clude at least one follow-on elective in probability and 1. Preparing Statisticians for Careers in Industry: Re- statistics. port of the ASA Section on Statistical Education. In addition to courses in the mathematical sciences, The American Statistician, 1980, pp. 65-80. a student preparing for graduate study in statistics 2. Preparing Statisticians for Careers; in Government: should: Report of the ASA Section on Statistics in Govern- Study in depth some subject where statistics is an ment. Paper presented at the Amserican Statistical important tool (physics, chemistry, economics, psy- Association meeting in August, 1980. chology, ...). In fact, a double major should be considered. Take as many courses as possible which are designed Panel Members: to enhance his or her communication skills. Statis- ticians in industry and government are often called RICHARDALO, CHAIR, Lamar University. upon to provide written reports and critiques; con- RICHARDKLEBER, St. Olaf College. sulting requires clear oral communications. DAVIDMOORE, Purdue University. A detailed discussion of preparation for a statistical MIKEPERRY , Appalachian State University. career in industry can be found in [l]A similar report, TIMROBERTSON, University of Iowa.

Discrete Mathematics

In the early 1980’~~as computer science enrollments because there ia no consensus as to the exact content ballooned on campuses across the country, the Mathe- of the course. Those suggestions offered by the Com- matical Association of America established an ad hoc mittee as to topics that should be considered for in- Committee on Discrete Mathematics to help provide clusion may have been loosely followed, but level and leadership to the rapidly ezpanding efforts to create attitude of books vary widely. A bibliography which a course an discrete mathematics that would meet the is current as of the end of 1988 appears in a report on needs of computer science and at the same time fit well discrete mathematics edited by Anthony Ralston (to ap- into the traditional mathematics program. Thi3 com- pear in the MAA Notes Series, 1989), as do final reports mittee issued a report in 1986 that conveyed their own of the Sloan Foundation funded discrete mathematics recommendations together with an appendia that in- projects. cluded reports from siz ezperimental projects supported New freshman-sophomore textbooks continue to ap- by the Alfred P. Sloan Foundation. This chapter con- pear. For the immediate future, it seems as if a one- tains the report of the Committee, without the appendiz, or two-semester discrete mathematics course, indepen- preceded by a new preface prepared by Committee Chair dent of (but at the same level as) calculus will be the Martha Siegel. typical one. Recent advances such as ISETL, a computer language for the teaching of discrete mathematics (Learning Discrete Mathematics with I‘SETL, by Nancy 1989 Preface Baxter, Ed Dubinsky, and Gary Levin), and the True In the years since 1986 when the Committee on Dis- BASIC Discrete Mathematics package(by John Kemeny crete Mathematics in the First Two Years published its and Thomas Kurtz, Addison-Wesley) may affect how report, there have been many changes in the attitudes the course evolves. Those few effork to incorporate of mathematics departments toward curricular change. discrete mathematics into the calcululs and the course- The Committee had found that faculty in disciplines ware that are being developed merit our attention for that required calculus were quite supportive of propos- the future. als to introduce more discrete mathematics into the first Who teaches discrete mathematics? Most mathe- two years. They frequently complained about the state matics departments have a course, though sometimes of calculus and encouraged us to get our house in better only on the junior-senior level. Sometimes the elemen- order. Many mathematicians also expressed dissatisfac- tary course is offered in the computer science depart- tion with the calculus sequence. The threat of replacing ment or in engineering. The 1985-86 CBMS survey in- some of the traditional calculus material with discrete dicates that more than 40% of all institutions require topics certainly helped to turn attention to the teach- discrete mathematics for computer science majors. Of ing of calculus. This movement toward a “calculus for universities and four-year colleges, about 60% require a new century” is exciting and timely. discrete mathematics or discrete structures. It is also It is disappointing, however, that there seem to be not uncommon that mathematics ma.jors are required only a few attempts to incorporate any significant dis- to take some discrete mathematics. crete mathematics into the revision of the curriculum The most recent accreditation standards issued by of the first two years. The discrete mathematics course the Computer Sciences Accreditation Board include a seems to be established in most schools as a separate discrete mathematics component. The most recent re- entity. It is encouraging that the National Council of port of the ACM Task Force on the Core of Computer Teachers of Mathematics has established a Task Force Science defines nine areas as the core of computer sci- on Discrete Mathematics to help teachers implement ence (Communications of ACM, Jan. 1989, 32:l). In curriculum standards for the inclusion of discrete math- all but a few areas, discrete matheinatics topics are ematics in the schools. listed as support areas. For algorithrnls and data struc- Many new textbooks for the standard (usually one tures, for example, students should be familiar with semester) discrete mathematics courses for the fresh- graph theory, recursive functions, recurrence relations, man or sophomore student have appeared or are in combinatorics, induction, predicate and temporal logic, press. Publishers seem to find the market troublesome among other things. Boolean algebra and coding the- 62 RESHAPINGCOLLEGE MATHEMAE ory are considered part of the architecture component. that might be considered for an elementary course:, two Students certainly would need discrete mathematics as workshop groups at the William Conference produced a prerequisite for many of the computer core courses. (in a very short time) a fairly remarkable set of two Graph theory, logic, and algebra appear in a significant course sequences: number of necessary support areas. 1. A two year sequence of independent courses, one Although the Committee on Discrete Mathematics in in discrete mathematics and one in a streaml.ined the First Two Years was dissolved after its 1986 report calculus, and was issued, concerns of the Committee have been incorporated into the mission of the MAA Committee on 2. A two year integrated course in discrete and contin- Calculus Revision and the First Two Years (CRAFTY). uous mathematics (calculus) in a modular form for Aside from concentration on the calculus initiatives service to many disciplines. across the country, CRAFTY is interested in continu- These course outlines were admittedly tentative and ing the effort of the earlier group to see discrete mathe- needed refinement and testing. At the same time, the matics become part of the typical freshman-sophomore CUPM-CTUM Subcommittee on Service Courses had curriculum in any of the mathematical sciences. The been examining the traditional service course offerings goal is to increase the effectiveness of the curriculum in of the first two years. The syllabi of these courses, in serving other disciplines while providing enough excite- which many freshman and sophomores are required to ment and challenge to attract talented undergraduates enroll, are studied periodically for their relevancy. Fi- to major in mathematics. nite mathematics, linear algebra, statistics, and calculus are considered to be essential to many majors, but with MARTHAJ. SIEGEL the importance of the computer, the Subcommittee on Towson State University Service Courses concluded that even the mathematics March, 1989 majors need mathematics of a new variety, not only so they can take computer science courses, but also so they can work on contemporary problems in mathematics. Introduction and History At that time, there were few or no textbooks or ex- The Committee on Discrete Mathematics in the First amples of such courses for the community to share.. At Two Years was established in the spring of 1983 for the the suggestion of the Subcommittee on Service Courses, purpose of continuing the work begun at the Williams the MAA agreed to help to develop the William courses College Conference held in the summer of 1982. That further through the Committee on Undergraduate Pro- conference brought to a forum the issue of revising the gram in Mathematics (CUPM) and the Committee on college curriculum to reflect the needs of modern pro- the Teaching of Mathematics (CTWM), standing com- grams and the students in them. Anthony Ralston mittees of the Association. That led to the estab- and Gail Young brought together 29 scientists (24 of lishment of the committee responsible for this report. whom were mathematicians) from both industry and Funds for the effort were secured from the Sloan Foun- academe to discuss the possible restructuring of the dation. The members of the committee were chosen first two years of college mathematics. Although the especially to reflect the communities who would even- growing importance of computer science majors as an tually be most affected by any changes in the traditional audience for undergraduate mathematics was an impor- mathematics curriculum. tant motivation for the Williams Conference, the con- In addition to the development of course outlines ference concerned itself quite broadly with the need to and plans for their implementation, the committee was revise the first two years of the mathematics curricu- also involved in the observation of a set of experi- lum for everyone-mathematics majors, physical sci- mental projects which also were begun as a result of ence and engineering majors, social and management the Williams Conference and the interest of the Sloan science majors as well as computer science majors. The Foundation. After the conference, the Sloan Founda- papers presented and discussed at the conference, and tion solicited about thirty proposals for courses which collected in The Future of College Mathematics [35], re- would approximate the syllabi suggested by the work- flect this breadth of view. shop participants. The call for proposals particularly The word used to describe what was needed was "dis- mentioned the need for the development of text :ma- crete" mathematics. Most of us knew what that meant terial and classroom testing and emphasized the hope approximately and respected the content as good math- that some schools would make the effort to try the inte- ematics. To illustrate the discrete mathematics topics grated curriculum. Six schools were ultimately chosen: DISCRETEMATHEMATICS 63

Colby College, Waterville, Maine. General Discussion 0 University of Delaware, Newark, Delaware. University of Denver, Denver, Colorado. In its final report, the committee has decided to 0 Florida State University, Tallahassee, Florida. present two course outlines for elementary mainstream Montclair State College, Montclair, New Jersey. discrete mathematics courses. Our unanimous prefer- St. Olaf College, Northfield, Minnesota. ence is for a one-year course, at the level of the calculus but independent of it. It is designed to serve as The committee, together with some of the committee a service course for computer science majors and oth- which chose the proposals to be funded (Don Bushaw, ers and as a possible requirement for mathematics ma- Steve Maurer, Tony Ralston, Alan Tucker, and Gail jors. The Committee on the Undergraduate Program in Young), monitored their progress for the two year pe- Mathematics (CUPM) has endorsed the recommenda- riod of funding, ending in August 1985. (Complete re- tion that every mathematics major take a course in dis- ports of these funded projects will appear in an MAA crete mathematics, and has agreed th(at the year course Notes volume on discrete mathematics to be published the committee recommends is a suitable one for math- in 1989.) ematics majors. It is expected that the course will be taken by freshmen or sophomores majoring in computer science so that they can apply the material in the first Summary of Recommendations and second year courses in their major. The ACM rec- 1. Discrete mathematics should be part of the first two ommendations [21, 221 for the first year computer sci- years of the standard mathematics curriculum at all ence course presume, if not specific topics, then cer- colleges and universities. tainly the level of maturity in mathematical thought 2. Discrete mathematics should be taught at the intel- which students taking the discrete mathematics course lectual level of calculus. might be expected to have attained. Hence, the Com- 3. Discrete mathematics courses should be one-year mittee recommends that the course be taken simulta- courses which may be taken independently of the neously with the first computer science course. The calculus. Committee understands that at some schools the first 4. The primary themes of discrete mathematics computer science course may be preceded by a course courses should be the notions of proof, recursion, strictly concerned with programming. At the very least, induction, modeling, and algorithmic thinking. the Committee expects that the discrete mathematics 5. The topics to be covered are less important than the course will be a prerequisite to upper-level computing acquisition of mathematical maturity and of skills in courses. For this reason, the Committee has tried to using abstraction and generalization. isolate those mathematical concepts that are used in 6. Discrete mathematics should be distinguished from computer science courses. The usual sequence of these finite mathematics, which as it is now most often courses might determine what should be taught in the taught, might be characterized as baby linear alge- corresponding mathematics courses. bra and some other topics for students not in the In addition to the Committee’s concern for computer “hard” sciences. science majors, there is a high expectation that math- 7. Discrete mathematics should be taught by mathe- ematics majors and those in most physical science and mat ic ians. engineering fields will benefit from the topics and the 8. All students in the sciences and engineering should problem-solving strategies introduced in this discrete be required to take some discrete mathematics as mathematics course. Subjects like combinatorics, logic, undergraduates. Mathematics majors should be re- algebraic structures, graphs, and network flows should quired to take at least one course in discrete math- be very useful to these students. In adldition, methods of ematics. proof, mathematical induction, techniques for reducing 9. Serious attention should be paid to the teaching of complex problems to simpler (previously solved) prob- the calculus. Integration of discrete methods with lems, and the development of algorithLms are tools to en- the calculus and the use of symbolic manipulators hance the mathematical maturity of all. Furthermore, should be considered. students in these scientific and mathematically-oriented 10. Secondary schools should introduce many ideas of fields will want to take computer science courses, and discrete mathematics into the curriculum to help will need some of the same mathematical preparation students improve their problem-solving skills and that the computer science major needs. prepare them for college mathematics. Thus, the Committee has agreed to recommend that 64 RESHAPINGCOLLEGE MATHEMATE the course be part of the regular mathematics sequence course are designed for the freshman level. Some top in the first two years for all students in mathematically- ics necessary for elementary computer science may have related majors. Our contacts with physicists and engi- to be taught at an appropriate later time, either in a neers reinforce the idea that their students will need junior-level discrete mathematics course or in the com- this material, but, of course, there is the concern that puter science courses. calculus will be short-changed. The Committee will make several suggestions regard- Needs of Computer Science ing the calculus, but individual institutions will best understand their own needs in this regard. We do not rec- What do the computer science majors need? In ommend that the third semester of calculus be cut from teaching the first year Introduction to Computer Sci- a standard curriculum. Serious students in mathemati- ence course, Tony Ralston kept track of mathematics cal sciences, engineering, and physical sciences need to topics he would have liked the students to have had know multivariable calculus. Many in the mathemati- before (or at least concurrently with) his course: cal communityrecognize that the content of the calculus Elementary Mathematics: Summation notation; should be updated to acknowledge the use of numeri- subscripts; absolute value, truncation logarithms, cal methods and computers, and promising initiatives trigonometric functions; prime numbers; greatest along this line are being taken. Engineers have been common divisor; floor and ceiling functions. especially anxious for this change. John Schmeelk sur- General Mathematical Ideas: Functions; sets iind veyed 34 schools and compiled suggestions for revising operations on sets. the standard calculus. (Schmeelk's survey was included * Algebra: Matrix algebra; Polish notation; congru- in the appendix to the original report of the Commit- ences. tee.) At some of the Sloan-funded schools and others, 0 Summation and Limits: Elementary summation there have been attempts to revise the calculus to incor- calculus; order notation, O(fn);harmonic numbers. porate some discrete methods and to use the power of Numbers and Number Systems: Positional notation; the symbolic manipulator packages. We describe these nondecimal bases. attempts later in the report. (A complete report on the Logic and Boolean Algebra: Boolean operators and Sloan-funded projects will be published in the MAA expressions; basic logic. Notes Series in 1989.) Probability: Sample spaces; laws of probability. There is, inherent in our proposal, the possibil- Combinatorics: Permutations, combinations, count- ity that some students may be required to take five ing; binomial coefficients, binomial theorem. semesters of mathematics in the first two years-a year Graph Theory: Basic concepts; trees. of discrete mathematics and the three semesters of cal- 0 Difference Equations and Recurrence Relations: culus. But, there is no reason why students cannot be Simple differential equations; generating functions. allowed to take one of the five in the junior year. We Many of the ideas are those that students should have point out that some linear algebra is included in the had in four years of the traditional high school curricu- year of discrete mathematics. Additionally, the use of lum. In addition, there are some ideas and techniques computers via the new and powerful symbolic manipu- that are probably beyond the scope of secondary school lation packages may reduce the amount of time needed mathematics. An elaboration of this list appears in the for the traditional calculus sequence. Appendix and in an article by Ralston in ACM Com- A one-semester discrete mathematics course will be munications [33]. described in the appendix to this report as a concession Many proposals have been coming from the computer to the political realities in many institutions. It has be- science community. Recommendations for a freshman- come obvious to the Committee over the last two years level discrete mathematics course from the Educational that at some colleges, there is a limitation on the num- Activities Board of IEEE probably are the most de- ber of new elementary courses that can be introduced manding. Students enrolled in the course outlined in at this time. the appendix are first semester freshman also enrollled The Committee believes strongly that mathematics in the calculus according to the IEEE recommendations should be taught by mathematicians. Although there published in December 1983, by the IEEE Computer are some freshman-sophomore courses in discrete math- Society [19]. ematics in computer science departments, the course Accreditation guidelines passed recently by ACM presented here should, the Committee believes, be and IEEE also require a discrete mathematics course. taught by mathematicians. The rigor and pace of this The recommendations for the mathematics component DISCRETEMATHEMATICS 65 of a program that would merit accreditation appear be- appropriate for freshman mathematics majors. Proofs low. The criteria appear in their entirety in an article will be an essential part of the course. by Michael Mulder and John Dalphin in the April 1984 Alfs Berztiss, a member of the Committee, led a Computer [28]. number of mathematics and computer science faculty at Certain areas of mathematics and science are funda- a conference at the University of Pittsburgh in 1983 at mental for the study of computer science. These areas which an attempt was made to formulate a high-quality must be included in all programs. The curriculum program in computer science which would prepare good must include one-half year equivalent to 15 semester students for graduate study in the field. Details are hours of study of mathematics. This material includes discrete mathematics, differential and integral calcu- available in a Technical Report (83-5) from the Univer- lus, probability and statistics, and at least one of the sity of Pittsburgh Department of Computer Science [8]. following areas: linear algebra, numerical analysis, Both that program and the new and extensive bach- modern algebra, or differential equations. It is rec- elor’s program in computer science at Carnegie-Mellon ognized that some of this material may be included in the offerings in computer science . . . . University depend on an elementary diiscrete mathematics course. Presentation of accreditation guidelines which re- In addition to the proposals for programs, the com- quire one and one-half years of study in computer sci- puter science community is in the process of revising el- ence, one year in the supporting disciplines, one year ementary computer science courses. Tlhough old courses of general education requirements, and one-half year of stressed language instruction, a more modern approach electives induced quick and angry response. The lib- stresses structured programming and a true introduc- eral arts colleges and the small colleges unable to offer tion to computer science. The beginning courses CS1 this number of courses or unwilling to require so many and CS2 are described by the ACM Task Force on CS1 credits in one discipline, have responded in many ways. and CS2. We quote from the article by Elliot Koffman, This Small College Task Force of the ACM issued its et al. [21] about the role of discrete mathematics in the own report, approved by the Education Board of the structure of these computer science courses. ACM [5]. We emphasize only the mathematics portion We are in agreement with many other computer sci- of those guidelines. entists that a strong mathematics foundation is an Many areas of the computer and information sciences essential component of the computer science curricu- rely heavily on mathematical concepts and techniques. lum and that discrete mathematics is the appropriate An understanding of the mathematics underlying var- first mathematics course for computer science majors. ious computing topics and a capability to implement Although discrete mathematics must be taken prior to that mathematics, at least at a basic level, will enable CS2, we do not think it is a necessary prerequisite to students to grasp more fully and deeply computer con- CSI. . . .We would . . .expect computer science ma- cepts as they occur in courses . . . . It seems entirely jors and other students interested in continuing their reasonable and appropriate, therefore, to recommend studies in computer science to take discrete mathe- a substantial mathematical component in the CSIS matics concurrently with the revised CS1. curriculum . . . . To this end, a year of discrete math- If high schools and colleges take the recommendation ematical structures is recommended for the freshman year, prior to a year of calculus. seriously, the student enrolled in CS1 would be enrolled in a discrete mathematics course concurrently. That The Sloan Foundation supported representatives of __ mathematics course would be required as a prerequi- a few liberal arts schools in their attempt to define a site for CS2. Of all the recommendat,ions, this is likely high-quality computer science major in such institu- to have the largest impact on enrollments in discrete tions. Again, we put the emphasis on the mathematics mathematics courses. component of the proposed program. From Model Curriculum for a Liberal Arts Degree in Computer Science by Norman E. Gibbs and Allen B. Syllabus Tucker [12]: The discrete mathematics course should play an im- What are the common needs of mathematics and portant role in the computer science curriculum . . . . computer science students in mathematics? The Com- We recommend that discrete mathematics be either mittee agrees that all the students need to understand a prerequisite or corequisite for CS2. This early posi- the nature of proof, and the essentialls of propositional tioning of discrete mathematics reinforces the fact that and predicate calculus. In addition, all need to under- computer science is not just programming, and that there is substantial mathematical content throughout stand recursion and induction and, related to that, the the discipline. Moreover, this course should have sig- analysis and verification of algorithms and the algorith- nificant theoretical content and be taught at a level mic method. The nature of abstraction should be part 66 RESHAPINGCOLLEGE MATEEMA= of this elementary course. While some of the Commit- tary nature of the subjects could be made clear by both tee supported the introduction to algebraic structures instructors. in this course, particularly for coding theory and fi- Algorithms are, of course, an integral part of the nite automata, others felt that those concepts were best course. There is still no general agreement on how to left to higher-level courses in mathematics. The basic express them in informal language. While a form of principles of discrete probability theory and elementary pseudocode might suit some people, others have found statistics might be considered to be as important and that an informal conversational style suffices. The Com- more accessible to students at this level. Professionals mittee would not want to make any specific recommen- in all disciplines cite the importance of teaching prob- dations except that the student be precise and convey lem solving skills. Graph theory and combinatorics are his/her methods. It is certainly not necessary to write excellent vehicles. All these students need some calcu- all algorithms in Pascal. Communication is the key. lus. The recommendations for a one-year discrete mathe- The Committee recommends the inclusion of as matics course are presented in several ways. An outline many of the proposed topics as possible with the un- of the course appears below. In the Appendix, the out- derstanding that taste and the structure of the curricu- line has been expanded to include objectives and sam- lum in each institution will dictate the depth and extent ple problems for each topic. The scope and level of to which they are taught. The ability of students in a the course can be appreciated best from the expanded course at this level must be considered in making these version. choices. While one of the goals of the course is to in- Discrete Mathematics crease the mathematical maturity of the student, some A One Year Freshman-Sophomore Course of the mathematical community who have communicated to the panel about their experiences teaching this (Preliminary Outline) course have indicated that there are prerequisite skills in reading and in maturity of thinking that really are Prerequisite: Four years of high school mathematics; needed, perhaps even more than in the calculus. may be taken before, during, or after calculus I and 11. The Committee recognizes that it might be some time before there is as much agreement on the content of 1. Sets. Finite sets, set notation, set operations, sub- a discrete mathematics sequence as there is now about sets, power sets, sets of ordered pairs, Cartesian the calculus sequence. In the meantime, diversity and products of finite sets, introduction to countably in- variety should be encouraged so that we may learn what finite sets. works and what does not. In any case, the Committee 2. The Number System. Natural numbers, integers, strongly endorses the notion that it is not what is taught rationals, reals, Zn, primes and composites, inhro- so much as how. If the general themes mentioned in duction to operations, and algebra. the previous paragraph are woven into the content of 3. The Nature of Proof. Use of examples to demon- the course, the course will serve the students well. Ad- strate direct and indirect proof, converse and con- equate time should be allowed for the students to do trapositive, introduction to induction, algorithms. a lot on their own: they should be solving problems, 4. Formal Logic. Propositional calculus, rules of logic, writing proofs, constructing truth tables, manipulating quantifiers and their properties, algorithms and symbols in Boolean algebra, deciding when, if, and how logic, simplification of expressions. to use induction, recursion, proofs by contradiction, etc. 5. Functions and Relations. Properties of order rela- And their efforts should be corrected. tions, equivalence relations and partitions, functions We have been asked about the role of the computer and properties, into, onto, 1-to-1, inverses, compo- in this course. To a person we have agreed that this sition, set equivalence, recursion, sequences, induc- is a mathematics course and that while students might tion proofs. be encouraged, if they have the background, to try the 6. Combinatorics. Permutations, combinations, bi- algorithms on a computer, the course should emphasize nomial and multinomial coefficients, counting sets mathematics. The skills that we are trying to teach will formed from other sets, pigeon-hole principle, algo- serve the student better than any programming skills rithms for generating combinations and permu ta- we might teach in their place, and the computer science tions, recurrence relations for counting. departments prefer it that way. Surely the ideal would 7. Recurrence Relations. Examples, models, algo- be that students be concurrently enrolled in this course rithms, proofs, the recurrence paradigm, solution and a computer science course where the complimen- of difference equations. DISCRETEMATHEMATICS 67

