This article was downloaded by: [Rich Shavelson] On: 16 September 2014, At: 16:20 Publisher: Routledge Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Educational Psychologist Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/hedp20 Teaching mathematics: Contributions of cognitive research Richard J. Shavelson a a Department of Education , University of California , Los Angeles, Calif., 90024 Published online: 01 Oct 2009. To cite this article: Richard J. Shavelson (1981) Teaching mathematics: Contributions of cognitive research, Educational Psychologist, 16:1, 23-44 To link to this article: http://dx.doi.org/10.1080/00461528109529229 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http:// www.tandfonline.com/page/terms-and-conditions
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This article was downloaded by: [Rich Shavelson]On: 16 September 2014, At: 16:20Publisher: RoutledgeInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK
Educational PsychologistPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/hedp20
Teaching mathematics: Contributions of cognitiveresearchRichard J. Shavelson aa Department of Education , University of California , Los Angeles, Calif., 90024Published online: 01 Oct 2009.
To cite this article: Richard J. Shavelson (1981) Teaching mathematics: Contributions of cognitive research, EducationalPsychologist, 16:1, 23-44
To link to this article: http://dx.doi.org/10.1080/00461528109529229
PLEASE SCROLL DOWN FOR ARTICLE
Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in thepublications on our platform. However, Taylor & Francis, our agents, and our licensors make no representationsor warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Anyopinions and views expressed in this publication are the opinions and views of the authors, and are not theviews of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should beindependently verified with primary sources of information. Taylor and Francis shall not be liable for any losses,actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoevercaused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content.
This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions
Teaching Mathematics: Contributionsof Cognitive Research
Richard J. ShavelsonUniversity of California, Los Angeles
Many elementary school teachers opt out of teaching mathematics forseveral reasons including an incomplete understanding of the structureof the subject matter and inadequate training in how to teach it. Thispaper outlines one possible way to think about teaching mathematics. Itdocs so by linking concepts and research on (a) subject-matter structure,(b) student cognitive structure, (c) teacher decision making, and (d)instructional context.
Many elementary school teachers opt out ofteaching mathematics. Rather, they select orhave selected for them the curriculum, viz. atextbook. If students arc fortunate, theirteachers will demonstrate, in clear, step-by-step fashion, how to solve a certain class ofproblems, assign exercises, and provideanswers to at least half the exercises so thechildren can immediately evaluate their under-standing. Most students arc not so fortunate.They arc assigned a lesson in their texts (theone which they arc not allowed to bringhome), told to do the odd numbered exercises,and not provided answers so as to avoidcheating (as if they couldn't figure out someother way to cheat if they were so disposed).
This reluctance to teach may, in part,account for why the Beginning TeacherEvaluation Study (BTES) (Fisher ct al., 1978)found that elementary school teachers, on theaverage, spent less than half the time on maththat they spent on reading. Elementary schoolteachers may avoid teaching mathematicsbecause they do not know how to teach it. Andmuch of the educational and psychologicalresearch on mathematics instruction has notprovided the necessary foundation.
This state of affairs can be remedied, in part,by providing a psychological basis for teachingmathematics. In doing so, four related topicsneed to be addressed: (a) the mathematical
This article is based on an invited address to the SpecialInterest Group in Mathematics Education at the meetingof the American Educational Research Association,Boston, Massachusetts, April 1980. The author wishes toexpress his thanks to Nick Branca and Bill Gesslin for theircritical comments on an earlier version of this paper.
The address of Richard J. Shavelson is Department ofEducation; University of California; Los Angeles, Calif.90024.
subject matter, (b) the student, (c) the teacher,and (d) the instructional context. As an initialstep, cognitive research is reviewed here. Thereview puts together bits and pieces of ideasabout and research on a psychological basis forteaching mathematics with no hope ofcompleting the mosaic. Addition and sub-traction serve as the primary examples butsome other topics could have been used aswell.
Subject Matter
In order to teach mathematics, teachers needto know the subject matter, contrary tofindings of very low correlations betweensubject-matter knowledge and teaching ef-fectiveness in past research on teaching (Gage,1963; Bcgle, Note 1). Indeed, one possiblereason many teachers avoid teaching math isbecause they have only a limited under-standing of what turns out to be a complexsubject matter — e.g., addition, subtraction,multiplication, division, measurement, andrational numbers.
Schwab (1962) defined subject-matter struc-ture as the bundle of facts and concepts, andtheir interrelations in a discipline. Workwithin this definition (e.g., Shavclson, 1972)has identified two closely related componentsof subject matter structure (cf. Grceno, 1978;Shavclson & Porton, 1979): * prepositionalstructure (sometimes called semantic structure)
1 Other components have also been identified but neednot concern us here (cf. Grceno, 1978; Shavelson &Porton, 1979).
Copyright 1981 by Division 15 of the American Psychological Association, Inc.
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24 RICHARD J. SHAVELSON
i s when
value
Figure 1. Representation of the meaning of a mathematical concept. (From Grccno, 1978. Usedhere with permission.)
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TEACHING MATHEMATICS 25
Table 1Verbal Problems Used by Carpenter and Moser [from Moser, Note 11, p. 2\. Used here with
permission.
Type of Problem
I. Increase ( + )
a. Action(Joining)
b. Static(Part-Part-Whole)
II. Decrease (-)
a. Action(Separating)
Example
Wally had 3 pennies. His father gavehim 6 more pennies. How many penniesdocs Wally have altogether?
Carol has 4 old shirts. She also has 7new shirts. How many shirts does Carolhave altogether?
Marie had 7 candles. She gave 5 candlesto Fran. How many candles did Marie have
b. Static
left?
There are 5 jars of paint. Three jarsarc red and the rest arc blue. How manyjars of blue paint arc there?
and procedural structure (sometimes calledalgorithmic structure).2
Propositional Structure
The prepositional structure of a subjectmatter refers to the meaning of mathematicalconcepts and operations. More accurately, itrefers to the verbal and visual representation ofmeaning. For example, the ideas that additioninvolves combining and subtraction involvesseparating arc part of the prepositionalstructure of mathematics. Figure 1 presentsGreeno's (1978, p . 266) representation of theprepositional structure for the "idea thatdivision is the inverse operation of multipli-cation." In this figure, "multiplication anddivision arc represented as actions that causechanges in the value of a quantity.''
