Arithmetic and Algebra in Early Mathematics Education Author(S): David W

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Arithmetic and Algebra in Early Mathematics Education Author(s): David W. Carraher, Analúcia D. Schliemann, Bárbara M. Brizuela and Darrell Earnest Source: Journal for Research in Mathematics Education, Vol. 37, No. 2 (Mar., 2006), pp. 87-115 Published by: National Council of Teachers of Mathematics Stable URL: http://www.jstor.org/stable/30034843 . Accessed: 05/03/2014 16:04

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This content downloaded from 137.123.124.4 on Wed, 5 Mar 2014 16:04:53 PM All use subject to JSTOR Terms and Conditions Journalfor Research in Mathematics Education 2006,Vol. 37, No. 2, 87-115

Arithmeticand Algebrain Early MathematicsEducation

David W. Carraher TERC AnalhciaD. Schliemannand Bfirbara M. Brizuela TuftsUniversity DarrellEarnest TERC

Algebrainstruction has traditionally been postponed until adolescence because of historicalreasons (algebra emerged relatively recently), assumptions about psycho- logicaldevelopment ("developmental constraints" and "developmental readiness"), anddata documenting thedifficulties that adolescents have with algebra. Here we provideevidence that young students, aged 9-10 years, can make use of algebraic ideas andrepresentations typically absent from the early mathematics curriculum and thoughttobe beyond students' reach. The data come from a 30-month longitudinal classroomstudy of four classrooms ina publicschool in Massachusetts, with students betweenGrades 2-4. The data help clarify the conditions under which young students canintegrate algebraic concepts and representations intotheir thinking. It is hoped thatthe present findings, along with those emerging from other research groups, will providea research basis for integrating algebra into early mathematics education. Keywords: Algebra; Children's strategies; Developmental readiness; Early algebra; Functions;Mathematics K-12

INTRODUCTION

Increasingnumbers of mathematicseducators, policymakers, and researchers believethat algebra should become part of theelementary education curriculum. The NationalCouncil of Teachers of Mathematics[NCTM] (2000) and a special commissionof the RAND Corporation(2003) have welcomedthe integration of algebrainto the early mathematics curricula. These endorsements do notdiminish theneed for research; quite the contrary, they highlight the need for a solidresearch base forguiding the mathematics education community along this new venture. This articlewill present partial findings from an investigationof eight- to ten-year-old students'algebraic reasoning during a 2 1/2year classroom intervention study. We

Thiswork was supportedby grant#9909591 from the National Science Foundationtothe project "Bringing Out the Algebraic Character ofArithmetic." Wethank Pat Thompson and Judah Schwartz for their contribution as consultants tothe project and Anne Goodrow and Susanna Lara-Roth for help in data collec- tion.Preliminary analyses of the present data appeared in Carraher, Schliemann, andBrizuela (2001) and in Carraher, Brizuela, and Earnest (2001).

This content downloaded from 137.123.124.4 on Wed, 5 Mar 2014 16:04:53 PM All use subject to JSTOR Terms and Conditions 88 Arithmeticand Algebra in Mathematics Education undertookthis investigation in thehopes of finding evidence that young students can learnmathematical concepts and representationsoften thought to lie beyond theirreach. Our approachto earlyalgebra has been guidedby theviews that generalizing lies at theheart of algebraic reasoning, arithmetical operations can be viewedas functions,and algebraic notation can lendsupport to mathematical reasoning even among youngstudents. We focus on algebra as a generalizedarithmetic of numbersand quantitiesin whichthe concept of function assumes a majorrole (Carraher,Schliemann, & Schwartz,in press). We view the introductionof algebrain elementaryschool as a movefrom particular numbers and measures towardrelations among sets of numbersand measures,especially functional relations. Functionshave deservedlyreceived increasing emphasis in middleand high schooltheorization, research, and curricula (e.g., Dubinsky & Harel,1992; Schwartz & Yerushalmy,1992; Yerushalmy & Schwartz,1993). We proposethat giving func- tionsa majorrole in the elementary mathematics curriculum will help facilitate the integrationof algebrainto the existingcurriculum. Key to our proposalis the notionthat addition, subtraction, multiplication, and divisionoperations can be treatedfrom the start as functions.This is consistentwith Quine's (1987) viewthat "a functionis an operator,or operation" (p. 72). Theidea is notto simply ascribe algebraic meaning to existing early mathematics activities,that is, to regard them as alreadyalgebraic. Existing content needs to be subtlytransformed inorder to bring out its algebraic character. To someextent, this transformationrequires algebraic symbolism. Even in early grades, algebraic nota- tioncan play a supportiverole in learning mathematics. Symbolic notation, number lines,function tables, and graphs are powerful tools that students can use tounder- standand express functional relationships across a widevariety of problem contexts. In thisarticle, we willfocus on thirdgraders' work with number lines and algebraic expressionsas theysolve problems in the domain of additive structures. We provide evidencethat young students can makeuse ofalgebraic ideas andrepresentations thatare typically omitted from the early mathematics curriculum and thought to be beyondtheir reach. Because we believethat functions offer a primeopportunity for integratingalgebra into existing curricular content, we also attemptto clarify what we meanby treating operators as functions. Beforewe presentthe results of our intervention study, we willreview selected mathematicaland psychological ideas relevantto thesuggestion that algebra has an importantrole in thepresent-day early mathematics curriculum.

EARLY ALGEBRA FROM MATHEMATICAL AND COGNITIVE PERSPECTIVES

Discussionsabout early algebra tend to focuson thenature of mathematics and students'learning and cognitivedevelopment. We willreview background issues alongthese two lines.

This content downloaded from 137.123.124.4 on Wed, 5 Mar 2014 16:04:53 PM All use subject to JSTOR Terms and Conditions D. W.Carraher, A. D. Schliemann,B. M. Brizuela, and D. Earnest 89

On theNature ofMathematics: Are Arithmetic and Algebra Distinct Domains?

The factthat algebra emerged historically after, and as a generalizationof, arith- meticsuggests to manypeople thatalgebra ought to followarithmetic in the curriculum.However obvious this claim may seem,we believethere are good reasonsfor thinking otherwise. Assume for the moment that arithmetic and algebra aredistinct topics. For example, let us assumethat arithmetic deals with operations involvingparticular numbers, whereas algebra deals withgeneralized numbers, variables,and functions. Such a distinctionallows for a tidyordering of topicsin thecurriculum. In elementaryschool, teachers can focus upon number facts, computationalfluency, and wordproblems involving particular values. Onlylater are lettersused to standfor any number or for sets of numbers. It is notsurprising that sucha sharpdemarcation leads to considerabletension along the frontier of arithmeticand algebra.It is preciselyfor this reason that many mathematics educators (e.g.,Filloy & Rojano,1989; Herscovics & Kieran,1980; Kieran 1985; Rojano, 1996; Sutherland& Rojano,1993) havedrawn so muchattention to the supposed transi- tionbetween arithmetic and algebra--a transition thought to occurduring a period in whicharithmetic is "ending" and algebra is "beginning."Transitional or "preal- gebra"approaches attempt to amelioratethe strains imposed by a rigidseparation ofarithmetic and algebra. However, "bridging or transitional proposals" are predi- catedon an impoverishedview of elementary mathematics-impoverished intheir postponementofmathematical generalization until the onset of algebra instruction. Studentsevidence difficulties inunderstanding algebra in their first algebra course. Butthere is reasonto believe that their difficulties are rooted in missedopportuni- tiesand notions originated in theirearly mathematics instruction that must later be "undone,"such as theview that the equals sign means "yields" (e.g., Kieran, 1981). Consider,for example, the opportunity to introducethe concept of functionin thecontext of addition. The expression"+3" can representnot only an operation foracting on a particularnumber but also a relationshipamong a setof input values anda setof output values. One can representthe operation of adding through stan- dardfunction notation, such asf(x) = x + 3, ormapping notation, such as x - x + 3. Adding3 is thustantamount to x + 3, a functionof x. Accordingly,the objects of arithmeticcan then be thoughtof as bothparticular (if n = 5, thenn + 3 = 5 + 3 = 8) andgeneral (n + 3 representsa mapping of Z ontoZ). Iftheir general nature is high- lighted,word stories need not be merelyabout working with particular quantities but withsets of possible values and hence about variationand covariation. Arithmeticcomprises number facts but also the general statements ofwhich the facts areinstances. We aresuggesting that arithmetic has an inherentlyalgebraic character in that it concernsgeneral cases and structuresthat can be succinctlycaptured in algebraic notation.We wouldargue that the algebraic meaning of arithmetical operations is notoptional "icing on thecake" but rather an essentialingredient. In thissense, we believethat algebraic concepts and notationneed to be regardedas integralto elementarymathematics.

