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Revista Latinoamericana de Investigación en
Matemática Educativa
ISSN: 1665-2436
Comité Latinoamericano de Matemática
Educativa
Organismo Internacional
Koukkoufis, Andreas; Williams, Julian
Semiotic Objectifications of the Compensation Strategy: En Route to the Reification of Integers
Revista Latinoamericana de Investigación en Matemática Educativa, núm. Esp, 2006, pp. 157-175
Comité Latinoamericano de Matemática Educativa
Distrito Federal, Organismo Internacional
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Semiotic Objectifications of the
Compensation Strategy:
En Route to the Reification of Integers
Andreas Koukkoufis 1
Julian Williams 1
RESUMEN
Reportamos aquí el análisis de una experiencia que reproduce el trabajo de investigación“Object-Process Linking and Embedding” (OPLE) en el caso de la enseñanza de laaritmética de los enteros, desarrollada por Linchevski y Williams (1999) en la tradiciónde la Educación Matemática Realista (realistic mathematics education (RME)). Nuestroanálisis aplica la teoría de la objetivación de Radford, con el propósito de aportar nuevaspistas sobre la forma en que la reificación tiene lugar. En particular, el método de análisismuestra cómo la generalización factual de la estrategia llamada de compensaciónencapsula la noción de “agregar de un lado es lo mismo que quitar del otro lado”; unabase fundamental de esto que será, más tarde, las operaciones con enteros. Discutimos,de igual modo, otros aspectos de la objetivación susceptibles de llegar a ser importantesen la cadena semiótica que los alumnos ejecutan en la secuencia OPLE, secuencia quepuede llevar a un fundamento intuitivo de las operaciones con los enteros. Sostenemosque es necesario elaborar teorías semióticas para comprender el papel vital de losmodelos y de la modelación en la implementación de las reificaciones en el seno de laEducación Matemática Realista (RME).
PALABRAS CLAVE: Enteros, semiótica, teorías del aprendizaje.
ABSTRACT
We report an analysis of data from an experimental replication of “Object-Process Linkingand Embedding” (OPLE) in the case of integer arithmetic instruction originally developedby Linchevski and Williams (1999) in the realistic mathematics education (RME) tradition.Our analysis applies Radford’s theory of semiotic objectification to reveal new insightsinto how reification is achieved. In particular the method of analysis shows how thefactual generalization of the so-called compensation strategy encapsulates the notionthat “adding to one side is the same as subtracting from the other side”: a vital groundingfor symbolic integer operations later. Other aspects of objectification are discussed thatare considered likely to be important to the semiotic chaining that students achieve inthe OPLE sequence that can lead to an intuitive grounding of integer operations. We
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Fecha de recepción: Febrero de 2006/ Fecha de aceptación: Abril de 2006
School of Education, the University of Manchester.1
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argue that semiotic theory needs to be elaborated to understand the vital role of modelsand modelling in leveraging reifications in RME.
KEY WORDS: Integers, Semiotics, Theories of Learning.
RESUMO
Reportamos aqui o análise de uma experiencia que reproduce o trabalho de investigação“Object-Process Linking and Embedding” (OPLE) en o caso da ensino da aritmética dosinteiros, desenvolvida por Linchevski e Williams (1999) na tradição da EducaçãoMatemática Realista (realistic mathematics education (RME)). Nossa análise aplica ateoria da objetivação de Radford, com o propósito de surgir novas pistas sobre a formaem que a reificação tem lugar. Em particular, o método de análise mostra como ageneralização factual da estratégia chamada de compensação encapsula a noção de“agregar de um lado é o mesmo que quitar do outro lado”; uma base fundamental dissoque será, mais tarde, as operações com inteiros. Discutimos, de igual modo, outrosaspectos da objetivação suscetíveis de chegar a ser importante na cadeia semióticaque os alunos executam na seqüência OPLE, seqüência que pode levar a um fundamentointuitivo das operações com os inteiros. Sustentamos que é necessário elaborar teoriassemióticas para compreender o papel vital dos modelos e da modelação naimplementação das reificações no seio da Educação Matemática Realista (RME).
PALAVRAS CHAVE: Inteiros, Semióticos, Teoria de Aprendizagem.
RÉSUMÉ
Nous rapportons ici l’analyse d’une expérience qui vise à reproduire le travail de recherche“Object-Process Linking and Embedding” (OPLE) dans le cas de l’enseignement del’arithmétique des entiers développé par Linchevski et Williams (1999) dans la traditionde l’Éducation Mathématique Réaliste (realistic mathematics education (RME)). Notreanalyse applique la théorie de l’objectivation sémiotique de Radford afin d’apporter denouveaux éclairages sur la façon dont la réification est accomplie. La méthode d’analysemontre, en particulier, comment la généralisation factuelle de la stratégie appelée decompensation encapsule la notion que « ajouter d’un côté, c’est la même chose qu’enleverde l’autre côté » : une base fondamentale de ce que sera plus tard les opérations avecdes entiers. Nous discutons également d’autres aspects de l’objectivation susceptibles dedevenir importants dans la chaine sémiotique que les élèves accomplissent dans la séquenceOPLE, séquence qui peut mener à un fondement intuitif des opérations sur des entiers.Nous soutenons qu’il est nécessaire d’élaborer des théorisations sémiotiques pourcomprendre le rôle vital des modèles et de la modélisation dans l’implémentation desréifications au sein de l’Éducation Mathématique Réaliste (RME).
MOTS CLÉS: Entiers, sémiotique, théories de l´apprentissage.