8. Graphs and Digraphs. Definitions, applications, ma- Others of us are not 80 sure. Increased use of algorith- trix representation of graphs, algorithms for path mic thinking in problem solving could be easily adapted problems, circuits, connectednesl, Hamiltonian and to many high school courses. Readers are encouraged Eulerian graphs, ordering relations-partial and lin- to read Steve Maurer’s article in the :September 1984 ear ordering, minimal and maximal elements, di- Mathematics Teacher for more on this subject. rected graphs. The Committee on Placement Examinations of the 9. Tmes. Binary trees, search problems, minimal span- MAA will be attempting to isolate those skills that seem ning trees, graph algorithms. to be needed by students taking discrete mathematics. 10. Algebraic Structures. Boolean algebra, semigroups, Although this study might not lead to the development monoids, groups, examples and applications and of a placement examination for the course, it will help proofs; or to explain what might be the appropriate preparation 11. Discrete Probability and Descriptive Statistics. for a successful experience in such a course. Events, assignment of probabilities, calculus of Year after year we face students who claim that they probabilities, conditional probability, tree diagrams, have never seen the binomial theorem, mathematical Law of Large Numbers, descriptive statistics, simu- induction, or logarithms before college. These used to lation. be topics taught at the eleventh or twelfth grade levels. 12. Algorithmic Linear Algebra. Matrix operations, What has happened to them? Students also say that relation to graphs, invertibility, row operations, they never had their papers corrected in high school so solution of systems of linear equations using ar- they never wrote proofs. Some of us have students who rays, algebraic structure under operations, linear cannot tell the hypothesis from the conclusion. programming-simplex and graphing techniques. Simple restoration of some of the classical topics and increased emphasis on problem solving might make the proposed course much easier for the student. As one Preparation for Discrete Mathematics studies the list of topics in the discrete mathematics course, it becomes clear that, in fact, there is little in A consideration of the topics listed in this course out- the way of specific prerequisites for such a course except line reveals that, while the course meets our objectives a solid background in algebra; nothing in the course re- of scope and level, this is a serious mathematics course. lies on trigonometry, number theory, or geometry, per The student will have to be prepared for this course by se. However, the abstraction and the emphasis on some an excellent secondary school background. Those of us formalism will shock the uninitiated and the mathemat- who have been teaching freshmen know that many stu- ically immature. dents are coming unprepared for abstract thinking and Recent experimentation at the Sloan-funded schools problem solving. We are aware that many secondary might tell us something about what we ought to require schools are doing a fine job of educating students to of students enrolling in this type of course. Results handle this work, but many more schools are not. It from these schools have not been completely analyzed, seems likely that courses ordinarily taught to mathe- but the failure rates seem consistent with those in the matically deficient first-year students to prepare them calculus courses. Some of the experimental group had for the calculus would also prepare them for this course. taken calculus first and others had not. There seemed In many cases, with only modest changes, these courses to be a filtering process in both cases so that results are can be adapted to be both prediscrete mathematics and not comparable from one discrete mat,hematics course precalculus. The Committee expects that the major in to another. One Sloan-funded correspondent reported computer science will include at least one year of calcu- that reading skills might be a factor in success and was lus so that at some time the student will surely reap the following through with a study to see if verbal SAT full benefits of these traditional preparatory courses. scores were any indicator of success. The additional question still remains unanswered- One of the concerns of the Committee throughout its what should be taught in the high schools or on the deliberations has been the articulation problem with a remedial level in the colleges to prepare students ad- course of this kind. We want to be clear that finite equately for this course? Our suggestion is tentative: mathematics courses in their present form are not the some of us feel that perhaps a revived emphasis on equivalent of this course. We have not t,otally succeeded the use of both formal and informal proof in geometry in communicating this in presentations at professional courses as a means for teaching methods of proof and meetings. The discrete and finite mathematics courses analytic thinking would be a step in the right direction. differ in several ways. First, the discrlete mathematics 68 RESHAPINGCOLLEGE MATHEMATE course is not an all-purpose service course. It has been al. The situation at this time in the two year colleges designed primarily for majors in mathematically-related is one of exploration, learning, and waiting. fields. It presumes at least four years of solid secondary Just as the calculus sequence at two year colleges is school mathematics and hence the level of the course is taught from the same texts and in the same manner as greater than or equal to the level of calculus. There is at the baccalaureate institutions, discrete mathematics inherent in this proposal a heavy emphasis on the use courses at two year schools are expected to conform of notation and symbolism to raise the students’ ability to requirements of four-year schools to which students to cope with abstraction. Secondly, a heavy emphasis hoped to transfer. Faculty at Florida State University, on algorithmic thinking is also recommended. in connection with one of the Sloan projects, introduced The pace, the rigor, the language, and the level are the discrete mathematics course at a nearby two year intended to differ from a standard finite mathematics college. The course was taught from the same text and course. We do not claim that this course can be taught in the same manner at both institutions. The students to everyone. Perhaps at some schools the computer sci- did well and project directors claim the results were ence majors are not very high caliber and college pro- “unremarkable .” grams naturally are geared to the needs of the students. Recent conferences of the American Mathematical There is nothing inherently wrong in requiring that such Association of Two Year Colleges (AMATYC) and as- students take the mathematics courses required of the sociations of two year college mathematics faculty in business majors: finite mathematics, basic statistics, many state organizations have been devoted to the spe- and “soft” calculus. Perhaps the finite mathematics cial problems of the two year schools with regard to courses can be improved and sections for some students discrete mathematics. The primary concern of most be enhanced by teaching binary arithmetic and elemen- schools is that they must wait for the four-year schools tary graphs. This is an alternative that many schools to indicate what type of course will be transferable. The will probably choose. It may reflect the reality on a Committee urges those teaching at four-year institu- campus where there is really no major in computer sci- tions to make a special effort to communicate their own ence, but a major in data processing or information requirements to the two year colleges that feed them. science which serves its students well. We have not What about discrete mathematics in the high attempted to define that kind of discrete mathemat- schools? Perhaps it will be an exciting change to see ics course. We specifically are defining a course on the the secondary schools place less emphasis on calculus intellectual level of calculus for science and mathemat- and more on some of the topics in the discrete mathe- ics majors. Our visits around the country indicate that matics. We understand that there is considerable pres- many schools need a course at the level of the present fi- sure from parents to have Junior (or Sis) take calculus nite mathematics offerings. Such courses are a valuable in high school. We are confident that that will change service to some students, but should not be considered as the first year of mathematics in the colleges becomes equivalent to the course we have described. more flexible to include either calculus or discrete mathematics at the same level. If the high schools continue the trend to teaching more computer science for ad- Two Year Colleges and High Schools vanced placement, then they will have to offer the dis- The mathematics faculty at two year colleges have crete mathematics to their students. The present A4d- been working through their own organizations and com- vanced Placement Examination in computer science is mittees toward curricular reform. The Committee on essentially for placement in CS2. To place above CS2, Discrete Mathematics has attempted to consider their there will probably be a level I1 examination which con- proposals in its own. Jerry Goldstein, Chairperson forms to the course outline for CS1 and CS2 as noted of CUPM and an ex-officio member of the Commit- in the Koffman report. An Advanced Placement Ex- tee on the Curriculum at the Two Year Colleges, has amination in discrete mathematics is some time in the been working to maintain articulation between the two future, as there is no universal agreement as to exactly groups. The Two Year College Committee began its what might be included at this time. deliberations after our Committee, so this report re- In January 1986 a Sloan-funded conference on cal- flects only preliminary conclusions from that source. A culus was held at Tulane University in New Orleans. “Williams”-like conference for the two year colleges took More than twenty participants presented papers and place in the summer of 1984 and proceedings are avail- participated in workshops on the state of calculus and able from Springer-Verlag in New Directions in Two its future. The Committee concurs with that group’s Year College Mathematics, edited by Donald Albers, et consensus that the goals of teaching (mathematics) are DISCRETEMATHEMATIC s 69 to develop increased conceptual and procedural skills, presenting a feasible solution as opposed to the ideal to develop the ability of students to read, write, and ex- solution. plain mathematics, and to help students deal with ab- Should the teaching of calculus reflect the tremen- stract ideas. These are the global concerns for all math- dously powerful symbolic manipulators now on the mar- ematics teaching. Secondary schools should be working ket? While the most powerful require mainframes, toward such goals too. some are available on minicomputers and muMath runs The Committee encourages faculty to get students on a personal computer. Can the time previously to work together to solve problems. From experience, spent in tedious practice of differentiation or integra- some of us have found that students cannot read a tion be better used to teach the power of the cal- problem-either they leave out essential words or do culus through problem solving and modeling? Two not know how to read the notation when asked to read of the Sloan-funded schools-Colby College and St. aloud. The word “it” should be banned from their vo- Olaf College-did experiment with the use of MAC- cabulary for a while. Students who use the word fre- SYMA, MAPLE, and SMP in the teaching of calcu- quently do so because they do not know what “it” re- lus. At Colby, in a course offered to1 those who had ally is. Correcting students’ homework has always been high school calculus, the computer packages were used one of the best ways of understanding their misconcep- to augment the one year single-variable and multivari- tions. In discrete mathematics courses this is even more ate calculus course. At St. Olaf, SIMP was used in so-concept and procedure vary from problem to prob- an elective course during a January Interim between lem. Students have to think and be creative. That’s the first and second semesters of the standard calcu- tough. They need the re-enforcement of the teacher’s lus course. Kathleen Heid writes in The Computing comments and the chance to try again. Working with Teacher [17] about her experience at the University other students should be encouraged because this forces of Maryland where she taught a section of the “soft” students to speak. This oral communication helps them calculus using muMath. The results of all these ex- to learn the terminology and helps them to present clear periments are quite favorable and indicate an impor- explanat ions. tant new consideration in our teaching of the subject. What about the use of the methoids introduced in The Impact on Calculus discrete mathematics in the other courses in the curriculum, including calculus and analysis? What of The concerns of some people that the introduction of difference equations? The Committee requested that discrete mathematics will cause a major change in the physical scientists and engineers respond to the idea calculus will probably prove to be unfounded. How- of changing the calculus. We mentioned the possi- ever, the Committee believes that there are several im- bility that calculus might contain ideas from discrete portant questions to be addressed. We should be ask- mathematics in the solving of traditional calculus-type ing ourselves if we are doing the best job of teaching problems. Several engineers and physicists have re- calculus. Some of our colleagues outside of mathemat- sponded to our query with some interesting endorse- ics who teach our calculus students have commented ments for change. Those who responded felt that the to the committee members that there are many as- present mathematical training we offer their professions pects of the calculus which seem to be ignored in the is inconsistent with what many of them were doing present courses. There is widespread dissatisfaction in their jobs-for they were using difference equations with the problem-solving skills of calculus students. and other discrete methods in their everyday applica- Problems that look even a little different from the ones tions. that they have solved in the standard course are of- We also should be asking what calculus the computer ten impossible for students. In addition, we are be- science major needs. Does the computer science student ing held responsible for our students lack of knowl- need the calculus to do statistics and probability? If so, edge of numerical techniques. The discrete aspect of how much rigor is needed? What background is needed the calculus was continually stressed by our respon- in numerical methods? Should matlhematics depart- dents. In fact, many commented that we were pro- ments be teaching numerical methods? Are the require- moting the idea of a dichotomy in mathematics where ments different from numerical analysis? Should we em- there is none by not proposing an integrated program phasize rigor, technique, or problem-solving skills? Do of discrete and continuous mathematics for the first two the traditional courses suffice to encourage integration years. The Committee admits that at this time it is of discrete and continuous mathematics? 70 RESHAPINGCOLLEGE MATHEMATE

Conclusion has been exciting. We have the opportunity to see what is successful. The Committee agrees that the next step This report is both incomplete and already out-of- in the development of the curriculum should be the in- date. Questions will continue to arise; answers are not tegration of the discrete and the continuous idea of easily found. Textbooks are now being published that mathematics into all courses. That would be ideal and are marketed as suitable for elementary discrete mathe- we encourage experimentation to that end. matics courses. Our annotated bibliography is undoubt- edly incomplete. We know of several forthcoming texts that are in manuscript form but which are unlisted be- Committee Me mb ers cause they could not be properly reviewed. There has been a great deal of interest, much of MARTHAJ. SIEGEL,CHAIR, Department of Mathemat- it enthusiastic, in the revitalization of the elementary ics, Towson State University; Member of CTUM. college-level mathematics curriculum. The committee ALFS BERZTISS,Department of Computer Science, members have had the opportunity to visit schools, University of Pittsburgh; Representative of the ACM speak at sectional and national meetings, and to speak Education Board. personally with hundreds of our colleagues. We are DONALDBUSHAW, Department of Pure and Applied wrestling with problems of ever-changing demands from Mathematics, Washington State University; Member other disciplines-some, as computer science, so young of CTUM and of CUPM, Chair of MAA Committee there is no standard curriculum. We need to adjust on Service Courses. our ideals to the realities of our own academic situa- JEROMEGOLDSTEIN , Department of Mathematics, 'h- tion. The Committee attempted to propose a course lane University; Chair of CUPM. with enough flexibility to allow institutions with differ- GERALDISAACS , Department of Computer Science, ent needs to follow the general course outline, putting Carroll College; Representative of the ACM Educa- emphases where they wanted. tion Board. The two year colleges and the high schools are deal- STEPHEN MAURER, Department of Mathematics, ing with demands of the four-year institutions, parents, Swarthmore College; then on leave at The Alfred P. and the College Entrance Examination Board. They Sloan Foundation. feel many pressures to keep calculus as the pivotal ANTHONYRALSTON, Department of Computer Sci- course. On the other hand, the proposal to integrate ence, State University of New York at Buffalo; Mem- discrete mathematics into the high school and even el- ber of MAA Board of Governors, and organizer of the ementary school curricula got considerable support at Williams Conference. the 1985 National Council of Teachers of Mathematics JOHN SCHMEELK,Department of Mathematics, 1%- (NCTM) meetings in San Antonio. ginia Commonwealth University; Member of The The recent publication of many discrete mathemat- American Society for Engineering Education (ASEE) ics textbooks suitable for the freshman-sophomore year Mathematics Education Committee. DISCRETEMATHEMATICS 71

Course Objectives and Sample Problems

1. Sets 3. Let A, = {k E P: k 5 n} for each nE P. 5 m 5 co STUDENTOBJECTIVES: Find A,, n A,, U A,, 1J A,. Understand set notation. n=l n=l n=l n:=l Recognize finite and infinite sets. Find A: and A, n A,,, for n, rn E P. Be able to understand and manipulate relations be- 4. For each n E N, let tween sets, and make proper use of such terms as: subsets A,, = { x E Q: x = m/3" for some m E Z). proper subsets supersets Describe the set Ao, All Az, and AIL\ Ao. equality m universe and empty set. Find n A,. n=O Understand and be able to manipulate indexed col- m lections of sets. Find n An. n=2 0 Understand and use the set-builder notation. 5. Sketch the following set S in N x N: Understand and manipulate operations on sets: intersection (finite and countable collections) union (finite and countable collections) (0,O) E S and if (m, n) E S then i(m,n + 1) E S, difference (m+ l,n+ 1) E S and (m+2,n+ 1) E S. symmetric difference complement Show S = { (m,n) : m 5 2n). Venn diagrams. 6. ShowthatifAC BandCc D,thenAxCC BxD. Understand the proofs of theorems and know the 7. Assume A, B, and C are subsets ad a universal set laws: U. Simplify commutative laws associative laws (An (B\ C))"u A. distributive laws DeMorgan's laws. * Understand Cartesian products of sets and power 2. The Number System sets. STUDENTOBJECTIVES: 0 Understand inductive (recursive) definitions of sets.

Understand a few applications: for example, gram- 0 Be able to define mars as sets. positive integers (P) Be able to do simple proofs by using Venn diagrams natural numbers (N) or elementary elementwise proofs. integers (Z) SAMPLEPROBLEMS: rational numbers (Q) 1. List the ordered pairs in the sets irrational numbers reals (as (-m, m)). A = {(m,n) E S x T: m < n} Be able to recognize subsets of N, P, Q, and Z. B = { (rn,n)E S x '2': m+ 1 = n} Be able to use interval notation. Understand the division algorithm and divisibility. where S = {I, 2,3,4) and T = {0,2,4,5}. Be able to do simple proofs about even and odd 2. True or false? numbers (e.g., the sum of two even integers is even). A\ (B UC) = (A \ B) u (A \ B). Know the definition of prime number, gcd and lcm. Be able to find the prime factorizat,ion of a number. Verify your answer (use elementwise argument, 0 Know how to write an integer given in base 10 as a Venn diagram and algebraic manipulation). numeral in base 2. 72 RESHAPINGCOLLEGE MATHEMATICS

SAMPLEPROBLEMS: Understand and be able to use the principle of math- 1. Find elements in the sets (if the set is infinite, list ematical induction. five elements of the set). @ See the necessity for the verification of algorithms. Do a substantial number of elementary proofs using { n 6 N : n2 = 4) simple examples from arithmetic. {n E P:n is prime and 15 n 5 20) SAMPLEPROBLEMS: { x E R : x2 = 4) 1. According to the Associated Press, a prominent { n E P : n2 = 3) public official recently said: "If a person is inno- cent of a crime, then he is not a suspect." What is { x E R : x2 5 4) the contrapositive of this quotation? { x E R : x2 < 0) 2. Prove or disprove the following statement about real { x E Q : x2 = 3) numbers x: {x E Q:2 < x < 3) If x2 = x, then x = 1.

2. Determine how many elements are in each set? What is the converse of this statement? Prove it or Write 00 if the answer is infinite. disprove it. 3. After considering some examples if necessary, guess (-11 11, {-LO), [-I, 11, [-I, 01, W), a formula that gives the sum of the interior angles P([-l,I]), P({--l,I)), {n E Z : -1 5 n 5 I), at the vertices of a convex polygon in terms of the {nE Z : -1 < n < I). number n of sides. Then prove the formula, if you can, by mathematical induction. 3. List elements in 4. Write an algorithm for finding the least common denominator of two fractions. Can you think of an- A = {n E Z : n is divisible by 2) other? B = {n E Z : n is divisible by 7) 5. Write an algorithm for finding the median of a list C=AUB consisting of n (an odd number) real numbers. 6. Prove: ifAuBc AnB, thenA=B. D=AnB 7. Prove or disprove: if A n B = A n C, then B = C. 4. Prove that the product of even integers is an even 8. Prove (by cases): for every n E N, n3 + n is even. integer. 9. Prove: for every n E P, 5. Use the Euclidean Algorithm to determine the 1 + 3 + 5 + ... + (2n- 1) = n2. greatest common divisor of 741 and 715. 6. Find the prime factorization of 4,978. 10. Prove (by contradiction): if x2 is odd, then x is osdd. 7. Determine the numeral in base 2 to represent 81 11. Prove: There are no integers u and b such that u2 = (base 10). 3b2. 12. Prove: n3 - n is divisible by 3 for every n E P. 3. The Nature of Proof 4. Formal Logic STUDENTOBJECTIVES: Be able to identify the hypothesis and the conclusion STUDENTOBJECTIVES: in sentences of various English constructions. Write English sentences for logical expressions and Understand the definition of a proposition, its con- vice versa. verse, its contrapositive. Complete the truth tables for the standard logical Understand the use of examples as an aid to finding connectives. a proof and the misuse of examples as proof. Give the truth values of simple propositions given Understand the use of counterexamples. in plain English. Be able to do direct proofs, including proof by cases. State the definitions of tautology and contradiction. Be able to do indirect proofs. Prove and use the standard logical equivalences: Understand the role of axioms and definitions. commutative, associative, distributive, and Understand the backward-forward method of con- idempotent properties; double negation; DeMor- structing a proof. gan laws. DISCRETEMATHEMATICS 73

Recognize computer language commands for stan- (c) If the bed is comfortable, Sally will sleep. The dard logical operations. bed is not comfortable. Will Sa1:ly sleep? 0yes, State and use logical implications, at least: modus 0no, 0not enough information. ponens, modus tolens, transitivity of + and H. (d) If the candidate is elected in Vermont, she will Negate P v q, P - Q, P A 4. be elected by the country. The candidate is not Identify the basic quantifiers, free and bound vari- elected by the country. Is the candidate elected ables, negations and the generalized DeMorgan in Vermont? 0yes, 0no, 0not enough infor- laws for quantified statements (e.g., iVzp(z) mation. 3XdX)). 12. Simplify the logic circuit below: Build logic circuits with AND, OR, NOT gates. Understand the terms consistency, inconsistency, completeness and decidability (optional). X SAMPLEPROBLEMS: Y 1. Use a truth table to prove that (pA q) + r is logically equivalent to p + (q - r). z 2. Prove that ipAr is logically equivalent to -(pV ir) without using truth tables. 3. If p = “cows bark”, q = “the Orioles are Baltimore’s baseball team” and r = “2 + 4 = 7”, find truth values of p A q, p + q, (pA q) - r. 5. Functions and Relations 4. Consider the proposition for x E R: STUDENTOBJECTIVES: If (x - 3)(x - 2) = 0 then either z = 3 or x = 2. a Be able to define “function” and “relation”. a) Write its converse. a Know the properties of relations: b) Write its contrapositive. reflexive c) Write its negation. transitive d) What is the truth value of the proposition, its symmetric converse, its contrapositive, its negation? antisymmetric. 5. Find the result of a Be able to identify order relations. a Be able to identify equivalence relations. [(0 AND 1) NAND 01 OR NOT [l IMP 01. a Understand the relationship between equivalence relations and partitions. 6. Draw a logic circuit representing (lpA q) V r. a Know the definitions of domain, codomain, image, 7. Prove the following logical argument: p A q, p r, - into, onto (or surjection), one-to-one (or injection), 7s q, and s + t imply r A t. - bijection. 8. Determine truth values of the following propositions. Assume the universe is N. a Be able to do simple proofs involving these defini- (a) Vrn3n[rn= n2] (b) 3mVn[m = n2] tions. a 9. Write in logical form: for every X, y E R, there exists Be able to work with composition and inverses of z E R such that x < z and z < y. relations and, in particular, of functions. 10. Negate VX[~(X)A -q(z)]. a Be familiar with recursive definitions of functions. 11. Answer each of the following in the appropriate box. a Be introduced to sequences as functions, again with If the book costs more than $20, it is a best- some emphasis on recurrence relations and recur- seller. The book costs more than $20. Is the sion. book a best-seller? 0yes, 0no, 0not enough a Be able to do proofs involving recursion. informat ion. a Be able to work with definitions of relations as or- If the kite is multicolored, it will fly. The kite dered pairs as opposed to as “rules”. flies. Is the kite multicolored? yes, 0 no, a Know the definition of the characteristic function of [7 not enough information. a set. 74 RESHAPINGCOLLEGE MATHEMAT~

SAMPLEPROBLEMS: @ Be able to use the binomial theorem. 1. Give at least one reason why each of the following Be able to do ball and urn type problems. does not define an equivalence relation on the set of Be able to state and apply the inclusion-exclusion integers: principle. a) z + y is odd; Be able to apply the pigeon-hole principle. b) z < 2y. @ Be familiar with combinatorial algorithms based on 2. Recall that a positive integer is prime if it has ex- recurrence relations, actly two positive integer divisors: itself and 1. Con- Be introduced to the basic ideas of intuitive discrete sider the relation defined on the set of all integers probability. greater than 1 by: “y is the smallest prime that is SAMPLEPROBLEMS: a divisor of x.” a) Explain why this relation is a function y = f(z). 1. In many states automobile license plates consist of b) What is the range off? three (capital) letters followed by three digits. Are c) List four elements of f-’(5). there any states in which this probably does not d) Prove that f o f = f. give enough different license plates even if discarded 3. When the prevailing rate of interest is lOOr%, an plate numbers can be reused? Are there any states account that has P dollars in it at the beginning of in which three letters followed by three digits or a year should have how much in it at the beginning three digits followed by three letters is probably not of the next year? Express your answer as a recur- enough? How many license plates are possible in sion formula, and solve it to find the size of such an your state? account after n years. 2. In a hypnosis experiment, a psychologist inflicts a 4. The factorial, usually denoted by n!, of a positive sequence of flashing lights on a subject. The psy- integer n is the product of all positive integers from chologist has red, blue and green lights available. 1 to n inclusive. Show how n! may be defined re- How many different ways are there to inflict 9 flashes cursively. if two are red, four blue and three green? 5. Prove that f(z) = 22 + 1 is one-to-one and onto 3. How many triangles are there using edges and di- from R to R. Is f one-to-one from Z to Z? Does f agonals of an n-sided polygon if the vertices of the map Z onto Z? Verify your answers. triangle must be vertices of the polygon? 4. Verify by induction and by a combinatorial argu- 6. List five elements in the sequence given by a0 = 1, and a, = 2an-1 for n 2 1). Give another formula ment that for a,, for any n E N. n 7. Let C = (a,b} and let C’ be the set of words over C(k,m) = C(n+ 1, m + 1). C. If w1 and w2 are elements of C’, define w1 5 w2 k =m if and only if length (w1) 5 length (wa). Is 5 a partial order? Why? What does this say about Pascal’s triangle? 8. Prove: If h(1) = 1 and h(n + 1) = 2 - h(n)+ 1 for 5. Evaluate n n 2 1, then h(n)= 2” - 1 for all n E P. k2C(n,k). 9. For m, n E N define 2 k=O m - n if and only if m2 - n2 is divisible by 3. 6. How many ways are there to take four distinguishable balls and put two in one distinguishable urn Prove that - is an equivalence relation on N. Find and 2 in another if 8 elements of each of the equivalence classes [O] and a) the order in which the balls are put in the urns [l].What is the partition of N induced by -? makes a difference; b) the order does not make a difference. 6. C ombinat orics 7. How many integers less than 105 are relatively prime to 105? STUDENTOBJECTIVES: 8. Use the algorithm which generates all permutations

@ Be able to apply the basic permutation and combi- of length n of 1,2,. . ., n where no digit can be re- nation formulas. peated to derive an algorithm to generate all permu- Be familiar with and be able to provide basic com- tations when any digit may be repeated an arbitrary binatorial identities using combinatorial reasoning. number of times. 7. Recurrence Relations Compute several terms. Find the pipttern and prove that it is correct. STUDENTOBJECTIVES: 8. Consumer loans work as follows. The Lender gives Have lots of exercise in the elements of recursive the Consumer a certain amount P, called the Prin- thinking (e.g., the recursive paradigm-solve prob- cipal. At the end of each payment, period (usually lems by jumping into the middle and working your each month) the consumer pays the Lender a fixed way out). amount p. This continues for a prearranged number Be familiar with recursive definitions of syntax. of periods (e.g., 60 months = 5 years). The value Be familiar with recursive algorithms. of p is calculated so that, at the end of the time, Understand what a difference equation is. the Principal and all interest due have been paid off See how difference equations can be used to model exactly. During each payment period the amount practical problems. owed by the Consumer increases by r, the period Understand the methods for the solution of linear, interest rate, but it also decreases iPt the end of the constant coefficient equations and first order differ- period by p. Let Pn be the amount owed after the ence equations. nth payment is made. Find a difference equation Be familiar with some applications of difference and boundary conditions for P,. equations. Be able to use the difference calculus (optional). 9. Solve SAMPLEPROBLEMS: 1. Compute the first ten terms of the sequence defined an+1 = 5an - 6an-lr a1 = !j, a2 = 7. by

if n = 1 or 2 then fn = 3 10. Find the general solution to else fn = f2-1+ fn-2. 2. What is f(1) iff is defined by if n 2 1000 f(n) = { ;vc + 6)) if n < 1000. 11. Solve 3. Describe the strings of characters defined by

::= l

with the standard definitions of digit and letter and where := is read “is defined to be”. 4. Consider the sum 12. In ternary search, t and u are the entries closest to 1/3 and 2/3 of the way through the list. Let 1 + 3 + 5 + 7+. . . + 2n - 1. the search word be w. If w < t, then search the first third of the list by ternary search. Similarly, Use the inductive and recursive paradigms to con- if t < w < u or u < w search, respectively, the jecture a closed form expression for this sum. middle and last thirds of the list. Write recurrence 5. Suppose we add to the usual form of the Towers relations for the worst and average case number of of Hanoi problem the rule that a disk can only be comparisons in ternary search. Also show that, if an moved from one peg to an adjacent peg (i.e., you appropriate sequence {ni} of list lengths is chosen, can never move a disk from peg 1 to peg 3 or vice then Wn, the number of compariaons in the worst versa). Devise an algorithm for solving this version case is given by of the problem. Display solutions when n = 2,3,4. 6. Display a recursive version of the Euclidean algorithm. wn = 2 log,(n + 1:). 7. Consider the recursion

n- 1 P,=I+~P~n>l, P,=I. (This problem assumes that students have seen a k=l similar analysis for binary search.11 76 RESHAPINGCOLLEGE MATHEMATICS

