DOCUMENT RESUME ED 069 496 24 SE 015 142 AUTHOR Lankford, Francis G., Jr. TITLE Some Computational Strategies of Seventh Grade Pupils. Final Report. INSTITUTION Virginia Univ., Charlottesville. School of Education. SPONS AGENCY National Center for Educational Research and Development (DHEW/OE) , Washington, D.C. BUREAU NO BR-2-C-013 PUB DATE Oct 72 GRANT OEG-3-72-0035 NOTE 96p. EDRS PRICE MF-$0.65 HC-$3.29 DESCRIPTORS *Algorithms; Fractions; Grade 7; Learning; *Mathematics Education; *Research; Secondary School Mathematics; Whole Numbers IDENTIFIERS *Computation; Diagnostic Interviews ABSTRACT One hundred seventy-six seventh grade students underwent a recorded interview where each was given a set of computational exercises and asked to say aloud his thinking as he worked them. The most frequently used strategies in computations with whole numbers and fractions are described in detail, an analysis of the nature of wrong answers is included, and characteristics of good and poor computers are listed and discussed. Thirteen conclusions are given, covering computational strategies, vertical vs. horizontal problem arrangement, mathematical vocabulary of students, estimating answers, and the technique of using recorded interviews in research. The computation problems given to the students are included in the report, and the appendices list all the wrong answers given with the accompanying verbal description by the student. (DT) Final Report Project number 2-C.013 Grant number OLG.-3-72.0035 Francis G. Lankford, Jr. School of Education University of Virginia Charlottesville, Virginia, 22903 Some Computational Strategies of Seventh Grade Pupils October, 1972 U.S. Department of Health. Education. and Welfare Office of Education National Center for Educational Research and Development (Regional Research Program) and The Center for Advanced Study The University of Virginia. RI:ST C:OPY CT" OL) FNAL REPORT 1=1 Project Amber 2-C-013 uJ Grant :umber nEG-3-72-0035 snmE COMPUTATIONAL STRATEGIESOF SEVENTH GRADE PUP' LS Francis G. Lankford, Jr. University of Virginia Charlottesville, Virginia 22903 October 1972 - - The research reported herein wasperformed pursuant to a grant with the Office ofEducation, U.S. Department of Health, Education, and '!el fare. Contractors under- taking such projects under GovernmentSponsorship are encouraged to express freely theirprofessional judge- ment in the conduct of the project. Points of view or opinions stated do not, therefore,necessarily represent official Office of Education position orpolicy. U.S. Department of Health, Education,and "elfare Office of Education National Center for Educational Researchand Development and The Center for Advanced Studies, TheUniversity of virginia 2 ACVOAEMEMPTS The principal investigator received much help and support in the work on the project renorted here. The Center for Advanced Studies at The University of Virninia awarded anassociateship which made it possible to devote an entire semester tothe project. The U. S. Office of Education granted funds,through its Regional. Research program, which extended the time thatcould be devoted to the project and covered other expenses. The teachers and principals in the severalschools arranged places for the pupil interviews and carefully scheduledthe pupils to appear one at a time. Central office officials gave the essential approval to use the schools. Many colleagues in the School of Education atthe University gave both help and encouragement. Special thanks are expressed to Dean Frederick R. Cyphert, and to ProfessorsRichard M. Brandt, Milton D. Jacobson, Donald Medley, H. Seaell, 'filliam C. Lowry, Donald M. Shoemaker, Earl P. Smith, as well as toresearch assistants "Mimi Vare and James Muir. Mr. Rod Collins in the Radio and Television Center at The University ofVirginia was especially helpful in preparing taped records of samplesof the pupil interviews. Professor Joseph N. Payne at the University ofMichigan gave valuable suggestions on the design and conduct of thestudy. He also read the first draft of the report and offeredhelpful reactions. Grateful acknowledgement of all the help that was so generously given is sincerely expressed. 