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Semiotic Objectifications of the Compensation Strategy:En Route to the Reification of Integers
The Need for a Semiotic Analysis
Based on the instructional methodology ofObject-Process Linking and Embedding(OPLE) (Linchevski & Williams, 1999;Williams & Linchevski, 1997), the dicegames instruction method for integeraddition and subtraction showed howstudents could intuitively construct integeroperations. This methodology, underpinnedby the theory of reification (Sfard, 1991;Sfard & Linchevski, 1994), was developedwithin the Realistic Mathematics Education(RME) instructional framework. Until veryrecently, the dice games method had notbeen analysed semiotically. We believe asemiotic analysis of students’ activities inthe dice games will illuminate students’meaning-making processes. It will alsoprovide some further understanding of thereification of integers in the dice games inparticular and more generally of the theoryof reification, which does not explain “whatspur[s] the students to make the transitionsbetween stages” (Goodson-Espy, 1998, p.234). Finally, it will contribute to thediscussion of the semiotic processesinvolved in RME, which are currentlyinsufficiently investigated (Cobb,Gravemeijer, Yackel, McClain, &Whitenack, 1997; Gravemeijer, Cobb,Bowers, & Whitenack, 2000). In this paperwe focus on the compensation strategy(Linchevski & Williams, 1999), a dice gamestrategy on which integer addition andsubtraction are grounded, and begin toaddress the following questions:
1. What are the students’ semioticprocesses of the compensation strategyin the reification of integers through theOPLE teaching of integers in the dicegames method?
2. What is the semiotic role of the abacusin the OPLE teaching of integers
through the dice games and what canwe generally hypothesise about thesignificance of models and modelling inthe RME tradition?
We found Radford’s semiotic theory ofobjectification (Radford, 2002, 2003) to bea particularly useful theoretical frameworkfor analysing students’ semiotic processesin the dice games, despite the very differentcontext in which it was developed.
The Object-Process Linking andEmbedding Methodology
Sfard (1991) reported as follows:
But here is a vicious circle: on theone hand, without an attempt at thehigher-level interiorization, thereification will not occur; on theother hand, existence of objects onwhich the higher-level processesare performed seemsindispensable for the interiorization– without such objects theprocesses must appear quitemeaningless. In other words: thelower-level reification and thehigher-level interiorization areprerequisites of each other! (p. 31)
In order to overcome this ‘vicious circle’,the Object-Process Linking andEmbedding (OPLE) pedagogy(Linchevski & Wil l iams, 1999) wasdeveloped: “children a) build strategiesin the situation, b) attach these to the newnumbers to be discovered, and finally c)embed them in mathematics byintroducing the mathematical voice andsigns” (Linchevski & Williams, 1999, p.144). The pedagogy can be bestunderstood through the dice gamescontext in which i t was developed(Linchevski & Williams, 1999), which
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aimed at overcoming the paradox ofreification described above for the caseof arithmetic of the integers.
The dice games instruction method(Linchevski & Williams, 1999) is an intuitiveinstruction of integer addition andsubtraction in the RME instructionalframework aiming at the reification ofintegers. The transition from the narrowerdomain of natural numbers to the broaderdomain of integers in the method isachieved through emergent modelling(Gravemeijer, 1997a, 1997b, 1997c;Gravemeijer et al., 2000) and takesadvantage of students’ intuition of fairness(Liebeck, 1990) for the cancellation ofnegative amounts by equal positiveamounts (Dirks, 1984; Linchevski &Williams, 1999; Lytle, 1994). Practically,the model – the double abacus (see figure1) – affords the representation andmanipulation of integers as objects beforethey are abstracted and symbolised assuch by the students (Linchevski &Williams, 1999):
The integer is identifiable in thechildren’s activity first as a processon the numbers already understoodby the children, then as a ‘report’ orscore recorded (concretised by theabacus). The operations on theintegers arise as actions on theirabacus representations, thenrecorded in mathematical signs.Finally, the operations on themathematical signs are encounteredin themselves, and justified by theabacus manipulations and gamesthey represent. Thus the integers areencountered as objects in socialactivity, before they are symbolisedmathematically, thus intuitively fillingthe gap formerly considered a majorobstacle to reification. (Linchevski &Williams, 1999, p. 144)
Figure 1: The abacus
Therefore, in the games the situatedstrategies are constructed in a realisticcontext which allows intuitions to arise.In this process the abacus model isutilized which “affords representation ofthe two kinds of numbers, and allowsaddition and subtraction (though clearlynot multiplication and division) of theintegers to be based on an extension ofthe children’s existing cardinal schemes”(Linchevski & Williams, 1999, p. 135).These strategies are linked to objects(yellow and red team points, see nextsection), thus allowing object-processlinking. Later, the formal mathematicallanguage and symbols enter the games.In the following section we present thegames more analytically.
The Dice Games Instruction
The method involves 4 games in eachof which two teams of two children arethrowing dice (e.g. a yellow and a reddie in game 1) and recording team pointson abacuses: the points for the yellowteam are recorded by yellow cubes onthe abacuses and those for the red teamare red cubes on the abacuses. Thestudents sit in two pairs, each having amember of each team and an abacus
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Semiotic Objectifications of the Compensation Strategy:En Route to the Reification of Integers
(see figure 2). On each pair’s abacus,points for both teams are being recordedand the team points on the two abacusesadd up. The students in turn throw thepair of dice, recording each time thepoints for the two teams on their abacus.When the two abacuses combine to giveone team a score of 5 points ahead oftheir opponents, that team wins thegame. For instance in game 1, if theyellow team at a certain point is 2 aheadand they get a score on the pair of dice,say 4 yellows and one red, then they canadd 3 yellows to their existing score of 2and so get 5 ahead, and they win. Butnote the complication that because wehave two abacuses for the two pairs, a‘combined score of 2 yellows’ mightinvolve, say 1 red ahead on the oneabacus and 3 yellows ahead on the otherabacus: so there are mult iple‘compensations’ of reds and yellowsgoing on in var ious combinat ions.Therefore, the important thing in thegames is not how many points a teamhas, but how many points ahead of theopponent: hence the nascent directivityof the numbers.