8. Graphs and Digraphs 9. Trees STUDENTOBJECTIVES: STUDENTOBJECTIVES: Understand the definition of the digraph and its use Know the definition of a tree. as the picture of a relation. * Be able to find the minimal spanning trees for a Be able to write the matrix representation of a di- given graph. graph. See the many applications of trees in search prob- See many applications of digraphs as natural models lems, with a complete introduction to binary search for networks in real life, such as systems of roads, trees. pipelines, airline routes. * See how to convert digraphs to trees. Know the definitions: connectedness, completeness, Know how to use digraph algorithms for cycles and complement. critical path analysis. Be introduced to path problems (and transitive clo- Prove the theorems on trees by induction and rely sure) and Warshall’s algorithm. on recursive algorithms for transversal problems on Be familiar with undirected graphs, the associ- trees. ated definitions and the classical problems of graph * Be familiar with sorting and searching algorithms. theory-the bridges of Koenigsberg, the four color See rooted trees and Polish notation as an applica- problem, Kuratowski’s theorem. tion. Be encouraged to solve interesting problems like SAMPLEPROBLEMS: Mastermind and Instant Insanity to see the usefulness of graph theory. 1. Show the equivalence of the following definition of Be using algorithms such as Kruskal’s algorithm and an undirected tree: (a) a connected graph without Dijkstra’s algorithm in solving problems. any circuits; (b) a connected graph that becomes Have the opportunity to see the applications to ac- disconnected on the removal of any one edge; (c) a tivity analysis (CPM and PERT). connected graph with its number of edges one less Be exposed to depth-first search algorithms and than its number of nodes. topological sorting. 2. Prove that the number of leaves in a binary tree with n internal nodes is at most n + 1. SAMPLEPROBLEMS: 3. Prove that for every nonnegative integer n it is pos- 1. Prove that a connected graph of n nodes contains sible to construct a binary tree with n leaves in at least n - 1 edges. which the outdegree of every internal node is 2.. 2. Prove that a digraph is disconnected iff its comple- 4. Find all spanning trees of a specific given graph. ment is connected. (The complement of a digraph 5. Find all simple cycles in a specific given graph. D is defined by the matrix obtained when in the 6. Given the drawing of a scheduling network, find the adjacency matrix of D every 0 is replaced by a 1, critical path(s) in this network. and every 1 by a 0.) 3. A digraph D = (A, R) is complete if for all a,b E A, 10. Algebraic Structures (a,b) E R implies (b, a) E R. With respect to this definition, is it true that (a,a) E R for all a E A, or STUDENTOBJECTIVES: is it true that (a,a) g! R for all a E A? Be able to define and recognize unary and binary 4. A digraph D = (A, R) is a tournament if, for all operations. (a,b) E A, (a,b) E R or @,a) E R whenever a # Be able to distinguish whether sets are closed with b, but (a,b) E R implies (b,a) # R. How many respect to a given operation. tournaments are there as a function of n,where n = Be familiar with a variety of operations on a variety IAl? Draw all tournaments for n = 3. of sets: 5. Prove (by induction) that every tournament con- arithmetic operations on N,PI Q, I,R tains a Hamiltonian path. set operations on P(S) 6. How many digraphs on n nodes are there? How logical operations on propositions many graphs? matrix operations on 2 x 2 matrices. 7. Find the shortest path from node 1 to every other * Be able to understand the general definition of an node in a specific given digraph. operation on a set via some unfamiliar rule or a 8. Find the transitive closure of the relation repre- table. sented by this same digraph. Be able to decide which of the properties hold: DISCRETEMATHEMATICS 77

commutative Have a working knowledge of descriptive statistics: associative populations vs. samples existence of identity simple graphing techniques existence of inverse for given operations on given calculation and meaning of mean, median and sets. mode Recognize semigroups, monoids, groups, and cosets. calculation and meaning of standard deviation Have an elementary knowledge of finite group codes (calculations may be restricted to ungrouped (need cosets). data). Be familiar with and be able to manipulate boolean Discuss the interpretation of sample data including algebras with many examples. exploratory data analysis. Have a rudimentary knowledge of lattices. 0 Know the meaning of expected value and variance Be able to apply the ideas of homomorphism and for random variables. isomorphism. Know how to use Chebyshev’s inelquality. SAMPLEPROBLEMS: 0 Know how to simulate some of the probability models discussed. 1. Determine if (N, +) is a group. Should be doing problems that deimonstrate the re- 2. Determine if (Z,o) is a semigroup, a monoid, a lationship between probability and difference equa- group, when a o b is defined to be a b - 2 whenever + tions, especially through classical problems like the a, b E Z. gambler’s ruin problem. 3. For X = 011010 and Y = 100100, find the Hamming distance from X to Y. SAMPLEPROBLEMS: 4* A ‘Ode has a minimum distance Of How many er- 1. Find the probability that in a random arrangement rors can it detect? How many errors can it correct? of n files, we find them to be in alphabetical order. 5. Determine if (Zs, *) is a group. 2. Simulate the gambler’s ruin problem assuming that 6. Let S = {a,b, c} and define the operation @ by the the coin being tossed is fair, that A wins a dollar table below. Determine whether (S,@)is a semi- from B when the coin lands heads up and gives a group. dollar to B otherwise. You may aissume that A begins the game with $3 and B begins with $2. De- termine the average length of a game and determine the frequency with which A wins Ithe game. c bac 3. Solve the gambler’s ruin problem using a probabil- 7. Prove that in every booleanr algebra, [B, +, 0,‘ , 0, 11, ity model. Use the situation given in problem two 202: = z and z+ 1 = 1 for any z E B. and compare your results here to the results of the 8. Show that the set of 2 x 2 matrices with integer simulation above. entries is a commutative monoid under matrix ad- 4. If scores on an examination are normally distributed dition. with p = 500 and Q = 75, find the probability that 8. Construct a logic network for the the boolean ex- a score exceeds 700. Find the 95th percentile for the pression scores. Find the probability that (a sample mean of 21 0 2; + (21 0 23)’. more than 510 is found for a random sample of 100 scores. 11. Discrete Probability and Descriptive Statistics 5. Given the data below, sketch a stern and leafdisplay, a frequency histogram, and find tlhe mean, median, STUDENTOBJECTIVES: mode and sample standard deviation. Determine Understand basic axioms, simple theorems of prob- the 75th percentile of these data. ability. 6. A prize has been put in 2% of all Sweeties cereal Understand conditional probability. boxes. Find the probability that the fourth box you 0 Understand, and be able to do problems involving open contains the first prize you find. Determine the discrete uniform, Bernoulli, binomial, Poisson the average number of boxes one needs to open in (optional), hypergeometric and geometric probabil- order to get one prize. Simulate the experiment, ity distributions and their random variables. also. 0 Understand the goal of random number generation. 7. If 80% of all programs fail to run on the first try, Understand the Law of Large Numbers. find the probability that in a grou:p of 100 programs 78 RESHAPINGCOLLEGE MATHEMA'TICS

at least 30 programs run on the first try. of paths of length 3 from 01 to 04. 8. Showb(z;n, 1-p) = b(n-z;n,p)where b(z;n,p)= (Z)P2(1 - P)"-". 9. Completion time on a standardized test is normally distributed with an average of 40 minutes and standard deviation 5 minutes. How much time should be allotted if the examiner wants 95% of the students to finish the test? What if the examiner wishes to leave 5 minutes for checking for 95% of the class? 2. A firm packages nut assortments: Fancy & Deluxe. 12. Algorithmic Linear Algebra The Fancy assortment contains 6 oz. cashews, 8 01. almonds and 10 oz. peanuts. It sells for $2.40. The Deluxe assortment contains 12 oz. cashews, 10 STUDENTOBJECTIVES: oz. almonds and 8 oz. peanuts. It is priced at $3.60. The supplier can provide a maximum of 3000 Understand matrix operations and their properties. oz. cashews, 3600 oz. almonds and 3200 oz. peanuts. Be able to determine whether a matrix is invertible, Find the number of boxes of each type that would and if so, be able to find the inverse. maximize revenue. Use the simplex method and a See the relationship of matrices to graphs. graphical method. * See the use of matrices in representation of linear 3. Determine if each of the systems systems. (a) 42+3y=7 (b) 62 + 3y + 72 = 4 Be able to use row operations to reduce matrices. 22 + 6y = 8 22 + 5y + 82 = 10 Be able to determine whether a system of linear equations has a solution, a unique solution or no has one solution, no solution, many solutions. Find solution. all solutions in each case. Be able to solve a system of linear equations, if a 4. Let solution exists. Have an understanding of linear inequalities, graph- A= ing them in the two variable case. [: :I Be able to solve linear programming problems using the simplex method (and the graphical method in the two variable case). Be able to use matrices to solve Markov chain mod- Determine if the following exist. If they do, find els. them; if not, explain. A + B, AB, A + D, A,-', * Use powers of incidence matrices to study connec- B-l, D-l, det A, det C. tivity properties of graphs or digraphs. 5. Given a Markov chain with transition matrix P, find See and use the recursive definition of the determi- the steady state probability vecotr. Let nant of a square matrix.

SAMPLEPROBLEMS:

1. Determine the matrix representation of the undi- (Here we would give a word problem with this tran- rected graph pictured below. Determine the number sition matrix.) DISCRETEMATHEMATICS 79

Bibliography

Textbooks tion to the theory of graphs; chromatic number, connectivity, and other graphical parameters; optimization problems. This is a list of textbooks that might be considered 5. Cohen, Daniel I.A. Basic Techniques of Combinato- for use in courses of the kind discussed in this report. rial Theory. New York: John Wiley and Sons; 1978; It is aa nearly complete as we could make it, but books ISBN 0-471-03535- 1. of this kind are still appearing often and we may well Assumes one semester of calculust; not inclined to have overlooked some good older ones. Inclusion in the use proof by induction. Contents: Introduction; bino- list thus does not imply endorsement, nor does omission mial coefficients; generating functione; advanced count- imply the opposite. Likewise, the “notes” are not meant ing numbers; two fundamental principles; p.ermutations; to be definitive in any way, but just remarks that users graphs. Appendix on mathematical induction. of this bibliography may find interesting. 6. Dierker, Paul; Voxman, William. Discrete Math- 1. Arbib, M.; Kfoury, A.; Moll, R. A Basis For The- ematics. San Diego: Harcourt Elrace Jovanovich; oretical Computer Science. New York: Springer- 1986; ISBN 0-15-517691-9. College algebra a prerequisite; primarily for freshmen Verlag; 1981; ISBN 0-387-90573-1. and sophomores. The theme of algorithms is a unify- An introduction to theoretical computer science. ing thread; otherwise, little independence between chap- Chapter 4 includes techniques of proving theorems. ters, so could be used as a text for one- or two-semester Contents: Sets, maps, and relations; induction, strings, courses. Contents: A first look at algorithms; number and languages; counting, recurrence, and trees; switch- systems and modular arithmetic; introduction to graph ing circuits, proofs, and logic; binary relations, lattices, theory; applications of graph theory; boolean algebra and infinity; graphs, matrices, and machines. and switching systems; symbolic logic and logic circuits; 2. Biggs, Norman L. Discrete Mathematics. New York: difference equations; an introduction to enumeration; el- Oxford University Press; 1985; ISBN 0-19-853252-0. ementary probability theory; generating functions; in- A mathematically-sound book with plenty of mate- troduction to automata and formal languages; appen- rial for a two-semester course. Three main sections dices on set theory, functions, matrices, and relations. are “Numbers and counting,” “Graphs and algorithms,” 7. Doerr, Alan; Levasseur, Kenneth. Applied Discrete and “Algebraic methods.” Although written from the Structures for Computer Science. Chicago: Science viewpoint of mathematics rather than computer science, it does pay a fair amount of attention to algorithms. Research Associates; 1985; ISBN 0-574-21755-X. Probably a bit too rigorous for freshmen and sopho- Aimed at freshman-sophomore computer science ma- mores. Contents: Graphs, combinatorics, number the- jors. Includes applications, some “Pascal notes.” Con- ory, coding theory, combinatorial optimization, abstract tents: Set theory; combinatorics; logic; more on sets; algebra. introduction to matrix algebra; relations; functions; recursion and recurrence relations; graph theory; trees; al- 3. Bogart, Kenneth. Introductory Combinatorics. gebraic systems; more matrix algebra; boolean algebra; Boston: Pitman; 1983; ISBN 0-273-01923-6. monoids and automata; group theory and applications; A fairly complete coverage of standard combinatorial an introduction to rings and fields. topics, but the treatment is essentially non-algorithmic; 8. Gersting, Judith. Mathematical Structures for e.g., there is no algorithm for permutations. Some- what sophisticated; probably requires a year of calcu- Computer Science. San Francisco: W.H. Freeman lus for maturity. Contents: Introduction to enumer- and Company; 1982; ISBN 0-7167-1305-5. ation; equivalence relations, partitions, and multisets; A fine text, but emphasis on computer science applica- algebraic counting techniques; graph theory; matching tions may be too great. An accessible reference on group and optimization; combinatorial designs; partially or- codes. Contents: How to speak mathematics: basic vo- dered sets. cabulary; structures and simulations; boolean algebra 4. Brualdi, Richard A. Introductory Combinatorics. and computer logic; algebraic structures; coding theory; finite-state machines; machine design and construction; New York, etc.: North-Holland; 1977; ISBN 0-7204- computability; formal languages. 8610-6. Sophomore level; calculus prerequisite. Sophisticated, 9. Grimaldi, Ralph P. Discrete and Combinatorial but not much algorithmic flavor. Contents: What is Mathematics. Reading, Mass.: Addison-Wesley; combinatorics? the pigeonhole principle; basic counting 1985; ISBN 0-201-12590-0. principles: permutations and combinations; the bino- Intended for sophomores and juniors. Contents: Fun- mial coefficients; the inclusion-exclusion principle; re- damental principles of counting; enumeration in set the- currence relations; generating functions; systems of dis- ory; relations and functions; languages; finite state ma- tinct representatives; combinatorial designs; introduc- chines; relations: the second time around; the system of 80 RESHAPINGCOLLEGE MATHEMATE

integers; the principle of inclusion and exclusion; rings torics; trees and hierarchies; graph theory: undirected and modular arithmetic; boolean algebra and switch- graphs; graph theory: directed graphs; discrete prob- ing functions; generating functions; recurrence relations; ability; automata and formal languages; boolean alge- groups, coding theory, and P6lya’s method of enumera- bras; logic: propositional and predicate calculus; algo- tion; finite fields and combinatorial designs; an introduc- rithms and programs. tion to graph theory; trees; optimization and matching. 15. Levy, Leon S. Discrete Structures of Computer Sci- 10, Hillman, Abraham P.; Alexanderson, Gerald L.; ence. New York: John Wiley and Sons; 1980; ISBN Grassl, Richard M. Discrete and Combinatorid 0-471-03208-5. Mathematics. New York: Dellen Publishing Com- A highly-personal statement on discrete structures for pany; 1986; ISBN 0-02-354580-1. computer science students. The presentation is very Sophomore-junior text. Contents: Sets and rela- sketchy. For a sophomore-junior course; leaps quickly tions; algebraic structures; logic; induction; combina- into abstraction and algorithms. Contents: An essay torial principles; digraphs and graphs; groups; polyno- on discrete structures; sets, functions, and relations; di- mials and rational functions; generating functions and rected graphs; algebraic systems; formal systems; trees; recursions; combinatorial analysis of algorithms; intro- programming applications. duction to coding; finite state machines and languages. 16. Lipschutz, Seymour. Discrete Mathematics. New 11. Johnsonbaugh, R. Discrete Mathematics, Revised York: McGraw-Hill (Schaum’s Outline Series); Edition. New York: Macmillan; 1984; ISBN 0-02- 1976; ISBN 0-07-037981-5. 360900-1. Contains an outstanding collection of (easy) worked Intended for a one-semester course for freshmen or examples and exercises. Vectors and matrices are regarded as an introductory topic. Contents: Set theory; sophomores. Mainly but not exclusively aimed at com- relations; functions; vectors and matrices; graph sthe- puter science students. Emphasizes an algorithmic ap- ory; planar graphs, colorations, trees; directed graphs, proach and does a considerable amount of algorithm finite-state machines; combinatorial analysis; algebraic analysis. Contents: Introduction; counting methods and systems, formal languages; posets and lattices; proposi- recurrence relations; graph theory; trees; network models and Petri nets; boolean algebras and combinatorial tion calculus; boolean algebra. circuits; automata, grammars, and languages. Appen- 17. Lipschutz, Seymour. Essential Computer Mathe- dices on logic and matrices. matics. New York: McGraw-Hill (Schaum’s Outline 12. Kalmanson, Kenneth. An Introduction to Discrete Series); 1982; ISBN 0-07-037990-4. Mathematics and Its Applications. Reading, Mass.: Again there is an excellent collection of examples and exercises. Includes discussion of representation of num- Addison-Wesley; 1986; ISBN 0-201-14947-8. bers and characters, linear algebra, and probability and Intended for freshmen and sophomores. Developed in statistics. Suitable €or technical mathematics course €or conjunction with Sloan-funded course at Montclair State data processing students; not appropriate as a text for College. Contents: Sets, numbers, and algorithms; sets, the discrete mathematics course proposed by the Com- logic and computer arithmetic; counting; introduction mittee. Contents: Binary number system; computer to graph theory; trees and algorithms; directed graphs codes; computer arithmetic; logic, truth tables; allgo- and networks; applied modern algebra; further topics rithms, flow charts, pseudocode programs; sets and rela- in counting and recursion; appendix with programs in tions; boolean algebra, logic gates; simplification of logic BASIC. circuits; vectors, matrices, subscripted variables; linear 13. Kolman, Bernard; Busby, Robert C. Discrete Math- equations; combinatorial analysis; probability; statis- ematical Structures for Computer Science. Engle- tics: random variables; graphs, directed graphs, machines. wood Cliffs, New Jersey: Prentice-Hall; 1984; ISBN 0- 13-215418-8. 18. Liu, C.L. Elements of Discrete Mathematics, Second “There are no formal prerequisites, but the reader is Edition. New York: McGraw-Hill; 1985; ISBN 0-07- encouraged to consult the Appendix as needed.” In- 038133-X. tended for a one- or two-semester course for freshmen A comparatively short, but well-constructed text. In- or sophomore computer science students. Contains rel- tended for a one-semester course but probably more ap- atively few algorithms. Approach on the informal side, propriate for juniors than for freshmen or sophomores, with not very many theorems or proofs. Contents: Fun- although there is no prerequisite beyond high school al- damentals; relations and digraphs; functions; order re- gebra. Induction and problem-solving are treated early. lations and structures; trees and languages; semigroups A rather traditional mathematical approach with em- and groups; finite-state machines and languages; groups phasis on combinatorics, relatively little on algorithms. and coding. Contents: Computability and formal languages; per- 14. Korfhage, Robert R. Discrete Computational Struc- mutations, combinations, and discrete probability; relations and functions; graphs and planar graphs; trees and tures, Second Edition. New York: Academic Press; cutsets; finite-state machines; analysis of algorithms; 1984; ISBN 0-12-420860-6. discrete numeric functions and generating functions; re- Contents: Entities, properties, and relations; arrays currence relations and recursive algorithms; groups and and matrices; graph theory: fundamentals; combina- rings; boolean algebras. DISCRETEMATHEMATICS 81

19. Liu, C.L. Introduction to Applied Combinatorial New York: John Wiley and Sons; 1985; ISBN 0-471- Mathematics. New York: McGraw-Hill; 1968;ISBN 80075-9. 0-0 7-038 1 24-0. Aimed at computer science majors; no college-level This is a good source for recurrence relations, and for prerequisites; theory with applications. Includes some P6lya’s theory of counting. It also contains introduc- proofs. Contents: Formal systems; functions and re- tions to linear and dynamic programming. Contents: lations; boolean algebras; boolean algebra and logic Permutations and combinations; generating functions; design; lattices and their applications; cardinality and recurrence relations; the principle of inclusion and exclu- countability; graphs and their use in computing; intro- sion; P6lya’s theory of counting; fundamental concepts duction to formal languages; computability. in the theory of graphs; trees, circuits, and cut-sets; pla- 25. Polimeni, Albert D.; Straight, H. Joseph. Foun- nar and dual graphs; domination, independence, and dations of Discrete Mathematics. chromatic numbers; transport networks; matching the- Monterey, Calif.: ory; linear programming; dynamic programming; block Brooks Cole Publishing Company; 1985; ISBN 0- designs. 534-03612-0. 20. Marcus, Marvin. Discrete Mathematics: A Compu- Intended for sophomores. Prerequisite: one year of college-level mathematics, including a semester of cal- tational Approach Using BASIC. Rockville, Mary- culus, and an introductory programniing course. Pascal land: Computer Science Press; 1983; ISBN 0- used throughout. Contents: Logic; riet theory; number 914894-38-2. theory and mathematical induction; relations; functions; Interesting approach; elementary. Complemented by algebraic structures; graph theory. a DOS 3.3 16-sector 5 1/4” floppy disk, DISCRETE 26. Prather, Ronald P. Discrete Mathematical Struc- PROGRAMS. Contents: Elementary logic; sets; relations and functions; some important functions; function tures for Computer Science. Boston: Houghton optimization; induction and combinatorics; introduction Mifflin; 1976; ISBN 0-395-20622-7(Solutions Man- to probability; introduction to matrices; solving linear ual; ISBN 0-395-20623-5). equations; elementary linear programming. A solid coverage of all the standard material. Boolean 21. Molluzzo, John L.; Buckley, Fred. A First Course in algebras are treated as lattices. Contents: Preliminaries, algebras and algorithms, graphs and digraphs, monoids Discrete Mathematics. Belmont, Calif.: Wadsworth and machines, lattices and boolean algebras, groups and Publishing Company; 1986; ISBN 0-534-05310-6. combinatorics, logic and languages. “. . .intended for non-mathematically-oriented students . . .first or second-year computer science or com- 27. Prather, Ronald P. Elements of Discrete Mathemat- puter information systems student.” Contents: Num- ics. Boston: Houghton Mifflin; 1986; ISBN 0-395- ber systems; sets and logic; combinatorics; probability; 35165-0 (Solutions Manual; ISBN 0-395-35166-9). relations and functions; vectors and matrices; boolean Suitable for a one-term course. “No prior program- algebra; graph theory; appendix on Pascal. ming experience is needed because a generic pseudocode 22. Mott, Joe L.; Kandel, Abraham; Baker, Theodore language is used to phrase algorithms.” Developed un- P. Discrete Mathematics for Computer Scientists. der a Sloan Foundation pilot project grant. Contents: Intuitive set theory; deductive mathematical logic; dis- Reston, Virginia: Reston Publishing Company; crete number systems; the notion of an algorithm; poly- 1983; ISBN 0-8359-1372-4. nomial algebra; graphs and combinatorics. Sophomore-junior course; programming experience desirable but not essential. For computer science au- 28. Roman, Steven. An Introduction io Discrete Mathe- dience (posets are defined on p. 17). Contents: Foun- matics. Philadelphia: Saunders College Publishing; dations; elementary combinatorics; recurrence relations; 1986; ISBN 0-03-064019-9. relations and digraphs; graphs; boolean algebras. Could be used for a one- or two-isemester course for 23. Norris, Fletcher R. Discrete Structures: An Intro- freshmen or sophomores in mathematics as well as computer science. A careful and not too hurried approach duction to Mathematics for Computer Scientists. but quite traditionally mathematicail with little atten- Englewood Cliffs, New Jersey: Prentice-Hall; 1985; tion to algorithms. Contents: Sets, functions, and proof ISBN 0-13-215260-6 (Instructor’s Manual; ISBN techniques; logic and logic circuits;, relations on sets; 2 15277). combinatorics-the art of counting; more on combina- Written for a one-semester course for freshmen and torics; an introduction to graph theory. sophomores. College algebra is the prerequisite. Con- 29. Ross, Kenneth A.; Wright, Charles R.B. Dis- tents: Propositions and logic; sets; boolean algebra; the crete Mathematics. Englewood Cliffs, New Jersey: algebra of switching circuits; functions, recursion, and induction; relations and their graphs; applications of Prentice-Hall; 1985; ISBN 0-13-215286-X. graph theory; discrete counting: an introduction to com- A large book certainly suitable for a two-semester binatorics; posets and lattices; appendices on the binary course for freshmen or sophomores in computer science number system and matrices. or mathematics. Considerable attention is paid to algorithms, but the approach is generally that of a math- 24. Pfleeger, Shari Lawrence; Straight, David W. In- ematician rather than a computer scientist. Contents: troduction to Discrete Structures. Revised Edition. Introduction to graphs and trees; sets; elementary logic 82 RESHAPINGCOLLEGE MATHEMATE

and induction; functions and sequences; matrices and ISBN 0-0 7-065 142-6. other semigroups; counting; more logic and induction; An interesting feature of this text is that its first hun- relations; graphs; trees; boolean algebra; algebraic sys- dred pages are devoted to logic. All in all, solid coverage tems. of the standard material. Boolean algebra is treated as a 30. Sahni, Sartaj. Concepts in Discrete Mathemat- subclass of lattices. The notation for algorithms can. become forbidding-see page 266 in particular. Contents: ics. Fridley, Minn.: Camelot Publishing Company; Mathematical logic; set theory; algebraic structures; lat- 1981; ISBN 0-942450-00-0. tices and boolean algebra; graph theory; introduction to The author says the book is for students of computer computability theory. science and engineering, with a bias towards the for- 35. Tucker, Alan C. Applied Combinatorics, Second mer, and contains needed topics not included in typical calculus and algebra courses. Algorithmic in flavor, Edition. New York: John Wiley and Sons; 1984; but moderately formal. Probably a year course at the ISBN 0-471-86371-8. sophomore-junior level. Many interesting examples not A text suitable for a wide range of audiences, from done elsewhere. Contents: Logic; constructive proofs sophomores to graduate students. Contents: Elements and mathematical induction; sets; relations; functions, of graph theory; covering circuits and graph coloring; recursion, and computability; analysis of algorithms; re- trees and selections; generating functions; recurrence currence relations; combinatorics and discrete probabil- relations; inclusion-exclusion; P6lya’s enumeration for- ity; graphs; modern algebra. mula; combinatorial modeling in theoretical comp.uter science; games with graphs; appendix on set theory and 31. Sedlock, James T. Mathematics for Computer Stud- logic, mathematical induction, probability, the pigeon- ies. Belmont, Calif.: Wadsworth Publishing Com- hole principle, and Mastermind. pany; 1985; ISBN 0-534-04326-7. Intended as first college mathematics course for computer science majors. Unsophisticated, at the level References of finite mathematics, without proofs or rigor. Con- tents: Introduction; computer-related arithmetic; sets, This bibliography lists some of the materials that the combinatorics, and probability; computer-related logic; computer-related linear mathematics; selected topics planner or instructor of a lower-division discrete math- (mathematics of finance, statistics, functions, induc- ematics course might want to consult. It includes books tion); introduction to advanced topics (graphs and trees, that may be too advanced or too specialized to belong semigroups, finite-state machines, languages and gram- in the BIBLIOGRAPHY:TEXTBOOKS. It also lists re- mars). ports and journal articles more or less pertinent to the 32. Skvarcius, Romualdas; Robinson, William. Discrete theme. Mathematics with Computer Science Applications. 1. ACM/IEEE Computer Society Joint Task Force. Menlo Park, Calif.: Benjamin Cummings Publish- “Computer science program requirements and ac- ing Company; 1986; ISBN 0-8053-7044-7. creditation.” Communications of the ACM, 1984, ”. . .intended audience is freshmen and sophomore 27 (4) 330-335. students who are taking a concentration in computer science . . .”. Contents: Introduction to discrete math- 2. ACM Curriculum Committee on Computer Science. ematics; logic and sets; relations and functions; com- “Curriculum ‘78: Recommendations for academic binatorics; undirected graphs; directed graphs; boolean programs in computer science.” Communications algebra; algebraic systems; machines and computations; of the ACM, 1979, 22 (3) 47-166. probability . 3. Aho, A.; Hopcroft, J.; Ullman, J. Data Structures 33. Stanat, Donald F.; McAllister, David F. Discrete and Algorithms. Reading, Mass.: Addison-Wesley, Mathematics in Computer Science. Englewood 1983. Cliffs, New Jersey: Prentice-Hall; 1977; ISBN 0-13- Primarily a book on data structures. Algorithms are 216150-8. presented in Pascal. An accessible reference for analysis A sophomore-junior level course; students will need of algorithms. some previous exposure to college-level mathematics. 4. Bavel, Zamir. Math Companion for Computer !Yci- The first discrete mathematics text to consider program verification in its coverage of mathematical reasoning. ence. Reston, Virginia: Reston Publishing Com- The text is essentially non-algorithmic, but contains a pany, 1982, ISBN 0-8359-4300-3or 0-8359-4299-6 special section on analysis of searching and sorting algo- (Pbk). rithms. Contents: Mathematical models; mathematical A manual, not a text. reasoning; sets; binary relations; functions; counting and 5. Beidler, John; Austing, Richard H.; Cassel, Lillian algorithm analysis; infinite sets; algebras. N. “Computing programs in small colleges.” Com- 34. Tremblay, Jean-Paul; Manohar, Ram. Discrete munications of the ACM, 1985, 28 (6) 605-611. Mathematical Structures with Applications to Com- Summary report of The ACM Small College Task puter Science. New York: McGraw-Hill; 1975; Force; outlines resources, courses, and problems for DISCRETEMATHEMATICS 83