Often there is more than one way torepresent prepositional structure. For example,Kiercn (in press) presented five models of theprepositional structure of the concept ofrational number (e.g., uninary operator,binary operator, and even a "fractioningmachine" analog). And a mathematicalproblem can be represented more accurately by
one form of prepositional structure thananother. From a cognitive point of view, thefact that alternative representations arcpossible has critical implications for whatstudents learn and the kinds of problems thatthey can solve. In particular, certain problemsmay be solved more readily from oneprepositional perspective than another.
For most students, then, the search forinstructional treatments that affect broadproblem solving capabilities may be pre-mature. The structure of the informationpresented to a student sets boundaries for mostof them which they usually do not cross. Inorder to understand the limits of problemsolving abilities, a taxonomy of mathematicaltasks is needed. In an adequate taxonomy,transfer would be expected within a welldefined class of tasks, but little if any transferwould be expected from one class of tasks toanother for most students without appropriateinstruction (cf. Scandura, 1977).
2Thc term algorithmic structure is avoided because itimplies that subject matter can be reduced to a finitenumber of steps leading to the solution of a class ofproblems. This is not the case in education ormathematics.
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26 RICHARD J. SHAVELSON
Studies by Carpenter and Moscr (Note 2)and Ncshcr (Note 3) demonstrated that thestructure of information in a task has aprofound influence on what students learn andwhat kinds of problems they can solve. In bothstudies, somewhat similar taxonomies forverbal addition and subtraction problems werepresented along with data bearing onelementary school students' performance ondifferent types of problems.
Carpenter and Moscr (Note 2) developed athree dimensional (2*) taxonomy with prob-lem characteristics varying as to whether theyincluded: (I) set inclusion, (II) action, and (III)an increase (or decrease) in number. Table 1presents the types of problems used in thestudy and locates the problems in thetaxonomy.
Problems varying on these three dimensionsas well as on the size of the numbers in theproblem (small = 5 to 9; large = 11 to 15)and the presence (or absence) of cubes(manipulatives) were given to 144 beginningfirst graders. Some of the results arc presentedin Figure 2.
The data in this figure suggest that (a)subtraction problems arc more difficult thanaddition problems, (b) action problems arcmore difficult than static problems in additionwhile just the reverse is true in subtraction, (c)accuracy decreases as the size of the numbersincrease, and (d) entering first graders knowhow to add small numbers reasonably well.While not shown in this figure, manipulativesimproved performance only on subtractionproblems with large numbers.
Neshcr (Note 3) also developed a threedimensional taxonomy for verbal addition andsubtraction problems with a text grammar forclassifying verbal problems in the taxonomy.The dimensions are: (a) a logical dimension(c.f. Carpenter and Moscr's increasing/decreasing plus a "constituent" for super-fluous information); (b) a semantic dimensionwith a contextual constituent (action/static/comparison) and a lexical constituent (words orphrases that cue or point to the appropriatearithmetic operation), and (c) a syntacticdimension (length of question, order ofwords).
Mean
Percent
Correct
90
80
70
60
50
40
30
20
10
0
SmallSize of Number
Large
Figure 2. Performance of 144 first gradcn on verbal addition (+) and subtraction (-) problemswith changes in sute caused by an action or no action (no manipulatives). (Data fromMoser, Note4, p. 5.)
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TEACHING MATHEMATICS 27
Of the three dimensions, Nesher found thatthe semantic dimension had the greatestinfluence on problem solving. Specifically, theinfluence of the contextual constituent of thesemantic dimension was different for additionproblems than for subtraction problems (seeTable 2, cf. Carpenter & Moser, Note 2). Themost difficult problems were comparisonproblems in addition and static problems insubtraction. "It seems then, that addition andsubtraction, despite the similarity in theirlogical structure, and despite the fact that bothemploy the same linguistic mechanism toencode their logical structure, arc different innature" (Ncsher, Note 3, p. 17).
Nesher also found that the lexical constit-uent of the semantic dimension influencedproblem solving, but the exact nature of thisinfluence depended upon the contextualconstituent of the verbal problem. Specifically,"the lexical constituent has a crucial role incomparison texts, but a small effect in dynamic(action) and static texts (for both addition andsubtraction problems). It might be that thedominant role of the VERBAL CUE inComparison texts is due to the fact that theverbal cue in such texts is not optional, butrather an integral part of the logical structure"(Ncsher, Note 3, p . 17).
Procedural Structure
Procedural structure refers to a set of rulesand heuristics that specify, at least partially,the stcp-by-stcp procedures leading from the
specification of a particular task (e.g., aproblem) to a goal state (e.g., a solution). Forexample, the "counting algorithm" foraddition is well known by many preschoolchildren (Gelman & Starkcy, Note 5;Ginsburg, Note 6). With this procedure,children arc taught to continuously countsubsets of objects or tally marks in order toobtain a sum. Procedural structure may bethought of as the application or performanceaspect of subject-matter knowledge.-3
An adequate representation of a subject-matter must include, but not be limited to,procedural structure. The new math, in part,grew out of the over-emphasis on proceduralstructure in traditional curricula. However, thestress placed on prepositional structure in thenew math was so great that certain importantperformance goals were not achieved. Over-emphasizing one component of subject-matterstructure at the expense of the other producespredictable outcomes, viz. the structuredictates, to a large extent, what people learnand how they apply it.
There is, of course, more than one procedurefor solving problems belonging to the sameclass. A procedure for subtraction withsequential borrowing — most often used inschool today — is presented in Figure 3.
The result of applying this procedure to theproblem 315. is shown in Figure 4a. An
-176
3The work of Scandura (e.g., 1977) and Landa (1976)focus on this attribute of the subject matter.
Table 2Mean Percent Correct on Verbal Problems [Data from Nesher, Note 3, p. 16]
*"Thc analysis of the addition texts, however, ended with only two distinct categories . . .Word-problems of addition containing verbs denoting disjointedness of the sets were notsignificantly different from the static texts" (Ncshcr, Note3,p. 16).