Figure 2: Students playing one of the dice games
In the first game (game 1) two dice areused, a yellow and a red one, giving pointsto the yellow and red team in each throw.Shortly after the beginning of game 1, oftenwith the urging of the researcher, thestudents intuitively understand that theycan cancel the team points on the dice,thus introducing an important gamestrategy, the cancellation strategy (notexamined in this paper). For example,according to this strategy, if a throw of thepair of dice shows 3 points for the yellowteam and 1 for the red team, this isequivalent to just giving 2 points for theyellow team. The rationale is that thedirected difference of the points of theyellow and red team (i.e. the amount ofpoints that the yellows are ahead or behindthe reds) will be the same anyway. As theabacus columns have only space for 10points for each team, a team column willoften be full before a team gets 5 pointsahead of the opponent. In order for thegame to go on, the compensation strategyis formulated, that is, if you can’t add pointsto one team, subtract the same amount ofpoints from the other, so as to maintain thecorrect directed difference of team points.
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This strategy is the second important gamestrategy and it is the one focused upon inthis paper. By the end of the games, thisstrategy will lead to the intuitive constructionof equivalences like: and .
Game 2 is similar to game 1, and isintroduced as soon as (and not before) thechildren are able to cancel the pair of diceinto ONE score quite fluently. In this gamean extra die is now thrown whose faces aremarked ‘add’ and ‘sub’ (subtract). From nowon this will be called the add/sub die. Theintroduction of this die allows for subtractionto come into play, instead of just addition,as in game 1. In analogy to game 1,according to the compensation strategy, ifyou have to subtract points from a teambut there are none on the abacus tosubtract, you can add points to the otherteam instead.
In game 3, formal mathematical symbolsfor integers are introduced. The add/sub dieis not used and the yellow and red die arereplaced with an integer die giving oneof the following results on each throw:–1, –2, –3, +1, +2, +3. Positive integers arepoints for the yellow team and negativeintegers are points taken from the yellows,thus they are points for the reds (for moredetails see Linchevski & Williams, 1999).Here the mathematical voice isencouraged, so that the children say “minus3” and “plus 2” etc.
In the final game (game 4), the add/sub dieis back into the game, allowing again forsubtraction to be concerned. In these twogames the cancellation strategy is nolonger needed and the compensationstrategy is transformed into a formalsymbolic, though still verbal, form: “addminus 3” etc. Once the students becomefluent in game 4, they begin recording thegames for a transition from verbal to written
+(+2) ≡ −(−2)+(−2)≡ −(+2)
use of formal mathematical symbols, butwe are not going to discuss this transitionfurther in this paper.
Some Earlier Analyses: Reification inthe Dice Games
Linchevski and Williams (1999) haveanalysed the dice games in terms ofreification. Through the instructionalmethodology of Object-Process Linkingand Embedding, they achieved the intuitivereification of integers and the constructionof processes related to integer addition andsubtraction through the manipulation ofobjects on a model (i.e. the yellow and redteam points). However, they did not providea semiotic-analytical account of thereification processes – their main concernwas to show that reification of integers waspossible through their method. We willdiscuss here the reifications taking placein the dice games, as we understand them,so that we can better appreciate the needfor a semiotic analysis of students’processes.
In relation to the reification of integers,according to Linchevski & Williams (1999),the object-process linking allows the intuitivemanipulation of integers as objects from thevery beginning of the dice games. As a resultof this methodological innovation, someelementary processes are obvious from thebeginning. These are, that if a team getspoints (or points are subtracted from it), thenew points add-up to (or are subtractedfrom) the points the team already has.These processes are intuitively obviousfrom the introduction of game 1 (and game2 respectively). However, one may arguethat the students still operate at the levelof natural numbers, not integers.
Integer processes begin to be constructed,though integers are not yet introduced
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explicitly, once the students focus on thescore of the game, that is, which team isahead and by how many points. Thecalculation of the score as the directeddifference of the piles of cubes of the pointsof the two teams is the first object-processlink to be constructed. The second object-process link to be achieved in the gamesis the cancellation of the team points onthe dice: i.e. if in a throw the yellows get 2points and the reds 1 point, you might aswell just give 1 point to the yellows. Thusthis link is possible through theestablishment of the so called cancellationstrategy. Further, the compensation strategy– according to which adding to one side ofthe abacus is the same as subtracting fromthe other side – needs to be introduced asan object-process link. Up to this point, allthe necessary object-process links are inplace. Next, at the beginning of game 3,integers are introduced into the games: theformal mathematical voice enters the games.Through the manipulation of the formalmathematical symbols of integers in theabove object-process links, integers arebeing reified and the addition and subtractionof integers are being established.
However, in the above analysis thefollowing significant question arises: Whatare the meaning-making processes(semiotic) involved in students’ integerreification in the dice games? We certainlydo not claim that we will exhaust this issuehere, but we will begin to address it throughthe vital component of the compensationstrategy.
Semiotics are Needed to ComplementReification Analyses
The theory of reification, drawing supportfrom a cognitivist/constructivist view oflearning, is mainly interested in the internalprocesses of students’ abstraction of
mathematical objects. It does not generallyrefer to the social semiotic means studentsused to achieve the abstraction of theseobjects, (e.g. in the dice games, the integers).The analysis of Linchevski & Williams (1999)did in fact go some way in providing a socialanalysis of the context as a resource forconstruction of the compensation strategy:they were excited mainly here by theaccessing of the socio-cultural resource of‘fairness’ in the games as a basis for anintuitive construction of compensation.Semiotic chaining was adduced to explainthe significance of the transition to the‘mathematical voice’, so that “two points fromyou is the same as two points to us” slidesunder a new formulation like “subtract minustwo is the same as adding two… plus two”.However, we will complement Linchevski &Williams’ (1999) study with a more detailedsemiotic analysis of the way that the abacus,gesture and deictics mediate children’sgeneralisations (after Radford’s, 2003, 2005methodology).