small colleges developing degree programs in comput- Addison-Wesley, 1969, ISBN 0-201-02787-9. ing. See especially “The Mathematics Component,” p. Deals almost exclusively with undirected graphs. Ex- 610. tensive bibliography (to 1968). Heavy emphasis on enu- 6. Bellman, Richard; Cooke, Kenneth; Lockett, Jo meration. Most exercises require considerable mathematical maturity. Ann. Algorithms, Graphs, and Computers. New York: Academic Press, 1970, ISBN 0-12-084840-6. 16. Hart, Eric W. “Is discrete mathematics the new math of the eighties?” Mathematics Teacher, 1985, 7. Berztiss, A.T. Data Structures: Theory and Prac- 75 (5) 334-338. tice, Second Edition. New York: Academic Press, 1975, ISBN 0-12-093552-X. 17. Heid, M. Kathleen. “Calculus with muMath: Im- Although dated, primarily by its dependence on FOR- plications for curriculum reform.” The Computing TRAN, can be used as a reference on representation of Teacher, 1983, 11 (4) 46-49. digraphs by trees, and on critical path analysis. 18. Hodgson, Bernard R.; Poland, John. “Revamping 8. Berztiss, Alfs T. Towards a Rigorous Curriculum for the mathematics curriculum: The influence of com- Computer Science. Technical Report 83-5, Univer- puters.” Notes of the Canadian Mathematical Soci- sity of Pittsburgh Department of Computer Science, ety, 1983, 15 (8) n.p. 1983. A product of a meeting of a working group of the 9. Bogart, K.P.; Cordiero, K.; Walsh, M.L. “What is a Canadian Mathematics Education Study Group. discrete mathematics course?” SIAM News, 1985, 19. IEEE Educational Activities Board, Model Pro- 18(1). gram Committee. The 1983 IEEE Computer So- 10. Deo, Narsingh. Graph Theory with Applications ciety Model Program in Computer Science and En- to Engineering and Computer Science. Englewood gineering. New York: The Institute of Electrical Cliffs, New Jersey: Prentice-Hall, 1974, ISBN 0-13- and Electronics Engineers, 1983. A section “Discrete Mathematics” appears on pp. 8- 363473-6. 12. This contains a rather demanding modular outline of An excellent source for applications of the theory of topics to be covered, and urges integration of the math- graphs, both undirected and directed, but somewhat ematical theory with computer science and engineering dated. Contains extensive bibliographies (to 1972). applications. 11. Dornhoff, Larry L.; Hohn, Frans E. Applied Modern 20. Knuth, Donald E. The Art of Computer Program- Algebra. New York: Macmillan Publishing Com- ming, Volume 1: Fundamental Algorithms. Read- pany, 1978, ISBN 0-02-329980-0. ing, Mass.: Addison-Wesley, 1973, ISBN 0-201- Deals in great detail with applications of algebra in 03809-9. computer engineering, but the complicated notation Still the standard reference on the tools for analysis of makes access to the application studies rather hard. algorithms. 12. Gibbs, Norman Tucker, Allen “Model curricu- E.; B. 21. Koffman, E.; Miller, P.; Wardle, C. “Recommended lum for a liberal arts degree in computer science.” curriculum for CS1, 1984: A report of the ACM Communications of the ACM, 1986, 29 (3). curriculum committee task force for CS1.” Com- A curriculum developed by computer scientists supported by a grant from the Sloan Foundation. The pur- munications of the ACM, 1985, 27 (10) 998-1001. pose was to define a rigorous undergraduate major in 22. Koffman, E.; Stemple, D.; Warclle, C. “Recom- computer science for liberal arts colleges. mended curriculum for CS2, 1984--A report of the 13. Gordon, Sheldon P. “A discrete approach to the cal- ACM curriculum committee task force for CS2.” culus.” Int. J. Math. Educ. Sci. Technol., 1979, Communications of the ACM, 19851, 28 (8) 815-818. 10 (1) 21-31. 23. Laufer, Henry. Applied Modern Algebra. Boston: A report on an experiment in using discrete calculus Prindle, Weber, and Schmidt, 19841, ISBN 0-87150- (finite differences and sums in place of derivatives and integrals) as an introduction to computer-based contin- 702- 1. uous calculus. Sophomore-junior level; intended to follow the description for an applied algebra course in the 1981 MAA- 14. Gries, David. The Science of Programming. New CUPM recommendations (see item 25 below). York: Springer-Verlag, 1981, ISBN 0-387-90641-X. This book deals with the development of correct pro- 24. Lidl, Rudolf; Pilz, Gunter. Applied Abstract Alge- grams. The introductory sections on logic achieve thor- bra. New York: Springer-Verlag, 1985, ISBN 0-387- oughness without becoming intimidating. Apart from 96035-X. these sections there is little relevance to the proposed Describes aspects of abstract algebra1 applicable to dis- discrete mathematics course. crete mathematics. 15. Harary, Frank. Graph Theory. Reading, Mass.: 25. MAA Committee on the Undergraduate Program 84 REs H A PING COLL E G E MATHEM ATE

in Mathematics. Recommendations for a General munications of the ACM, 1980, 23 (2) 67-70. Mathematical Sciences Program. Washington, DC: 35. Ralston, A.; Young, G.S. (Eds.). The Future of The Mathematical Association of America, 1981. College Mathematics. New York: Springer-Verlag, 26. Manna, Zohar; Waldinger, Richard. The Logi- 1983, ISBN 0-387-90813-7. cal Basis for Computer Programming, Volume 1: Proceedings of the 1982 “Williams Conference” that Deductive Reasoning. Reading, Mass.: Addison- added impetus to the movement to add more discrete mathematics to the lower-division curriculum. The Wesley, 1985, ISBN 0-201-18260. ideas presented in the book are far from exhausted. This book is in two parts. The first is a thorough exposition of logic. In the second, data types are treated 36. Reingold, Edward M.; Nievergelt, Jurg; Deo, Nars- as theories. Although the authors see the material as ingh. Combinatorial Algorithms: Theory and Prac- ultimately replacing calculus, it probably could not do tice. Englewood Cliffs, New Jersey: Prentice-Hall, so as presented-the departure from established prac- 1977, ISBN 0-13-15244-7. tice is too radical, and the style is too austere for an undergraduate text. 37. Roberts, Fred S. Applied Combinatorics. Engle- 27. Maurer, Stephen B. “The algorithmic way of life is wood Cliffs, New Jersey: Prentice-Hall, 1984, ISBN best.” The College Mathematics Journal, 1985, 16 0-13-039313-4. Junior-senior level; wide range of applications; algo- (1) 2-5. rithms. This is the lead article for a “forum” in which others contribute their own very diverse points of view on the 38. Roberts, Fred S. Discrete Mathematical Models with general theme. The whole exchange occupies pp. 2-21. Applications to Social, Biological, and Environm.en- 28. Mulder, Michael C.; Dalphin, John. “Computer tal Models. Englewood Cliffs, New Jersey: Prentice- science program requirements and accreditation.” Hall, 1976, ISBN 0-13-214171-X. Computer, 1984, 17 (4) 30-35. 39. Shaw, M. (Ed.). The Carnegie-Mellon Curriculum 29. Nijenhuis, Albert; Wilf, Herbert S. Combinatorial for Undergraduate Computer Science. New York: Algorithms. New York: Academic Press, 1975, Springer-Verlag, 1984. ISBN 0- 12-519250-9. 40. Smith, Douglas; Eggen, Maurice; St. Andre, A study of aspects of computers, algorithms, and Richard. A Transition to Advanced Mathematics. mathematics, and of the relations between them, based Monterey, Calif.: Brooks-Cole, 1983, ISBN 0-534- on an examination of programs in FORTRAN. 01249-3. 30. Preparata, Franco P.; Yeh, Raymond T. Introduc- This book is intended to pave the way from calculus to tion to Discrete Structures for Computer Science more advanced mathematics. Its core chapters deal with and Engineering. Reading, Mass.: Addison-Wesley, logic, proofs, sets, relations, functions, and cardinality. 1973, ISBN 0-201-05968-1. 41. Solow, Daniel. How To Read and Do Proofs. New Junior level. Boolean algebras are arrived at via lat- York: John Wiley and Sons, 1982, ISBN 0-471- tices. Solid coverage, but the material on graphs needs 86645-8. to be supplemented. For use as a supplement if the discrete mathematics 31. Ralston, A. “Computer science, mathematics, and course emphasizes proof. the undergraduate curricula in both.” American 42. Wand, Mitchell. Induction, Recursion, and Pro- Mathematical Monthly, 1981, 88 (7) 472-485. gramming. New York: North Holland, 1980, ISBN 32. Ralston, A. “The really new college mathematics 0-444-00322-3. and its impact on the high school curriculum.” In This text is addressed specifically to computer science students. It is based on the thesis that a program is a Hirsch, Christian R.; Zweng, Marilyn J. (Eds.), The mathematical object. Correctness of programs is then Secondary School Mathematics Curriculum: 1985 established by reasoning about these mathematical ob- Yearbook. Reston, Virginia: National Council of jects in an appropriate language. Prior exposure to sets Teachers of Mathematics, 1985, 29-42, ISBN 0- and functions is assumed. 87353-217-1. 43. Whitehead, Earl Glen, Jr. Combinatorial Algo- 33. Ralston, A. “The first course in computer science rithms. New York: Courant Institute of Mathemat- needs a mathematical corequisite.” Communica- ical Sciences, New York University, 1973. tions of the ACM, 1984, 27 (10) 1002-1005. 44. Wirth, Niklaus. Algorithms and Data Structures. 34. Ralston, A,; Shaw, M. “Curriculum ‘78-1s com- Englewood Cliffs, New Jersey: Prentice-Hall, 1986, puter science really that unmathematical?” Com- ISBN 0-13-022005-1. Calculus Transition: From High School to College

From 1983 to 1986, the CUPM Panel on Calcu- Introduction lus Articulation studied problems associated with college transition for students who had studied calculus in high There is a widespread and growing dissatisfaction school. The report of the CUPM Panel originally ap- with the performance in college calculua courses of many peared under the title UTransitionfrom High School to students who had studied calculus in high school. In College Calculus” in the AMERICANMATHEMATICAL response to this concern, in the fall of 1983, the Com- MONTHLY,October, 1987. It is reprinted here with mi- mittee on the Undergraduate Program in Mathematics nor editorial changes. (CUPM) formed a Panel on Calculus Articulation to undertake a three-year study of questions concerning the transition of students from high school calculus to 1989 Preface college calculus and submit a report to CUPM detail- ing the problems encountered and proposals for their The importance of the problems identified in solution. the panel’s report has been underscored by sev- The seriousness of the issues involved in the Panel’s eral recent international and national assessments study is underscored by the number of students involved of mathematics education (e.g., The National Re- and their academic ability. During the ten-year period search Council’s report Everybody Counts: A Report 1973 to 1982, the number of students in high school to the Nation on the Future of Mathematics Educa- calculus courses grew at a rate exceeding 10% annually. tion). Of the 234,000 students who passed a high school cal- Since the panel’s report was written in 1986, the culus course in 1982, 148,600 received a grade of B- or severity of the transition problems from high school higher [2]. Assuming a continuation of the 10% growth to college calculus has increased both qualitatively rate and a similar grade distribution there were approx- and quantitatively. In 1987 there were 59,123 stu- imately 200,000 high school students in the spring of dents who took an Advanced Placement Calculus Ex- 1985 who received a grade of B- or higher in a calculus amination, an increase of 85% from 1982. Although course. Thus possibly a third or more of the 500,000 this increase does not necessarily indicate a similar college students who began their college calculus pro- growth rate in the number of students studying cal- gram (in Calculus I, Calculus 11, or Cakulus 111) in the culus in high school, it does document a large in- fall of 1985 had already received a grade of B- or higher crease in the number of students entering college cal- in a high school calculus course. culus having earned an Advanced Placement Calculus The students studying calculus in high school consti- score of three or less. This increase serves to inten- tute a large majority of the more mathematically capa- sify the Report’s recommendation that colleges and uni- ble high school students. (In 1982, 55% of high school versities develop special calculus courses for these stu- students attended schools where calculus was taught dents. [2].) Students who score a 4 or 5 on an Advanced The development of Computer Algebra Systems de- Placement (AP) Calculus examination normally do well signed for classroom teaching introduces a new com- in maintaining their accelerated mathematics program ponent into the transition problems students may en- during the transition from high school to college. How- counter in going from high school to college cal- ever, this is a very small percentage of the students who culus. It is vitally important to expand com- take calculus in high school. For exaimple, in 1982, of munication between college and high school teach- the 32,000 students who took an Advanced Placement ers in regards to the development of this tech- calculus examination, just over 12,000 received scores of nology, particularly with respect to pedagogical is- 4 or 5, which represents only 6% of all high school stu- sues. dents who took calculus that year. The primary concern Donald B. Small of the Panel was with the transition difficulties associ- Colby College ated with the remaining almost 94% of the high school March, 1989 calculus students. 86 RESHAPINGCOLLEGE MATHEMATE

Problem Areas course perform in the bottom 25% in this international survey. The poor performance in geometry by both the Past studies and the Panel's surveys of high school precalculus and calculus students correlates well with teachers, college teachers, and state supervisors suggest the statistic that 38% of the students were never taught that the major problems associated with the transition the material contained in the geometry section of the from high school calculus to college calculus are: test [9, p. 591. The test data underscores the concern 1. High school teacher qualifications and expectations. expressed by many college teachers that more emphasis 2. Student qualifications and expectations. needs to be placed on geometry throughout the high 3. The effect of repeating a course in college after school curriculum. This data does not, however, indi- having experienced success in a similar high school cate that accelerated programs emphasize geometry less course. than non-accelerated programs. 4. College placement. The success of the accelerated programs in com- 5. Lack of communication between high schools and pleting the "normal" four year high school mathemat- colleges. ics program by the end of the eleventh grade presents These problems were addressed by first considering schools with both an opportunity and a challenge for a accelerated programs in general, high school calculus "fifth" year program. There are two acceptable options: (successful, unsuccessful), and the responsibilities of the 1. Offer college-level mathematics courses that would colleges. continue the students' accelerated program and thus provide exemption from one or two semesters of college mathematics. Accelerated Programs 2. Offer high school mathematics courses that would Accelerated mathematics programs, usually begin- broaden and strengthen a student's background and ning with algebra in eighth grade, are now well estab- understanding of precollege mathematics. lished and accepted in most school systems. The suc- Not offering a fifth year course or offering a watered- cess of these programs in attracting the mathematically down college level course with no ezpectation of students capable students was documented in the 1981-82 test- earning advanced placement are not considered to be ac- ing that was done for the "Second International Math- ceptable options. ematics Study." The Summary Report [9] states with A great deal of prestige is associated with offering reference to a comparison between twelfth grade precalculus as a fifth year course. Communities often view calculus students and twelfth grade calculus students the offering of calculus in their high school as an in- in the United States: dication of a quality educational program. Parents, school board officials, counselors, and school adminis- We note furthermore that in every content area (sets trators often demonstrate a competitive pride in their and relations, number systems, algebra, geometry, elementary functions/calculus, probability and statis- school's offering of calculus. This prestige factor can tics, finite mathematics), the end-of-the-year average easily manifest itself in strong political pressure for a achievement of the precalculus classes was less (and in school to offer calculus without sufficient regard to the many cases considerably less) than the beginning-of- qualifications of teachers or students. the-year achievement of the calculus students. It is important that this political pressure be resisted The report continues: and that the choice of a fifth year program be made It is important to observe that the great majority of by the mathematics faculty of the local school and be U.S. senior high school students in fourth and fifth made on the basis of the interest and qualifications of year mathematics classes (that is, those in precalculus the mathematics faculty and the quality and number of classes) had an average performance level that was at accelerated students. School officials should be encour- or below that of the lower 25% of the countries. The aged to develop public awareness programs to extend end-of-year performance of the students in the calculus classes was at or near the international means for the the prestige and support that exists for the calculus various content areas, with the exception of geometry. to acceleration programs in general. This would help Here U.S. performance was below the international diffuse the political pressure as well as broaden school average. support within the community. Thus those students in accelerated programs cul- Schools that elect the first option of offering a col- minating in a calculus course perform near the inter- lege level course should follow a standard college course national mean level while their classmates in (non- syllabus (e.g., the Advanced Placement syllabus for cal- accelerated) programs culminating in a precalculus culus). They should use placement test scores along CALCULUSTRANSITION 87 with the college records of their graduates as primary High School Calculus measures of the validity of their course. There are many valid reasons why i%fifth year pro- For schools that elect the second option, a variety of gram should include a calculus course. Four major rea- courses is possible. The following course descriptions sons: (1) calculus is generally recognized as the starting represent four possibilities. point of a college mathematics prograia, (2) there ex- ANALYTICALGEOMETRY. This course could go well ists a (nationally accepted) syllabus, (3) the Advanced beyond the material normally included in second year Placement program offers a nation-widle mechanism for algebra and precalculus. It could include Cartesian and obtaining advanced placement, and (4) there is a large vector geometry in two- and three-dimensions with top- prestige factor associated with offering calculus in high ics such as translation and rotation of axes, characteris- school. Calculus, however, should not be offered un- tics of general quadratic relations, curve sketching, po- less there is a strong indication that the course will be lar coordinates, and lines, planes, and surfaces in three- successful. dimensional space. Such a course would provide specific preparation for calculus and linear algebra, as well as give considerable additional practice in trigonometry Successful Calculus Courses and algebraic manipulations. The primary characteristics of a successful high PROBABILITYAND STATISTICS.This course could school calculus course are: be taught at a variety of levels, to be accessible to 1. A qualified and motivated instructor with a math- most students, or to challenge the strongest ones. It ematics degree that included at least one semester could cover counting methods and some topics in dis- of a junior-senior real analysis course involving a crete probability such as expected values, conditional rigorous treatment of limits, continuity, etc. probability, and binomial distributions. The statistics 2. Administrative support, including provision of ad- portion of the course could emphasize exploratory data ditional preparation time for the instructor (e.g., as analysis including random sampling and sampling dis- recommended by the North Central Accreditation tributions, experimental design, measurement theory, Association). measures of central tendency and spread, measures of 3. A full year program based on the Advanced Place- association, confidence intervals, and significance test- ment syllabus. ing. Such an introduction to probability and statistics 4. A college text should be used (not a watered-down would be valuable to all students, and for those who high school version). do not plan to study mathematics, engineering, or the 5. Advanced placement for students (rather than mere physical sciences, probably more valuable than a calcu- preparation for repeating calculus in college) is a lus course. major goal. DISCRETEMATHEMATICS. This type of course could 6. Course evaluation based primarily on college place- include introductions to a number of topics that are ei- ment and the performance of its graduates in the ther ignored or treated lightly within a standard high next higher level calculus course. school curriculum, but which would be stimulating and 7. Restriction of course enrollment t,o only qualified widely useful for the college-bound high school student. and interested students. Suggested topics include permutations, combinations, 8. The existence of an alternative fifth year course that and other counting techniques: mathematical induc- students may select who are not qualified for or in- tion; difference equations; some discrete probability; el- terested in continuing in an accelerated program. ementary number theory and modular arithmetic; vec- The bottom line of what makes a hig;h school calculus tor and matrix algebra, perhaps with an introduction course successful is no surprise to anyone. A qualified to linear or dynamic programming; and graph theory. teacher with high but realistic expectations, using some- MATRIXALGEBRA. This course could include ba- what standard course objectives, and students who are sic arithmetic operations on matrices, techniques for willing and able to learn result in a successful transi- finding matrix inverses, and solving systems of linear tion at any level of our educational process. Problems equations and their equivalent matrix equations using appear when any of the above ingredients are missing. Gaussian elimination. In addition, some introduction to linear programming and dynamic programming could Unsuccessful Calculus Courses be included. This course could also emphasize three- dimensional geometry. Two types of high school calculus courses have an 88 RESHAPINGCOLLEGE MATHEMAT= undesirable impact on students who later take calculus Calculus program by the granting of one semester ad- in college. vanced placement with credit in calculus for students One type is a one semester or partial year course that with a 4 or 5 score on the AB examination, and two presents the highlights of calculus, including an intuitive semesters of advanced placement with credit in calculus look at the main concepts and a few applications, and for students with a 4 or 5 score on the BC examination. makes no pretense about being a complete course in The studies reviewed by the Panel do not indicate the subject. The motivation for offering a course of this any clear conclusions concerning performance in sub- kind is the misguided idea that it prepares students for sequent calculus courses by students who have scared a redcourse in college. a 3 on an Advanced Placement Calculus examination. However, such a preview covers only the glory and The treatment of these students is a very important thus takes the excitement of calculus away from the col- transition problem since approximately one-third of all lege course without adequately preparing students for students who take an Advanced Placement Calculus ex- the hard work and occasional drudgery needed to un- amination are in this group and many of them are quite derstand concepts and master technical skills. Profes- mathematically capable. sor Sherbert has commented: “It is like showing a ten It is therefore recommended that these students be minute highlights film of a baseball game, including the treated on a special basis in a manner that is appro- final score, and then forcing the viewer to watch the priate for the institution involved. For example, sev- entire game from the beginning-with a quiz after each eral colleges offer a student who has earned a 3 on an inning.” Advanced Placement Calculus examination the oppor- The second type of course is a year-long, semi- tunity to upgrade this score to an “equivalent 4” by serious, but watered-down treatment of calculus that doing sufficiently well on a Department of Mathemat- does not deal in depth with the concepts, covers no ics placement examination. Another option is to give proofs or rigorous derivations, and mostly stresses me- such students one semester of advanced placement with chanics. The lack of both high standards and emphasis credit for Calculus I upon successful completion of Cal- on ‘understanding dangerously misleads students into culus 11. A third option is to give one semester of ad- thinking they know more than they really do. vanced placement with credit for Calculus I and provide In this case, not only is the excitement taken away, a special section of Calculus I1 for such students. but an unfounded feeling of subject mastery is fostered Other important transition problems are associ- that can lead to serious problems in college calculus ated with students who have studied calculus in high courses. Students can receive respectable grades in a school, but have not attained advanced placement ei- course of this type, yet have only a slight chance of pass- ther through the Advanced Placement Calculus pro- ing an examination. Those who place into second term gram or effective college procedures. These students calculus in college will find themselves in heavy com- pose an important and difficult challenge to college petition with better prepared classmates. Those who mathematics departments, namely: How should these elect (or are selected) to repeat first term calculus be- students be dealt with so that they can benefit from lieve they know more than they do, and the motivation their accelerated high school program and not succumb and willingness to learn the subject are lacking. to the negative and (academically) destructive attitude problems that often result when a student repeats a course in which success has already been experienced? There are three major factors to consider with respect College Programs to these students. Several studies ([l],[3], [5], [6], [7]) have been con- 1. The lack of uniformity of high school calculus ducted on the performance in later courses by students courses. The wide diversity in the backgrounds of who have received advanced placement (and possibly the student8 necessitates that a large review com- college credit) by virtue of their scores on Advanced ponent be included in their first college calculus Placement Calculus examinations. The studies show course to guarantee the necessary foundation for fu- that, overall, students earning a score of 4 or 5 on either ture courses. the AB or BC Advanced Placement Calculus examina- 2. The mistaken belief of most of these students that tion do as well or better in subsequent calculus courses they really know the calculus when, in fact, they do than the students who have taken all their calculus in not. Thus they fail to study enough at the beginning college. It is therefore strongly recommended that col- of the course. When they realize their mistake (if leges recognize the validity of the Advanced Placement they do), it is often too late. These students often CALCULUSTRANSITION 89

become discouraged and resentful as a result of their the above criteria. For example, Col.by College has poor performance in college calculus, and believe successfully developed a two semester calculus course that it is the college course that must be at fault. that fulfills the four conditions. The clourse integrates 3. The "Pecking Order" syndrome. The better the multivariable with single variable calcuIlus, and thereby student, the more upsetting are the understandable covers the traditional three semester program in two feelings of uncertainty about his or her position rel- semesters [lo]. ative to the others in the class. Although this is a Of course, the introduction of a new course entails common problem for all college freshmen, it is com- an accompanying modification of college placement pro- pounded when the student appears to be repeating grams. However, providing new or alternative courses a course in which success had been achieved the pre- should have the effect of simplifying placement issues ceding year. This promotes feelings of anxiety and and easing transition difficulties that nlow exist. produces an accompanying set of excuses if the student does not do at least as well as in the previous year. Recommendations

The uncertainty of one's position relative to the rest 1. School administrators should develop public aware- of the class often manifests itself in the student not ness programs with the objective of extending the asking questions or discussing in (or out of) class for support that exists for fifth year callculus courses to fear of appearing dumb. This is in marked contrast accelerated programs including all of the fifth year to the highly confident high school senior whose options. questions and discussions were major components 2. A fifth year program should offer a student a choice in his or her learning process. of courses (not just calculus). The unpleasant fact is that the majority of students 3. The choice of fifth year options should be made by who have taken calculus in high school and have not the high school mathematics facu1t:y on the basis of clearly earned advanced placement do not fit in either their interest and qualifications and the quality and the standard Calculus I or Calculus I1 course. The stu- number of the accelerated students. dents do not have the level of mastery of Calculus I 4. If a fifth year course is intended as a college level topics to be successful if placed in Calculus I1 and are course, then it should be treated as a college level often doomed by attitude problems if placed in Calculus course (text, syllabus, rigor). I. In modern parlance, this is the rock and hard place. 5. A fifth year college level course should be taught An additional factor to consider is the negative effect with the expectation that successful graduates (B- that a group of students who are repeating most of the or better) would not repeat the course in college. content of Calculus I has on the rest of the class as well 6. A fifth year program should provide an alternative as on the level of the instructor's presentations. option for the student who is not qualified to con- What is needed are courses designed especially for tinue in an accelerated program. students who have taken calculus in high school and 7. A mathematics degree that includes at least one have not clearly earned advanced placement. These semester of a junior-senior real analysis course in- courses need to be designed so that they: volving a rigorous treatment of limit, continuity, 1. Acknowledge and build on the high school experi- etc., is strongly recommended for anyone teaching ences of the students; calculus. 2. Provide necessary review opportunities to ensure an 8. A high school calculus course should be a full year acceptable level of understanding of Calculus I top- course based on the Advanced Placement syllabus. ics; 9. The instructor of a high school calculus course 3. Are clearly diflerent from high school calculus should be provided with additional preparation time courses (in order that students do not feel that for this course. they are essentially just repeating their high school 10. High school calculus students should take either the course); AB or BC Advanced Placement cidculus examina- 4. Result in an equivalent of one semester advanced tion. placement. 11. The evaluation of a high school calculus course Altering the traditional lecture format or rearrang- should be based primarily on college placement and ing and supplementing content seem to be two promis- the performance of its graduates in the next level ing approaches to developing courses that will satisfy calculus course. 90 RESHAPINGCOLLEGE MATHEMATE