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We are notthe first to suggestthat algebra be viewedas an integralpart of the earlymathematics curriculum. Davis (1985, 1989) arguedthat algebra should beginin Grade 2 or3. Vergnaud(1988) proposed that instruction inalgebra or preal- gebrastart at the elementary school level to better prepare students to deal withthe epistemologicalissues involved in the transition from arithmetic toalgebra; his theo- rizingabout conceptual fields provided the rationale for a mathematicseducation whereconcepts are treated as intimatelyinterwoven instead of separate. Schoenfeld (1995), in thefinal report of theAlgebra Initiative Colloquium Working Groups (LaCampagne,1995), proposesthat instead of appearingin isolatedcourses in middleor highschool, algebra should pervade the curriculum. Mason (1996) has forcefullyargued for a focuson generalizationat theelementary school level. Lins and Gimenez(1997) notedthat current mathematics curricula from K-12 provide a limitedview of arithmetic. Kaput (1998) proposedalgebraic reasoning across all gradesas an integratingstrand across the curriculum and the key for adding coher- ence,depth, and power to school mathematics, eliminating the late, abrupt, isolated, and superficialhigh school algebra courses. Similar arguments have been devel- opedby Booth (1988), Brownand Coles (2001), Crawford(2001), Henry(2001), andWarren (2001). In keepingwith researchers' and educators' calls, the NCTM, throughThe AlgebraWorking Group (NCTM, 1997) and theNCTM Standards (2000), proposethat activities that will potentiallynurture children's algebraic reasoningshould start in thevery first years of schooling. Butdo youngstudents have the capacity for learning algebraic concepts? Let us lookat what research tells us aboutlearning and cognitive development as itrelates to thelearning of algebra.This brief review will set the stage for the presentation and analysisof our own data.

On theNature of Students' Learning and CognitiveDevelopment: ClaimsAbout Developmental Constraints

Whenstudents experience pronounced difficulties in learningalgebra (see, for example,Booth, 1984; Da Rocha Falcio, 1993; Filloy& Rojano, 1989; Kieran, 1981,1989; Kuchemann, 1981; Resnick, Cauzinille-Marmeche, & Mathieu, 1987; Sfard& Linchevsky,1994; Steinberg, Sleeman, & Ktorza,1990; Vergnaud, 1985; Vergnaud,Cortes, & Favre-Artigue,1988; and Wagner,1981), one naturally wonderswhether this is due to developmentalconstraints or whetherthe students have simplynot achieved the necessary preparation. (Developmental constraints are impedimentsto learningthat are supposedlytied to insufficientlydeveloped mentalstructures, schemes, and general information-processing mechanisms. They are termed"developmental" to implythat they are intimatelytied to gradually emergingstructures that serve a widevariety of functions in mentallife.) Developmentalconstraints refer to presumedrestrictions in students'current cognitivecompetence (i.e., "Untilstudents have reached a certaindevelopmental levelthey presumably cannot understand certain things nor will they be ableto do so inthe near future"), not simply their performance (i.e., "they did not use the prop-

This content downloaded from 137.123.124.4 on Wed, 5 Mar 2014 16:04:53 PM All use subject to JSTOR Terms and Conditions D. W.Carraher, A. D. Schliemann,B. M. Brizuela, and D. Earnest 91 erty").They are associated with the expressions "(developmental) readiness" and "appropriateness."Algebra has sometimesbeen thoughtto be "developmentally inappropriate"for young learners, lying well beyond their current capabilities. Attributionsofdevelopmental constraints have been made by Collis (1975), Filloy and Rojano (1989), Herscovicsand Linchevski(1994), Kuchemann(1981), and MacGregor(2001), amongothers. Filloy and Rojano (1989) proposedthat arith- meticalthinking evolves very slowly from concrete processes into more abstract, algebraicthinking and that there is a "cut-pointseparating one kind of thought from theother" (p. 19). Theyrefer to thiscut as "a breakin thedevelopment of opera- tionson theunknown" (p. 19). Alongthe same lines, Herscovics and Linchevski (1994) proposedthe existence of a cognitivegap betweenarithmetic and algebra, characterizedas "thestudents' inability to operatespontaneously with or on the unknown"(p. 59). Althoughthey recognized that young children routinely solve problemscontaining unknowns (e.g., "5 + ? = 8"), theyargued that students solve suchproblems without having to representand operateon theunknowns; instead, theysimply use countingprocedures or the inverse operation to producea result. Althoughsome (e.g., Sfard & Linchevski,1994) have considered use ofthe inverse operationas evidenceof earlyalgebraic thinking, others have consideredthis procedureas merelyprealgebraic (e.g., Boulton-Lewis et al., 1997).

HistoricalSupport for the Idea ofDevelopmental Constraints Parallelsbetween historical developments and the learning trajectories ofstudents have providedsome support to thenotion that students' difficulties with algebra reflectdevelopmental constraints. For example,researchers have used Harper's (1987) insightfulanalysis of thehistorical evolution of algebra-throughrhetor- ical,syncopated, and symbolic stages-to framethe evolution of student algebraic competence.Sfard (1995) andSfard and Linchevski (1994) havefound connections betweenhistorical and individualdevelopments in mathematicsin theirtheory of reification,which attempts to clarifythe psychological processes underlying the developmentof mathematical understanding, including algebra. Likewise, Filloy and Rojano (1989) providedhistorical evidence for their idea of a "cut-point" separatingarithmetic from algebra and arguedthat something analogous to this occursin present-daymathematics education at thelevel of individual thought.

Re-examiningAssumptions About Young Students' Capabilities Faced withhistorical analysis and empiricalevidence, it would be easy to concludethat students face a long,difficult journey to algebra.However, history can be misleading.Negative numbers were the subject of heateddebate among professionalmathematicians less than 2 centuriesago, yet they are standard fare in curriculadesigned for today's preadolescent and adolescentstudents. This is not to denythat negative numbers are challenging for many students. In fact,many of theobstacles that students face may indeed be similarto those faced by earlier math- ematicians.However, when new mathematical and scientific knowledge have been