We wish to clarify at this point that we do notreject the reification analyses. Instead, weagree with Cobb (1994) who takes anapproach of theoretical pragmatism,suggesting that we should focus on “whatvarious perspectives might have to offerrelative to the problems or issues at hand”(p. 18). We propose that in this sensesemiotic social theories can becomplementary to constructivist ones. Moreprecisely, we propose that Radford’s theoryof objectification (Radford, 2002, 2003) canbe seen as complementary to the theory ofreification (Sfard, 1991; Sfard & Linchevski,1994): while Sfard (1991) provides a modelfor the cognitive changes taking place,Radford (2002, 2003) provides the meansto analyse these changes on the social,‘intermental’ plane.
Radford addresses the issue of semioticmediation through his theory of
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Radford’s Semiotic Theory ofObjectification
Objectification is “a process aimed at bringingsomething in front of someone’s attention orview” (Radford, 2002, p.15). It appears inthree modes of generalization: generalizationthrough actions, through language andthrough mathematical symbols. These arefactual, contextual and symbolicgeneralization (Radford, 2003).Objectification during these generalizationsis carried out gradually through the use ofsemiotic means of objectification (Radford,2002):
…objects, tools, linguistic devices,and signs that individuals intentionallyuse in social meaning-makingprocesses to achieve a stable form ofawareness, to make apparent theirintentions, and to carry out theiractions to attain the goal of theiractivities, I call semiotic means ofobjectification. (Radford, 2003, p. 41)
Factual generalization, a generalization ofactions (but not of objects), is described asfollows:
… A factual generalization is ageneralization of actions in the formof an operational scheme (in a neo-Piagetian sense). This operationalscheme remains bound to theconcrete level (e.g., “1 plus 2, 2 plus3” …). In addition, this schemeenables the students to tackle virtuallyany particular case successfully.(Radford, 2003, p. 47)
The formulation of the operational schemeof factual generalization is based on deicticsemiotic activity, e.g. deictic gestures, deicticlinguistic terms and rhythm. The students rely
objectification (Radford, 2002, 2003). Thistheory, presented in some detail in thefollowing section, analyses students’dependence on the available semioticmeans of objectification (SMO) (Radford,2002, 2003) to achieve increasinglysocially-distanced levels of generality.Radford explains this reliance on SMOthrough reference to Frege’s triad: thereference (the object of knowledge), thesense and the sign (Radford, 2002). TheSMO refer to Frege’s sense, that is, theymediate the transition from the referenceto the sign. Moreover, Radford extended thePiagetian schema concept to include asensual dimension, as Piaget’s emphasison the process of reflective abstraction canlead to an inadequate analysis of the roleof signs and symbols (Radford, 2005).
The schema …is … both a sensualand an intellectual action or a complexof actions. In its intellectual dimensionit is embedded in the theoreticalcategories of the culture. In its sensualdimension, it is executed or carried outin accordance to the technology ofsemiotic activity… (Radford, 2005, p.7)
Given this extended schema definition, theprocess of abstraction of a newmathematical object needs to beinvestigated in relation to the semiotic activitymediating it. This investigation shouldexpose students’ meaning makingprocesses in the objectifications takingplace in the dice games, which allow theconstruction of integers as newmathematical objects, i.e. their reification inSfard’s sense.
In the next section we present analyticallyRadford’s theory of objectification (Radford,2002, 2003), which will then be applied inthe section following it to some of our datafrom the instruction through the dice games.
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on the signification power provided by deicticsto refer to actions on non-generic physicalobjects. These are perceivable, non-abstractobjects which can be manipulatedaccordingly. In the example from Radford(2003) below, the students had to find thenumber of toothpicks for any figure in thefollowing pattern.
The elaboration of the operational schemein this case can be seen in the followingsection of an episode provided by Radford(2003).
1. Josh: It’s always the next. Look! [andpointing to the figures with the pencil hesays the following] 1 plus 2, 2 plus 3 […].(Radford, 2003, p. 46-47)
Josh constructed the operational scheme forthe calculation of the toothpicks of any figurein the form “1 plus 2, 2 plus 3”, while pointingto the figures. Moreover, he used the linguisticterm always to show the general applicabilityof this calculation method for any specificfigure and the term next which “emphasizesthe ordered position of objects in the spaceand shapes a perception relating the numberof toothpicks of the next figure to the numberof toothpicks in the previous figure” (p. 48).Hence, in factual generalization:
…the students’ construction of meaninghas been grounded in a type of socialunderstanding based on implicitagreements and mutual comprehension
that would be impossible in a nonface-to-face interaction. … Naturally, somemeans of objectification may be powerfulenough to reveal the individuals’intentions and to carry them through thecourse of achieving a certain goal.(Radford, 2003, p. 50)
In contextual generalization the previouslyconstructed operational scheme isgeneralised through language. Its generativecapacity lies in allowing the emergence ofnew abstract objects to replace the previouslyused specific concrete objects. This is thefirst difference between contextual andfactual generalization: new abstract objectsare introduced (Radford, 2003). Its seconddifference is that students’ explanationsshould be comprehensible to a “genericaddressee” (Radford, 2003, p. 50): relianceon face-to-face communication is excluded.Consequently, contextual generalizationreaches a higher level of generality. Morespecifically, in Radford (2003) the operationalscheme “1 plus 2, 2 plus 3” presented abovebecomes “You add the figure and the nextfigure” (p. 52). Therefore, the pairs of specificsucceeding figures 1, 2 or 2, 3 become thefigure and the next figure. These two linguisticterms allow for the emergence of two newabstract objects, still situated, spatial andtemporal (Radford, 2003). Reliance on face-to-face communication is eliminated, anddeictic means subside. However, thepersonal voice, reflected through the wordyou, still remains.