12. Only interested students who have successfully com- sophomores who graduated in 1982." National Council pleted the standard four year college preparatory of Education Statistics, April 1984 (LSB 84-4-3). program in mathematics should be permitted to [3] P.C. Chamberlain, R.C. Pugh, J. Schellhammer. "Does take a high school calculus course. advanced placement continue throughout the undergraduate years?" College and Univerrity, Winter 1968. 13. Colleges should grant credit and advanced place- [4] "Advanced Placement Course Description, Mathemat- ment out of Calculus I for students with a 4 or 5 ics." The College Board, 1984. score on the AB Advanced Placement calculus ex- [5] E. Dickey. "A study comparing advanced placement and amination, and credit and advanced placement out first-year college calculus students on a calculus achieve- of Calculus I1 for students with a 4 or 5 score on ment test." Ed.D. dissertation, University of South Car- the BC Advanced Placement calculus examination. olina, 1982. Colleges should develop procedures for providing [6] D.A. Frisbie. "Comparison of course performance of AP and non-AP Calculus students." Research Memorandum special treatment for students who have earned a No. 207, University of Illinois, September 1980. score of 3 on an Advanced Placement calculus ex- [7] D. Fry. "A comparison of the college performance amination. in calculus-level mathematics courses between regular- 14. Colleges should individualize as much as possible progress students and advanced placement Students." the advising and placement of students who have Ed.D. dissertation, Temple University, 1973. taken calculus in high school. Placement test scores [8] C. Jones, J. Kenelly, D. Krcider. "The advanced place- and personal interviews should be used in determin- ment program in mathematics-Update 1975." Mathe- matics Teacher, 1975. ing the placement of these students. [9] Second Internotional Mathematics Study Summary Re- 15. Colleges should develop special courses in calculus port for the United Stater. Champaign, IL: Stipes Pub- for students who have been successful in accelerated lishing, 1985. programs, but have clearly not earned advanced [lo] D. Small, J. Hosack. Calculus: An Integrated Approach. placement. McGraw-Hill, 1990. [ll] D.H. Sorge, G.H.Wheatley. "Calculus in high school- Colleges have an opportunity and responsibility to de- At what cost?" American Mathematical Monthly 84 velop and foster communication with high schools. In (1977) 644-647. particular: [12] D.M. Spresser. "Placement of the first college course." International Journal Mathematics Education, Science, 16. Colleges should establish periodic meetings where and Technology 10 (1979) 593-600. high school and college teachers can discuss expectations, requirements, and student performance. 17. Colleges should coordinate the development of en- Panel Members richment programs (courses, workshops, institutes) for high school teachers in conjunction with school DONALDB. SMALL,CHAIR, Colby College. districts and state mathematics coordinators. GORDONBUSHAW, Central Kitsap High School, Sil- verdale , Washington. JOHNH. HODGES,University of Colorado. References DONALDJ. NUTTER,Firestone High School, Akron, [l] C. Cahow, N. Christensen, J. Gregg, E. Nathans, H. Ohio. Strobel, G. Williams. Undergraduate Faculty Council of RONALDSCHNACKENBURG, Steamboat Springs High Arts and Sciences Committee on Curriculum; Subcom- School, Colorado. mittee on Advanced Placement Report, Trinity College, DONALD SHERBERT,University of Illinois. Duke University, 1979. R. [2] C. Dennis Carroll. "High school and beyond tabula- BARBARASTOTT, Riverdale High School, Jefferson, tion: Mathematics courses taken by 1980 high school Louisiana. Curriculum for Grades 11-13

The following report was approved in 1987 by the additional study. Board of Directors of the National Council of Teach- Although it is tempting to make no statements dur- ers of Mathematics and by the Board of Governors of ing this time of rapid evolution, we are persuaded that the Mathematical Association of America. The joint school and university students, parents, and teachers NCTM-MAA committee that prepared the report was will welcome some clarification from the professional chaired by Joan Leitzel, Associate Provost of The Ohio associations. The joint statement of the NCTM and State University. It is printed here in its entirety for MAA on college preparation has not been updated since the first time. it was released in the mid-70s. Thus, although we do not in any way want to suggest that the status quo is The Joint Task Force on Curriculum for Grades 11-13 acceptable for education in mathematics, we do wish was formed in Spring 1986 by John Dossey, President to summarize prevailing opinions in the hope that some of the National Council of Teachers of Mathematics, guidance will be helpful to students and teachers at this and Lynn Steen, President of the Mathematical Associ- time. ation of America. The Task Force was charged to focus This Task Force was formed to consider curricular on curriculum for the mainstream of students who take matters. Even though it is beyond tlhe charge of the a college preparatory program in high school and who Task Force, we are compelled to mention that attention go on in college to a standard freshman mathematics to curriculum will be meaningless without concurrent course such as calculus, finite mathematics, statistics, attention to other matters. One such concern is the in- or discrete mathematics. The Task Force was asked to creasing shortage of qualified teachers at all levels, es- advise on the need for new recommendations to high pecially in the middle grades and the secondary schools. schools and colleges concerning curriculum for students In addition, colleges and universities frequently depend who intend to pursue careers that depend on mathe- heavily on teaching assistants and part-time teachers. matics. Secondly, it is difficult to discuss cuririculum in grades The Task Force has studied and synthesized recent 11-13 without reference to the mathematics of the ear- national reports on the state of mathematics education, lier grades. The Task Force has been reminded fre- as well as the recommendations of many national and quently of the critical role that middle school mathe- state boards, and the reports of several recent curricu- matics and the first year of algebra play in a student’s lum projects. It is clear that the present situation in success in college preparatory mathematics. Continued mathematics education is dynamic and that significant attention needs to be given to the cui:riculum of these changes in curriculum in grades 11-13 may occur in the grades. Indeed, some of the recommendations we make next few years. The Task Force has been impressed and in this report for grades 11-13 apply as well to earlier excited by the quality and scope of curricular projects grades. currently underway. We note particularly the K-12 Curriculum Project undertaken by the Mathematical Sciences Education Board (MSEB), the frameworks of Issues of Apparent Consensus which will be released during 1989, and the collegiate mathematics project undertaken jointly by MSEB and After reviewing numerous reports and talking with the Board of Mathematical Sciences of the National Re- teachers, mathematicians, and mathematics educators, search Council (The Mathematical Sciences in the Year the Task Force believes that there is agreement within 2000: Assessment for Renewal in U.S. Colleges and Uni- the profession on many issues related to curriculum in versities). grades 11-13 for those students who are directed toward It is not the purpose of this report to resolve present mathematics-dependent collegiate programs. From this conflicts or to determine new directions for mathemat- consensus, it appears that the following recommenda- ics education. Rather, it is the purpose of this report tions can be made: to summarize those areas related to grades 11-13 cur- 1. College-bound students should Lake mathematics riculum in which it appears the profession now has a in all years of secondary school; Ithis mathematics level of consensus, and also to summarize those issues should include the content of intermediate algebra where there is lack of consensus and apparent need of and geometry. 92 RESHAPINGCOLLEGE MATHEMA-

2. Geometry in both two and three dimensions, coor- ematics as these applications develop. In addition, dinate geometry and the development of geometric students should understand that mathematics itself perception are essential parts of college preparatory is a developing discipline, and, where possible, stu- mathematics. dents should come to appreciate new developments 3. Although computer programming is an important in mathematics. tool for all college-bound students, courses in com- 11. The curriculum of grades 11-13 should contain units puter programming should not be regarded as sub- in statistics, probability, and fundamental topics in stitutes for college preparatory mathematics. discrete mathematics for all college-bound students, 4. Students should learn the content of a full four years of college preparatory mathematics before taking a 12. Mathematics in grades 11-13 should have goals be- calculus course. Students directed toward twelfth yond the acquisition of computational techniques. grade calculus need an enriched program as early as Mathematical understanding and analytical rea- grades 7 and 8. soning are basic goals for mathematics at this 5. A full program of college preparatory mathematics level. Problem solving strategies should be stressed, should be provided in every secondary school. and manipulative and computational techniques, al- 6. Calculus, if taught in the schools, should be equiva- though important, should not predominate. lent to college-level calculus. NCTM and MAA have 13. Teachers of mathematics in grades 11-13 should em- made a joint recommendation to the schools that ploy strategies that encourage student reading, writ- students should expect to write proficiency exam- ing, and reflection. Assignments and examinations inations (either Advanced Placement or university should be designed to help students become more in- credit exams) to establish that they have learned dependent learners of mathematics and to increase beginning calculus, and to obtain college credit for their abilities to discuss both orally and in writing high school calculus courses. Watered-down calcu- the mathematical ideas they are learning. Courses lus courses in the secondary schools stressing manip- should not cover an excessive number of topics at ulations but slighting subtle processes do not help the expense of reflection and independent learning students. Introducing polynomial calculus in pre- on the part of students. Teachers should take advan- calculus courses uses time better spent on other top- tage of mathematics competitions and science fairs ics. to encourage independent learning in students. 7. If mathematics is required in a student’s college 14. While there is need for meaningful review within program, the student should enroll in mathemat- new mathematics, the amount of time spent on re- ics courses as a college freshman. There should not view at the beginning of a course in grades 11-13 be a gap of a year or more during which a student should not be excessive. Typically, review should takes no mathematics. be integrated into the learning of new mathematics. 8. Students should expect to make use of calculators and computers in their mathematics courses in 15. Expectations of students with regard to homework, grades 11-13. Calculators and computers should be examinations, and knowledge of previous courses used as tools to enhance and expand the learning of need to be raised in many grade 11-13 programs. mathematics in these grades. These expectations should include daily homework, 9. The availability of technology should permit com- cumulative examinations, and examination ques- putational approaches to college preparatory math- tions that require problem-solving skills. ematics that result in this mathematics being acces- 16. To overcome the effects of socialization that discour- sible to more students. Further, the availability of ages girls and American minorities from studying technology should permit students in these grades mathematics, these groups of students should be es- to investigate problem situations that are not ap- pecially encouraged in the study of mathematics in proachable without computational tools, and to be grades 11-13,and efforts should be made to identify introduced to different mathematics than was pos- applications of mathematics that hold particular in- sible without calculators and computers. terest for under-represented groups. Furthermore, 10. Mathematics in grades 11-13 needs to have a clear the perceived preponderance of negative attitudes connection to real world problems, and students toward mathematics in this country should be stud- should be expected to acquire a growing ability to ied to determine what aspects of curriculum and use mathematics to model real situations. Students what features of culture contribute most heavily to should become aware of new applications of math- these attitudes. CURRICULUM FOR GRADES11-13 93

Recommendat ion heavily manipulative techniques are postponed to later The Task Force is persuaded that improvement in the courses. In his paper, “Why Elementa.ry Algebra Can, curriculum of grades 11-13 and in student performance Should, and Must be an 8th Grade Course for Average in this curriculum requires strong collaborative effort Students,” Zalman Usiskin argues that in other coun- among mathematicians, university faculty in mathe- tries algebra is usually done with all students at grades matics education, teachers, school leaders, counselors, 7 and 8 and that with proper curriculum in grades 1-6, students, and parents. The Task Force recommends algebra could be mastered in grade 8 ‘by U.S. students that funding be sought so that the curricular issues cited as well. above and other essential related information can be In The Underachieving Curriculum: Assessing U.S. communicated to these groups. Particularly targeted School Mathematics fiom an International Perspec- brochures and flyers need to be developed and circu- tive, a report on the Second International Mathematics lated to teachers, school leaders, counselors, students, Study, the authors recommend: and parents. The content of the mathematics curriculum needs to We have included at the end of the report a fuller de- be re-examined and revitalbed. The domination of scription of the rationale and content of these proposed the lower secondary school curriculum by the arithmetic of the elementary school has resulted in a pro- communiques and recommendations for the desired col- gram that, from an international point of view, is very laboration. lean. The curriculum should be broa-dened and enriched by including a substantial treatment of topics such as geometry, probability, statistics and algebra, Issues Requiring Further Study as well as promoting higher-level process goals such as In addition to the issues listed above on which the estimation and problem-solving. profession appears to have a consensus, we have iden- In “Let’s Not Teach Algebra to Eighth Graders!” tified many issues where there is lack of consensus. In (Mathematics Teacher, November 1985), Fernand Pre- many cases, studies, reviews, and experimentation are vost provides evidence that offering algebra in eighth underway in a variety of projects. Where we know of grade has unwanted consequences. Hiis study of New such projects, we have cited them here. In some cases it Hampshire schools showed that “only about half of the appears that more study and discussion will be needed students who take algebra as eighth graders continue to clarify these issues. their study of mathematics through a fifth year.” Pre- vost recommends an enriched program rather than al- Gifted Students: Acceleration or Enrichment? gebra, with only the top 3 to 5 percent being truly ac- The types of courses that students are capable of tak- celerated. Two very relevant questions regarding his ing in grades 11-13 will depend upon the preparation of study are: these students in earlier years. Should the very best stu- 1. Do the sixth and seventh grade programs ad- dents be accelerated or should their programs instead equately prepare students for a:lgebra in eighth be enriched with topics they may otherwise miss? This grade? question is still a matter of debate among mathematics 2. Would the retention rate be greater if there were educators. alternative twelfth grade courses? Advocates of early introduction of algebra usually Several letters of rebuttal were submitted by readers argue that the current curriculum for grades 7 and 8 is in response to Prevost’s article. mainly just a review of topics taught in previous grades There has been specific concern about students who with little new material introduced. The Report of the study calculus in grade 11. ETS reports that 4,000 of MSEB Task Force on Curriculum Frameworks for K-12 the 60,000 students now taking AP calculus exams are Mathematics (Draft, October 1986) states that: in eleventh grade and lower. Very special arrangements We applaud the current attempts to make algebra an eighth grade subject. There is ample evidence from are needed to guarantee that these students have ap- other countries that eighth graders can handle algebra. propriate mathematics in grade 12. More generally, we think that grades 7 and 8 should The NCTM’s position statement on Provisions for look forward to high school mathematics as much or Mathematically Talented and Gifted Students (October more than they look backward to elementary school 1986) contends that: mathematics. The needs of mathematically talented and gifted stu- The University of Chicago School Mathematics Pro- dents cannot be met by programs of study that only gram is being designed for the general school popula- accelerate these students through the standard school tion. Its eighth grade course is mainly algebra, but curriculum, nor can they be met by programs that al- 94 RESHAPINGCOLLEGE MATHEMATICS

low students to terminate their study of mathematics 1986) states: before their graduation from high school. Not only do we believe that an integrated curricu- The NCTM paper goes on to recommend that: lum offers the possibility of a richer, more coherent program than the alternative but, further, we believe All mathematically talented and gifted students that the introduction of a variety of new subject mat- should be enrolled in a program that provides a broad ter into the secondary school mathematics curriculum and enriched view of mathematics in a context of will inevitably signal the demise of the segregated cur- higher expectation. Acceleration within such a pro- riculum if only for logistics reasons. gram is recommended only for those students whose interests, attitudes, and participation clearly reflect How inevitable this is remains to be seen. One fac- the ability to persevere and excel throughout the en- tor that must always be taken into consideration when tire program. making curriculum decisions in the U.S. is the mobility of our population. It must be admitted that the ability Greater Integration of Topics to transfer a credit in algebra or a credit in geometry has It can hardly be disputed that the curriculum for simplified matters for many students. Unified curricula grades 11-13 must be closely related to the curriculum may make the process of transferring more traumatic. of the preceding years. An issue on which there is no Such curricula may also cause much rethinking on the apparent consensus and which therefore requires further part of colleges that are accustomed to accepting well study is the extent to which the secondary school math- defined units of credit, although placement by examina- ematics curriculum should be integrated or unified. tion may suffice. An additional concern is that teach- In most countries mathematics is not compartmen- ing unified courses requires a breadth of understanding talized into algebra, geometry, etc., as is conventional beyond what many teachers have been prepared to :pro- in the U.S. Since World War I1 there have been numer- vide. ous attempts to break down these compartments in this It should be pointed out that there is yet another country. One such attempt in the 1950's was Florida's aspect of integration that needs further study. This is Functional Mathematics Program which was short-lived the possibility of integrating mathematics with other mainly because of the advent of the School Mathematics disciplines, particularly science. Study Group (SMSG) and the "new mathematics." In the late 1960's and early 1970's, the Secondary School Role of Statistics & Discrete Mathematics Mathematics Curriculum Improvement Study (SSM- There appears to be agreement that topics in discrete CIS), directed by Howard Fehr at Columbia, developed mathematics and in statistics and probability should a unified program intended for the top 15 to 20 percent be included in the mathematics curriculum for college- of secondary school students. bound students. However, there is lack of agreement on In 1984, after several years of experimentation, the the appropriate place for these subjects in the mathe- New York State Board of Regents adopted an integrated matics curriculum and on the number of hours of study approach as the high school Regents program in mathe- required, especially for students who study calculus in matics with intentions of gradually phasing out the tra- grades 12 or 13. The traditional mathematics cur- ditional program. At the present time, the University riculum from elementary algebra, geometry, interme- of Chicago School Mathematics Project is developing diate algebra, through precalculus is largely a calciilus an applications-oriented curriculum that is integrated preparatory curriculum. Usually these courses do not to some extent although algebra prevails in grade eight include substantial study of discrete mathematics or of and geometry receives the emphasis in grade nine. statistics and probability even though foundational top Despite these moves toward unification, it is still the ics in these areas are often in the back of textbooks used case in the U.S. that most college preparatory mathe- in the courses. matics programs begin with a year of algebra, followed by a year of geometry and another year of algebra. The STATISTICS AND PROBABILITY Second International Mathematics Study Summary Re- Nearly every major committee making recommenda- port for the United States (1985) contends: tions on the high school curriculum has said that famil- It is plausible that the "fragmentation" and "low in- iarity with the basic concepts of statistics and statis- tensity" found in many of our mathematics programs tical reasoning should be a fundamental goal for high could be allayed by a more integrated approach to the school mathematics (new state frameworks in Califor- high school mathematics curriculum. nia, Illinois, Wisconsin, and New York; The College The Report of the MSEB Task Force on Curricu- Board's Academic Preparation in Mathematics; MSEB lum Frameworks for K-12 Mathematics (Draft, October Task Force Draft Report on Curriculum Frameworks CURRICULUM FOR GRADES11-13 95 for K-12 Mathematics; A Nation at Risk from the Na- dents (Travers, et al.), and four bookl.ets published by tional Commission on Excellence in Education). Typi- the ASA-NCTM Joint Committee on the Curriculum cal is the statement in Educating Americans for the 21st in Statistics and Probability. Century (1983) from the National Science Board Com- Arguments against including mo:re statistics are mission on Precollege Education in Mathematics, Sci- raised by educators who feel that students in grades ence and Technology: “Elementary statistics and prob- 11-13 need to spend their time on “basic” mathemat- ability should now be considered fundamental for all ics topics. For example, the integrated New York State high school students.” Similarly, after studying the per- mathematics program is criticized by the IEEE Long formance of 12th grade college preparatory mathemat- Island Section because the introduction of new subjects ics students on the Second International Mathematics such as statistics and probability reduces the time spent Study, the U.S. National Committee recommends that, on basic algebra, geometry, and trigonometry. “The curriculum should be broadened and enriched by including a substantial treatment of topics such as ge- DISCRETEMATHEMATICS ometry, probability, statistics and algebra, as well as Recently, topics in discrete mathematics have been promoting higher-level process goals such as estimation recommended for inclusion in the high school curricu- and problem-solving.” lum in several reports including the Report of The Committees have been making recommendations for MSEB Task Force on Curriculum Fraineworks for K-12 the inclusion of statistics for thirty years. In 1959, the Mathematics (Draft, October 1986). The CBMS (“The Commission on Mathematics of the CEEB in its report, Mathematical Sciences Curriculum K-.12: What Is Still Program for College Preparatory Mathematics, recom- Fundamental and What Is Not”) recommends that dis- mended a one-semester course in probability and statis- crete mathematics now be regarded as “fundamental.” tics as an alternative for grade twelve. The Commission While certain topics (induction, matrices, discrete published an experimental text, Introductory Probabil- probability, and combinatorics) are found at the back ity and Statistical Inference, the same year. In 1975, of many high school textbooks, they are not always the Euclid conference sponsored by the NIE identified taught. Additional topics such as gra.ph theory, differ- probability and organization and interpretation of nu- ence equations, recurrence relations, and game theory merical data as two of ten basic goals for mathematics are also recommended by some. No high school curricu- education. Also in 1975, the Conference Board of the lum has yet been standardized in discrete mathematics. Mathematical Sciences National Advisory Committee Some relatively short units, such as the HiMap mod- on Mathematical Education (NACOME) reported: ules, are now available and finite mathematics texts are While probability instruction seems to have made sometimes adapted for this instruction. some progress, statistics instruction has yet to get off Many colleges and universities now offer a lower divi- the ground . . . . sion course in discrete mathematics particularly suited The situation has not changed much since this NA- to students in computer science. There has been dis- COME report. For example, Bruce Williamson in The cussion of how students should be prepared for such a Statistics Teacher Network Newsletter (1983) reported course. To quote from the preliminary report (1984) of that a study of approximately 350 high schools in Wis- the MAA Panel on Discrete Mathematics in the First consin found that the percentage of schools which allot Two Years: more than three weeks in the total high school program What should be taught in the high schools or on the to statistics declined from 26% in 1975 to 23% in 1983. remedial level in the colleges to prep

2. The course should contain a broad review compo- suggest where attention may need to be focused: nent designed to provide depth missing in most high If this problem is to be corrected, more! attention must school courses. be paid to the development of a student’s reasoning 3. The course should assume a high school calculus ability at all levels, and the instruction should focus experience and build on it. on inquiry and discovery rather than an over-riding emphasis on mastery of a body of rules and techniques. 4. The course, when completed successfully, should Is the following statement true or fahe? How can we provide one semester of beginning calculus credit decide? If true, why is it true? How can we see if the in addition to the credit for the course. statement is a consequence of ideas that have been Colby College has developed a two-semester calcu- discussed before? lus sequence integrating the treatment of one and sev- The emphasis on reasoning shoulld focus on intuitive understanding rather than a development of eral variables that has these characteristics. The course formal logic. The structure of implication, however, is offered for identified students instead of the regular should be stressed. What is the nature of mathemati- three-semester calculus course. cal implications? What is the hypothesis and what is On the other side, many argue that if colleges develop the conclusion of an implication? What is the diffet- special courses for students who take calculus in high ence between a statement and its converse? school but do not master it at a college level, students Traditionally, the geometry course was the first course in which students met the notion of mathemati- will be encouraged to be satisfied with less than full cal proof. Proofs were very formal but, since there was mastery of their high school calculus and high schools never an attempt to extend this form of reasoning to will be encouraged to offer watered-down courses in cal- other mathematics courses, very little in the way of culus in the 12th grade. (The MAA and the NCTM reasoning skills was retained when stmudentscame to have prepared a joint statement for the schools indicat- college. There are many areas where mathematical implication can be stressed: solution of equations and ing that students who take high school calculus should inequalities in algebra as well as certain elementary expect to establish college credit either through the AP ideas from number theory, use of coordinate geom- exam or through a college proficiency exam.) These ed- etry and vector techniques in geometry, mathemati- ucators maintain that alternatives to calculus should be cal induction, and counting arguments and elementary developed in grade 12 for students who are not ready probability in advanced algebra or discrete mathematics. to master calculus, and that it is not a good use of stu- To develop mathematical reasoning skills takes dent time to spend one and a half to two years on the time and should be one of the primary goals of the high content of first-year calculus. school program. Ideas should be developed leisurely; the focus on the program should not be just to make The Place of Deductive Reasoning students ready for calculus in the 12th grade. To ac- complish this development the program needs to be University and college faculty complain that too of- opened up at all levels. In particulstr, the necessity ten students enter college with the view that mathe- for inordinate amount of review present at all lev- matics is just a collection of rules and algorithms to be els should be lessened. Rule bound students are on used to attack a variety of standard problems. They as- a mathematical dead end. The high school program should be able to do more. sert that students’ abilities to reason either deductively or inductively have not been developed by their mathematics courses, and that their ability to attack prob- Curricular Impact of Calculators lems of an unfamiliar nature has not been developed. If The introduction of single variable calculus as the these statements are true, there is need to determine the “desired” 12th grade mathematics course in high school cause. Some argue that the rush to streamline the high is thought by some to have resulted irk a streamlining of school program so that calculus can be taught in the the program prior to calculus. The reriulting lack of sea- 12th grade, jettisoning things deemed unimportant to soning or maturity on the part of many incoming college this goal and omitting end-of-the-text topics, bears pri- freshmen is perhaps the biggest complaint of college fac- mary responsibility. Others claim that the emphasis on ulties. Students are said to know much more than stu- the formal structure of mathematics has stifled the abil- dents of a generation ago as far as calculus-related ideas ity to reason intuitively. Also it is not clear how much are concerned, but geometric intuition, skill with coor- better entering college students a generation ago were dinate geometry, the ability to organize applied prob- at mathematical reasoning. There is general agreement lems and the ability to construct mathematical argu- that strengthening students’ reasoning ability is a goal ments are all too often reported to be lacking. Students in grades 11-13, but little evidence that this is happen- may have an algorithmic facility with mathematics, but ing. The following comments by Phil Curtis, UCLA, many lack a true understanding of thle subject. 98 RESHAPINGCOLLEGE MATHEMATE