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28 RICHARD J. SHAVELSON
Find rightmostcolumn. Marki t as Activecolumn.
ctiveColumn
Is bottomtop7
N0-
H
Active column.Write 1 in frontof top digit.
YES
Active columnsubtract.
I
Move one columnlef t in top num-eral. Mark i t asborrow column.
Active column.Write resultbelow line.
Borrow column.Change 0 to 9.
Borrow column.Reduce top digitby 1.
M
Move borrowmarker 1 columnto l e f t .
'More \ NO(columns)
Read answerbelow line.
Move active markerone column to left .
Figure 3. Procedure for subtraction with borrowing (from Resnick, Note 12. Used here withpermission.)
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TEACHING MATHEMATICS 29
alternative procedure would be the "Bor-rowing decision procedure" where all theborrowing details are carried out before anysubtraction is done (sec Figure 4b). Perhaps aneven more reasonable procedure with theavailability of pocket calculators would be onethat gives a rough estimation of the problemsolution at each step so the result shown by thecalculator could be readily evaluated (Figure4c).
The important point is that differentprocedures evoke slightly different ideas aboutsubtraction and how it can be used to solveproblems. Or, put another way, differentprocedures fit better with different representa-tions of the prepositional structure of a subjectmatter or a task. If students can translate averbal problem (with its contextual and lexicalconstituents; cf. Neshcr, Note 3) into aprepositional structure which points to aprocedure, the probability that they will beable to solve the problem should increase. (The
linguistic structure of many simple, verbal,addition and subtraction problems readilypermits this translation while algebra wordproblems do not.) If the student cannottranslate the verbal problem into a structurethat matches his prepositional structure for(say) subtraction, an inappropriate procedureor no procedure may be used, leading to anerror. Moreover, when the information in averbal problem cannot be fitted easily to astudent's prepositional structure, most stu-dents cannot change the structure of the task tofit what they know about subtraction in orderto make the problem meaningful to them.
In summary: 1. The structure of the subjectmatter or problem greatly influences what islearned or what procedures arc used to solveproblems. There is more than one way torepresent a subject-matter structure anddifferent representations lead to differentunderstandings and different classes ofproblems that can be solved. While all possible
30*5•17 6
31015- 1 7 6
2V5- 1 7 6
2V5- 1 7 6
(a)
2V5-17 6
(b)
21115- 1 7 61 3 9
(O
Figure 4. Three procedures for subtraction with Borrowing: (a) sequential borrowing, (b)decision borrowing, and (c) calculator borrowing.
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30 RICHARD J. SHAVELSON
representations cannot be taught to children,teachers certainly should be familiar with themajor ones. Also, children invent conceptionsof mathematics and algorithms for solvingproblems. Some of these inventions areaccurate and creative; some arc not. Byinference, teachers should have sufficientunderstanding of arithmetic to recognize andreward creative inventions and to fix inaccurateones.
2. An adequate representation of thestructure of an area of mathematics wouldcontain components for prepositional structureand procedural structure along with the linksbetween the two. Omitting one or the otherwould produce curricula (and students) strongin one but weak in the other, much like thedifferences in old and new math. But moreimportantly, this omission would distort therepresentation of the subject matter structure.
3. Attempts arc being made to develop ataxonomy of tasks in school mathematics. Thework of Carpenter and Moser (Note 2), Ncsher(Note 3), and others (e.g., Scandura, 1977;Vergnaud, 1979, Note 7) represents abeginning. But rather than developing ataxonomy of problems, we need to develop ataxonomy which includes different representa-tions of prepositional structure (e.g., additionas a unary operation, binary operation) linkedto the procedural structures associated withthem. These categories would, in turn, belinked to classes of problems for which theyoffer solutions.
4. While teaching students to be generalproblem solvers in mathematics is a noble goal,we arc far from realizing the goal. Rather,learning and problem solving seem to be taskspecific and transfer occurs within a problemclass rather than between classes. Once ataxonomy is achieved such as that describedabove, the processes underlying generalproblem solving may be better understood.Once the taxonomy has been validated, it maybe possible to develop instructional treatmentsthat foster general problem solving abilities.
The Student
In teaching mathematics, dearly the teacherneeds to know something about the student.Research on teaching suggests that teachersthink about students in terms of their ability,motivation and classroom behavior (e.g.,
Shavelson & Borko, 1979; Shavclson, Note 8).This section focuses on student ability, albeitsomewhat differently than do teachers, since itfocuses on some important elements of aninformation-processing model of the student.(Motivation and classroom behavior aretouched on in the section on Context.)Specifically, three questions arc addressedbelow: (a) What arc some of the informationprocessing capabilities and limitations ofstudents? (b) How do students view thestructure of the task presented to them? And(c) how do students link their perception of thetask to procedural knowledge?
Information-processing Capabilities
Teachers need to know what they canreasonably expect their students to learn anddo. Much of the teacher's knowledge of thisson is gathered, first-hand, through ex-perience. Another source of this information isresearch on the information-processing charac-teristics of humans, especially that researchwith a developmental focus. There is spaceenough to touch on only two points. First,young children know more than often they aregiven credit for. Second, knowledge of theinformation processing limitations of childrenallows us to predict effective design ofinstruction for lessons we have not previouslyexperienced.
Children's mathematical knowledge. Re-search by Gclman and Starkey (Note 5) andGinsberg (Note 6) has shown that childrenknow more about mathematics than they arcgiven credit for. For example, Gelman andStarkey (Note 5) report that preschoolers: (a)count with considerable skill; (b) use a"counting algorithm" for addition (cf.Carpenter & Moser, Note 2); (c) understandthe directional effects of addition (i.e.,increase) and subtraction (i.e., decrease) onnumerosity; (d) understand, at least implicity,the principle of inversion — that subtractioncan cancel the effect of addition and vice versa;and (e) understand, at least implicitly, theprinciple of compensation — that if anumerical relationship between two quantitiesis changed by increasing (decreasing) onequantity, this relationship can be reinstated byincreasing (decreasing) the other quantity (cf.Gclman & Gallistel, 1978).