Figure 3: First three ‘Figures’ of the ‘toothpick pattern’, labelled ‘Figure1’, ‘Figure 2’, ‘Figure 3’ byRadford (the picture in the box was taken from Radford, 2003, p. 45)
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In symbolic generalization, the spatial andtemporal limitations of the objects ofcontextual generalization have to bewithdrawn. Symbolic mathematical objects(in Radford’s case algebraic ones) shouldbecome “nonsituated and nontemporal”(Radford, 2003, p.55) and the students loseany reference point to the objects. Toaccomplish these changes, Radford’s(2003) students excluded the personalvoice (such us you) from theirgeneralization and replaced the genericlinguistic terms the figure and the nextfigure with the symbolic expressions and correspondingly. Hence, theexpression you add the figure and the nextfigure became . Still, Radford(2003) points out that for the students thesymbolic expressions n and (n+1)remained indexed to the situated objectsthey substituted. This is evident in students’persistent use of brackets and their refusalto see the equivalence of the expressions
and . Summarising,the mathematical symbols of symbolicgeneralization were indexes of the linguisticobjects of contextual generalization, whichin turn were indexes of the actions onconcrete physical objects enclosed in thefactual generalization operational scheme.
The Compensation Strategy – FactualGeneralization
In this section we analyse the objectificationof the compensation strategy in terms offactual, contextual and symbol icgeneralization. We present excerpts ofthe discourse contained in the games,which we analyse in terms of theircontr ibut ion to the progressiveabstraction of integers through the meansof objectification. We also discuss theSMO involved in students’ processes. Theanalyses of factual, contextual andsymbolic generalization are presented
n(n +1)
n + (n +1)
(n + n) +1n + (n + 1)
separately, but first we provide someinformation about the students and theepisodes in this paper.
The study, part of an ongoing PhDresearch, involves year 5 students inGreater Manchester, who had not yet beentaught integer addition and subtraction.The PhD involves two experimentalmethods (respectively containing 5 and 6groups of 4 students) from 2 separateclasses and a control group from a thirdclass. In each experimental method classthe students were arranged by theirteacher in mixed gender and abilitygroups, which were taught for three one-hour lessons. In this paper we focusedon a microanalysis of one group of oneof the methods – the dice games asoriginally applied by Linchevski andWilliams (1999).
Radford’s factual generalization is quitea clear-cut process based on action onphysical objects formulated into anoperational scheme through deictic activity.However, in our investigation of thecompensation strategy, we find a multi-stepprocess of semiotic contraction happeninginside it. The three following episodes co-constitute in our view the factualgeneralization. In these episodes, occurringduring game 1 (in lesson 1), the studentswere faced with a situation where they hadto add cubes/points to one of the two teams,but there was no space on the abacus. As aresult, a breakthrough was needed for thescoring to continue.
Episode 1 (Minutes 14:30-14:50, lesson 1):Umar had to add 1 yellow cube on theabacus but, as there was no space in therelevant column, he got stuck. Fayproposed taking away 1 red cube instead.““…” indicates a pause of 3 sec or more,and “.” or “,” indicate a pause of less than3 sec” (Radford, 2003, p. 46).
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Fay: You take 1 off the reds [pointing tothe red column on her abacus]. […]Because then you still got thesame, because you’re going backdown [showing with both her handsgoing down at the same level]‘cause instead of the yellowsgetting one [raising the right handat a higher level than her left hand]the red have one taken off [raisingher left hand and immediatelymoving it down, to show that thistime the reds decrease].
Fay’s proposal for the subtraction of a redcube instead of the addition of a yellow oneis the first articulation of the compensationstrategy in the games for this group ofstudents. We especially noticed theanalytical explanation of the proposedaction, which allows the process ofcompensation to be introduced for the firsttime. Deictic activity was associated bothwith the proposed action of taking away ared cube and with the justification followingit. Fay used pointing to the red cubes onthe abacus, as well as a gesture with bothher hands indicating the increase/decreaseof the pile of cubes in each team’s column.Moreover, the names “the yellows” and “thered” have a deictic role. We also notice thephrase “you still got the same”, stressingthat something (obviously important)remains unaltered: either we add a yellow
point/cube or subtract a red point/cube.This signi f icant unaltered gamecharacteristic, which we call the directeddifference of the points of the yellow andred team, still cannot be articulated as ithas not yet acquired a name.
Episode 2 (Minutes 20:15-20:43, lesson1): The yellows’ column was full and thereds’ only had space for 1 cube.Compensation was needed and as Zenoncould not understand, Jackie explainedas follows.
Jackie: It’s still the same, like … [a verycharacteristic gesture (see figure4): she brings her hands to thesame level and then she begins tomove them up and down inopposite directions, indicating thedifferent resulting heights of thecubes of the two columns of theabacus] because it’s still 2, theyellows are still 2 ahead [she doesthe same gesture while she talks]and the reds are still 2 below, soit’s still the same… [again thegesture] … em like… [closing hereyes, frowning hard] … I don’tknow what it’s called but it’s still thesame… score [the gesture ‘same’again before and while articulatingthe word “score” – indicating ‘same’score on her abacus].
Figure 4: Jackie’s gesture (this sequence of action performed fast and repeated several times)
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In episode 2, we noticed the repeated useof the phrase “it’s still the same”, the word“still” followed by the difference in teampoints (i.e. “still 2”, “still 2 ahead”), as wellas the accompanying characteristicgesture. The gesture, too, emphasized theimportance of the unaltered directeddifference of the cubes of the two teams.We also noticed Jackie’s difficulty in findinga proper word for this important unalteredgame characteristic: “em like… [closing hereyes, frowning hard] … I don’t know whatit’s called but it’s still the same… score”(extract from episode 2 above). We believethe articulation of the word score, meaningwhat we call the directed difference of teampoints, as well as Jackie’s gesture werevery important for the factual generalizationprocess, because they achieved thesemiotic contraction (Radford, 2002) of theprocess originally established in episode 1.From this point onward, the students do notneed to provide an analytical semioticjustification of the proposed action, as Fayneeded to in episode 1. Just saying thatthe score will be the same is enough. Asimilar effect was accomplished by the worddifference in a different group (Koukkoufis& Williams, 2005).
Episode 3 (Minutes 21:27-21:57, lesson 1): There’s only space for 2 yellow cubes, butFay has to add 3 yellows and 1 red.