It is often expressed that when hand-held calculators The Sloan Foundation has funded projects at seven are generally available at all grade levels the narrowing colleges and universities to consider the potential im- of the curriculum may get worse. This apprehension is pact of computer algebra systems on the teaching of cal- typified by the remark: "If students are to use calcu- culus. The National Science Foundation has announced lators, why should we teach all of this arithmetic (and a major initiative in the area of calculus curriculum that possibly algebra) that can be done so much easier on a can be expected to include some projects where calcu- calculator?" Experience in other countries, e.g., Aus- lators and computers play a central role. tralia, shows that narrowing of the curriculum need not be a serious problem. Alternatives to Remedial Courses Indeed, many argue that an introduction of calcu- An ever increasing fraction of resources is now de- lators can encourage a broadening of the curriculum voted to remedial courses at both the college and high rather than a further narrowing of it. At the elemen- school levels. Algebra and precalculus courses are com- tary level, students can handle numerical data in much mon at colleges for students who previously have been greater amounts and in wider practical situations than exposed to this material, Failure rates of 40-60% are would be possible without calculators. A sense of 'rea- reported in beginning algebra courses at the 10th grade sonableness of solution' to more complicated calcula- level. At both levels, part of the problem seems to tional problems can and should be a goal of instruction. be that the necessary prerequisite material was not There is an increased opportunity to develop a student's learned; and the remedy often is to teach material over mental arithmetic and estimation skills. again in the same way it was taught the first time. Some Rather than decrease students' algebraic skills, these argue that there are alternatives. proponents argue, the ability to confront practical prob- Initially, students who have no prospect of success lems of much greater computational complexity should should not be placed in a beginning algebra course. be an impetus for development of the algebra skills nec- There are two problems here: the need for an effec- essary to organize the problems so they are amenable tive predictor test and the need for middle grade cur- to numerical calculation. All of the calculations involv- ricula to better prepare students for algebra. The Cal- ing compound interest are possible, with the associ- ifornia Mathematics Diagnostic Testing Project is de- ated opportunity to do manipulations with geometric veloping an algebra readiness test. Several curriculum series. Transcendental equations in trigonometry are projects for the middle grades are seeking to strengthen easily solved. This could be a spur to confront more curriculum at that level. A common assumption in meaningful applied problems in trigonometry involving these projects is that imaginative use of hand-held cal- manipulation and solution of trigonometric equations; culators, with the tremendous increase in calculational computations that were impossible in the past. Certain power they give students, should play a central role in topics of course can be dropped; for example, depen- stimulating student interest and providing different per- dence on tables for teaching trigonometry as well as the spectives on the abstract ideas, for example that of a use of logarithms to perform multiplication and division variable, that students will encounter. calculations. There are proposals that more flexible scheduling Curricular materials are under development now on could also ease the remediation problem. These pro- several levels and at several locations that make central posals argue that students who fail the first semester use of calculators and computers. The answers to many of beginning algebra should have the opportunity of re- questions concerning the effect of technology on the cur- peating it in the second semester and not be forced to riculum will emerge as these materials become widely wait until the following year to begin again. However, available. For grades 11-13 the content of intermediate this flexibility is seldom present in school schedules. In algebra, precalculus, discrete mathematics, and calcu- some districts, students who fail an entire year are given lus are all likely to be affected. Ronald Douglas has the opportunity of making the course up in a summer noted in the Introduction to Toward a Lean and Lively session. When a student attempts to learn a year of al- Calculus (MAA, 1987): gebra I, geometry, or algebra I1 in a shortened summer Anyone who has seen hand-held calculators which out- session, there is little chance the student will become put the graph of an equation usually realizes that we proficient with the fundamental ideas of the course. can and, indeed, we will have to change what we ask Some geometry texts attempt to help avoid remedi- students to learn and what we test them on. And this is just a start of developments that include the grow- ation, or excessive review, at the intermediate algebra ing availability of programs for symbolic and algebraic level by providing diagnosis in the geometry course: of methods as well as for numerical methods. what algebra skills have been retained and what have CURRICULUM FOR GRADES11-13 99 not. A renewed emphasis on these topics parallel to mathematics course, rather than providing a route cov- the development of geometry can be used to achieve ering necessary preparatory material. The latter then readiness for the second course in algebra. Coordinate places the student in the course more properly prepared. geometry is a possible framework for renewal work in Other misuses of tests are commonly cited: standard- algebra. ized tests designed for comparative assessment purposes In some colleges and universities, algebra and precal- that “drive the curriculum,” failure to allow for margin culus courses avoid being just a review of high school of error in interpreting scores, excessive use of multiple material by making actual use of computers and calcu- choice tests for classroom assessment. lators and by blending precalculus topics with an intro- The Mathematical Sciences Education Board has de- duction to the notion of limit and other fundamental signed a comprehensive testing study that will survey ideas from calculus. testing practices, assess how test results are used, and the effects of testing on curriculum and teaching behav- Use of standardized Test Scores ior. This study should provide a foundation for future Standardized tests are widely used in high school decisions related to test construction a.nd use. and beginning college mathematics programs throughout the United States. These tests range from achieve- The Form Geometry Should Taker ment examinations, such as the mathematics achieve- There is general agreement that many entering col- ment tests of the College Board and the New York State lege students lack geometric intuition and the ability Regents Examinations, to mathematics aptitude tests to visualize geometric situations. This would seem to such as the SAT, and various assessment instruments indicate that a strengthening of the geometric content designed at the state level and often used at the 12th of the high school program is sorely needed. But there grade and other levels. In addition there are diagnostic is not a consensus as to how this should be done. Nor instruments designed to assess readiness for the next is there agreement on the place of such topics as trans- level of the program and placement examinations, such formations and vectors or the amount of emphasis that as those available from the MAA, often used at the col- should be placed on formal proof or the desirability of lege freshman level and in high school. introducing students to formal logic within the geome- These tests are often criticized on various grounds. try course. First, since they are usually multiple choice exams, they What ought to be taught in high school geometry are criticized for supposedly not testing higher order has been debated for at least a half century. In an thinking or problem solving skills. Secondly, since the address to the 1958 annual meeting of the NCTM in general assessment exams are used to compare schools Cleveland (published in the Mathematics Teacher as and programs, there can be considerable political pres- “The Nature and Content of Geometry in the High sure “to get those scores up,” with the result that there Schools”), Julius Hlavaty referred to the “continuing can be a narrow concentration by the teacher on just crisis in the teaching of geometry” that had endured for those basic skills covered in the examination. “fully 25 years.” Titles of other articles in the Math- Also, tests designed for one purpose are often used ematics Teacher in the last two decades substantiate for another. Diagnostic examinations are used as assess- the existence of an ongoing debate (Adler, “What Shall ment instruments to compare the performance of classes We Teach in High School Geometry?” March 1968; Al- and schools. When this is done the pressure to narrow lendoerfer, “The Dilemma in Geometry,” March 1969; the curriculum to just those basic skills necessary for Fehr, Eccles, and Meserve, “The Forum: What Should success in the next course can become great. On the Become of the High School Geometry Course?” Febru- other hand if this comparative pressure is removed, di- ary 1972). The introduction to the 1973 NCTM Year- agnostic tests can be quite effective in indicating areas book Geometry in the Mathematics Curriculum is enti- of the curriculum which are not being retained by the tled “Disparities in Viewing Geometry.” The book con- student but which are absolutely necessary for success tains chapters discussing several different approaches to at the next level of the program. Strategies for dealing high school geometry: conventional, coordinate, trans- with these deficiencies can then be constructed which formation, affine, vector, and an eclectic approach. In should result in a broadening of the curriculum rather the latest yearbook of the NCTM, Learning and Teach- than a narrowing of it to a concentration on just basic ing Geometry K-12 (1987), Usiskin writes on “Resolv- skills. ing the Continuing Dilemmas in School Geometry,” and Placement exams can be misused if they are used as Niven on “Can Geometry Survive in the Secondary Cur- a barrier which prevents students from taking a given riculum?” Usiskin offers suggestions for resolving the 100 RESHAPINGCOLLEGE MATHEMATICS dilemmas, and Niven proposes recommendations that counselors; the home, primarily the parents; the mathe- he thinks will make geometry a more attractive sub- matics professional community, primarily the classroom ject. Other chapters in the Yearbook discuss various teachers; and of course, the students. Collaboration geometric topics and applications. within the schools needs to be linked also to colleges and The proliferation of computers, the growing popu- universities. In these institutions, in addition to mathe- larity of Logo, and the development of software such as matics faculty members, admissions personnel play key “Geometric Supposer” are adding new elements to the roles. Some prototype regional programs for collabora- debate on geometry. Can they perhaps help to resolve tion include the Bay Area Mathematics Project, and the issues that have been around for quite some time? Mathematics and Science Education Network of North Carolina. The more recently formed American Math- ematics Project is chaired by R.O. Wells, Jr., of Rice Collaborative Efforts University. It aims at encouraging and extending lo- In recent years there have been many studies and re- cal cooperative efforts involving elementary, junior high ports released calling for significant changes and reforms school, and high school teachers, college and university in American education. While there has been consider- faculty, and professionals in industries. able consensus in the identification of the ills within this In the paragraphs which follow, several issues and system, there has not been universal agreement as to questions relative to each of the essential groups will be what the solutions should be in order to remediate the addressed to provide evidence for a need of such part- identified problems and shortcomings. Such has been nerships. Through the raising and addressing of such the situation with issues involving education in mathe- issues it is apparent that educators and professional or- matics, particularly those related to curriculum. ganizations will need to take advantage of many oppor- Issues related to curriculum have occupied an im- tunities in collaborative efforts in order to affect cur- portant position in many of the released studies and ricular changes. It is also evident that curricular issues reports dealing with education in mathematics in the in and of themselves cannot be considered apart from United States. In particular, discussions and recom- factors which will have a significant effect upon any cur- mendations pertaining to the mathematics curriculum ricular proposals or modifications. of grades 11-13 have been prominent in many of the There are numerous issues related to the leadership published reports and studies. Some of the consensus within the schools today, and there is research related to and lack of consensus items pertaining to the mathe- the role of leadership, particularly the role of principals, matics curriculum have been identified and dealt with within successful learning environments. Administra- in other portions of this report. While a listing of these tors, as agents of Boards of Education, need to become items is the major thrust in this report, it is necessary to partners with mathematics educators in dealing with identify briefly several other considerations which per- curriculum-impacting issues such as the following: tain to the total mathematics experience for 11-13 grade 1. The recruitment and retention of qualified staff students. members. Many of the recent reports have cited the success of 2. The development and maintenance of educational mathematics students in other countries, particularly settings and environments which promote quality those from Japan. The reports emphasize that the suc- instruction, effective learning, and the maintenance cess of the students is due in part to the collaborative of academic standards. efforts of the school and family, and due to the empha- 3. The providing of support and encouragement for sis, status, and importance accorded to learning and professional growth, and the upgrading of content education. Enhanced success in mathematics educa- competencies and professional teaching skills. tion is dependent on extensive collaborative efforts in- 4. The providing of support which enables the oppor- volving several subgroups. We have recommended that tunities within schools for curricular change. targeted brochures and flyers be developed and circu- 5. The providing of settings which enable participation lated to these groups. We wish here to discuss further in leadership by those having expertise related to the various roles these groups must play in the desired education in mathematics. collaboration and also the essential role of professional 6. The involvement of administrators as advocates for organizations in creating a climate for change. appropriate changes in the mathematics curriculum. Extensive partnership and collaborative efforts A second area where collaborative efforts need to be should involve school leadership, primarily the prin- fostered involves the school guidance personnel. Ef- cipal; public guidance personnel, primarily guidance forts are needed to provide aid and support to coun- CURRICULUM FOR GRADES11-13 101 selors who have significant impact upon students. (In 5. Parents themselves need to be educated about the March, 1987 the NCTM made several recommendations learning of mathematics. To many adults the study concerning the advising of students in mathematics in of mathematics is thought to be the performance the statement, “Counseling Students in Planning Their of arithmetic computations. It needs to be stressed Mathematics Programs.”) Often, guidance counselors that the learning of mathematics is a continuous, do not have, nor can they be expected to have, a com- sequential process to be pursued over a period of prehensive understanding of the content or importance time. Parents need to understand toheneed for reg- of the 11-13 grade mathematics curriculum. Mathemat- ular practice in the learning of mathematics. ics teachers and guidance counselors need to address the Another component in the collaborartive partnership following issues together: is the teacher. Until the teacher becomes convinced 1. How should students be placed in appropriate that there is a need to improve the 11-1.3 grade curricu- courses? lum, advances in the quality of 11-13 grade mathemat- 2. Which mathematics courses do students need in ics programs will be limited. Professional organizations preparation for other mathematics courses, careers, will have significant opportunities to provide leadership, and future academic work? encouragement, and training to teachers. Significant 3. What communication with students and parents is thought and attention will be requiredl to bring teach- necessary in order to direct students into appropri- ers into the process of affecting change in mathematics. ate mathematics courses? Students need to be involved fully in the educational 4. What support should counselors be able to provide process. They need to be made aware that an ap- to encourage students in their studies of mathemat- propriate mathematics background is a necessity and ics? that mathematics is a “hands-on activity” and a “do- 5. What support should counselors be able to provide it-yourself’ activity. The teacher can be an aid or fa- to students who are struggling in their mathematics cilitator of learning, but cannot do the learning for the studies? students. In the era of instant gratificartion it is impor- 6. What efforts are needed to encourage all groups of tant that students realize that solving a mathematical students, regardless of gender or race, to take more problem will not always be quick nor will it be easy; in mathematics courses? fact, not every problem will have a solution. Students Many of the recently released reports attribute the need to develop persistence and to realize that progress success of students in mathematics to the attitudes and will often seem slow, especially in the reading of math- the values developed in the home. Parental influence ematics. Students also need to understand that it is has been shown to have a significant impact upon the important to keep up with their assignments and that successes of the child in school. There are numerous class attendance is essential for succesis in the study of issues and questions which need be addressed in de- mathematics. veloping stronger partnerships involving parents. (In In the preceding paragraphs several subgroups have Spring 1987 the National PTA sent a special mailing been identified as being necessary components for effec- to the 25,000 local PTA presidents describing compar- tive partnerships in addressing the issues and questions ative data from international studies in mathematics associated with the 11-13 grade mathematics curricu- and calling on parents to be more involved in finding lum and instructional program. The primary responsi- solutions to problems in mathematics education in the bility of this task force study has been to focus upon schools.) Items for consideration include the following: the areas of consensus or lack of connensus as identi- 1. Parents need to provide an appropriate environment fied in prior reports and studies. Nevertheless, it seems which encourages home study and the completion of appropriate to address some of those issues which ul- homework. timately will determine much of the success or failure 2. Educators and parents need to work together to establish attitudes and priorities where learning be- related to those curricular items for which there is con- comes more valued. sensus. There needs to be collaboration of school lead- 3. There needs to be a support system within which ers, counselors, parents, teachers, and students-and class attendance is a high priority. linking with college and university personnel-in order 4. Parents need to become better informed about the to improve mathematics programs for the 11-13 grade need for appropriate mathematics backgrounds for student. their children. It is not enough that students take This Task Force has recommended that the NCTM mathematics courses. They should take mathemat- and MAA prepare brochures and flyer:%targeted specif- ics courses appropriate for their career goals. ically at the various groups described in this section. 102 RESHAPINGCOLLEGE MATHEMATICS

There are additional roles that the professional organi- Task Force Members zations will need to play if the collaborative efforts of these groups are to succeed. We urge the organizations Executive Committee to continue to enable greater communication among the various groups in these ways: JOANR. LEITZEL,CHAIR, The Ohio State University. * To increase and expand the design of workshops and PHILIPC. CURTIS,University of California, Los Ange- programs for policy makers, high school principals, les. and post high school level administrators in order CHARLESHAMBERG , Illinois Mathematics and Science to address effectively the issues relating to the re- Academy. cruitment and retention of qualified instructors and DONOVANR. LICHTENBERG,University of South teachers of mathematics. Florida. * To expand efforts to work at national, state, and ANNWATKINS, Los Angeles Pierce College. local levels with and through the affiliated groups on issues related to the recruitment and retention Reading Committee of qualified mathematics teachers and instructors. 0 To expand their roles in working with affiliated and BETTYEANNE CASE, Florida State University. non-affiliated groups in providing forums to address DON CHAMBERS,Wisconsin State Department of Ed- such issues a8 curriculum change, uniform standards ucation. of quality instruction, equity of opportunity, and DWIGHTCOBLENTZ, San Diego, California. expectations in mathematics education. STEVECONRAD, Roslyn High School, New York. To focus further on ways of being more effective in RONALDG. DOUGLAS,State University of New York, communicating the solutions to identified problems Stony Brook. beyond their membership in order to provide the ap propriate and necessary impact upon policy makers, WADEELLIS, JR., West Valley College. administrators, counselors, teachers, parents, and DALEEWEN, Parkland College. students. JEROMEGOLDSTEIN, Tulane University. To undertake actions and assume a significant role KATHLEENHEID, Pennsylvania State University. in developing programs where articulation with DAVIDJOHNSON, Nicolet High School, Wisconsin. guidance counselors becomes a major priority. KATHERINEP. LAYTON,Beverly Hills High School. To work with school administrators, counselors, and PETERLINDSTROM, North Lake College. teachers in establishing programs which will en- WILLIAMLUCAS Claremont Graduate School. hance and increase parental involvement and sup , port in the educational process. IVANNIVEN, University of Oregon. To develop strategies that will involve the students DONALDSMALL, Colby College. more deeply in the educational process. THOMASTUCKER, Colgate University. Minimal Mathematical Competencies for College Graduates

This chapter contains the report of the CUPM Panel idea that seems to have been carried into action in very on “Minimal Mathematical Competencies for College few places. Gradutes, ” reprinted from the AMERICANMATHEMAT- One would like to think that need for the “remedial” ICAL MONTHLY,89 (April 1982) 266-272. Donald course sketched in the report has declined, or will soon Bushaw, chair of the panel, has prepared a new preface decline, because of nation-wide attention to weaknesses relating issues addressed by the panel to many themes in the precollege mathematics curriculum. In any case, that are part of today’s debates about higher education. the course itself should still be useful for whatever remains of the clientele for which it was intended. Courses in mathematics appreciat#ionmeeting the 1989 Preface standards implied in the report are probably still rare, On Thursday, December 15, 1977, the Carnegie although courses of similar intent are not uncommon. Foundation for the Advancement of Teaching released If a survey of persons from the Combined Member- its famous report “Missions of the College Curriculum.” ship List were redone today, the responses might show This report, which received a great deal of attention at more interest in discrete mathematics, and might show the time, described general education in U.S. colleges effects of the rapid progress in the design and dissemi- and universities as “a disaster area,” and expressed spe- nation of calculators and microcomputers in the inter- cial concern about the neglect of mathematics and En- vening years; but the responses given almost ten years glish composition. ago tended to be conservative, and a new round of re- The following Monday, Henry L. Alder, then Presi- sponses would probably tend to be conservative too. dent of the MAA, wrote a letter challenging the MAA’s Thus the report, though neither iradical nor volu- Committee on the Undergraduate Program in Mathe- minous, presents some worthwhile ideas that are still matics (CUPM) to take up the matter, suggesting as far from commonplace, and which, if widely adopted, one possibility the formation of a “new CUPM panel could contribute significantly to the mathematical com- or subcommittee” to “consider the problem of general petence and maturity of coming generations. education in mathematics for all or most college students.” Donald W. Bushaw At the CUPM meeting of January 8, 1978, Chairman Washington State University William F. Lucas appointed a subcommittee (“panel” March, 198!3 in the then current nomenclature) to do just that. Af- ter a considerable amount of study and discussion, and several diverse surveys, the panel presented its brief and Introduction temperate report to CUPM, which approved it. Too many people know too little mdecimal, a mathematics, and for them those opportunities did not one-word memory, and very little else. A device for work. Therefore, the college remedial course should not projecting the face of the instructor's calculator on a be a mere rehash, and certainly not an accelerated one, screen would be useful. There should also be a large of the traditional secondary or even elementary course. collection of advertisements, newspaper and magazine 106 RESHAPINGCOLLEGE MATHEMATICS articles, sales and credit agreements, and so on, the in- Mat hematics Appreciation terpretation or use of which would require some of the topics listed below. These might be complemented by While the panel does not insist that a knowledge reasonable imaginary examples, but the illustration of of the cultural side of mathematics should be required no topic should depend entirely on artificial applica- of all college students, its Recommendations D and E tions. If no genuine examples can be found, why should above suggest that attractive and accessible courses the topic be included? In some topics, however, a step dealing especially with that aspect should be offered. should be taken beyond the evidently practical. This section of the report contains some reasons for this Students should be supplied with a single page of position and some comments on how it might be real- formulas, sufficient for the whole course. ized. The grading policy should be compassionate but Mathematics has played a central role in the devel- firm. Tests should be frequent and repeatable at least opment of modern civilization. It has been essential not once. They should be straightforward, but only high only to the growth of science and technology, but has scores should be considered passing. Mastery should had profound effects on philosophy and other forms of be recognized irrespective of the number of attempts thought as well. needed to show it, within limits, but outstanding per- There was certainly no doubt in past centuries that formance should be recognized. If possible, permanent every college graduate, to be an educated person, had to records of students who need to repeat the course should know some mathematics. In medieval times, for exam- not show the unsuccessful tries. ple, four of the seven traditional liberal arts were largely One list of topics for such a course is given below. or wholly mathematical. The importance attached to Additions and modifications should be made in response mathematics was evident in courses of study in the nine- to real-world needs and to experience in offering the teenth century, and this carried over into the twentieth. course. Now, however, it is possible to graduate from many col- 1. Positive decimals; conversion of fractions to deci- leges without any contact with mathematics beyond the mals with the calculator. most elementary high-school courses. 2. Pencil-and-paper arithmetic with signed whole While high-school mathematics is important, it does numbers. tend to emphasize development of skills. The same, un- 3. Pencil- and-paper arithmetic with signed fractions. fortunately, may be said of most college courses whose (There should be no three-or-more digit numerators mission is primarily remedial or preprofessional. But an or denominators, except powers of ten.) educated, well-informed person should know something 4. Calculator arithmetic with signed decimals. about mathematics beyond skills. 5. Rounding off. To many, the distinction between mathematicians 6. Estimation; orders of magnitude. and accountants is not clear. People who are alert and 7. Scientific notation. informed about many things, even colleagues in a uni- 8. Units of measurement; elements of the metric sys- versity, sometimes assume that mathematicians are con- tem. stantly doing arithmetic and are surprised to hear that 9. Percent, there is such a thing as mathematical research. Their 10. What is a formula? What is a function? experiences with school mathematics left them with the 11. Times, distance, and rates. impression that mathematics is ancient and immutable, 12. Area and volume. and consists of rules and formulas for unfortunate school 13. What is an algorithm? Flowcharting. children to memorize. 14. Statistics and its dangers. The great mathematicians do not occupy their right- 15. When is an argument correct? ful place in the public consciousness. In his New Yorker 16. Compound interest. article on mathematics (February 19, 1972), Alfred 17. Exponential change. Adler rightly observed that This list should not give rise to hideous visions of . . .it would be astonishing if the reader could identify workbooks filled with drill exercises. Games, problems more than two of the following names: Gauss, Cauchy, of obvious everyday interest, opportunities for creativ- Eulery Iiiemann. It be ishing if he should be unfamiliar with the names of ity, and occasional attention to general problem-solving Mann, Stravinsky, de Kooning, Pasteur, John Dewey. strategies should contribute to a cheerful and progres- The point is not that the first five are the math- sive atmosphere and a positive experience. ematical equivalents of the second five. They are MINIMALMATHEMATICAL COMPETENCIES 107

not. They are the mathematical equivalents of Tol- prove the obvious can convince people that the whole stoy, Beethoven, Rembrandt, Darwin, Freud. The ge- endeavor is silly. ometry of relativity-the work of Riemann-has had Applications are appealing to many students and consequences as profound as psychoanalysis has . . . . should be included. There are convenient sources of Many college graduates know a great deal of mathe- authentic applications of mathematics at every level matics; most of them have had to take mathematics in of difficulty. Applications, however, should not be al- preparation for their work. But how many of these, or lowed to upstage the real star of the show, mathematical how many mathematics majors, for that matter, could thought itself. Calculators and computing might have tell much about Abel or Jacobi? More important, how their place in the course, and some time could profitably many of them could comment plausibly on the relation be spent on the place of computers in modern society. of mathematics to other disciplines? Serious study of computer science, however, is probably The point here is not that mathematics and mathe- best left to other courses. maticians should be glorified but that a reasonable per- The course should give students copious evidence spective on the place of mathematics in the human en- that mathematics has not only played1 a great part in terprise should be more widely shared. human history, but continues to thrive in the service A course designed specifically to improve this per- of other fields and as an independent source of intel- spective would ideally give some idea of what sorts of lectual excitement and aesthetic appeal. Mathemati- problems mathematicians consider and how such prob- cal “current events,” such as the solution of the four lems are attacked. The object would be to promote color problem and the discovery of new large primes mathematical literacy, interpreted to include an aware- should be mentioned. Something might be said about ness among future colleagues in colleges and univer- Hilbert’s problems and the Fields medals. Carefully se- sities, in business, in industry, in government, and in lected readings from Scientific American, The Mathe- many other callings of what mathematics is, why it is matical Intelligencer, and similar publications can help. important, and how it might serve them. Some history The choice of faculty for an appreciation course is should be covered along the way, but a straight course critical. It is an extraordinary teaching assistant who in the history of mathematics is not recommended for would have the experience and breadth of outlook to this purpose; it can have meaning only if the students teach such a course. It should usually be taught by se- already have some understanding of the mathematical nior faculty, and if appropriate faculty cannot be found, ideas whose development is traced. the course should not be taught at all. And it is better The course could include, for example, a discussion of that it be taught by the right faculty in larger sections the Euler formula for polyhedra-and the names of Eu- than by reluctant or inept instructors in small ones. ler, Descartes, and Cauchy already would have entered The course mentioned in Recommendation E offers the discussion. An account of non-Euclidean geome- further opportunities. It is still too easy for mathe- try would be appropriate, and provide an occasion for matics and science majors to complete their programs introducing Gauss and Riemann as well as Bolyai and without knowing that research is done in mathematics, Lobachevski, and for commenting on the element of ar- that mathematics has deep and productive relationships bitrariness in mathematical modeling of reality. Neither with many fields, and that mathematics has a rich and of these topics requires any high level of algebraic skill. fascinating history. A mathematics aplpreciation course A discussion of the insolubility of the quintic equation for students with good technical proficiency in mathe- might involve more algebra but would refer to the work matics can do much to take care of this and be a mem- of Lagrange, Galois, and Abel--and the important idea orable experience for all concerned. of mathematical impossibility would have arisen. There As has already been said in Recommendation D, are many other topics that bring up important mathe- these observations about separate mathematics appreci- matical ideas and events but do not require much back- ation courses should apply, to some eztent, to all math- ground. ematics instruction, even remedial. In a perfect world Axiomatics, though obviously important, should not every mathematics course would be a mathematics ap- be overemphasized. Axiomatic systems should not be preciation course. The world, however, is not perfect. presented in detail unless one obtains by their use some interesting results that were not intuitively obvious from the start. Elementary graph theory offers Appendix some nice opportunities here, as well as a great variety The panel began by consulting the pertinent litera- of easily understood applications. Laborious efforts to ture; officers of organizations represented in the Council 108 RESHAPINGCOLLEGE MATHEMATICS of Scientific Society Presidents or the Conference Board pressed.) of Mathematical Sciences, and a sample of mathemati- All respondents, academic or not, were asked to mark cians drawn at random from the 1978-1979 Combined in a forty-item list of mathematical topics those they Membership List. Summaries of the results may be ob- thought should be required of all college graduates. The tained from the chairman of the panel. following topics were marked by at least half of the re- A general announcement and appeal for information spondents: and ideas also appeared in Notices of the American Basic arithmetic skills (94.6%) Mathematical Society, Change Magazine, The Mathe- Area and volume of common figures (76.4%) matics Teacher, The Chronicle of Higher Education, Linear equations (71.3%) The Two- Year College Mathematics Journal, SIAM 0 Algebraic manipulations (63%) News, and The American Mathematical Monthly. Elementary statistics (55.5%) From the first two surveys mentioned, the panel Graphing of elementary functions (54.9%) learned not much more than that no national organi- Integer and fractional exponents (54.3%) zation in this country, the MAA itself not excepted, 0 Elementary plane geometry (51.9%) has ever taken a position on what college graduates in Next in order were: elementary probability (49%), general should know of mathematics. general problem-solving skills (heuristic) (49%), quad- The survey based on the Combined Membership List ratic equations (47.5%), mathematics in business (CML) and the appeal in periodicals, though more pro- (46.9%), and radicals (43.9%). Computer program- ductive, did not provide as much unambiguous guidance ming was marked by 33.l%, just after elementary logic as the panel had hoped to get. The CML survey yielded (35.5%) and systems of equations (35.2%). 335 usable responses from a thousand questionnaires. The question about what standard courses should 226 were from persons at colleges and universities. Of be required elicited a wide variety of answers, many of these, 105 (39.5%) were from institutions where a math- which were in fact far from standard. College algebra ematics requirement for graduation was in force. These (mentioned by 51 respondents) led the list, and was 105 respondents were asked about the nature of the re- followed by probability and statistics (47), calculus (45), quirement, whether they favored it, and whether they elementary or intermediate algebra (44), and computer thought it was effective. In the great majority of cases programming or appreciation (30). (91 or 86.7%) the requirement could be satisfied by one About 45% of the respondents accepted an invita- or more courses. Seven of these respondents reported tion to comment further. Many merely expanded on that the requirement could be satisfied by examination; earlier answers, but some submitted careful statements five others said both courses and an examination were of their views. These statements, though not easy to required. summarize, were carefully studied by the panel. One hundred (95.2%) of the 105 said they favored Responses to the appeal in periodicals were inter- the requirement, and 75 (71.4%) said they thought it esting too, but they are even less reducible to a brief was at least partially effective. summary. The median course requirement, where one existed, The panel met three times and also conducted a vo- was between 3 and 4 semester hours. A specific course luminous correspondence within itself and with others. or sequence of courses was seldom required; indeed, ac- It completes this report with high respect for the com- ceptable courses were remarkably diverse. plexity of the problem, but hopes that its proposals will The 161 respondents in colleges and universities be of some use in finding solutions. which had no general mathematics requirement were asked whether they favored such a requirement. In re- ply, 148 expressed a preference, and of these 104 (70.3%) Panel Members favored some kind of a requirement. When the two groups are combined, one finds that DONALDW. BUSHAW,CHAIR, Washington State Uni- 204 of 253 (80.6%) of those college- or university- versity. affiliated mathematicians in the sample who expressed GERALDL. ALEXANDERSON,University of Santa any preference favored some general graduation require- Clara. ment in mathematics. The panel did not expect this ROBERTJ. BUMCROT,Hofstra University. fraction to be so high. (Unfortunately, the question- JUANITA J. PETERSON,Laney College. naire did not ask for reasons for the preference ex- EDWINH. SPANIER,University of California, Berkeley. Mathematics Appreciation Courses