This content downloaded from 137.123.124.4 on Wed, 5 Mar 2014 16:04:53 PM All use subject to JSTOR Terms and Conditions 92 Arithmeticand Algebra in Mathematics Education systematizedand workedinto the corpus of existingknowledge, it maybecome surprisinglyapproachable. Historicaldevelopments in mathematicsare importantfor understanding the dilemmasand difficultiesstudents may encounter. But decidingwhether certain ideas and methodsfrom algebra are withinthe grasp of youngstudents requires empiricalstudies with young students who have had access to activitiesand chal- lengesthat involve algebraic reasoning and algebraicrepresentation. As Booth (1988) has suggested,students' difficulties with algebra may result from the limited waysthat they were taught about arithmetic and elementary mathematics. The classroomstudies by Davydov's team (see Bodanskii,1969/1991; Davydov, 1969/1991)show that Russian children who received instruction inalgebraic repre- sentationof verbal problems from Grades 1 to4 performedbetter than their control peersthroughout later school years and showed better results in algebraicproblem solvingwhen compared to sixthand seventhgraders in traditionalprograms of 5 yearsof arithmetic followed by algebra instruction from Grade 6. Otherpromising resultsconcerning work on equationscome from interview studies in Brazil. Brito Limaand da RochaFalcio (1997) foundthat first- to sixth-grade Brazilian children can developwritten representations for algebraic problems and, with help from the interviewer,solve linearequation problems using different solution strategies. Lins Lessa (1995) foundthat after only one instructionalsession, fifth-grade students(11- to 12-year-olds)could solve verbal problems or situations presented on a balancescale thatcorresponded to equations, such as x + y + 70 = 2x + y + 20 or 2x + 2y + 50 = 4x + 2y + 10. She also showedthat the children's solutions in a posttestwere based on thedevelopment of writtenequations and, in more than60% ofthe cases, the solutions were based on theuse ofsyntactical algebraic rulesfor solving equations. In ourown work, we havefound that even 7-year-olds can handle the basic logic underlyingadditive transformations in equations (Schliemann,Carraher, & Brizuela,in press;Schliemann, Carraher, Brizuela, & Jones,1998). Evidencethat elementary school children in U.S. classroomscan reasonalge- braicallyhas beenbuilding up overthe years as a resultof reform in mathematics educationthat led to theintroduction of discussionson generalizationof number patternsin elementary school. Carpenter and Franke (2001) andCarpenter and Levi (2000) showedthat fairly young children who participated in classroomactivities thatexplore mathematical relations can understand, for instance, that "a + b - b = a" forany numbers a andb. Schifter(1999) describedcompelling examples of implicit algebraicreasoning and generalizations by elementaryschool children in classrooms wherereasoning about mathematical relations is thefocus of instruction. Blanton and Kaput(2000) furthershowed third graders making robust generalizations as theydiscuss operations on even and odd numbers and consider them as placeholders or as variables. Anotherset of studiesexamined young children's generalizations and their understandingof variablesand functions.Davis (1971-1972) and his colleagues inthe Madison Project developed a seriesof classroom activities that could be used

This content downloaded from 137.123.124.4 on Wed, 5 Mar 2014 16:04:53 PM All use subject to JSTOR Terms and Conditions D. W.Carraher, A. D. Schliemann,B. M. Brizuela, and D. Earnest 93 tointroduce, among other things, concepts and notation for variables, Cartesian coor- dinates,and functions in elementary and middle school. These tasks were success- fullypiloted in Grades 5 to9 and,as Davis stressed,many of the activities are appro- priatefor children from Grade 2 onward.In a previousclassroom intervention study, we havefound that, given the proper challenges, third graders can engagein algebraicreasoning and work with function tables, using algebra notation to represent functionalrelations (Brizuela, 2004; Brizuela & Lara-Roth,2001; Brizuela, Carraher,& Schliemann,2000; Carraher,Brizuela, & Schliemann,2000; Carraher, Schliemann,& Brizuela,2000; Schliemann,Carraher, & Brizuela,2001, in press; Schliemann,Goodrow, & Lara-Roth,2001). Evidenceof algebraicreasoning has beenfound even among first and second graders who participated inEarly Algebra activitiesinspired by Davydov's (1975/1991)work (Dougherty, 2003; Smith, 2000). Morerecently, we found(Brizuela & Schliemann,2004) thatfourth graders (9- to 10-year-olds)who participated in ourEarly Algebra activities can use algebraicnotation to solveproblems of an algebraicnature.

WhyResearch in Early Algebra Is StillNeeded It mayseem that the major issues of Early Algebra Education were settled when Davydov'steam of researchersshowed success in introducingalgebra to young learners(Davydov, 1969/1991). We regardtheir work as groundbreaking,but view it as openingrather than closing the field of early algebra. It highlightedmany of themeans by which algebraic concepts could be madeaccessible and meaningful to youngchildren, but there is stillmuch to do. Althoughthe Soviet work gives a straightforwardlook at earlyalgebra from the teachers' perspective, it is vitally importanttounderstand how students make sense of the issues. What are their questions?What sorts of conflicts and multiple interpretations do they generate? How do theirinitially iconic drawings eventually evolve into schematic diagrams and notation?What sorts of intermediary understandings do childrenproduce? In addition,the Davydov team has tended to downplay the potential of arithmetic as a basisfor algebraic knowledge. At times, they even argue that arithmetic be intro- ducedinto the curriculum after algebra. The authors do makea goodcase forusing unmeasuredquantities in orderto encouragestudents to reflectupon quantitative relationsand to makeit difficultfor them to bypasssuch reflection by resorting directlyto computations. However, it is difficultto conceiveof children developing strongintuitions about number lines, for example, without ever having used metrics andwithout having a rudimentarygrasp of addition and subtractionfacts. Finally,we needto have a betterunderstanding of howfunctions can be introducedin arithmeticalcontexts. As we notedelsewhere (Carraher, Schliemann, & Brizuela,2000), one of the best-kept secrets of early mathematics education is that additionis a function,or at least can be viewedas a function.Of course one can go a longway without considering addition a function,and this is whatthe traditional curriculumdoes. Tables of numberscan be thoughtof as functiontables (Schliemann,Carraher, & Brizuela,2001). But sincechildren can fillout tables

This content downloaded from 137.123.124.4 on Wed, 5 Mar 2014 16:04:53 PM All use subject to JSTOR Terms and Conditions 94 Arithmeticand Algebra in Mathematics Education correctlywithout making explicit the functional dependence of the dependent variable on theindependent variable(s), they may merely be extendingpatterns. There needsto be an additionalstep of making explicit the functional dependence under- lyingsuch patterns. This demands that students make generalizations in language, algebraicnotation, or other representations such as graphsand diagrams. We triedto deal with the issues above in a 30-monthlongitudinal study with chil- drenbetween 8 and 10 yearsof age (middleof second to endof fourth grade). We developedand examined the results of a seriesof activities aimed at bringingout thealgebraic character of arithmetic (see Brizuela& Schliemann,2004; Carraher, Brizuela,& Earnest,2001; Carraher, Schliemann, & Schwartz,in press; Schliemann & Carraher,2002; Schliemann, Carraher, Brizuela, Earnest, et al., 2003; Schliemann, Goodrow,et al., 2001). In this article, we describethe outcomes of two of the lessons we implementedinthird grade. Our aim is toexemplify how young children, as they learnaddition and subtraction, can be encouragedto integratealgebraic concepts andrepresentations into their thinking. Studentsare often introduced to algebra through first-order equations of the form ax + b = cx + d (or a variantsuch as ax + b = d). Unfortunately,this introduces far too manynew issues at once andfurther encourages students to viewvariables as havinga singlevalue. These problems can be largelyavoided by givingstudents theopportunity towork extensively with functions before encountering equations. In ourapproach, they first encounter "additive offset" functions, a subclass of linear functionsof the form x + b. Because theconstant of proportionality is 1, issuesof proportionalgrowth are temporarily suppressed in orderto highlightthe additive constantaspect of linear functions. This has certain distinct advantages. Children's initialintuitions about order, change, and equality first arise in additivesituations. And,as we will show,as childrenwork with the number line and witha variable numberline, they can come to effectively deal with variables and functional covari- ationto approachproblems involving additive relations. Additionally, first-order equationscan be finallyintroduced as a specialcondition in whichtwo functions havebeen constrained to be equal.

THE CLASSROOM STUDY

YoungChildren Doing "AlgebrafiedArithmetic"

The presentdata come from a longitudinalstudy with 69 students,in four classrooms,as theylearned about algebraic relations and notation, from Grade 2 to 4. Studentswere from a multiethniccommunity (75% Latino)in theGreater Boston area.From the beginning of their second semester in second grade to the end of their fourthgrade, we implementedand analyzed weekly activities in their classrooms. Each semester,students participated in six to eightactivities, each activitylasting forabout 90 minutes.They worked with variables, functions, algebraic notation, functiontables, graphs, and equations. The algebraicactivities related to addition, subtraction,multiplication, division, fractions, ratio, proportion, and negative

This content downloaded from 137.123.124.4 on Wed, 5 Mar 2014 16:04:53 PM All use subject to JSTOR Terms and Conditions D. W. Carraher,A. D. Schliemann,B. M. Brizuela,and D. Earnest 95 numbers.All theactivities were videotaped. Here we providea generaloverview oflessons employed during Grade 3. Oursequence of activities does not constitute a fullydeveloped Early Algebra curriculum and is includedmainly to providea contextfor the subsequent analysis. Duringthe first term of their third grade, when the children were eight and nine yearsof age, we held eight90-minute weekly meetings in each of fourclasses, workingwith additive structure problems and representations. We willfirst provide a broaddescription ofhow the students were introduced tonumber lines from Lesson 3 to Lesson6. We willthen describe in moredetail the activities we developedin Lesson 7 and Lesson 8 in one of the classrooms.The instructors,David and B~rbara,are coauthors in thepresent article. The lessonsgenerally did notbegin witha polishedmathematical representation orwith a problemsupporting merely onecorrect response. Children were instead presented with an open-endedproblem involvingindeterminate quantities. After holding an initialdiscussion about the situation,we askedstudents to expresstheir ideas in writing.We thendiscussed their representationsand introducedconventional representations; the conventions we choseto introduce had connections both to the problem we wereworking with and to thestudents' own representations.