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TEACHING MATHEMATICS 31
Information-processing limitations. A wellknown fact is that humans' span of attention islimited. Research on human informationprocessing has shown that an adult, withoutspecial training, is able to remember about fiveto seven different pieces of information at onetime (e.g., Miller, 1956).4 Children also havelimited short-term memories. Case (Note 10,p. 18) argued that:
(1) as children progress through each [Piagetian]stage, they pass through a sequence of substages inwhich basic operations of which they arc capablebecome embedded in strategies of increasingcomplexity; (2) the addition of a new element . . .to a strategy requires the active mediation of thechild's attention; (3) a general requirement forsubstage trasition is an increase in the span ofattention or short term storage space; (4) a specificrequirement for stage transition is exposure to thesituation, with the rate of transition being affectedby the frequency, cuc-sailience, and developmentalmatch of the task to the child's current level.
Case's (Note 10) research has severalimportant implications for teaching mathema-tics. One implication is that the mathematicaltasks presented in instruction should fit theinformation processing capabilities of stu-dents. For example, procedures for solving acertain class of problems should not requiremore memory space for execution than thechild has available. One way of reducingdemands on memory is to let children use"external memories" in the form of blocks or"fingers." Some teachers and math educatorshave reached the same conclusion, butprobably on different grounds. The impor-tance of Case's work is that it provides a theorywhich predicts what may occur in situationsunfamiliar to teachers. A second implication isthat, in order to move students to morecomplex conceptions and operations, practiceon and ovcrlcarning of operations is essential.Practice automatizes operations and so reducesdemands on memory. In this way, storagespace for new information can be madeavailable in memory.
In sum, recent research on the informationprocessing capacity of children providesinformation about when to teach childrencertain mathematical concepts and proceduresand how to do it. Now there is a need tosystematically catalogue this rapidly growingbody of knowledge and to examine itsimplications for mathematics curricula andteaching.
Propositional Structure
In teaching mathematics, information abouta student's propositional knowledge is impor-tant for at least two reasons. First, thisinformation can be used in deciding how tolink new concepts with what the studentalready knows when planning and carrying outinstruction. Second, this information can beused for interpreting the student's misconcep-tions and errors in problem solving, and inplanning instruction to correct them.
In representing a student's propositionalknowledge, a number of diverse elements mustbe considered. Perhaps the most importantelement is the context in which this knowledgeis acquired and used. This article assumes thatthe context is young children learning to addand subtract. Other elements arc: (a)developmental level (sec above); (b) knowl-edge of key concepts such as combining,separating, and base 10 (with columnscorresponding to position codes); (c) priorexperience with counting objects, combiningobjects, separating objects, expanding objects,and shrinking objects; (d) general knowledgeabout and experience with the world; and (e)context in which addition and subtraction willbe applied. The research cited here and belowmay illuminate how some of these elementscome into play when information aboutpropositional structure is used in teachingchildren.
My colleagues and I have attempted toexamine students' propositional knowledge ofkey concepts in a mathematical (operational)system. In this research, a representation of astudent's propositional structure is obtained byasking her to indicate the distances between allpairs of key concepts in a subject matter andthen clustering the key concepts on the basis ofdistances. A representation of this structure ina subject matter is obtained by a linguisticanalysis of text which produces a directedgraph of the distances between key concepts.These distances arc submitted to clusteranalysis in order to represent structure. Thenthe correspondence between a representationof the (average) student's propositional
4With special training and practice, immediate memorycan be enlarged considerably (e.g.. Miller, 1956; Neisser,1976). For example, Hatano (Note 9) showed that twoabacus masters could remember from 5 to 7 words readfrom a list of unrelated words. However, these same twomasters could remember about 16 unrelated numbers (!).
structure and a representation of this structurein the subject matter (sec Figure 5a-c) isexamined.
We have found, as expected, that thecorrespondence gets increasingly closer asinstruction progresses (Sec Figure 5b-c; secBranca, 1980; Shavelson & Stanton, 1975; seealso Gecslin & Shavelson, 1975a, 1975b for the
subject matter of probability theory). There arehowever, several limitations of this research,the major one being that the diagnositc valueof such representations for individual studentsor even groups of students is unknown. Futurestudies might attempt to adapt instruction toindividual differences in students' similaritystructures (cf. Shavelson & Porton, 1979).
Figure 5. Hierarchical scaling representations of (a) content structuere, (b) cognitive structure ofstudents at pretest, and (c) cognitive structure of students at posttest (Date fromBranca, 1980, pp. 40-42)
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TEACHING MATHEMATICS 33
Another approach to representing proposi-tional structure is to observe students' problemsolving behavior and ask them what they arcdoing. For example, Moscr (Note 11, p. 2)observed first graders solving simple subtrac-tion problems and reported a hierarchy ofsolution strategics.' 'The lowest level is the useof sets of physical objects to represent problementities together with enumeration of thosesets. The second level is counting and the thirdis the use of unobserved mental processes.'' Atthe lowest stage for example.
The child begins the solution process by setting outphysical models to represent, on a one-to-one basis,one or both of the given sets in the problem . . .when manipulative! (cubes) were available, theywere utilized to do the modeling. In otherinstances, fingers were used as models [particularlywith small numbers]. After modeling, some actionwas performed. (Moser, Note 11, p. 2)
This description of the student's perceptionof the problem points up two problems withthis technique. First, it is much more difficultto model good problem solvers than poor ones(in math, physics, etc.) because the processesused by good problem solvers arc often too fastto be observed and the good problem solvershave difficulty explaining what processes theydid use. The second problem is that the child'srepresentation of the problem may tell us moreabout the task structure than her propositionalstructure. For example, Carpenter and Moser(Note 2, p. 30) found that "the dominantfactor in determining children's strategy [forsolving a subtraction problem] was thestructure of the problem. The strategy used bythe great majority of children modeled theaction or relationship described in theproblem."
Procedural Structure
The fact that children use procedures insolving (say) addition and subtraction prob-lems has been established by research showingthat "a substantial portion of children's errorsare the result not of random mistakes in (forexample) subtraction 'facts,' but of systematicfollowing of the wrong procedures" (Resnick,Note 12, p . 4). Information about a student'sprocedural knowledge is important because ittells us something about the cause of his errorsin solving problems and something about thenature of his propositional knowledge.