Fay: Add 2 on [she adds 2 yellowcubes] and then take 1 of theirs off[she takes off a red cube] and thenfor the reds [pointing to the reddice] you add 1, so you add the redback on [she adds 1 red cube].
Researcher: […] Does everybody agree?(Jackie and Umar say“Yeah”).
Finally, in the above episode furthersemiotic contraction took place. In fact, no
justification of the proposed action wasprovided, as it seemed to be unnecessary– indeed Jackie and Umar agreed with Faywithout further explanation. We argue thatthe further semiotic contraction happeningin episode 3 completed the factualgeneralization of the compensationstrategy.
To sum up, we see in the three episodesprovided up to this point a continuum asfollows: in episode 1 Fay presented aproper action and an analytical process tojustify it; in episode 2 again a proper actionwas presented but the process justifying itwas contracted; finally in episode 3 thepresentation of the proposed action wassufficient, therefore further semioticcontraction took place and the process forresulting in this proposed actiondisappeared.
The Compensation Strategy –Contextual Generalization
Contextual generalization, in whichabstraction of new objects throughlanguage takes place, has not yet beencompleted in this case. If we had had acontextual generalization of thecompensation strategy, we would have ageneralization like this: if you can’t add anumber of yellow/red points, you cansubtract the same number of red/yellowpoints instead. Similarly for subtraction, thegeneralization would be similar to this: ifyou can’t subtract a number of yellowpoints/red points, you can add the samenumber of red/yellow points instead.However, our students did notspontaneously produce such ageneralization, neither does theinstructional method demand it; thereforewe did not insist that the students produceit. We believe that the lack of articulationof the compensation strategy through
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generic linguistic terms, and thus theincompleteness of the production of acontextual generalization, has to do withthe compensation strategy being toointuitively obvious. On the contrary, in thecase of the cancellation strategy(Linchevski & Williams, 1999) which wasnot so obvious, the same studentsproduced a contextual generalization asfollows (Fay, minutes 38:17-38:40, lesson1, 5 reds and 2 yellows): “you find thebiggest number, then you take off thesmaller number”. In the case of thecontextual generalization of thecancellation strategy, we notice that newabstract objects (“the biggest number”, “thesmaller number”) enter the discourse, asin Radford (2003). However, we will notdiscuss the contextual objectification of thecancellation strategy here.
The Compensation Strategy –Symbolic Generalization
Despite the incompleteness of the contextualgeneralization, we found that symbolicgeneralization was not obstructed! In thissection we discuss the symbolicgeneralization of the compensation strategy,which presents some differences from thatof the case presented by Radford (2003).
To begin with, in Radford (2003) symbolicgeneralization remained indexical throughoutthe instruction. In our case, the studentsbegan using symbolic generalization non-indexically. For convenience, we presentindexical and non-indexical symbolicgeneralization separately.
Indexical Symbolic Generalization
The elaboration of a symbolicgeneralization for the compensationstrategy demands the replacement of pre-symbolic signs with symbolic ones.
Therefore, the reference to yellow and redteam points has to be substituted byreference to positive and negative integers.According to the dice games method, thisis achieved in the beginning of game 3,when the red and the yellow die arereplaced by the integer die. Analytically, thenumbers +1, +2 and +3 (on the integer die)are points for the yellow team. Further, –1,–2 and –3 (on the integer die) are pointstaken away from the yellow team, thus theyare points for the red team. Of course,similarly one can say that +1, +2 and +3are point taken away from the red team.Conclusively, when it is “+” it is yellowpoints, while when it is “–” it is red points.In the following episode we witness thetransition from the pre-symbolic signs of“yellow team points” and “red team points”to the symbolic signs of “+” and “–” (positiveand negative integers).
Episode 4: Minutes: 20:45-21:55, lesson 3.
Researcher: +1. Who is getting points?Jackie: The yellowsResearcher: […] Who is losing points?Jackie, Umar: The redsFay: […] reds are becoming
called minuses and then the yellows are becoming
called plus.
As a result of the above introduction of theformal mathematical symbols for integers,positive integers are used to indicate yellowteam points and negative integers are usedto indicate red team points. Here lies thefirst difference from Radford’s symbolicgeneralization, which is soon to becomeevident.
In Radford (2003), the symbolic signs/expressions used in symbolicgeneralization were indexes of thecontextual abstract objects of contextualgeneralization. Hence, the expressions n
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and n +1 indicated the generic linguisticterms the figure and the next figure.Instead, in the dice games the formalmathematical symbols of integers wereindexes not of the generic linguistic termsof contextual generalization (which wasnever completed), but of the concreteobjects of factual generalization. Forexample, +2 is an index of “2 more foryellows” as well as of “2 yellow points”, asin episode 5.
Episode 5 (Minutes: 33:15-33:53, lesson 3)
Researcher: […] you get –2. What wouldyou do? (Fay takes 2 yellowcubes off) […] What if youhad +3?
Umar: You take away 3 of the reds.Zenon: … or you could add 3 to the
yellow.Fay, Jackie: … add 3 to the yellow.Researcher: Oh, 3 off the reds or 3 to the
yellows. (All the studentsagree)
Indeed, the students read +2 on the die,the researcher articulates it as “plus 2”,but then the students’ discussion is interms of reds and yellows. If symbolicsigns were being used non-indexically atthat point, Umar would have said “minus3” instead of saying “3 of the reds” (as inthe phrase “take away 3 of the reds”).Also the others would have said “plus 3”instead of “3 to the yellow” (as in thephrase “add 3 to the yellow”). It becomesclear that in our case, we witnessed adirect transition from factual to indexicalsymbolic generalization, without thecompletion of contextual generalizationbeing necessary. This transition wasafforded due to the RME context and theabacus model.
In indexical symbolic generalization, thoughthe operational scheme of factual
generalization is reconstructed through theuse of symbolic signs instead of concretephysical objects, i t is not a simplerepetition of factual generalization insymbolic terms that takes place. Nosemiotic contraction needs to take placefor the establ ishment of thecompensation strategy in symbolic terms.The students know right away thatinstead of adding +2 (2 yellow points)they can subtract 2 red points.