IntroducingNumber Lines

By thetime our students had reachedLesson 7 and 8 in thefall semester, they had alreadyspent several hours working with number lines.

Lesson3 Theirfirst encounter took place in Lesson 3 whenwe strungtwine across the room towhich large, easily readable numbers were attached at regular intervals; this physical numberline offered their first look at negative numbers (it ranged from -10 to +20 overapproximately 10meters). It also allowedus tocarry out discussions about howchanges in measuredand counted quantities-age, distance, money, candies, and temperature-aswell as purenumbers mapped onto number line representations.For such activities, children represented diverse values by locating themselves at variouspositions along the physical number line. They learned, with pleasure oftenbordering on glee,to interpret displacements on the number cord-line in terms of theirgrowing older, getting warmer or colder,earning and spendingmoney (which severalreferred to as "wastingmoney," perhaps from the ambiguous Spanishverb, gastar'). The contextserved to supportvigorous discussions about therelationships among physical quantities and the order of numbers. Considerations aboutdebt were crucial to clarifying what negative numbers mean and for helping thestudents realize, for example, the difference between having $0 andhaving -$2 despitethe fact that one's pocketswere probably empty in eithercase.

1 A largepercentage of the students in theclasses we teachcome from immigrant Spanish-speaking homes.

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In oneclass, David raised various questions about the number line ("How many numbersare thereon theline? Do thenumbers only begin at -10 and proceedto +20 or are theremore? How farcan the numberline go?"). Studentsat first suggestedthat the only numbers on the line were those for which there were printed labels (-10 through+20). David askedwhether those were all thenumbers that existed.A studentsuggested that there were "50 numbers,"adding "You can only go as faras thewall." David suggestedthat the students ignore the wall; the impor- tantthing was tomake sure that all thenumbers were included. From that moment, thestudents began to suggestlocations to whichthe number line wouldextend: acrossthe playground, to otherregions of the country, and eventually outer space itself.Each timethat students mentioned a new location, David askedwhether all thenumbers were now accounted for. Eventually several students suggested that thenumber line went "to infinity," and explained that it "kept going on and on."

Lesson4

A weeklater, David askedthe students to explain what the number line was that theyhad been discussing.In one classroom,a childmentioned having to behave "likeghosts" when using the number line, because one penetratedwalls to reach thedesired numbers. Another child referred to thenumber line as "a timemachine": it allowed himto proceedbackward and forwardin timewhen he treatedthe numbersas referringto his age. One studentobjected, arguing that people cannot go backwardin time, to which another responded, "You can inyour imagination." In a subsequentlesson, we movedfrom the number line made of twine to diagrams on paperand projectedonto a screenfrom an overheadprojector. We introduced arrowslinking points on the number line to represent changes in values. When several arrowswere connected on the number line, students learned that they could simplify theinformation by shortcuts that went from the tail of the first arrow to the head of thelast arrow. They also learned to express such shortcuts or simplifications through notation:for example, "+7 - 10" couldbe representedas "-3," sinceeach expres- sionhad the same effect. They could show the similarities and differences between thetwo by making trips along the number line, in front of the class or on paper.The large-scalenumber line, projected or strungin frontof thewhole class, allowed studentsand teacher to discuss mathematical operations in a forumwhere students who weremomentarily not talking could nonetheless follow the reasoning of the participants.This helped students deal with a rangeof issues, including the immensely importantone of distinguishingbetween numbers as pointsand numbers as intervals. Operandscould be treatedas pointsor as intervals(from 0 to theendpoint), butoperations such as "+6" referredto thenumber of unit spaces between positions andnot to the number of fence posts or markers, so tospeak, lying between numbers.

Lesson6 In Lesson 6, thefourth session in whichwe dealtwith number lines, we intro- duceda "variablenumber line" as a meansof talking about operations on unknowns

This content downloaded from 137.123.124.4 on Wed, 5 Mar 2014 16:04:53 PM All use subject to JSTOR Terms and Conditions D. W.Carraher, A. D. Schliemann,B. M. Brizuela, and D. Earnest 97

(see Figure1). "N minus4" couldbe treatedas theresult of displacement of four spacesleftward from N, regardlessof what number N stoodfor. With an overhead projector,we sometimesemployed two number lines: the variable number line and a standardnumber line with an originat 0. By placingone line over the other (they sharedthe same metric),students could determinethe value of N; it was the integeraligned with N. Theyalso graduallyrealized they could inferthe values of,say, N + 3 fromseeing that N + 7 sat above4 on theregular number line. The connectionsto algebraicequations should be obviousto thereader.

N-3 N-2 N-1 N N+1 N+2 N+3 N+4 N+5

Figure1. TheN-number line.

WorkingWith Unknown Quantities Lesson 7

Figure2 showsthe problem we presentedto studentsin Lesson 7. The problem didnot specify the amounts of money that Mary and John have in their piggy banks at thebeginning of the story; it merely stated that they have equal amounts.In the subsequentparts of the problem, the students learned about changes that occurred in theamounts. In thefinal part, the students learned how muchMary had in her piggybank on Thursday. From this information, the students ultimately determined how muchthe protagonists had at thebeginning and how much they had on each

Maryand Johneach have a piggybank. On Sunday,they both had thesame amountin their piggy banks. On Monday,their Grandmother comes to visit them and gives$3 toeach of them. On Tuesday,they go togetherto the bookstore. Mary spends $3 on Harry Potter'snew book. John spends $5 on a 2001 calendarwith dog pictureson it. On Wednesday,John washes hisneighbor's car and makes$4. Maryalso made $4 babysitting.They run to puttheir money in their piggy banks. On Thursday,Mary opens herpiggy bank and findsthat she has $9.

Figure2. ThePiggy Bank problem.

This content downloaded from 137.123.124.4 on Wed, 5 Mar 2014 16:04:53 PM All use subject to JSTOR Terms and Conditions 98 Arithmeticand Algebra in Mathematics Education dayin the story. We initiallydisplayed the problem as a whole(except for the line on whathappened on Thursday)so thatthe studentscould understandthat it consistedof a numberof parts. Then we coveredup all daysexcepting Sunday.

Representingan UnknownAmount Afterreading what happened each day,students worked alone or in pairs, trying torepresent on paper what was describedin the problem. During this time, members ofthe research team asked individual children to explain what they were doing and questionedthem in waysthat encouraged them to developmore adequate representations.In whatfollows we attemptto describe,on thebasis of whatwas depictedin the videotapes and in the children's written work, the content of the class- roomdiscussion that followed and children'sinsights and achievementsas they attemptedto representand solve the various steps of the problem. Sunday:After Kimberley read the Sunday part for the whole class, B irbara,the researcherrunning this class, asked students if they knew how much money each ofthe characters in the story had. The childrenstated in unison,"No," anddid not appearto be botheredby that.A fewuttered "N," and Talik stated,"N, it's for anything."Other children shouted "any number" and "anything." WhenBirbara asked the children what they are goingto showon theiranswer sheetsfor this first step in theproblem, Filipe said, "You couldmake [sic] some moneyin them,but it has to be thesame amount." Birbara reminded him that we do notknow what the amount is, and he thensuggested that he couldwrite N to representthe unknown amount. Jeffrey immediately said thatthat is whathe was goingto do. The childrenstarted writing in their handouts, which contained infor- mationonly about Sunday and a copyof the N-number line. Birbara reminded the studentsthat they could use theN-number line (a numberline with N at theorigin anda metricof one unit) on theirpaper if they wanted. She also drewa copyof the N-numberline on theboard. The studentsworked for about 3 minutes,drawing piggy banks and representing theamounts in each of them. Four children attributed specific values for Mary and Johnon Sunday.Five representedthe amount as N, usuallyinside a drawingof a piggybank. Two childrenplaced a questionmark inside or next to each piggy bank. Andfive children drew piggy banks with no indicationof what each would contain. Jennifer,one ofthe students who used N to representthe initial amount in each bank(see Figure3), drewtwo piggy banks, labeling one for Mary, the other for John, and wrotenext to thema largeN afterthe statement "Don't know?"In a one-on- oneinteraction with Jennifer, David (present but not serving as instructorinthe class) pointsto theN on herhandout and asked: David: So,what does it say over here? Jennifer: N. David: Whydid you write that down? Jennifer: Becauseyou don't know. You don't know how much amount they have. David: So,does N... Whatdoes that mean to you?