This chapter contains the report of the CUPM Panel molding nonscientists’ opinions of mathematics and its on Mathematics Appreciation Courses. The Panel’s re- role in society, CUPM decided that it should call atten- port was originally published in two parts in the AMER- tion to the importance of these courses and offer some ICAN MATHEMATICALMONTHLY, Vol. 90 (1983): the suggestions on how they may be organized and taught tezt appeared on pp. 44-51, while the references ap- effectively. peared in the Center Section. For this reprinting’ the This is the report, approved by CUPM, of the CUPM report has been re-edited to include the references in the Panel on Mathematics Appreciation Courses. While the tezt of the report. Jerome Goldstein, Chair of the Panel, panel has many guidelines and recommendations to of- has prepared a brief new preface for this reprinting. fer, it does not feel that a particular selection of topics or teaching strategy should be universally adopted for 1989 Preface mathematics appreciation courses. A imain goal of such Mathematicians generally share the view that all courses is to get students to appreciate the significant well-educated people should be mathematically liter- role that mathematics plays in society, both past and ate. As a result, “mathematics appreciation courses” present. All material presented in such courses should continue to be offered with regularity to students in be well motivated and related to the role of mathemat- the fine arts, in the humanities, in some social sciences, ics in culture and technology. and in education. All students who receive college-level training in mathematics deserve to have well-conceived Philosophy courses, centered around significant mathematics. In particular, students should get a glimpse of what it is The inclusion of a mathematics appreciation course that attracts mathematicians to their subject. in the undergraduate curriculum is calmmon in the na- The CUPM Panel on Mathematics Appreciation tion’s colleges and universities. This trend is a direct Courses emphasized the course objectives rather than result of an underlying belief, held by most mathe- the intended audience, and stressed philosophy and maticians, that every well-educated person should be teaching strategies rather than specific content. In fact, mathematically literate. Whether or not a mathemat- the comments in the Panel’s report, with their heavy ics course is required at a particular institution often emphasis on attitudes and teaching strategies, have uni- depends, among other things, upon the extent to which versal appeal and can be read with profit by all college this belief is shared by the general .faculty. But the mathematics teachers. ultimate success of an “appreciation” course in mathe- The report is as timely now as when it was written. matics should not depend upon mandatory enrollment. Rather, the value and importance of such a course JEROMEA. GOLDSTEIN should be directly attributable to the: care and under- Tulane University standing with which it is conceived and taught. March, 1989 If as mathematicians we accept the notion that an educated person should know something about mathemat- Introduction ics, then we must also accept the responsibility for con- In 1977 the Committee on the Undergraduate Pro- scientiously providing appropriate tr

live and work. Examples of these influences abound, library research because they have become accus- and should form a major part of any mathematics tomed to these in their humanities courses. There appreciation course. Historical developments and is much good mathematics that can be learned in the evolution of mathematical concepts should be this way, and assignments can be arranged that uti- properly emphasized. lize these familiar learning tools. 3. The nature of contemporary mathematics. The mathematics known by most humanities students Things to Avoid is ancient mathematics-the geometry of ancient Greece, and the algebra of the early renaissance; not 1. Do not leave the assignment of an instructor in surprisingly, such students have the impression that the mathematics appreciation course until the last mathematics is dead. Showing them that it is in minute, and do not assign it on the sole basis of fact a vigorous, growing discipline with considerable availability. The course requires more planning influence in contemporary society is an important and preparation than almost any other mathematics aspect of any course in mathematics appreciation. course if it is to be successful. 4. The recent emergence of several mathematical sci- 2. Do not simply allow the students to sit back and ences. While the mathematics appreciation course listen. It is important that they be involved actively. should not be devoted solely to one Umodernllarea But this need not take the form of daily homework. such as statistics, computer science, or operations In fact, drill type assignments should be avoided. research, it surely provides an opportunity to use The involvement could take the fiorm of projects, these fields as illustrations of the panoply of con- papers, book reports, “discovering” mathematics in temporary mathematical science. class, participating in class discussions. 5. The necessity of doing mathematics to learn math- 3. Do not over-emphasize the history of mathematics. ematics. While some parts of the mathematics ap- While the history of mathematics could and should preciation course can and should be about mathe- be used to enliven the topics covered, a student who matics, it is essential that some parts actually en- knows (and cares) nothing about a mathematical gage the students in doing mathematics. Only in topic is not likely to be interested in its history. this way can they gain a realistic sense of the pro- 4. Do not stress remedial topics. While many of the cess and nature of mathematics. Of course it is vi- students in a mathematics appreciation course may tally important that the instructor have appropriate need remedial work, any such material that is cov- respect for the students’ interest and abilities, and ered must be presented as part of a topic that fits that exercises be selected so as to maintain rather into the scope of the course as a whole. than destroy their enthusiasm. 5. Do not make a fetish of rigor; in particular do not 6. The role of mathematics as a tool for problem solv- prove things that are self-evident to the students. ing. As the language of science and industry, mathe- For example, a rigorous presentation of the real matical models are the tool par ezcellence for solving numbers in which one proves the uniqueness of zero problems. Students in mathematics appreciation is entirely inappropriate in courses of this type. courses should be exposed to contemporary mathe- 6. Do not cover topics you do not yourself find interest- matical modelling, to gain some appreciation both ing and important. It is hard to fool these students, of its power and its limitations. and if the teacher does not care, ithey will not see 7. The verbalization and reasoning necessary to under- why they should. stand symbolism. While symbols provide the math- 7. Do not be condescending. While the students in ematician and scientist with great power, they ob- such courses may not be mathematically inclined, scure the meaning of mathematics from the uniniti- this does not mean that they asre unintelligent. ated. A great service the teacher of a mathematics Many who take mathematics appreciation courses appreciation course can provide is to enable students are outstanding, creative students who have sim- to overcome their fear of symbols, to learn to think ply concentrated in the nonquantitative areas of the through arguments apart from the traditional sym- curriculum. The attitude of the teacher can help ei- bols in which they are expressed. ther to open or to close their minda to the material. 8. The ezistence of a large body of interesting writing 8. Do not cover topics which you cannot relate in some about mathematics. Students in mathematics ap- way to ideas familiar to the studeints. Clock arith- preciation courses generally feel comfortable with metic and symbolic logic, for example, are of little assignments such as term papers, book reports, and value to mathematics appreciation courses unless 112 RESHAPINGCOLLEGE MATHEMATICS

you can find applications the students can appre- Examples of Topics ciate and understand. 9. Do not make the course too easy. The material The topics available for courses in mathematics ap should not be way over the heads of the students, preciation are as diverse as mathematical science itself. but it should not be trivial either. Standard textbooks offer a rather traditional assort- 10. Do not accept anyone else’s blueprint for a mathe- ment of topics: probability, graph theory, finite differ- maticians appreciation course. If you can communi- ence equations, computers, matrices, statistics, expo- cate, in your own way, why you believe that math- nential growth, set theory, and logic seem to dominate. ematics is beautiful and important, the course will But there are numerous other themes that can be used fulfill its purpose. for large or small components of courses. Here are a few of the many possible examples. 1. Understanding how to use the buttons on a pocket Course Organization calculator. It used to be that the number e was a complete mystery to those who had not studied cal- There are nearly as many ways to teach a course in culus, and that %in” had for humanities students mathematics appreciation as there are teachers of these more the connotation of theology than of mathe- courses. While some strategies will work superbly in matics. But no more. Virtually everyone has, or some contexts, none can be recommended for all; the has seen, inexpensive hand calculators with but- teacher’s enthusiasm for what is being done as well as tons that perform operations involving exponen- the appropriateness of the strategy for the students in tial, trigonometric, and basic statistical functions. the course is generally more important than the actual Teaching a class what these buttons do is an ex- strategy adopted. Nevertheless, to encourage flexibil- citing new way to explore some traditional parts of ity, we list below some approaches to teaching mathe- classical mathematics. matics appreciation that have been effective in certain REFERENCES:See the handbooks for various cal- contexts. culators.._ A sampler approach, featuring a variety of more Tracing the modern descendants of classical mathe- or less independent topics. The advantage of this matical ideas can illustrate the power of mathemat- method is that it covers many areas without requir- ics to influence the real world, as well as its remote- ing a sustained continuity of interest; students who ness from it. For example, classical Greek geome- fall behind or simply fail to comprehend one topic try involving conic sections led to models for plan- always know that they have a chance for a fresh etary motion, and ultimately to the possibility of start in a few days. The disadvantage is that of all space flight. And probability, which had its origins survey courses: not enough time spent on any one in seventeenth-century discussions about gambling, thing to ensure long-term learning. now dominates actuarial and fiscal policy, influenc- A single-thread approach, built around a common ing government and corporate budgets, thus affect- theme, for example, 2 x 2 matrices, or algorithms, ing the level of interest, of unemployment, and the or patterns of symmetry. Doing this takes care- health of the entire economy. ful planning, and runs the risk of alienating some REFERENCES: Much of this is in standard text- of the class who find the thread incomprehensible. books. Morris Kline’s Mathematics in Wedern Cul- But it guarantees a solid example of the intellectual ture and George P6lya’s Mathematical Models in coherence that is so much a part of contemporary Science are helpful sources. mathematics, that ideas arising in one context find applications in others, and that a common abstract Connecting mathematics with Nobel prizes. No- structure underlies them all. bel prizes are not given in mathematics (and the apocryphal reasons for this are quite amusing). But A Socratic approach, in which the instructor works the work that led to Nobel prizes (e.g., of Libby, carefully to let the students develop their own reasoning. This works well in small classes with a of Arrow, of Lederberg, and others) often has an intrinsically-mathematical basis. The study of this highly-motivated instructor. While the content of scientific work provides an opportunity to show how such courses is hard to guarantee in advance, the mathematics is important in the most profound dis- achievement for students who are able to think for coveries of modern science. themselves, perceiving patterns where others simply see chaos, is a worthy objective for a course in REFERENCES: Libby’s work is briefly discussed mathematics appreciation. in several elementary texts on ordinary differential MATHEMATICSAPPRECIATION COURSES 113

equations, e.g., in Differential Equations with Ap- offshoot theories of voting, the recently discovered plications and Historical Notes by George F. Sim- problems associated with apportionment of legisla- mons. For some work of Arrow see Edward Bender, tures, and strategies of fair voting in multiple can- An Introduction to Mathematical Modeling, Wiley, didate elections. New York, 1978. Lederberg published an article in REFERENCES: See Bender, An Introduction to this Monthly entitled "Hamilton circuits of convex Mathematical Modeling, Wiley, New York, 1978; the trivalent polyhedra (up to 18 vertices)" in Vol. 74, articles by William Lucas in Vol. 2 of the forth- pp. 522-527. coming Modules in Applied Mathematics (Springer- 4. Applying exponential growth models. The applica- Verlag, New York); M. Balinski aind H.P. Young, tions of traditional topics from elementary mathe- Proc. Nat. Acad. Sci. U.S.A. 77 (January 1980) matics can often be explored more fully than has 1-4; H. Hamburger, J. Math. Sociology 3 (1973) usually been the case. Exponential growth and de- 27-48; David Gale, UMAP Module 317, 1978; W. cay models provide a striking example. Simple non- Stromquist, this Monthly 87 (1980) 640-644; and calculus approaches to models of growth provide a George Minty's article in M.D. Thoimpson, ed., Dis- basis for discussion not only of interest and infla- crete Mathematics and its Applications (Indiana tion, but also of such things as radio-carbon dating, University, Bloomington, 1977). cooling and heating of houses, population dynam- 7. Exploring the powers and limitatioins of mathematics, strategies for controlling epidemics, and even ical models. Each of the modern. social sciences detection of art forgeries. abounds with applications of elementary mathemat- REFERENCES: See any modern text on ordinary ics. All of the examples mentio:ned above, and differential equations. A particularly good one is many more, involve the use of mathematical mod- Martin Braun, DifferentialEquations and Their Ap- els. Sometimes these models are quite accurate and plications, Springer-Verlag, New York, 1975. sometimes they are not. But even in the latter 5. Relating traditional mathematics to new applica- case the model can help clarify one%thinking about tions. A discussion of beginning probability the- the underlying problem. An example of this use of ory can quickly lead to a treatment of the Hardy- mathematical modelling is the prisoner's dilemma Weinberg law of genetics and a calculation of the argument of game theory and its :possible connec- probability of winning state lotteries. An introduction with U.S.-U.S.S.R. relations. tory treatment of statistics can quickly lead to a discussion of political polls, the design and inter- Two-Year Colleges pretation of surveys and of related decision-making Many courses that ought to follow the "mathemat- problems. Modern applications of elementary net- ics appreciation" philosophy are taught, in two-year col- work theory include recent work in computational leges. Innovative approaches and curriculum develop- complexity and almost unbreakable codes. ment by some two-year faculty are reflected by their REFERENCES: The Hardy-Weinberg law appears texts and articles in this area. Although the preceding in several texts on finite mathematics, e.g., Applied sections of this report are applicable to mathematics Finite Mathematics by Anton and Kolman. How to appreciation courses in all colleges, this separate sec- Lie with Statistics by Darrel Huff and Irving Geis, tion appears because of the special problems created Norton, 1954, and other texts contain situational in two-year colleges by generally heavy teaching loads, mathematics which can be discussed according to by staffing in some cases by faculty whose mathemat- the interests of the audience. For the two topics ical experiences are not sufficient to rnake them com- mentioned last, see Scientific American, Jan. 1978 fortable with the broad range of topi'cs demanded by and Aug. 1977. these courses, and by the regrettable frequency of ad- 6. Introducing problems involving decision making. ministrative procedures which allow students needing There are many situations described by elementary remediation to enroll in these courses. The following mathematics in which one must choose "rationally" suggestions may help to overcome these impediments to among possible options. One can discuss quantify- two-year college implementation of the goals of mathe- ing risk and uncertainty, fair division schemes, ap- matics appreciation courses. plications of network flows, pursuit and navigation 1. When there is a choice among faculty members for problems, game theory, and numerous other topics. assignment to the mathematics appreciation course, Political science is full of unexpected but usually in- only those having a broad range of mathematical teresting topics, including Arrow's theorem and its experiences and expressing interest in the course 114 RESHAPINGCOLLEGE MATHEMATICS

should teach it. Mathematics program adminis- ences with mathematics and, if there is a general ed- trators should provide extra guidance to faculty ucation mathematics requirement which may be sat- teaching this course for the first time. In the two- isfied by either a mathematics appreciation course year college there will usually be one text used by or a pre-calculus/calculus course, they will often all teachers, often supplemented by a reading list elect the mathematics appreciation course. Well and/or other texts; a description of special uses of into a successful term, the student may begin to these materials, as well as sample course outlines, think realistically about mathematics requirements supplementary and classwork materials, and tests, of various university majors. Since most majors out- will be helpful. Entrance and/or exit requirements side the humanities will necessitate at least some may be matters of policy and should be explained. mathematics at a technical level rarely achieved in Lists of applicable resource material owned by the the typical two-year college mathematics apprecia- school should be provided to new teachers, along tion course, an important service of this course can with knowledge of the schools' firm rental policies. be to channel these students back into regular se- 2. Special attention should be paid to the needs of this quence mathematics courses. course by the library, audio-visual, and computer Without violating the spirit of a mathematics appre- facilities. The mathematics program administrator ciation course, it is possible to include a topically orga- should be sure these courses are adequately sup nized unit requiring the review and use of elementary ported. algebra and graphing techniques; this may give the stu- 3. Since mathematics appreciation courses, properly dent a successful experience in doing mathematics that taught, take an enormous amount of preparation serve as encouragement to return to regular sequence time, any load relief possible would be appropriate. mathematics courses. (A linear programming unit, for In a suitable lecture room, the course can be effec- example, requires the students to review or acquire facil- tively taught to "double" sections of 60-90 students ity with graphing and algebra techniques. Many of the if doing so would leave the teacher several hours topics suitable for a mathematics appreciation course more preparation time each week. (Such a load can be handled in this way.) Students with the experi- might be counted as two or three sections, corre- ence will frequently place higher in the sequence courses sponding to the grading load.) than they would have upon original enrollment, and will 4. Sharing materials and ideas and perhaps team- go on as solid, though late-blooming, students. teaching would be reasonable for mathematics ap- Acknowledgements preciation courses. One teacher at a school might Many people, including nonmathematicians, re- be most qualified for teaching, say, a computer unit, sponded to the Panel's request for information and ad- and might "rotate" across several sections. Many vice. An open meeting of the panel at the MAA meet- more "hand-out" materials seem to be necessary ing in Biloxi, Mississippi, in January 1979, was attended for mathematics appreciation courses than for tradi- by more than sixty people who had much to contribute. tional courses; these might be used by several teach- The panel particularly wishes to thank Professors Henry ers in a given term, or re-used in succeeding terms. Alder, Dorothy Bernstein, Donald Bushaw, William Lu- Faculty teaching mathematics appreciation courses cas, David Penney, William M. Priestley, David Roselle, seem to enjoy sharing materials and methods. and Alice Schafer for their helpful comments. 5. In-course remediation should be avoided. If students are enrolled who cannot handle elementary operations at the level needed for the work of the Panel Members course, a "math lab" facility might be used to design JEROME A. GOLDSTEIN,CHAIR, Tulane University. and administer individual remediation programs. It BETTYEANNE CASE, Florida State University. cannot be over-emphasized that a mathematics appreciation course cannot fulfill its goal if it degener- JOHNCONWAY, Indiana University. ates into the teaching of arithmetic computations or RICHARDJ . DUFFIN, Carnegie-Mellon University. pre-algebra skills, or if it is limited to a topic such ELAINEKOPPELMAN, Goucher College. as "consumer mathematics." KENNETHREBMAN, California State University at 6, A large proportion of students enter two-year col- Hayward. leges with little realistic expectation concerning ma- LYNNARTHUR STEEN, St. Olaf College. jors. Many of these students have had poor experi- JAMESVINEYARD, Laney College. MATHEMATICSAPPRECIATION COURSES 115

Reference Section

Films Congruent Triangles. (1978; 7 Min; Color). International Film Bureau. Since students in mathematics appreciation courses Conic Sections. (1968; 11 Min; Color) :BFA Educational frequently have little experience in sustaining interest in Media. regular mathematics lectures, it is usually appropriate Conics. (1979; 10 Min; Color) Wards ]Modern Learning in these courses to provide a variety of class activities. Aids. Films are a useful but under-utilized medium for mathe- Conics. (1978; 25 Min; Color). University Media. matics instruction generally. They are especially useful Constructing an Algorithm. (25 Min; BW) Open Univer- for the mathematics appreciation course. sity. The following selection of films about mathematical Cosmic Zoom. McGraw-Hill Films. subjects features those that are suitable for lay audi- Curves. (1968; 17 Min; Color) A.I.M.S. (Reviews: Amer. ences. (Distributor addresses are listed at the end of Math. Monthly 83 (1976) 71-72;Math. ‘reacher 64 (1971) the list.) Further information on these and other films 525.) is available in the booklet Annotated Bibliography of Curves of Constant Width. (1971; 16 Min; Color) Interna- Films and Videotapes for College Mathematics by David tional Film Bureau. (Reviews: Amer. Math. Monthly 78 (1971) 539; Math. Teacher 65 (1972) 234.) Schneider (M.A.A., 1980). Cycloidal Curves or Tales From the Wanklenberg Woods. A Non-Euclidean Universe. (1978; 25 Min; Color). Univer- (1974; 22 Min; Color) Modern Film Rentals. sity Media. Dance Squared. (1963; 4 Min; Color) International Film A Time for Change-The Calculus. (1975; 25 Min; Color) Bureau. Review: Math. Teacher 64 (1971) 627.) University Media. Dihedral Kaleidoscopes. (1966; 13 Min; Cador) International Accidental Nuclear War. (1976; 8 Min; Color) Pictura Film Bureau. (Review: Math. Teacher 66 (1973) 51.) Film. Dimension. (1970; 13 Min; Color) A.I.M.S. (Reviews: Adventures in Perception. (1973; 22 Min; Color) BFA Ed- Amer. Math. Monthly 83 (1976) 71-72; Math. Teacher ucational Media. (Reviews: Amer. Math. Monthly 84 64 (1971) 525.) (1977) 582.) Donald in Mathmagicland. (1960; 26 Min; Color) Walt Dis- An Historical Introduction to Algebra. Modern Film ney Educational Media Company. Rentals. Dr. Posin’s Giants: Isaac Newton. Indiana University Au- An Introduction to Feedback. (1960; 11 Min; Color) Ency- diovisual Library. clopedia Britannica Educational Corporation. Dragon Fold... And Other Ways to Fill Space. (1979; 7 1/2 Anatomy of Analogy. (25 Min; BW) Open University. Min; Color) International Film Bureau. Area and Pi. (1969; 10 Min; Color) Modern Film Rentals. Equidecomposable Polygons. (10 1/2 Min; Color) Interna- Auto Insurance. (1976; 8 Min; Color) Pictura Film. tional Film Bureau. (Reviews: Amer. Math. Monthly 82 Caroms. (1971; 9 1/2 Min; Color) International Film Bu- (1975) 687-688;Math. Teacher 65 (1972) 734.) reau. (Reviews: Amer. Math. Monthly 82 (1975) 417; Errors That Die. (25 Min; BW) Open University. Math Teacher 66 (1973) 51.) Flatland. (1965; 12 Min; Color) McGraw-Hill Films. (Re- Centml Perspectivities. (1971; 13 1/2 Min; Color) Interna- view: Math. Teacher 64 (1971) 44-45.) tional Film Bureau. (Reviews: Amer. Math. Monthly 82 (1975) 419; Math. Teacher (1972) 733.) Functions and Graphs. (1978; 25 Min; Color) University Media. Centml Similarities. (1966; 10 Min; Color) International Film Bureau. (Reviews: Amer. Math. Monthly 82 (1975) Geodesic Domes: Math Raises the Roof. (1979; 20 Min; 418; Math. Teacher (1972) 643-644.) Color) David Nulsen Enterprises. Challenge in the Classroom. (1966; 55 Min; Color) Modern Geometric Vectors-Addition. (1971; 17 lain; Color) Inter- Film Rentals. national Film Bureau. (Review: Amer. Math. Monthly Circle Circus. (1979; 7 Min; Color) International Film Bu- 82 (1975) 420.) reau. Geometry: Inductive and Deductive Reaaroning. (1962; 12 Common Generation of Conics. (4 Min; Color). Educa- 1/2 Min; Color) Coronet Films. tional Solutions. Good for What? (25 Min; BW) Open University. Complez Numbers. (1978; 25 Min; Color). University Me- Gottingen and New York. (1966; 43 Min; Color) Modern dia. Film Rentals. Computer Perspective. (1972; 8 Min; Color) Pyramid How Far is Around? (1979; 7 1/2 Min; Color) International Films. Film Bureau. 116 RESHAPINGCOLLEGE MATHEMATICS