Psychological research on children's proce-dures for solving addition and subtractionproblems has progressed far enough so thatdifferent procedures have been catalogued. Forexample, Woods, Resnick, and Groen (1975)found that most children used one of twoprocedures for solving single digit subtractionproblems of form, m - n, where m is greaterthan or equal to n: (a) set a "mental counter"to m and then decrement it m - n times; or(b) either set the counter to n and increment itm - n times or set the counter to m and thendecrement it n times, depending on whichprocedure requires the fewest steps.
Children also invent correct proceduresunlike those that they were taught. Forexample, Groen and Resnick (1977) providedevidence of invention by young childrenlearning simple addition when exposure toinstruction was controlled. The children weretaught to add two numbers, m + n, by settinga "mental counter" to zero, incrementing it mtimes, then without resetting, incrementing itn more times. However, many childrenapparently set the counter to whichever of thetwo numbers is greater and then incrementedthis number by the smaller addend. Theinvented procedure is more complex but muchmore efficient than the one taught.
Research on children's procedures hasprogressed far enough so that crrorful or"buggy" procedures have also been cata-logued (e.g., Brown & Valehn, Note 13;Resnick, Note 12). With the "smaller fromlarger" procedure for subtraction for example,the student subtracts the smaller digit from thelarger digit in a column regardless of which oneis on top. For the subtraction problem used inthe last section — 3 1 5 - 1 7 6 — this wouldproduce the following result: 261. Othercommon "buggy" procedures arc given inTable 3.
links between Propositional and ProceduralStructures
Probably the most important area ofresearch on students' mathematical knowledgeis the research on links between propositionaland procedural structures. This is because, forexample, "children arc likely to know a gooddeal about the base [ten] system, butnevertheless be unable to use this semanticknowledge in support of written (symbolic)
procedures for arithmetic" (Resnick, Note 12,p. 12).
Resnick (Note 12, p. 25) proposed linkingtwo procedural structures with the intent ofindirectly influencing prepositional structure.The link between two procedural structureswas called mapping: "The extent to whichchildren had connected the knowledge theyhad about the base system, as displayed intheir work with the concrete materials, and theknowledge they had about the written(symbolic, e.g., 201 — 159) procedures forsubtraction and addition." She identifiedthree types of mapping: (a) code mapping —the ability to represent written numbers inconcrete form and vice versa; (b) expectationmapping — the extent to which the childexpects procedures in the symbolic system toyield the same answer as procedures usingconcrete forms; and (c) operation mapping —the extent to which a child can identifyequivalent operations in symbolic and concreteform.
While code mapping was observed in eachof her four subjects, none of the childreninsisted on abolute maintenance of the mapbetween written and concrete form. Forexample, when they had added two numberswith blocks or chips and had more than tenunits (e.g., 5 tens, 13 units), they wouldcount: "Fifty, fifty-one, fifty-two, fifty-three,. . . fifty nine [sic], sixty, sixty one [sic], sixtytwo [sic], sixty-three,' and then write thenumerals 6 and 3 to compose 63" (Resnick,Note 10, pp. 26-27).
There were distinct, individual differencesin expectation mapping. Two children "gaveclear evidence of expecting blocks and writingto yield the same answers" (Resnick, Note 12,p. 27). In contrast, "Andy . . . seemed to beundisturbed by getting different answers inblocks than in written subtraction" (p. 27).With respect to operation mapping,. childrendid not examine the steps they used to solveproblems with concrete forms as a way ofhelping themselves to solve written problems.
Table 3Some Common "Buggy" Subtraction Procedures Used by Students [from Resnick, Note 12],Used here with permission.
Procedure
Borrow from Zero:To borrow from a top digitof 0, change the 0 to 9.
0-N-N:If 0 is on top, theanswer is the lowernumber.
0-N = 0:If 0 is on top, theanswer is 0.
Stop Borrow at Zero:Add 10 to active column;0 - N = NorO - N = 0.
Ignore Zero in Borrowing:Borrow from nearest columnwith top digit 5* 0;0 - N = NorO - N = 0.
502-125
709-125
709-125
502-125
502-125
Example
5O9l2-12 5
709-125
624
709-125
604
50*2-12 5
47 2
*t0l2-12 5
32 7
OR
OR
50*2
40 7
4*0! 2-12 530 7
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TEACHING MATHEMATICS 35
An important problem for future research tosolve is how to represent the student'sperception of a possible solution to a problem.Such a representation would include the givensto a problem, the child's prepositionalstructure relevant to the problem, and theprocedures the child chooses among in order toreach a solution to the problem. This is thekind of information the teacher needs in orderto plan and carry out instruction, and to correctbuggy procedures (cf. Vergnaud, 1979)-
In summary: 1. Psychological research isproviding both useful theory and facts aboutchildren's capabilities and limitations inlearning arithmetic. The word useful describesthe theory and facts because they translate intohypotheses about how to teach arithmetic tochildren. Research and practical teachingexperience are needed to determine just howuseful these findings are.
2. Children learn procedures for solvingproblems and invent their own procedures. Insome cases, partial learning or faulty inventionlead to "buggy" procedures. When these"buggy" procedures arc contrasted with theprocedural structure of the subject matter,errors can be found and teachers have some ofthe information needed to correct theseprocedures.
3. The link between children's prepositionalstructures and procedural structures is atenuous one. It is greatly influenced by thestructure of the task (lesson, problem)confronting them. However, this link is themost critical if they arc to learn the subjectmatter.
The Teacher
I assume that mathematics teaching is aprocess by which teachers make and carry outreasonable decisions with the intent ofmaximizing students' knowledge of mathe-matics and their ability to solve problems(Shavelson, 1976, Note 8; Shavclson &Atwood, 1977; Shavclson & Borko, 1979;Shulman & Elstcin, 1975). While teachers'decision making docs not always match thisdescription,5 it applies to many goal-orientedteaching situations. In this context, the teacheris seen as an active agent with many teachingskills and techniques at his disposal to helpstudents reach some goal. In order to choosefrom this repertoire, he must integrate a large
amount of information about students from avariety of sources. And this information mustsomehow be combined with his beliefs andgoals, the nature of the instructional task, theconstraints of the situation, and so on.