This content downloaded from 137.123.124.4 on Wed, 5 Mar 2014 16:04:53 PM All use subject to JSTOR Terms and Conditions D. W. Carraher,A. D. Schliemann,B. M. Brizuela,and D. Earnest 99

Figure3. Jennifer'sinitial representation forthe problem.

Jennifer: N meansany number. David: Do theyeach have N, or do they have N together? Jennifer: [Doesnot respond.] David: Howmuch does Mary have? Jennifer: N. David: Andhow about John? Jennifer: N. David: Is thatthe same N ordo they have different Ns? Jennifer: They'rethe same, because it said on Sunday that they had the same amount ofmoney. David: Andso, if we say that John has N, is itthat they have, like, $10 each? Jennifer: No. David: Whynot? Jennifer: Becausewe don't know how much they have. Fromthe very beginning of this class, the children themselves proposed to use N torepresent an unknown quantity. We hadintroduced the convention before in other contexts,but now it was makingits way into their own repertoire of representational tools.Several children appear to be comfortablewith the notation for an unknown as wellas withthe idea that they can work with quantities that may remain unknown.

TalkingAbout Changes in UnknownAmounts Monday:When the children read the statement about what happened on Monday, thatis, that each child received $3 fromtheir grandmother, they inferred that Mary andJohn would continue to have the same amount of money as eachother, and that theyboth would have $3 morethan the day before: A child: Nowthey have three more than the amount that they had. Bdrbara: Do youthink that John and Mary still have the same amount of money?

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Children: Yeah! Bdrbara: Howdo you know? Talik: Becausebefore they had the same amount of money, plus three, they both hadthree more, so it'sthe same amount. David: Thesame amount as beforeor the same amount as eachother? Talik: Thesame amount as eachother. Before, itwas the same amount. David: Anddo they have the same amount on Monday as theyhad on Sunday? Talik: No. Anotherchild: Youdon't know! Bdrbara: Whatis thedifference between the amount they had on Sundayand the amountthey had on Monday? Children: Theygot three more. Talik: Yeah.They have three more. They could have a hundreddollars; Grandma comesand gives them three more dollars, so it'sa hundredand three.

Bfirbaranext asked the children to propose a wayto show the amounts of money inthe piggy banks on Monday. Nathan was thefirst to propose that on Monday they wouldeach have Nplus 3, explaining,"Because we don'tknow how much money theyhad on Sunday,and they got plus, and they got three more dollars on Monday." Talikproposed to draw a pictureshowing Grandma giving money to thechildren. Filiperepresented the amount of money on Mondayas "? + 3." Jeffreysaid that he wrote"three more" because their Grandmother gave them three more dollars. The drawingsin Figure4 areJeffrey's spontaneous depictions of N + 3. In each case, the3 unitsare individually distinguished atop a quantity,N, of unspecified amount. Jamesproposed and wrote on hispaper that on Sundayeach would have "N + 2" andtherefore on Mondaythey would have N+ 5. It is notclear to us whyhe chose N + 2 as a startingpoint. Carolina wrote N + 3. Jenniferwrote N + 3 in a vertical arrangementwith an explanationunderneath: "3 morefor each." Talik wrote N+ 3= N+ 3. Carolina,Adriana, and Andy wrote N+ 3 insideor next to each piggy

Figure4. Jeffrey'srepresentations forwhat happened on Sunday and on Monday.

This content downloaded from 137.123.124.4 on Wed, 5 Mar 2014 16:04:53 PM All use subject to JSTOR Terms and Conditions D. W.Carraher, A. D. Schliemann,B. M. Brizuela, and D. Earnest 101 bankunder the heading Monday. Jeffrey wrote N + 3 forMonday and explained thatthat is becausetheir Grandma gave them 3 dollarsmore. But when David asks himhow much they had on Sunday,Jeffrey answered, "Zero." Max, sittingnext to him,then says, "You don'tknow." Jimmy, who first represented the amounts on Sundayas questionmarks, wrote N + 3, withconnections to Maryand John's schematicrepresentation of piggybanks, and explained,"Because whenthe Grandmothercame to visit them they had like, N. Andthen she gave Mary and John $3. That'swhy I say [pointingto N+ 3 on paper]Nplus three." B6rbaracommented on Filipe'suse ofquestion marks, and he andother children acknowledgedthat N is anotherway to showthe question marks. She thentold the classthat some of the children proposed specific values for the amounts in the piggy bankson Sunday.Speaking against this approach, Filipe stated that "nobody knows [howmuch they have]" and James said that these other children "are wrong" to assign specificvalues. Jennifer clarified that it could be one ofthe suggested amounts. At thispoint, several children seemed comfortable with the notationfor an unknownand with the idea thatthey could work with quantities that might remain unknown.Their written work showed that, by the end of the class, 11 ofthe 16 chil- drenhad adopted N + 3 forthe amounts each would have on Monday.Only one of thechildren continued to use specificamounts in his worksheet, and four produced drawingsthat could not be interpretedorwritten work that included Nbut in incor- rectways such as N + 3 = N.

Operatingon UnknownsWith Multiple Representations Tuesday:When they considered what happened on Tuesday,some of thechil- drenappeared puzzled and uncomfortable as they wondered whether there would be enoughmoney in the piggy banks to allow for the purchases. A studentsuggested thatthe protagonists inthe story probably had $10. Othersassumed that there must be atleast $5 intheir piggy banks by the end of Monday, otherwise John could not havebought a $5 calendar. Birbararecalled for the class whathappened on Sundayand Monday. The chil- drenagreed that on Mondayeach ofthe children had thesame amounts. She then askedthese questions: Bdrbara: Adriana,what happens on Tuesday? Adriana: OnTuesday, they had different amounts of money. Bdrbara: Whydo they have different amounts of money? Adriana: Becausethey spent, Mary spent $3, and John spent $5. Bdrbara: So,who spent more money? Adriana: John. Bdrbara: So,on Tuesday, who has more money on Tuesday? Adrianaand otherchildren: Mary. Jenniferdescribed what happened from Sunday to Tuesday and concluded that, on Tuesday,Mary ended up withthe same amountof moneythat she had on