Non-indexical Symbolic Generalization
Up to now the formal symbolic signs forintegers are being used indexically, but theintended instructional outcome is thatstudents will eventually be using thesesymbols non-indexically. We do not implythat the symbols should drop theirconnection to the context though. Indeedit is essential that students can go back tothe contextual meanings of these symbolsin the dice games, so as to draw intuitivesupport regarding integers. We justemphasize that the students shouldbecome flexible in using the formal symbolsof integers either indexically or non-indexically. A non-indexical use of integersymbols would mean explicit referencesolely to pluses and minuses (i.e. +2, –3etc). Therefore, the compensation strategyshould be constructed only based on theformal symbols of integers, excluding thepre-symbolic signs of yellow and red teampoints.
In order to target non-indexical symbolicgeneralization, we encouraged students toarticulate the symbols on the dice as “+”(plus) and “–” (minus), in an attempt tofacilitate the connection of the verbalizationplus/minus to the symbolic signs +/–.Though in the beginning most studentsneeded to be reminded to use the “proper”names of the signs, by the time the studentshad played game 4 for a while they were
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able to refer to integers in a formal manner,as can be seen in the following examplesof student verbalizations. We believe thatthe introduction of the add/sub die in game4 obliged the students to refer correctly tothe integers with their formal names, so asto be able to perform the actions of additionand subtraction on these symbols. Forexample (brackets added), Fay said: add[minus 3], subtract [2 of the minuses];Zenon said: add [2 to the pluses]; Jackiesaid: add [minus 2]. Umar was stil lstruggling with the verbalization andsometimes said [minus 1] add or add[subtract 2] etc.
Finally, we checked if students hadspontaneously produced a more generalverbalization in a form like “if you can’t addpluses/minuses, you can subtract minuses/pluses” or the other way around. In thisgroup, such a generalization did not takeplace. We believe, however, that this willnot necessarily be the case for other groupsof students, and indeed that it may bedesirable to encourage this in the teaching.
The Semiotic Role of the Abacus Model
As may be clear by now, the abacus modeland the RME context of the dice games arevery significant for the reification of integersand the instruction of integer addition andsubtraction through the dice gamesmethod. Up to now we have referred to thesemiotic processes, but we have notreferred to the abacus model: thoughRadford’s theory of objectification has beencrucial in the analyses so far, we contendit needs to be complemented by an analysisof the role of the abacus in affording thesesemiotics. We claim that analysing thecontribution of the model in students’semiosis will afford some primarydiscussion of phenomena such as (i) theembodiment of semiotic activity, (ii) the
incompleteness of the contextualgeneralization and (iii) the direct transitionfrom factual to symbolic generalization.
The abacus model in the games seems inmany ways to be the centre of the activity:the abacus is in the centre of a ‘circle ofattention’, as we are all sitting around theabacuses (see figure 2 again); it affordsthe representation of the yellow and redteam points through their red and yellowcubes; it is the constant point of referenceabout which team is ahead. It was onlynatural that the abacus, being in the centreof the spatial arrangement and creditedwith allowing the students to keep thescore, became the focus of semioticactivity. What is even more important: theabacus mediated in some cases thesemiotic activity.
This can be seen in several features of thegames. To begin with, the team pointsreferred to the above episodes as “pointsfor the yellow/red” (or as “yellow/redpoints”) were concretized or ‘objectified’from the start: they were yellow and redcubes. That is, the points were embodiedinto the cubes. This allows, as Linchevskiand Williams (1999) point out, for integersto be introduced in the discourse as objectsfrom the very beginning: the studentsspeak about the general categories ofyellow and red points from the beginning.Additionally, the directed difference wasembodied on the abacus, as the differenceof yellow and red points can be seen witha glance at the abacus, and the sign isevidently that of the larger pile of cubes:i.e. in figure 1 the yellows on that abacusare 2 points ahead. This convenientreference to the directed difference in thetwo piles of cubes afforded the associationof semiotic activity to it, which made theestablishment of the compensationstrategy possible. Such semiotic activity isFay’s gesture in episode 1 in which the
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movements of her hands were matchedwith a verbal manipulation of the differenceof the two piles of cubes (i.e. “you’re goingback down”) to show that the directeddifference remained the same. AlsoJackie’s quick movement of her hands upand down in episode 2 again indicates thedifference in the two piles of cubes, in otherwords it points to the directed differenceas it is embodied on the abacus. Wesuggest that this embodiment is crucialbecause it mediates the emergence ofdeictic semiotic activity such as that of Fayand Jackie in episodes 1 and 2 and henceallows the objectification to take place. Wemay even consider the toothpick figures inRadford (2003) to afford the same role.
As we noted above, the embodiment of theyellow and red team points through thecubes allowed the introduction of points forthe yellow and points for the red as generalabstract categories. We mentioned earlierthat the students did not complete thecontextual generalization of thecompensation strategy to produce ageneralization like “if you can’t add anumber of yellow/red points, you cansubtract the same number of red/yellowpoints instead”. However, the embodimentof the yellow and red team points of theabacus had already introduced genericsituated objects into the discourse, eventhough this was not achieved throughlanguage. Consequently, the studentscould obviously see that the operationalscheme of the factual generalization canbe applied for any number of points for ateam. This is an additional reason to the onepresented earlier for the incompleteness ofcontextual generalization. Hence, thisembodiment of the team points in a senseshapes the semiotic activity in thecompensation strategy, providing one morereason why the completeness of contextualgeneralization was unnecessary in thiscase.
Further, the semiotic role of the abacus wascrucial in the direct transition from factualto symbolic generalization. As we haveseen in episode 4, the yellow pointsbecame “plus” and the red points are now“minuses”. We say that this direct transitionwas afforded through the construction of achain of signification (Gravemeijer et al.,2000; Walkerdine, 1988), in the form of atransition from the embodiment of yellowand red points through the abacus cubesto the embodiment of positive and negativeintegers. As a consequence of thistransition, the formal symbols could beembedded into the operational scheme forthe compensation strategy establishedthrough the factual generalization. Quitenaturally then, the embedding of the formalsymbols in the operational schemeperformed on the abacus produced thesymbolic generalization directly fromfactual generalization.