A schematic representation of these ideasabout teaching is shown in Figure 6. Thisfigure identifies some important factors whichmay affect teachers' pedagogical decisions.Teachers have available a large amount ofinformation about their students from manysources (e.g., their own informal observations,anecdotal reports of other teachers, test scores,school records). In order to handle theinformation "overload" teachers itegratc thisinformation into a few estimates about thestudent's cognitive, affective, and behavioralstates (cf. Shavclson & Atwood, 1977). Theseestimates about students arc expected toinfluence a teacher's plans for instruction andthe decision the teacher makes, consciously orunconsciously, during instruction.
Individual differences between teachers suchas educational beliefs (e.g., Kerlingcr &Pedhazur, 1968) and cognitive styles (e.g.,Witkin, Moore, Goodenough & Cox, 1977)influence both their identification andintegration of information about students andtheir pedagogical decisions. For example,teachers' beliefs about education are expectedto affect their decisions directly by limiting thetypes of instructional strategies which will beconsidered when making a decision. Thesebeliefs may also affect decisions indirectly by.influencing the types of estimates aboutstudents which are used to integrate thisinformation.
The nature of the instructional task, thesubject matter or problem to be solved,themselves, present decision problems. Forexample, given alternative curricula that couldbe used for a semester's math lessons, how isone chosen? Further, the curriculum is alsoexpected to influence decisions about instruc-tion directly and indirectly by limiting thealternative instructional strategics which teach-ers consider when planning a lesson. Thenature of the instructional- task may alsoindirectly affect these decisions by influencingthe information about students to which theteacher attends and the estimates about
5 In many cases, and I suspect math teaching is a primeexample, teachers make decisions so as to minimizepsychological stress. Hence, seat work is commonly foundin many classrooms while the teacher grades papers, etc.
Figure 6. Some factors contributing to teachers' pedagogical judgments and decisions.
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TEACHING MATHEMATICS 37
students that arc considered relevant to thedecision.
Finally, the availability of alternativeteaching strategies and materials is alsoexpected to influence decisions directly bylimiting the set of possible alternatives fromwhich the teacher is able to choose.Availability may be influenced by the natureof the instructional task, the teacher'srepertoire of strategies, constraints imposed bythe institutional environment (e.g., school andclassroom, policies) and so on.
In linking these vague ideas to concreteteaching situations, two examples have beenchosen because they illustrate well some of theelements in the decision schema (sec Figure 6).The first example is the diagnosis of a"buggy" procedure and the second is aninstructional sequence that links prepositionalknowledge with procedural knowledge.
Diagnosis of a' 'Buggy" Procedure^
Teachers arc often confronted with studentswho come up with a creative, albeit erroneous,solution to a problem. In order to decide howto correct the student's "buggy" procedure,the teacher must estimate the student'sprobable state of mind (cf. Figure 6). In thiscase, the estimate would be the most probable"buggy" procedure or conceptual flaw thatwould explain the observed error. In ascer-taining the nature of the error, the teachermust compare the incorrect solution with (a)the steps that would be used in the applicationof a correct procedure and (b) the prepositionalstructure underlying the procedure.
In order to transport you to a diagnosticteaching frame of mind so you can see how thedecisions schema works, here's one of Charlie'screations: 17 . What might have given rise
+213
to this mistake? One possible explanation isthat Charlie added 7 + 5 and got 2 carry 1. Hethen added 2 + carry 1 and got 3 indicating aprocedural error, a conceptual error or both.Hence, 17 + 5 = 13. This reasoning is shownin Figure 7a. Suppose now you decide to testyour estimate of Charlie's state of mind bygiving him two additional problems. Theresult is encouraging:
18 43+ 6_ +7915 "23"
At this point, most teachers would decide tostop the testing and decide how to fix the"buggy" procedure, the conceptual error, orboth. This decision, however, illustrates animportant characteristic of human decisionmaking (Einhorn & Hogarth, 1978) observedin teaching. People attempt to confirmhypotheses rather than disconfirm them.Hence the two subsequent problems chosen totest Charlie were similar in form to the fustproblem, rather than being quite different,such as 51 + 1701. Confirmation is not themost efficient way to treat the decisionsituation because the teacher may miss thedisconfirming case and so attempt to fix thewrong buggy procedure. Figure 7b summarizesthe discussion to this point.
Suppose that instead of beginning remedialinstruction, an additional problem was givento Charlie which differed just slightly from theform of problem given before: 21 + 39 = ?.Charlie's answer was 15! According to thepresent hypothesis about the state of Charlie'smind, he should have answered 51. Now whatis Charlie's state of mind?
One possible procedure that will producethe answers to all of the problems given toCharlie so far is to add each of the digits in theproblem. For example, 21 + 39 = 2 + 1 + 3+ 9 = 15 (!). Figure 7c summarizes thediscussion to this point. In order to test thenew hypothesis, suppose two very differentaddition problems were given to Charlie withthe following result:
95+ HS
20
At this point, the teacher has probablycorrectly estimated the student's state of mindand is now in a position to decide how to fixthe buggy procedure.
In addition to illustrating teacher's decisionmaking, this simple example highlights thecomplexity of the teacher's job. In decidinghow to help Charlie, the teacher must do morethan simply ask another student to provide theanswer, as is commonly done in classrooms (cf.Good & Grouws, 1977). The teacher mustdiagnose the underlying cause — the "buggy''procedure along with Charlie's misconception
^This section draws on the work of Brown (e.g.. Brown& Burton, 1977).
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38 RICHARD J. SHAVELSON
of base 10 and arithmetic syntax — in order todecide how to help him.
The task is further complicated for theteacher by its structure. The teacher's firsthypothesis is most likely to include aborrowing bug since that is what the taskdemands. And the teacher, like most people,is prone to try to confirm the hypothesis. Afterall, he docs not have all day to figure outCharlie's problem. Only by chance or by
systematically trying to disconfirm his hypoth-esis could the teacher reach the correctdiagnosis. Given the correct diagnosis, theprobability that the teacher will successfully fixthe buggy procedure increases.