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Sunday,"because she spends her $3." Atthis point, B~irbara encouraged the chil- drento use theN-number line to represent what has beengoing on fromSunday toTuesday. Always dialoguing with the children and getting their input, she drew arrowsgoing from N to N + 3 and thenback to N againto showthe changes in Mary's amounts.She showedthe same thingwith algebraic notation, narrating thechanges from Sunday to Tuesday,step by step,and gettingthe children's inputwhile she wroteN + 3 - 3. She thenwrote a bracketunder +3 - 3 and a 0 below it.She commentedthat +3 - 3 is thesame as 0, andextended the notation to N + 3 - 3 = N + 0 = N. Jenniferthen explained how the$3 spentcancel out the$3 givenby the Grandmother: "Because youadded three, right, and then she took,she spentthose three and she has thenumber she startedwith." Birbarathen led thechildren through John's transactions on theN-number line, drawingarrows from N toN + 3, thenN- 2, foreach stepof her drawing. During thisprocess she used algebraic script to register the states and transformations, and withthe students' input kept track of thestates and transformations,eventually writingN + 3 - 5 to expressJohn's amount at theend of Tuesday. Some children suggestedthat this amount is equalto "N minus2," an inferencethat B~rbara regis- teredas N+ 3 - 5 = N- 2. Birbaraasked Jennifer to approachthe number line and showthe difference betweenJohn and Mary's amounts on Tuesday.Jennifer atfirst pointed vaguely to theinterval between N- 2 andN. WhenBsarbara asked her to showexactly where thedifference starts and ends,Jennifer correctly pointed to N- 2 and to N as the endpointsof the segment. David askedJennifer how much one would have to give Johnso thathe had thesame amountof moneyas he had on Sunday.Jennifer answeredthat we wouldhave to give $2 to Johnand explains,showing on the numberline, that if he is atN- 2 andwe add 2, he goes backto N. Birbararepre- sentedwhat Jennifer has said as N- 2 + 2 = N. Jennifertook the marker from Biirbara'shand, drew brackets around the expression "-2 + 2," andwrote 0 under it.B~irbara asked why "-2 + 2" equals0 and,together with Jennifer, went through thesteps corresponding toN- 2 + 2 on theN-number line showing how N- 2 + 2 endsup atN. Talikshowed how this works when N has thevalue of 150. Barbara usedTalik' s exampleof N = 150to show how one ends up atvalue N onthe number line. Nathan'sdrawing in Figure5 depictsSunday (top), Monday (bottom left), and Tuesday(bottom right). For Tuesday,he firstdrew iconic representations of the calendarand the book next to the values $5 and$3, respectively, the icons and dollar valuesconnected by an equals sign.During his discussionwith an in-classinter- viewer,he wrotethe two equations N + 3 - 5 = N- 2 andN+ 3 - 3 = N, usingthe N-numberline as supportfor his decisions. Later, when he learnedthat N was equal to 5 (afterreceiving the information in theproblem about Thursday), he wrote8 nextto N + 3 on theMonday section of his worksheet. Wednesday:Filipe read theWednesday step of the problem.Birbara asked whetherMary and John will end up withthe same amount as on Monday.James said,"No," and Adrianathen explained that Mary will have N + 4 and Johnwill

This content downloaded from 137.123.124.4 on Wed, 5 Mar 2014 16:04:53 PM All use subject to JSTOR Terms and Conditions D. W.Carraher, A. D. Schliemann,B. M. Brizuela, and D. Earnest 103

Figure5. Nathan'srepresentation ofthe problem.

haveN+ 2. Birbaradrew an N-number line and asked Adriana to tell the story using theline. Adriana represented the changes for John and for Mary on theN-number line,much as shewould on a regularnumber line. Bairbara wrote N + 4 = N+ 4 and thenN- 2 + 4 = N + 2. Talikvolunteered to explainthis. He said thatif you take 2 fromthe 4, itwill equal 2. To clarifywhere the 2 comesfrom, Birbara represented thefollowing operations on a regularnumber line: -2 + 4 = 2. Birbarathen asked if anyone could show why N+ 3 - 3 + 4 equalsN+ 4. Talik volunteeredto do so, crossedout the subexpression +3 - 3, and said,"We don't needthat anymore." Jennifer stated that this is thesame as 0. Thisis a significant moment,because we had notintroduced the procedure of strikingout the sum of a numberand its additive inverse (although we hadused brackets to simplify sums). Talik's strikinginstead of bracketing shows his understanding that a numberand itsbracketed inverse yield 0. Birbaraproposed to write out an equationconveying whathappened to John'samounts throughout the week. The studentshelped her go througheach of thesteps in thestory and buildthe equation N + 3 - 5 + 4 = N+ 2. Butthey did notinitially agree upon the expression for the right-hand side ofthe equation. Barbara helped them visualize the operations on theN-number line. She askedJennifer to showhow the equation could be simplified.While Jennifer pondered,Birbara pointed out that this problem was harderthan the former and asked herto startout with+3 - 5; Jenniferanswered, "Minus two." Then they bracketedthe second part at -2 + 4, andJennifer, counting on herfingers, said it is +2 and wrote.Talik came up to thefront and explained, "N is anything,plus 3, minus5, is minus2; N minus2 plus 4, equals [whilecounting on his fingers] N plus 2." Talikthen tried to groupthe numbers differently, first adding 3 and4 and thentaking away 5. Birbarashowed that when grouping +3 + 4, one arrives at +7. Next,+7 - 5 resultsin +2. Hence,the answer is 2, regardlessof theorder of theoperations.

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Discoveringa ParticularValue and InstantiatingOther Values Thursday:When Amir read the Thursday part of the problem, where it stated that Maryended with $9, severalchildren suggested out loud thatN has to be 5. So Bairbaraasked the children, "How muchdoes Johnhave in hispiggy bank [at the end of Thursday]?"Some studentsclaimed that John (whose amount was repre- sentedby N+ 2) has"two more," apparently meaning "two more than N." Jennifer, James,and other children said that he has 7. Someof the students apparently deter- minedthis by adding5 + 2. Othersdetermined it fromMary's final amount (9): BecauseN + 2 (John'samount) is 2 less thanN+ 4 (Mary'samount), John would haveto have 2 less thanMary (known to have 9). Birbaraconcluded the lesson by workingwith the students in fillingout a 2 x 4 tabledisplaying the amounts that Maryand Johnhad acrossthe 4 days.Some studentssuggested expressions with unknownvalues; others suggested using the actual values, as inferredafter the infor- mationabout Thursday's events had been disclosed.

A New Context:Differences Between Heights Lesson8

The followingweek, in Lesson 8 we askedthe same group of students to work on theproblem shown in Figure 6 (see Carraher,Schliemann, & Brizuela,2000, in press, for a previousanalysis of the same problemby anothergroup of students).The problemstates the differences in heightsamong three characters withoutrevealing their actual heights.The heightscould be thoughtto vary insofaras theycould take on a setof possible values. Of course that was ourview. The pointof researchingthe issue was to see whatsense the students made of such a problem. Afterdiscussing each statementin theproblem, the instructor encouraged the studentsto focus on the differences between the protagonists' heights (see Carraher, Brizuela,et al., 2001, for details on thisfirst part of the class), and the students were

Tomis 4 inchestaller than Maria. Mariais 6 inchesshorter than Leslie. DrawTom's height, Maria's height, and Leslie'sheight. Showwhat the numbers 4 and 6 referto.

Maria Maria'sHeight

Figure6. TheHeights problem.

This content downloaded from 137.123.124.4 on Wed, 5 Mar 2014 16:04:53 PM All use subject to JSTOR Terms and Conditions D. W. Carraher,A. D. Schliemann,B. M. Brizuela,and D. Earnest 105 askedto represent the problem on individualworksheets. Most of the students used verticallines to showthe three heights (see examplein Figure7). To oursurprise, one ofthe students (Jennifer) chose to representthe heights on a variablenumber linemuch like the one theyhad been working with during previous meetings (see Figure8). Blirbarathen adopted Jennifer's number line as a basis fora full-class

Figure7. Jeffrey'sdrawing and notation for the Heights problem.

Tomis 4 inchestaller than Maria. Mariais 6 inchesshorter than Leslie. Draw: Tom'sheight Maria'sheight Leslie'sheight Showwhat the numbers 4 and 6 referto inyour drawing.

Figure8. Jennifer'sdrawing (notches) showing differences but no origin.She also uses a variablenumber that forms the basis of subsequentdiscussion.

This content downloaded from 137.123.124.4 on Wed, 5 Mar 2014 16:04:53 PM All use subject to JSTOR Terms and Conditions 106 Arithmeticand Algebra in Mathematics Education discussionof the relationsamong the heights.She furtheradopted Jennifer's assumptionthat Maria is locatedat N on thevariable number line (see themiddle numberline in Figure9).

Figure9. Threevariable number line representations (onan overheadprojector) used by studentsand teacher to discuss the cases where Maria (middle) is attributedtheheight of N; Leslie(bottom) is assignedthe height of N; andTom (top) is assigneda height of N.