3 TABLE OF COTITENTS Page Introduction Related Research 3 Methods and Procedures 5 The Pupils Interviewed 7 Exercises Used in Interviews 15 Results 16 Strategies Frequently Used inFour Operations with '!hole numbers 20 Strategies Frequently Used in Four Operations with Fractions 24 lature of Wrong AnswersWhole Numbers 27 Mature of Wrong Answers--Fractions 30 Reasons Given for Wrong Answers to Comparison Exercises with Fractions 33 Some Characteristics of Good and Poor Computers 36 Conclusions 40 Recommendations 42 Appendix A. Wrong AnswersMole Numbers A-1 Appendix B. !irong Answers -- Fractions B-1 Appendix C. Wrong Answers--Comparison Exercises and Reasons for Choices C-1 SOIT. COMPUTATIOWAL ST1ATMESOF SEvEIJIl 'RAGE PUPILS Introduction It is encouraging that currently thereis widespread interest in improving OF,computational skills ofpupils-- especially in the elementary ;anetarly secondary years. This interest has found expression in numerous experimentalefforts to individualize instruction in computational skills. It may have been stimulated by the calls to "accountability" in theschools and the adontion of performance contracting.' The relatively low scoresof many pupils on the computational skills part ofstandardized acW2ve- ment test batteries which have been seenin many schools may also have supported this interest. 4 iiany of the experimental programs havestarted with a detailed analysis of the tasks, or skills,in6'imiletation-- placing them in many categories related to thecontent of the operations involved.' Often the next step in the,design of pre-test to locate the particular tasks orskill's a pupil does incorrectly. Remedial exercises--associatedwith variously designed programmed instruction--are keyed tothe pre-test. .34 Then follows a post-test--similar to thepre-test--intended to show whether or not the Pupil can now perform at asatisfactory ;4 level. 1 of An example may be taken from thework of Patrick Suppes Stanford University who has done extensiveresearch in computer assisted Instruction. In a Teacher's Handbook forCAI Courses (Technical Report No. 175, Sept. 1, 1971, Institutefor Tiathe- matical Studies in the Social Sciences, StanfordUniversity) 14 "strands" in arithmetic are listed. One of these is the-Hori- zontal Subtraction Strand."The content of this strandis sum- maried as follows. (P.13) Differences in canonical and noncanonicalaformat. Maximum minuend is 9 Maximum minuend is10 Problems with two-digit minuends in canonicalformat Problems with one-digit minuends innoncanonical format Problems with two-digit minuends equal to orless than 19 in noncanonical format Problems with minuends greater than 19 innononnonical format Problems with a difference on both sidesoceoual sign (a) The problem 4 - 0 = ? is in canonical format The problem 7 - ? = 7 is in noncanonicalformat. 1 An example of an exr:rinent insuch an individualized nro- gram is the .1:1:ii,..ijiueljze,i.",th7naticsSystem' develmed at The Regional Education Laboratory for the Carol nasand Virginia. In this system the elementary schoolcurric_lum is divided into 11 topics; each topic is divided into ninelevels of difficulty, and for each level of difficulty there are anumber of snecific skills. 'By dividing the work into units of skills,mathematics is presented as a series of small, relatedtasks. A placement test determines the topic onwhich a pupil begins work, and at what level of difficulty. 'Progress is evaluated by short check-up tests following eachskill folder, and then by a post-test covering all theskills assigned for the unit." Commercially available diagnostic tests arealso often arranged by type or class of computationalexercise. Scores may reveal, for example, that a nupil is weak inadding fractions of any one or more of these classes: (1) denominators the same, (2) one denominator, the common denominator,(3) neither denomi- nator, the common denominator, (4) mixednumber plus a fraction, (5) mixed number plus mixed number. The study reported here was promptedby the belief that pupils vary not only in the types of computationalexercises they can do successfully but also inthe computational strategiesthey employ. It was thought that these strategies arehighly indi- vidualized and that they are often not revealed inthe pencil and paper answers to test items ordiagnostic exercises. If some insight could be gained into the thinking sequencefollowed by a pupil as he computes, perhaps someclues might be obtained as to when and how this thinking becomesfaulty. It was expected that such "patterns of thinking" might oftenbe quite unorthodox but for a particular pupil a satisfactorysubstitute for orthodoxy; while for another pupil a favorite strategycould be faulty arithmetically and destined to yield incorrect answers. it was further expected that computationalstrategies of success- ful computers differ substantially from those ofunsuccessful computers. Problems with a regroup_in canonical format Problems without a regroup in noncanonicalformat Problems with difference on both sides ofequal sign. 1The Individualized Nathematics System. Regional Education Laboratory for -the-taiOlinas and VirginiatChapel Pill and Duke Streets, Durham, dorth Carolina. 2 In brief this study was designed to examine thesebasic assumptions. 1. Patterns of thinkingcom)utational strategieswhich pupils develop in their study