Linking Propositional and Procedural Structure
The link between students' propositionalknowledge and their procedural knowledge is
Information:17 + 5 = 13
Task Structure:Relevant propositionalknowledge (e.g., base 10)and procedural knowledge(addition procedure).
Estimate of StudentState:m + n = z, carry;z + carry = ans.
Decision:*1. Fix bug (most
would do)?2. Test some more?
(b)
Information:21 + 39 = 15
Task Structure:See (a)
Estimate of StudentState:? or add all digitstogether
Decision:1. Test some
more?2. Seek help?
(c)
Information:51 + 1707 = 3633 + 99 = 2465 + 6 = 20
Task Structure:See (a)
Estimate of StudentState:
+ * | j = a + b + c + d
Decision:Choose instructionalsequence.
(d)
Figure 7. Sequence of decisions in discovering "buggy" procedure.
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TEACHING MATHEMATICS 39
often quite tenuous. Resnick (Note 12, p . 30)has developed a teaching procedure that"explicitly forces a mapping — at theoperational level between block subtractionand written subtraction. The procedurerequired that Leslie perform the same problemusing [multibasc blocks] and writing, alter-nating her operations between the tworepresentations."
In this study, Resnick's goal was to helpLeslie relate her prepositional knowledgeabout the base 10 system (represented by heruse of multibase blocks) to a written procedurefor doing subtraction with borrowing. Resnickhad observed bugs in Leslie's subtraction; infact, Leslie was in need of rememdial work onsubtraction with regrouping when she partici-pated in the study. While Leslie had fairlyaccurate knowledge of the base 10 system,Resnick found that this prepositional knowl-edge had not been linked to subtraction
procedures. So she decided to force this linkinstructionally. In doing so, she assumed thatLeslie's use of multibasc blocks revealedsomething of her knowledge of the base 10system and that subtraction meant separatingor decrementing. This led to an instructionalprocedure which:
(a) made use of a blocks routine specificallyorganized so as to match, almost one-for-one, thewritten subtraction routine — that is for everymove in blocks there was a corresponding move inwriting and vice versa; (b) forced her attention tothis correspondence by requiring that each step inthe blocks procedure be immediately notated inthe written problem; and (c) eventually "faded"the blocks away leaving her to work with only thewritten routine — presumably now enriched byincorporation of the semantics inherent in theblocks (Resnick, Note 12, p. 31).
An example of this instructional procedure,with Leslie solving the problem, 85 - 47 = , ispresented in Table 4.
Table 4Example of Instructional Procedure for the Problem, 8531-33]
47 = ? [from Resnick, Note 12, pp.
1. Represent the 85 with [multibase] blocks
2. Start in the units column and try to remove the 7 blocks shown in the subtrahend.
3. There aren't enough blocks there, so go to the tens column and "borrow" (remove) aten-bar.
4. On the written problem, cross out the 8, and write 7, to show the change in blocks in thetens column: 7jET5
5. Trade the ten-bar for 10 units-cubes and piacc them in the units column.
6. On the written problem, represent this by writing a 1 that changes the 5 to 15:•4 7
7. Now remove the number of blocks shown in the ones column of the subtrahend.
8. Count the number of ones blocks remaining, and write the answer in the ones column of thewritten problem: 7jgrl5
-4 78
9. Go to the tens column and try to remove the number of blocks shown in the subtrahend.
10. Since there arc enough blocks, complete the operation, count the blocks remaining andwrite the answer in the tens column of" the written problem: 7^15
Figure 8. Decision analysis of teaching Leslie to link propositional knowledge with proceduralknowledge.
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TEACHING MATHEMATICS 41
This description of Rcsnick's study has beencast into a teacher's decision schema as shownin Figure 8. Notice that the teacher'sknowledge of subject matter structure plays akey role in both the diagnosis of Leslie'sproblem and in the development of theinstructional procedure. Also notice thatdiagnosis, on the basis of information aboutLeslie, is essential in order to decide how toremediate Leslie's problem.
The teaching experiment was successful.Rcsnick (Note 12, p . 39) concluded herdiscussion of the experiment by pointing outthat "we need to know whether it works formore children, and for what kinds ofpresenting difficulties with subtraction andaddition [i.e., states of mind regardingsubtraction and addition]." She also notedthat while Leslie made great progress, she maystill have difficulties with subtraction. And shepointed out that there may be certain bugsinherent in the blocks that "may unwittinglyalso be adopted into the writing algorithm"(p. 39). Finally, she suggested other possiblerepresentations — sticks, abacus — that maybe equally or more powerful than the blocks.
In summary: 1. Teaching is a complex,decision-making and implementing process. Itrequires a thorough knowledge of the structureof the subject matter, mathematics in this case.It also requires a thorough knowledge of howstudents process information and solveproblems. By combining subject matterknowledge with an information processingmodel of the student, instruction can bereasonably planned and revised.
2. The schema for teachers' decision-makingintegrates much of what has been discussed upto this point in the article. It also suggeststopics that have been left out (e.g., individualdifferences, alternative instructional strategies,peer influences, school policies).
Context
Context refers to the environment in whichthe teacher teaches the subject matter to thestudent. It is a hierarchical concept withcentral, proximal and distal aspects. Thecentral aspects of context arc the teacher'spurposive manipulations of the classroomenvironment which directly mediate instruc-tion to the student. Examples of central aspectsof context arc the use of multibase blocks to
teach subtraction with regrouping, the use ofsmall groups to foster problem solving, or thechoice of textbook and lesson to teachmultiplication of fractions. The proximalaspects of context include materials (e.g., mathgames and other <a#a»cf instructional material)and personnel (classroom aide) which can beused in support of a lesson. And distal aspectsof context refer to school or district policieswhich indirectly affect instruction. Policiesregarding textbook adoption, ability tracking,and allocation of time for math instruction areexamples of distal aspects of context.
The most frequent decisions teachers makein planning instruction concern context(Peterson & Clark, 1978), especially the choiceof content presentation. Context is theteacher's stock in trade — how to manipulatethe instructional environment so as to makeinstruction meaningful to students while alsoproviding an accurate representation of thesubject matter.