Bdrbara: Nowif Maria's height was N, what would Tom's height be? Students: N plusfour. Bdrbara: Why? Students: Becausehe would be four inches taller. Bdrbara: Mm,hmm. And what would Leslie's height be? Nathanand students: N plussix. Nathan: BecauseLeslie is sixinches taller. Itis remarkablethat Jennifer realized that a representationaltool introduced in earlier classeswould help to clarifythe problem at hand.It is equallyimpressive that the remainingstudents appeared comfortable with this idea and easily inferred Tom and Leslie's heights(N + 2, N+ 4, respectively)from Maria's (N). Birbarawondered to herselfwhether the students realized that the decision to call Maria's heightN

This content downloaded from 137.123.124.4 on Wed, 5 Mar 2014 16:04:53 PM All use subject to JSTOR Terms and Conditions D. W.Carraher, A. D. Schliemann,B. M. Brizuela, and D. Earnest 107 was arbitrary.So sheasked the students to assume instead that Leslie's heightwas N. The studentsanswered that Tom's heightwould be N- 2 andMaria's would be N- 6 (see thebottom number line in Figure 9). Theyinferred this not by acting on thealgebraic script but rather by making the appropriate displacements on the variable numberline. Next,B&irbara asked the students to assumethat Tom's heightwas N. Max went tothe front of the class and placed Leslie at N+ 3 (see thetop number line in Figure 9; thereis an erasureunder N+ 3 whereMax hadfirst incorrectly put Leslie's name). Max realizedthat the difference between Tom and Leslie is 2,but nonetheless placed her3 unitsto the right of Tom. (This is an exampleof the "fence post" issue. Students are well accustomedto theidea thata numberrefers to thecount of elements in a set,that is, a set'scardinality. However, the issue before children often is "What shouldI count?"On a numberline, two sorts of elements suggest themselves. One can countthe number of intervals or one can countthe number of "fence posts," or notches.In Max's case,he seemedto have counted the "fence posts" lying between N andN + 3, thedelimiters.) Other students correctly stated that Leslie shouldbe placedunder N+ 2. Finally,when Birbara asked Amir to show where Maria's name shouldbe located,he placedit, without hesitation, under N -4. Ifwe focustoo much on the occasional errors made by students like Max, we may failto see thelarger picture; namely, that by the end of the lesson the students are relatingthe given numericaldifferences to algebraicnotation, line segment diagrams,number lines (including variable number lines), subtraction, counting, andnatural language descriptions. The fluidity with which students move from one representationalform to anothersuggests that their understanding offunctions of theform x + b is robustand flexible. Their willingness to use Nto representthe height of any of thethree characters in the story(as long as therelations among the heightsof the three protagonists are kept invariant) shows an encouragingdegree ofrobustness in their thinking.

DISCUSSION

WereThese Students Doing Algebra?

Do theactivities documented here qualify as algebra?Some mightbe tempted toargue that students had solved the Piggy Bank problem through procedures, such as "undoing,"deemed merely arithmetical or prealgebraic(Boulton-Lewis et al., 1997; Filloy& Rojano,1989). Othersmight note that the students did not"solve forx" inthe traditional sense of applying syntactical rules to written forms, without regardto theirmeaning, to producea uniquevalue for the unknown. However, it is importantto recognizewhat the students did achieve. They used letters to mean- ingfullyrepresent variables. They used algebraic expressions such as N+ 3 torepre- sentfunctions. Furthermore, they used knowledge about the changes in quantities toformulate new algebraic expressions. They understood the relations between the dailyamounts of each protagonist in thestory problem; they also understoodhow

This content downloaded from 137.123.124.4 on Wed, 5 Mar 2014 16:04:53 PM All use subject to JSTOR Terms and Conditions 108 Arithmeticand Algebra in Mathematics Education theamounts on each dayrelated to thestarting amounts. In thediscussion of the Heightsproblem (i.e., Lesson 8), theydisplayed a cleargrasp of functionalrela- tionsamong indeterminate quantities; they were working with variables while maintainingan invariant relationship between them. And they generated appropriate expressionsfor the heights of the remaining protagonists regardless of which actor had been assignedthe initial value N. As such,it appearsthat the students were workingwith functions, a fundamentalobject of studyin algebra(Schwartz & Yerushalmy,1994). In severalsenses the lessons described above are typical of the 32 earlyalgebra lessonswe implementedduring 1 semester(eight lessons implemented in each of fourthird-grade classrooms). At the beginning, most children relied on instantiating unknownsto particular values. But over time, in each lesson, and across the lessons, thestudents increasingly came to use algebraicnotations and numberline repre- sentationsas a naturalmeans of describingthe eventsof problemsthey were presentedwith (see Carraher,Schliemann, & Schwartz,in press, for further analysis ofthis evolution in fourthgrade). Althoughstudents expressed their personal understandings in drawingsand explanations,we do notsuggest that their behavior was completelyspontaneous. Clearly,their thinking was expressed through culturally grounded systems, including mathematicalrepresentations ofvarious sorts. Number line representations and the use ofletters to representunknown amounts or variablesare examples of cultural representationswe explicitlyintroduced to thestudents. The issueis notwhether theyinvent such representations fully on theirown but rather whether they embrace themas theirown-that is, whetherthey incorporate them into their repertoire of expressivetools. Findingssuch as thesehave persuaded us that,given the proper experiences, chil- drenas youngas eightand nine years of age can learnto comfortably use lettersto representunknown values and can operateon representations involving letters and numberswithout having to instantiate them. To concludethat the sequence of operations"N + 3 - 5 + 4" is equal toN + 2, andto be able toexplain, as manychildren wereable to, that Nplus 2 mustequal 2 morethan what John started out with, what- ever thatvalue mightbe, is a significantachievement-one that many people wouldthink young children incapable of. Yet suchcases werefrequent and not confinedto any particular kind of problem context. It would be a mistaketo dismiss such advancesas mereconcrete solutions, unworthy of the term"algebraic." Childrenwere able to operate on unknownvalues and draw inferences about these operationswhile fully realizing that they did not know the values of the unknowns. We havealso foundfurther evidence that children can treat the unknowns in additive situationsas havingmultiple possible solutions. For example,in a simple comparisonproblem (Carraher, Schliemann, & Schwartz,in press)wherein we describedone childas havingthree more candies than another, our students from Grade3 wereable to proposethat one childwould have N candiesand theother wouldhave N + 3 candies.Furthermore, they found it perfectly reasonable to view a hostof ordered pairs-(3, 6), (9, 12), (4, 7), (5, 8)-as all beingvalid solutions