Inferential Statistics, Part I: Sampling and Estimation. Modelling Pollution. (1978; 25 Min; Color). University Me- (1977; 19 Min; Color) Media Guild. dia. Inferential Statistics, Part IZ: Hypothesis Testing. (1977; 25 Modelling Surveys. (1978; 25 Min; Color). University Me- Min; Color). Media Guild. dia. Infinity. (1972; 17 Min; Color) A.I.M.S. (Review: Amer. Modmath. (14 112 Min; Color) International Film Bureau. Math. Monthly 83 (1976) 71-72.) Mr. Simplez Saves the Aspidistra. (1966; 33 Min; Color) Inversion. (12 Min; Color) International Film Bureau. Modern Film Rentals. (Reviews: Amer. Math. Monthly 83 (1976) 71; Math. Networks and Matrices. (1978; 25 Min; Color) University Teacher (1972) 644.) Media. Isaac Newton. (1959; 13 112 Min; Color) Coronet Films. New Worlds From Old. (1975; 25 Min; Color) University Isn’t That the Limit! (1980; 17 Min; Color) David Nulsen Media. Enterprises. Newton’s Equal Area. (1968; 8 Min; Color) International Isometries. (1967; 26 Min; Color) International Film Bu- Film Bureau. (Reviews: Amer. Math. Monthly 79 (1972) reau. (Review: Math. Teacher 66 (1973) 51-52.) 1054; Math. Teacher 63 (1970) 449.) Iteration and Convergence. (1978; 25 Min; Color) University Nim and Other Oriented Graph Games. (1966; 63 Min; BW) Media. Modern Film Rentals. John won Neumann, A Documentary. (1966; 63 Min; BW) Notes on a Triangle. International Film Bureau. (Review: Modern Film Rentals. (Review: Amer. Math. Monthly Math. Teacher 63 (1970) 363.) 75 (1968) 435.) Numbers Now and Then. (1975; 25 Min; Color) University Journey to the Center of a Triangle. (1977; 8 112 Min; Media. Color) International Film Bureau. Orthogonal Projection. (1965; 13 Min; Color) International Let Us Teach Guessing. (1966; 61 Min; Color) Modern Film Film Bureau. (Reviews: Amer. Math. Monthly 82 (1975) Rentals. (Review: Amer. Math. Monthly 75 (1968) 219.) 419-420; Math. Teacher (1972) 643.) Limit Curves and Curves of Infinite Length. (1979; 14 Min; Paradoz Boz. Scientific American. Silent; Color) International Film Bureau. Pits, Peaks, and Passes (Part 1). (1966; 48 Win; Color) Limit Surfaces and Space Filling Curves. (1979; 10 112 Min; Modern Film Rentals. Silent; Color) International Film Bureau. Plateau’s Problem. A Film by Sr. Rita Ehrmann. Limits. (25 Min; BW) Open University. Points of View: Perspective and Projection. (1975; 25 Min; Linear Programming. (1969; 9 Min; Color) Macmillan Films. Color) University Media. Look Again. (1970; 15 Min; Color) A.I.M.S. (Reviews: Possibly So, Pythagoras. (1973; 14 Min; Color). Interna- Amer. Math. Monthly 83 (1976) 71-72; Math. Teacher tional Film Bureau. (Review: Math. Teacher 64 (1971) 64 (1971) 525.) 626.) Love Song. (1976; 11 Min; Color) Pictura Film. Powers of Ten. (1978; 9 Min; Color) Pyramid Films. (Re- view: Math. Teacher (1979) 388.) Mathematical Curves. (1977; 10 Min; Color) Churchill Films. Predicting at Random. (1966; 43 Min; Color) Modern Film Rentals. Mathematical Induction. (1960; 62 Min; Color) Modern Film Rentals. Probability. (12 Min; Color) McGraw-Hill Films. Mathematical Induction. (1978; 25 Min; Color) University Professor George Po’lya and Students, Parts I and 11. (1972; Media. 60 Min; Color) University Media. Mathematical Peep Show. (1961; 11 Min; Color) Encyclope- Professor George Po’lya Talks to Professor Mazim Bruck- dia Britannica Educational Corporation. (Review: Math. beimer. (1972; 60 Min; Color) University Media. Teacher 64 (1971) 625.) Projective Generation of Conics. (16 Min; Color) Interna- Mathematician and the River. (1959; 19 Min; Color) No tional Film Bureau. (Reviews: Amer. Math. Monthly 82 distributor. (1975) 538-539; Math. Teacher 66 (1973) 51.) Mathematics of the Honeycomb. (1964; 13 Min; Color) Quaterniona: A Herald of Modern Algebra. (1975; 25 Min; Moody Institute of Science. (Review: Math. Teacher 64 Color) University Media. (1971) 334.) Rational Numbers and the Square Root of 2. (1978; 25 Min; Matrices. (9 Min; Color) Macmillan Films. Color) University Media. Matrioska. Indiana University Audiovisual Library. Regular Homotopies in the Plane: Part I: (1975; 14 Min; Maurits Escher, Painter of Fantasies. (1970; 26 112 Min; Color); Part II: (1975; 18 112 Min; Color) International Color) Coronet Films. (Review: Amer. Math. Monthly Film Bureau. (Review: Amer. Math. Monthly 85 (1978) 83 (1976) 495.) 212. Mean, Median, Mode. McGraw-Hill Films. Root Two: Geometry or Arithmetic? (1975; 25 Min; Color) Modelling Drug Therapy. (1978; 25 Min; Color). University University Media. Media. Sampling. (25 Min; BW) Open University. MATHEMATICSAPPRECIATION COURSES 117

Sets, Crows, and Infinity. (12 Min; Color) BFA Educational The Nature of Digital Computing. (25 Min; BW) University Media. Media. Shaking the Foundations. (1975; 25 Min; Color) University The Perfection of Matter. McGraw-Hill Films. Media. The Search for Solid Ground. (1963; 62 Min; BW) Modern Shapes of the Future: Some Unsolved Problems in Geome- Film Rentals. try: Part Z: Two Dimensions (1975; 22 Min; Color); Part The Seven Bridges of Ko’nigsberg. (1965; 4 Min; Color) In- ZZ: Three Dimensions (1970; 21 Min; Color) Modern Film ternational Film Bureau. Rentals. (Review: Amer. Math. Monthly 79 (1972) 1052- The Structure of a Computer. (25 Min; BNW) Open Univer- 1053.) sity. Sierpinski’s Curve Fills Space. (1979; 4 1/2 Min; Color) Topology. (1972; 9 Min; Color) Macmillari Films. International Film Bureau. Topology. (1966; 30 Min; BW) Modern Film Rentals. (Re- Similar Triangles. (1976; 7 1/2 Min; Color) International view: Amer. Math. Monthly 75 (1968) 790.) Film Bureau. Topology: Some Historical Concepta. (2:L 3/4 Min; Color) Space Filling Curves. (1975; 25 1/2 Min; Color) Interna- Richard Cline Film Productions. tional Film Bureau. (Review: Math. Teacher 69 (1976) Trio for Three Angles. (8 Min; Color) International Film 164-165.) Bureau. Sphere Eversions. (1979; 7 1/2 Min; Silent; Color) Interna- Turning a Sphere Inside Out. (1976; 23 Min; Color) Inter- tional Film Bureau. national Film Bureau. (Reviews: Amer. Math. Monthly Spheres. International Film Bureau. 86 (1979) 511-512; Math. Teacher 70 (1977) 55.) Statistics At A Glance. (1972; 28 Min; Color) Media Guild. View From the People Wall. (1964; 14 lain; Color) Ency- (Review: Amer. Math. Monthly 82 (1975) 312.) clopedia Britannica Educational Corporation. Statistics and Probability Z. (15 Min; BW) Open University. Weather by the Numbers. University of Indiana Audiovisual Statistics and Probability ZZ. (25 Min; BW) Open Univer- Library. sity. What is a Limit? (25 Min; BW) Open University. Statistics and Probability ZZZ. (25 Min; BW) Open Univer- What is a Set? Part Z El ZZ. (1967; 15 Min; Color) Modern sity. Film Rentals. (Review: Amer. Math. Monthly 75 (1968) Symbols, Equations and the Computer. (1978; 25 Min; 324.) Color) University Media. What is Mathematics and How Do We Teach it? (1966; 60 Min; BW) Modern Film Rentals. Symmetries of the Cube. (1971; 13 1/2 Min; Color) Inter- national Film Bureau. (Review: Math. Teacher 65 (1972) Zooms on Self-Similar Figures. (1979; 8 Min; Color) Inter- 733.) national Film Bureau. The Algebra of the Unknown. (1975; 25 Min; Color) Univer- ADDRESSESOF DISTRIBUTORS: sity Media. A.I.M.S., 626 Justin Avenue, Glendale, CA 91201. The Binomial Theorem. (1978; 25 Min; BW) University International Film Bureau, 332 South Michigan Avenue, Media. Chicago, IL 60604. The Butterfly Catastrophe. (1979; 4 1/2 Min; Silent; Color) McGraw-Hill Films, McGraw-Hill Book Company, 330 West International Film Bureau. 42nd Street, New York, NY 10036. The Delian Problem. (1975; 25 Min; Color) University Me- Modern Film Rentals, 2323 New Hyde Park Road, New dia. Hyde Park, NY 11040. The Dot and the Line. Indiana University Audiovisual Li- Moody Institute of Science, 12000 East Washington Boule- brary. vard, Whittier, CA 90606. The Geometry Euclid Didn’t Know. (1979; 16 Min; Color) Indiana University Audiovisual Library, Bloomington, IN David Nulsen Enterprises. (Reviews: Amer. Math. 47401. Monthly 86 (1979) 600; Math. Teacher (1979) 300.) Ward’s Modern Learning Aids Division, P.O. Box 1712, The Great Art-Solving Equations. (1975; 25 Min; Color) Rochester, NY 14603. University Media. University Media, 118 South Acacia, Box 881, Solana Beach, The Hypercube: Projections and Slicing. (1978; 12 Min; CA 92075. Color) Banchoff-Strauss Productions. Pyramid Films, P.O. Box 1048, Santa Monica, CA 90406. The Kakeya Problem. (1962; 60 Min; Color) Modern Film Educational Solutions, Inc., 80 Fifth Avenue, New York, NY Rentals. 10011. The Majestic Clockwork. (1974; 52 Min; Color) Time Life Banchoff-Straws Productions, Inc., P.O. Box 2430, East Multimedia. Side Station, Providence, RI 02906. The Marriage Theorem, Parts Z Ei ZZ. (1974; 46 Min. and Macmillan Films, Inc., 34 MacQuesten Parkway South, Mt. 47 Min; BW) Modern Film Rentals. Vernon, NY 10550. The Music of the Spheres. (1974; 52 Min; Color) Time Life Encyclopedia Britannica Educational Corporation, 425 Multimedia. North Michigan Avenue, Chicago, IL 60611. 118 RESHAPINGCOLLEGE MATHEMATICS

David Nulsen Enterprises, 3211 Pic0 Boulevard, Santa Mon- LaPine Scientific Company, Department B43, 6005 South ica, CA 90405. Knox Avenue, Chicago, IL 60629; 373 Chestnut Street, The Media Guild, 118 South Acacia, Box 881, Solana Beach, Norwood, NJ 07648; 920 Parker Street, Berkeley, CA CA 92075. 94710; Box 95, Postal Station U., Toronto, Canada M8Z Coronet Films, 65 East South Water Street, Chicago, IL 5M4. 60601. An extensive offering of models, teaching aids, games, Pictura Film Co., 111 8th Avenue, New York, NY 10011. and audio-visual materials for elementary, high school BFA Educational Media, 2211 Michigan Avenue, P.O. Box and beginning college mathematics. Mathematics cata- 1795, Santa Monica, CA 90406. logue exceeds 100 pages. Time-Life Multimedia, Time and Life Building, New York, Geyer Instructional Aids Co., Inc., P.O. Box 7306, Fort Wayne, IN 46807. NY 10020. A large collection of models, games, books and classroom Churchill Films, 662 North Robertson Boulevard, Los An- aids for high school and elementary college courses. geles, CA 90069. Creative Publications, 3977 East Bayshore Road, P.O. Box Walt Disney Educational Media Company, 500 South Buena 10328, Palo Alto, CA 94303. (415-968-3977). Vista Street, Burbank, CA 91521. An attractive 100-page catalogue of mathematical books, Contemporary-McGraw Hill Films, 1221 Avenue of the models, games, posters, construction sets, and puzzles Americas, New York, NY 10020. for all grade levels. The premier source for mathematics enrichment material. Inquiry Audio-Visuals, 1754 West Farragut Avenue, Classroom Aids Chicago, IL 60640. Filmstrips and transparencies for high school algebra top Certain topics treated in mathematics appreciation ics. courses are particularly amenable to demonstration Educational Audio Visual, Inc., Pleasantville, NY 10570. with physical or geometric devices. Useful exhibits can Transparencies and games for algebra, geometry, calculus often be seen at NCTM meetings. A list of major sup- and statistics. pliers of mathematics classroom devices is given below: W.H. Freeman and Company, 660 Market Street, San Fran- cisco, CA 94104. The Math Group, Inc., 396 East 79th Street, Minneapolis, Offprints of Scientific American articles. MN 55420. David Nulsen Enterprises, 3211 Pic0 Boulevard, Santa Mon- Unique puzzles and card games. Designed for elemen- ica, CA 90405. tary school enrichment, many are flexible enough to be Three 15-minute, 16mm color films (“Curves,” “Dimen- of interest to adults as well. sion,” “Look Again”) for rent or purchase. Lano Company, 9001 Gross Road, Dexter, MI 48130. (313- Popular Science Audio-Visuals, Inc., 5235 Ravenswood Av- 426-4860). enue, Chicago, IL 60640. Mathematical visual aids (solids, transparencies, graph- Filmstrips and overhead transparencies, principally in ing aids) together with various games and probability high school mathematics (geometry, algebra, elementary models. functions). Math Shop, Inc., 5 Bridge Street, Watertown, MA 02172. Time Life Films, 43 West 16th Street, New York, NY 10011. A full range of curriculum enrichment material for ele- (212-691-2930). mentary, junior and senior high school mathematics. In- Numerous BBC-produced 20-minute 16mm B & W films cludes several games and puzzles of use in introductory on topics ranging from inequalities to matrices. college mathematics. Edmund Scientific Company, 1985 Edscorp Building, Bar- International Film Bureau, Inc., 332 South Michigan Av- rington, NJ 08007. (609-547-3488). enue, Chicago, IL 60604. The major American distributor of mathematics films. The current list includes nearly 50 films at the high school References and college level, each of which may be either purchased or rented. Since many of the topics that arise in mathematics Yoder Instruments, East Palestine, OH 44413. (216-426- appreciation courses occur nowhere else in the mathe- 3612; 216-426-9580). matics curriculum, it is quite important that instructors Two unique geometric construction sets for plane and be aware of the expository literature of mathematics solid geometry. that treats its relations to science and society. Stu- Cuisenaire Co. of America, Inc., 12 Church Street, New Rochelle, NY 10805. dent term papers in courses on mathematics apprecia- Sensory apparatus designed for elementary schools, some tion typically tax the instructor’s knowledge of the lit- of which (e.g., geoboards, polyhedral structures) would erature more than any other course in the mathematics be suitable to courses in mathematics appreciation. curriculum. MATHEMATICSAPPRECIATION COURSES 119

To aid instructors of mathematics appreciation National Research Committee on Support of Research in courses, we prepared a list of major references that the Mathematical Sciences (COSRIMS). The Mothemati- would be suitable for background reading, and as cal Sciences- A Collection of Essays. MIT Press, Cam- sources for special projects. This list does not include bridge, Massachusetts, 1969. Newman, James R., ed. The World of Mathematics, 4 vols. textbooks, partly because we do not wish to endorse Simon and Schuster, New York, 1956-60. some books over others, and partly because texts go in Saaty, Thomas L. and Weyl, F. Joachim, eds. The Spirit and out of print much more rapidly than the reference and User of the Mathematical Sciencer. ‘McGraw-Hill, New classics. York, 1969. Schaaf, William L., ed. Our Mathematical Heritage, New, Survey Monographs Revised Edition. Collier Books, New York, 1963. Boehm, George A.W. The New World of Mathematics. The Steen, Lynn Arthur, ed. Mathematics Today: Twelve Infor- Dial Press, New York, 1959. mal Essays. Springer-Verlag, New York, 1978. Courant, Richard and Robbins, Herbert. What is Mathe- matics? Oxford University Press, New York, 1941. Nature of Mathematics Dantzig, Tobias. Number, the Language of Science, 4th ed. Adler, Alfred. “Reflections-mathematics and creativity.” Free Press, New York, 1967. New Yorker 47 (February 19, 1972) 39-45. Goarding, Lars. Encounter with Mathematics. Springer- Bronowski, Jacob. “The music of the spheres.” in J. Verlag, New York, 1977. Bronowski, The Ascent of Man. Little, Brown, and Com- Herstein, I.N. and Kaplansky, I. Matters Mathematical. pany, Boston, Massachusetts, 1973, pp. 154-187. Harper and Row, New York, 1974. Bronowski, Jacob. “The logic of the mind.” Amer. Scientist Khurgin, Ya. Did You Say Mathematics? MIR Publishers, 54 (1966) 1-14. Moscow, Russia, 1974. Bruter, C.P. Sur la nature des mathe‘matiques. Gauthier- Kline, Morris. Mathematics: The Loss of Certainty. Oxford Villars, Paris, France, 1973. University Press, New York, 1980. Cartwright, Mary L. “The mathematical mind.” Math. Pedoe, Daniel. The Gentle Art of Mathematics. Macmillan, Spectrum 2 (1969-70) 37-45. Riverside, New Jersey, 1958, 1963. Cartwright, Mary L. “Mathematics and thinking mathemat- Rademacher, Hans and Toeplitz, Otto. The Enjoyment of ically.” Amer. Math. Monthly 77 (1970) 20-28. Mathematics. Princeton University Press, Princeton, New Davis, Philip J. and Hersh, Reuben. The Mathematical Ez- Jersey, 1957. perience. Birkhiiuser , Cambridge, Massachusetts, 1981. Sawyer, W.W. Introducing Mathematics, 4 vols. Penguin Fisher, Charles S. “Some social characi.eristics of mathe- Books, New York, 1964-70. maticians and their work.” Amer. J. Sociology 78 (1973) Singh, Jagjit. Great Ideas of Modern Mathematics, Their 1094-1118. Nature and Use. Dover, New York, 1959; Hutchinson and Grabiner, Judith V. “Is mathematical truth time-depen- Company, London, England, 1972. dent?” Amer. Math. Monthly 81 (1974) 354-365. Stein, Sherman K. Mathematics, The Man-made Universe: Hadamard, Jacques. Psychology of Invenrtion in the Mathe- An Introduction to the Spirit of Mathematics, Third Edi- matical Field. Dover, New York, 1945. tion. W.H. Freeman, San Francisco, California, 1976. Halmos, Paul R. “Mathematics as a creative art.” Amer. Steinhaus, Hugo. Mathematical Snapshots, 2nd ed. Oxford Scientist 56 (1968) 375-389. University Press, New York, 1969. Hahn, Hans. “Geometry and intuition.” Scientific Ameri- Stewart, Ian. Concepts of Modern Mathematics. Penguin can 190 (April 1954) 84-91,108; also in M. Kline. Math- Books, New York, 1975. ematics in the Modern World. W.H. Freeman, San Fran- Whitehead, Alfred North. An Introduction to Mathematics. cisco, California, 1968, pp. 184-188,399. Oxford University Press, New York, 1958. Hardy, G.H. A Mathematician’s Apology. Cambridge Uni- versity Press, Cambridge, Massachusetts, 1940; 1967; ex- Collections of Essays cerpted in J.R. Newman. The World cd Mathematics, V. Kline, Morris, ed. Mathematics in the Modern World. W.H. 4, Simon and Schuster, New York, 1956, pp. 2027-2038. Freeman, San Francisco, California, 1968. Helitzer, Florence. UA conversation with three mathemati- Kline, Morris, ed. Mathematics: An Introduction to Its cians.” University: A Princeton Quarterly 59 (Winter Spirit and Use. W.H. Freeman, San Francisco, California, 1974) 1-5, 28-30. 1979. Henkin, Leon. “Are logic and mathematics identical?” Sci- LeLionnais, F., ed. Great Currents of Mathematical ence 138 (1962) 788-794. Thought, 2 vols. Dover, New York, 1971. Hilton, Peter J. “The art of mathematics.” Univ. of Birm- Messick, David M., ed. Mathematical Thinking in Behav- ingham, 1960. ioral Sciences. W .H. Freeman, San Francisco, California, Iliev, L. “Mathematics as the science of models.” Russian 1968. Math. Surveys 27:2 (1972) 181-189. 120 RESHAPINGCOLLEGE MATHEMATICS

Jones, Landon Y., Jr. “Mathematicians: They’re special.” MacLane, Saunders. “Mathematical models of space.” Think 40:4 (1974) 32-35. Amer. Scientist 53 (1980) 252. Kapur, J.N. Thoughts on the Nature of Mathematics. Atma Meserve, Bruce E. “New mathematics.” Encyclopedia Ram, Delhi, India, 1973. Americana 20 (1976) 202-205. Lefschetz, Solomon. “The structure of mathematics.” Meserve, Bruce E. ”Number systems and notation.” Ency- Amer. Scientist 38 (1950) 105-111. clopedia Americana 20 (1976) 536f-536j. Newman, M.H.A. “What is mathematics? New answers to Murray, Francis J. and Ford, Lester R. “Mathematics as a an old question.” Math. Gazette 43 (1959) 161-171. calculatory science.“ Encyclopaedia Britannica, 15th ed., Otte, Michael. Mathematiker tiber die Mathematik. 1974, Macropaedia V. 11, pp. 671-696. Springer-Verlag, New York, 1974. Richards, Ian. “Impossibility.” Math. Magazine 48 (1975) Poincard, Henri. “Mathematical creation.” Scientific Amer- 249-262. icon179 (August 1948) 54-57; also in M. Kline. Mathemat- Stone, Marshall H. “The future of mathematics.” J. Math. ics in the Modern World. W.H. Freeman, San Francisco, SOC.Jap. 9 (1957) 493-507. California, 1968, pp. 14-17; and in J.R. Newman, The Temple, G. “The growth of mathematics.” Math. Gazette World of Mathematics, V. 4, Simon and Schuster, New 41 (1957) 161-168. York, 1956, pp. 2041-2050. Weil, AndrC. “The future of mathematics.” Amer. Math. RCnyi, Alfred. “A Socratic dialogue on mathematics.” Monthly 57 (1950) 295-306; also in F. LeLionnais. Great Canad. Math. Bull. 7 (1964) 441-462; also in A. RCnyi. Currents of Mathematical Thought, V. 1. Dover, New Dialogues on Mathematics. Holden-Day, San Francisco, York, 1971, pp. 321-336. California, 1967, pp. 3-25. Stein, Sherman K. “The mathematician as an explorer.” Advanced Exposition Scientific American 204 (May 1961) 148-158, 206. Abbott, J.C. The Chauuenet Papers: A Collection of Prize- Stone, Marshall H. “The revolution in mathematics.” Lib- Winning Ezpository Papers in Mathematics, 2 vols. Math- era2 Education, 47 (1961) 304-327; also in Arner. Math. ematical Association of America, Washington, D.C., 1978. Monthly 68 (1961) 715-734. Aleksandrov, A.D., Kolmogorov, A.N., Lavrent’ev, M.A. von Neumann, John. “The mathematician.” in R.B. Hey- Mathematics, Its Content, Methods, and Meaning, 3 vols. wood, The Works of the Mind. University of Chicago MIT Press, Cambridge, Massachusetts, 1969. Press, Chicago, Illinois, 1947, pp. 180-196; also in J.R. Behnke, H., et al. Fundamentals of Mathematics, 3 vols. Newman, The World of Mathematics, V. 4, Simon and MIT Press, Cambridge, Massachusetts, 1974. Schuster, New Yok,1956, pp. 2053-2063. Saaty, Thomas L. Lectures on Modern Mathematics, 3 vols. Weidman, Donald R. “Emotional perils of mathematics.” John Wiley, New York, 1963-1965. Science 149 (1965) 1048. Biography and Autobiography Weissinger, Johannes. “The characteristic features of mathematical thought.” in T.L. Saaty and F.J. Weyl, The Spirit Bell, Eric Temple. Men of Mathematics. Simon and Schus- and Uses of the Mathematical Sciences. McGraw-Hill, New ter, New York, 1937. York, 1969, pp. 9-27. Box, Joan Fisher. R.A. Fisher, The Life of a Scientist. Weyl, Hermann. “The mathematical way of thinking.” Sci- Wiley, New York, 1978. ence 92 (1940) 437-446; also in Studies in the History of Dauben, Joseph Warren. Georg Cantor: His Mathematics Science. University of Pennsylvania Press, 1941, pp. 103- and Philosophy of the Infinite. Harvard University Press, 123. Cambridge, Massachusetts, 1979. Weyl, Hermann. “Insight and reflection.” in T.L. Saaty Grattan-Guinness, Ivor. Joseph Fourier, 1768-1830. MIT and F.J. Weyl. The Spirit and Uses of the Mathematical Press, Cambridge, Massachusetts, 1972. Sciences. McGraw-Hill, New York, 1969, pp. 281-301. Halmos, Paul R. “Nicolas Bourbaki.” Scientific American Wilder, Raymond L. “The role of the axiomatic method.” 196 (May 1957) 88-99, 174. Amer. Math. Monthly 74 (1967) 115-127; also in Math. Halmos, Paul R. “The legend of John von Neumann.” Amer. Teaching 41 (1967) 32-40. Math. Monthly 80 (1973) 382-394. Hardy, G.H. Ramanujan. Chelsea Publishing, New York, Survey Papers 1968. Hoffman, Banesh. Albert Einstein, Creator and Rebel. Bochner, Salomon. “Mathematics.” McGraw-Hill Encyclo- Viking Press, New York, 1972. pedia of Science and Technology 8 (1960) 175-180. Infeld, Leopold. Whom the Gods Love. Whittlesey House, DieudonnC, Jean A. “Recent developments in mathematics.” 1948; NCTM, Reston, Virginia, 1978. Amer. Math. Monthly 71 (1964) 239-248. Kovalevskaya, Sofya. Sofya Kovalevskaya: A Russian Child- Eves, Howard W. “Mathematics.” Encyclopedia Americana hood. Springer-Verlag, New York, 1978. 18 (1976) 431-434. Mahoney, Michael S. The Mathematical Career of Pierre de Ficken, F.A. “Mathematics and the layman.” Amer. Scien- Fermat. Princeton University Press, Princeton, New Jer- tist 52 (1964) 419-430. sey, 1973. MATHEMATICSAPPRECIATION COURSES 121

Meschkowski, Herbert. Ways of Thought of Great Mathe- Wilder, Raymond L. "The origin and groswth of mathemat- maticians. Holden-Day, San Francisco, California, 1964. ical concepts." Bull. Amer. Math. Soc. 59 (1963) 423- Morgan, Bryan. Men and Discoveries in Mathematics. 448. Transatlantic Arts, Inc., Levittown, New York, 1972. Wilder, Raymond L. Evolution of Mathsmatical Concepts. Morse, Philip M. In at the Beginnings: A Physicist's Life. Halsted Press, New York, 1974. MIT Press, Cambridge, Massachusetts, 1977. Wilder, Raymond L. "History in the mathematics curricu- Ore, Oystein. Niels Henrik A bel, Mathematician Eztraor- lum: Its status, quality, and function." Amer. Math. dinary. University of Minnesota Press, Minneapolis, Monthly 79 (1972) 479-495. Minnsota, 1957; Chelsea Press, New York, 1974. Zaslavsky, Claudia. Africa Counts: Number and Pattern Osen, Lynn M. Women in Mathematics. MIT Press, Cam- in African Culture. Prindle, Weber and Schmidt, Boston, bridge, Massachusetts, 1974. 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