While research and folklore abound on thetopic of context, one example of recentresearch should suffice to highlight theimportance. Webb (1980a, 1980b, in press)taught eleventh graders "to calculate analgebraic expression for the n*h polygonalnumber. The »'^ polygonal number isithetotalnumber of dots in an array of polygons, inwhich the outermost polygon has » dots ineach side" (Webb, 1980b, p . 11). Figure 9presents an array for the nth triangularnumber. The number of dots in the figureforms an arithmetic scries: 1 + 2 + 3 + . . . .The n*h triangular number is the sum of thisseries: (n + n^)/2 where n is the number ofdots on a side.
The study was carried out in three phases. Inphase 1, students worked individually,learning the components of the algorithmneeded to solve complex problems like the onedescribed above. In phase 2, students workedin groups of four. Half the groups werehomogeneous with regard to ability (4 high, or4 middle, or 4 low) while the other half washeterogeneous (1 high, 2 middle, 1 low). As agroup, students were asked to find the n*"pentagonal number, the »** hexogonalnumber, and the n*" octogonal number. Inphase 3, students returned to their seats andwere asked to find the nf& triangular numberand on a delayed test, the n*h square number.
Webb found that the high- and low-abilitystudents performed better in hcterogenous
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42 RICHARD J. SHAVELSON
Figure 9. Array for the «th triangular number.
groups while the middle-ability studentsperformed better in homogeneous groups. Ingeneral, "Better performance was associatedwith active verbal participation in the group.In mixed ability groups (for example),high-ability and low-ability students interactedwith each other; highs helped the lows. Unlessmedium-ability members aggressively askedfor explanations or took part in the explaining,they were ignored" (Webb, 1980b, p. 10).
The methods Webb used in studying theeffects of different grouping strategies onproblem solving were just as important arc herfindings. In addition to the usual comparisonof group means, she collected extensiveinteraction data. From these, she was able todescribe the roles students took in the groups(Webb, 1980b) and the algorithms groupsfollowed to solve problems (Webb, 1980a, inpress). For example, in solving for the ti*6pentagonal number, the following roles wereassumed, by members of one of theheterogeneous groups. High and Middle-1assumed responsibility for solving the problemwhile Middle-2 and Low watched. High andMiddle-1 "argued about whether the layers ofdots in the array included dots in the previouslayers" (Webb, 1980b, p. 12) with Highcorrectly prevailing. No other errors arose inthe problem solution. Not until the problemwas solved did High and Middle-1 offer anexplanation to Middle-2 and Low. Then theyturned to Low and offered an explanation. ' 'Atno time during work on the first problem didanother group member ask Middle-2 whethershe understood how to solve the problem"(Webb, 1980b, p. 12).
Research, then, suggests that the context inwhich mathematics is taught greatly affects: (a)who learns the materials, (b) which proceduresare used to solve problems (and so whichproblems can be solved), and (c) the meaning(prepositional structure) of the mathematicsfor the student. Research, however, has nottaken us beyond this generalization because, todate, accumulation of findings on context hasbeen difficult to achieve. Nevertheless, inorder to learn how to teach mathematics,knowledge, perhaps in the form of ataxonomy, must be accumulated about theeffects of different contexts on differentstudents, on the procedures they use to solveproblems, and on the meanings the mathe-matics has for them.
The difficulty of accumulating findingsarises for several reasons. First, much of theresearch has focused on comparisons of meandifferences between groups of students taughtin different contexts almost to the exclusion ofan examination of the processes (group ormental) which arise in each context. Second,different contexts may give rise to differentmeanings for the mathematics and differentprocedures for solving problems. Often in theresearch, this effect of context has not beencontrolled. Hence, comparisons may notreflect comparable treatments. Third, asWebb's data showed, different contexts mayaffect different students in different ways.Careful attention should be paid to thepossibility of aptitude-treatment interactionsin naturally occurring classroom settings.
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TEACHING MATHEMATICS 43
Conclusion
In teaching mathematics, and in doingresearch on mathematics teaching, carefulattention needs to be paid to the fourcommonplaces of education: subject matter,students, teachers, and contexts. The fourcommonplaces are inextricably intetwined andthe potential contribution of cognitive psy-chology is clear.
Reference Notes
1. Begle, E. G. Teacher knowledge and student achieve-ment in algebra. (SMSG Report No. 9). Stanford,CA: School Mathematics Study Group, 1972.
2. Carpenter, T. P., & Moser, J. M. The development ofaddition and subtraction problem solving skills. Paperpresented at the Wingspread Conference on Number,Racine, Wisconsin, 26-29 November, 1979.
3. Nesher, P. Levels of description in the analysis ofaddition and subtraction. Paper presented at theWingspread Conference on Number, Racine,Wisconsin, 26-29 November, 1979.
4. Moser, J. How well do young children solve verbalproblems? (Interpretive report No. 2). MathematicsWork Group, Wisconsin R & D Center, 1980.
5. Gelman, R., & Starkey, P. Addition and subtractionalgorithms in preschool children. Paper presented atthe Wingspread Conference on Number, Racine,Wisconsin, 26-29 November. 1979.
7. Vergnaud, G. A classification of cognitive tasks andoperations of thought involved in addition andsubtraction problems. Paper presented at theWingspread Conference on Number, RacineWisconsin, 26-29 November, 1979.
8. Shavelson, R. J. A model of teacher decision making.Paper presented at the meeting of the AmericanEducational Research Association, Toronto, Ontario,31 March, 1978.
9. Hatano, G. Learning to add and subtract: A Japaneseperspective. Paper presented at the WingspreadConference on Number, Racine, Wisconsin, 26-29November, 1979.
10. Case, R. Intellectual development, maturation, andthe timing of instruction. Paper presented at theWingspread Conference on Number, Racine,Wisconsin, 26-29 November, 1979.
11. Moser, J. A first look at the solving of verbal additionproblems. (Interpretive report No. 3). MathematicsWork Group, Wisconsin R. & D. Center, 1980.
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13. Brown, J. S., & Vanlehn, K. Towards a generativetheory of "bugs". Paper presented at the WingspreadConference on Number, Racine, Wisconsin, 26-29November, 1979.
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