This content downloaded from 137.123.124.4 on Wed, 5 Mar 2014 16:04:53 PM All use subject to JSTOR Terms and Conditions D. W.Carraher, A. D. Schliemann,B. M. Brizuela, and D. Earnest 109 forthe case at hand,even though they knew that in anygiven situation only one solutioncould be true.They even were able to express the pattern in a tableof such pairsthrough statements such as "thenumber that comes out is alwaysthree larger thanthe number you start with." When children make statements of such a general nature,they are essentially talking about relations among variables and not simply unknownsrestricted to singlevalues. We havefound that eight- and nine-year-old childrencan notonly understand additive functions but also meaningfullyuse algebraicexpressions such as "n - n + 3" and"y = x + 3" (see Carraher,Schliemann, & Brizuela,2000; Schliemann,Carraher, & Brizuela,in press). Cases suchas thoseabove mayseem strange to peopleaccustomed to thinking ofvariation in terms of changes in the values of a singleentity over time. Variation is actuallya broaderconcept than change. Sometimes, variability occurs across a setof unrelated cases, as inthe variation of heights in a populationor as inthe covari- ationof heights and weights. The exampleof one child having three more candies thananother can be understoodas variationwithin and, even moreimportant, invarianceacross a setof possibilities. The invarianceacross cases can formthe basisof discussions with students about function tables containing many "solutions," one foreach row.In thiscase, columnone wouldcorrespond to theamount of candiesthe first child has; columntwo would correspond to the amount of candies thesecond child has. Each rowcontains a validsolution, insofar as itis consistent withthe information given. Once students have filled out and explored such a func- tiontable, the issue is toexplain it. What properties stay the same regardless of the row?This is no trivialmatter, and it gives rise to general statements about number patternsthat are the essence of algebra. Suchforms of variation are important because they allow one to reason both about particularvalues and sets of possible values. They allow one to consider unknowns as variables.This is preciselythe spiritwith which many students viewed the PiggyBank problem before information was discovered,regarding Thursday, that finallyallowed them to disregard the multiple possibilities and focus on the values to whichthe problem was nowconstrained. Students who feel the need to instan- tiatevariables from the beginning can do so andparticipate in the classes from their ownperspective, restricting their attention to one possiblescenario from the start. Thisshould not be a reasonfor concern, for we havefound that such students learn fromothers and from class discussions,and within a fewweeks they comfortably welcomealgebraic representations into their own personal repertoire of expressive tools.Were their initial reliance upon instantiation due to developmental constraints, therelatively quick learning we witnessedfrom the piggy bank lesson to the lesson on heightscould not have taken place. The studentsdemonstrated that they had begun to handle fundamental algebraic concepts.Still, there is muchfor them to learn.Algebra is a vastdomain of mathematics,and the progress shown by studentsin thepresent study is thebeginning ofa longtrajectory. In thisinitial stage, students benefit immensely from working in richproblem contexts that they use to helpstructure and check their solutions. As theybecome increasingly fluent in algebra,they will be able to relyrelatively

This content downloaded from 137.123.124.4 on Wed, 5 Mar 2014 16:04:53 PM All use subject to JSTOR Terms and Conditions 110 Arithmeticand Algebra in Mathematics Education less on thesemantics of the problem situation to solve problems. Algebraic expres- sionswill not only capture but, more and more, will help guide their thinking and problemsolving. With time, they will hopefully be ableto derivevalid inferences byacting on thewritten and graphical forms themselves, without having to reflect back on therich problem contexts in orderto successfullyproceed; that is, their semanticallydriven problem solving will become increasingly driven by the syntax ofthe written expressions.

ConcludingRemarks

Ourwork has been guided by the ideas that (1) children'sunderstanding ofaddi- tivestructures provides a fruitfulpoint of departure for an "algebrafiedarithmetic"; (2) additivestructures require that children develop an earlyawareness of negative numbersand quantities and their representation innumber lines; (3) multipleprob- lemsand representations forhandling unknowns and variables, including algebraic notationitself, can and shouldbecome part of children'srepertoires as earlyas possible;and (4) meaningand children's spontaneous notations should provide a footingfor syntactical structures during initial learning, even though syntactical reasoningshould become relatively autonomous over time. Theremay be manyreasons for viewing algebra as moreadvanced than arithmeticand therefore placing it after arithmetic in themathematics curriculum. But thereare more compelling reasons for introducing algebra as anintegral part of early mathematics.There are good reasons for considering the abstract and concrete as interwovenrather than fully distinct (Carraher & Schliemann,2002). Additionand subtraction,multiplication and division are operations, but they are also functions andso areamenable to description through algebraic notation. If we dwelltoo much on theconcrete nature of arithmetic, we run'therisk of offering students a superfi- cial viewof mathematics and of discouraging their attempts to generalize. Although computationalfluency is important(even crucial) for allowing students to reason algebraically,itdoes not assure that students will be attentiveto the patterns under- lyingarithmetic and arithmeticalrelations. Algebraic notation (as well as tables, numberlines, and graphs) offers a meansfor expressing such patterns clearly and succinctly.If introducedin meaningful ways, it offers the virtue of bringing together ideas thatotherwise might remain fragmented and isolated. Manyhave argued that young children are incapable of learning algebra because theydo nothave the cognitive wherewithal tohandle concepts such as variablesand functions(Collis, 1975; Filloy& Rojano,1989; Herscovics& Linchevski,1994; Kuchemann,1981; MacGregor, 2001). Ourclassroom studies suggest that children can handlealgebraic concepts and use algebraicnotation somewhat earlier than commonlysupposed. There may be noneed for algebra education to wait a supposed "transitionperiod" after arithmetic. As othershave shown (Carpenter & Franke,2001; Carpenter& Levi,2000; Schifter, 1999), there are many opportunities forintroducing algebraicconcepts into the curriculum during the early years of mathematics education.Moreover, we alsoshow, as Bodanskii(1991) and Davis (1971-1972) did before

This content downloaded from 137.123.124.4 on Wed, 5 Mar 2014 16:04:53 PM All use subject to JSTOR Terms and Conditions D. W.Carraher, A. D. Schliemann,B. M. Brizuela, and D. Earnest 111 us, thatit is possibleto introducealgebraic notation in theearly grades. Our data furtherexpand our understanding ofhow young children come to appropriate algebra notationas theyrepresent open-ended problems, and we provideexamples of how theteaching and learning of arithmetic operations can be relatedto functions. Lest we give themistaken impression that any mathematicalconcept can be learnedat any time, let us setthe record straight. By arguingthat the algebraic char- acterof arithmeticdeserves a place in earlymathematics education, we are not denyingthe developmental nature of mathematicalskills. Number concepts, the abilityto use algebraicnotation, to interpret graphs, model situations, and so forth, developover the course of many years. Even in so "simple"an areaas additivestructures,children need to be ableto reify differences so thatthey can be treatedas bona fidequantities with their own propertiesand subjectto arithmeticaloperations. Childrenwho are just beginning to workwith addition and subtractionmay inter- preta statementsuch as "Tomis 4 inchestaller than Maria, and Maria is 6 inches shorterthan Leslie" as meaningthat one of the children is 4 inchestall while another is 6 inchestall (Carraher, Schliemann, & Brizuela,2000; Schliemann,Carraher, & Brizuela,in press).They may confusea heightwith a differencebetween two heights.When children get beyond this issue, this does notsignify that they will no longerhave troubles with additive differences. When one changesthe context toone aboutmoney or introduces a number line, new problems arise (for example, "How does a differencein heightsmanifest itself when two line segments are used torepresent people's heights?"). Thompson (1993) foundthat fifth-grade students may confusesecond-order additive differences ("the differencesbetween the heightsof twobrother-sister pairs") with first-order differences ("the differences betweena brotherand sister"). Issues involving concepts as richas additivediffer- ences,ratio and proportion, division, and so on,crop up againand again in the course of one's life,and it wouldbe naiveto assumethat the challenges are conquered, once andfor all, at a particularmoment in time,least of all, whenone learnshow toperform calculations with addition and subtraction inearly schooling. One could takea pessimisticview of such a conclusion:We willnever cease tostumble when confrontedwith variations of mathematicalproblems that we have encountered before.But thissame situationprovides reason for hope, for it signifiesthat the schemesthat have begun their evolution very early in life, perhaps as earlyas when a babybegins to playwith nesting cups, will later prove useful to tasksthat they werenever designed to handlebut which nonetheless succumb to metaphorical opportunism. EarlyAlgebra Education is by no meansa well-understoodfield. Surprisingly littleis knownabout children's ability to makemathematical generalizations and to use algebraicnotation. As faras we can tell,at the present moment, not a single majortextbook in the English language offers a coherentalgebrafied vision of early mathematics.We view algebrafiedarithmetic as an excitingproposition, but one forwhich the ramifications can onlybe knownif a significantnumber of people undertakesystematic teaching experiments and research. It maytake a longtime forteacher education programs to adjustto thefact that the times have changed.

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We hope thatthe mathematics education community and its sourcesof funding recognizethe importance of this venture.

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Authors

David W. Carraher, TERC, 2067 Massachusetts Avenue, Cambridge, MA 02140; [email protected] Anahicia D. Schliemann,Department of Education,Tufts University, Paige Hall, Medford,MA 02155; [email protected] BairbaraM. Brizuela,Department of Education, Tufts University, Paige Hall, Medford,MA 02155; [email protected] DarrellEarnest, TERC, 2067 Massachusetts Avenue, Cambridge, MA 02140;[email protected]

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