Conclusion
Beginning with a presentation of the OPLEmethodology and the dice games instruction,we argued the need for a finer grained,semiotic analysis of objectifications to explainhow reification is accomplished.
We have applied Radford’s theory ofobjectification to fill this gap in understandingthe case of the compensation strategy, a vitallink in the chain of significations necessaryto OPLE’s success: thus we were able toidentify relevant objectifications applyingRadford’s semiotic categories ofgeneralisation. This work began to revealthe significance of the abacus itself, whichaffords, and indeed shapes the semiosisin essential ways. We have also shownhow the effectiveness of the pedagogybased on OPLE can be explained assemiotic chaining using multiple semioticobjectifications and begun to discuss the
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significance of models and modelling in thedice games, and hence in the OPLEmethodology. Finally, we suggest that ourdiscussion over the semiotics of the abacusmodel might be the route to understandingthe significance of models and modellingin the RME tradition more generally. Wesuggest that the role of the abacus as amodel in this case might be typical of othermodels in RME. Indeed, Williams & Wake(in press) provide an analysis of the role ofthe number line in a similar vein.
In applying Radford’s theory in a very
different context we are bound to point outcertain differences in the two cases: forinstance, the differing roles of contextualgeneralisation in the two cases. Though theadaptation of the theory was necessary atsome points, we have shown that thistheory can be a powerful tool of analysis.The question arises as to whether theinstruction method adopted here gains orperhaps loses something by elidingcontextual generalisation: thus we suggestthat Radford’s categories might in fact beregarded as raising design-related issuesas well as providing tools of analysis.
References
Cobb, P. (1994). Where Is the Mind? Constructivist and Sociocultural Perspectives onMathematical Development. Educational Researcher, 23(7), 13-20.
Cobb, P., Gravemeijer, K., Yackel, E., McClain, K., & Whitenack, J. (1997). Mathematizingand Symbolizing: The Emergence of Chains of Signification in One First-Grade Classroom.In D. Kirshner & J. A. Whitson (Eds.), Situated Cognition. Social, Semiotic, andPsychological Perspectives (pp. 151-233). Mahwah, N.J.: Lawrence Erlbaum Associates.
Dirks, M. K. (1984). The integer abacus. Arithmetic Teacher, 31(7), 50–54.
Goodson-Espy, T. (1998). The roles of reification and reflective abstraction in thedevelopment of abstract thought: transitions from arithmetic to algebra. EducationalStudies in Mathematics, 36(3), 219-245.
Gravemeijer, K. (1997a). Instructional design for reform in mathematics education. In K.P. E. G. M. Beishuizen, and E.C.D.M. van Lieshout (Ed.), The Role of Contexts andModels in the Development of Mathematical Strategies and Procedures (pp. 13-34).Utrecht: CD-ß Press.
Gravemeijer, K. (1997b). Mediating Between Concrete and Abstract. In T. Nunez & P.Bryant (Eds.), Learning and Teaching in Mathematics. An International Perspective (pp.315-345). Sussex, UK: Psychology Press.
Gravemeijer, K. (1997c). Solving word problems: A case of modelling? Learning andInstruction, 7(4), 389-397.
Gravemeijer, K., Cobb, P., Bowers, J., & Whitenack, J. (2000). Symbolising, Modellingand Instructional Design. Perspectives on Discourse, Tools and Instructional Design. In
173
Relime
P. Cobb, E. Yackel & K. McClain (Eds.), Symbolizing and Communicating in MathematicsClassrooms (pp. 225-273). Mahwah, NJ: Lawrence Erlbaum Associates.
Koukkoufis, A., & Williams, J. (2005). Integer Operations in the Primary School: A SemioticAnalysis of a «Factual Generalization». Paper presented at the British Society for Researchin to the Learning of Mathematics Day Conference November 2005, St. Martin’s College,Lancaster.
Liebeck, P. (1990). Scores and forfeits – an intuitive model for integer arithmetic.Educational Studies in Mathematics, 21(3), 221–239.
Linchevski, L., & Williams, J. (1999). Using intuition from everyday life in ‘filling’ the gapin children’s extension of their number concept to include the negative numbers.Educational Studies in Mathematics, 39(1-3), 131-147.
Lytle, P. A. (1994). Investigation of a model based on neutralization of opposites to teachintegers. Paper presented at the Nineteenth International Group for the Psychology ofMathematics Education, Universidade Federal de Pernambuco, Recife, Brazil.
Radford, L. (2002). The seen, the spoken and the written. A semiotic approach to theproblem of objectification of mathematical knowledge. For the Learning of Mathematics,22(2), 14-23.
Radford, L. (2003). Gestures, speech, and the sprouting of signs. Mathematical Thinkingand Learning, 5(1), 37-70.
Radford, L. (2005). The semiotics of the schema. Kant, Piaget, and the Calculator. In M.H. G. Hoffmann, J. Lenhard & F. Seeger (Eds.), Activity and Sign. Grounding MathematicsEducation (pp. 137-152). New York: Springer.
Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections onprocesses and objects as different sides of the same coin. Educational Studies inMathematics, 22, 1-36.
Sfard, A., & Linchevski, L. (1994). The gains and pitfalls of reification: the case of algebra.Educational Studies in Mathematics, 26, 87–124.
Walkerdine, V. (1988). The mastery of reason. London: Routledge.
Williams, J. & Wake, G. (in press). Metaphors and models in translation between Collegeand workplace mathematics. Educational Studies in Mathematics.
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Andreas KoukkoufisSchool of EducationThe University of ManchesterUK
E-mail: [email protected]
Julian WilliamsSchool of EducationThe University of ManchesterUK
E-mail: [email protected]
