Learning Mathematics: Increasing the

Value of Initial Mathematical Wealth

Adalira Sáenz-Ludlow 1

A sign may recall a certain concept or combination of concepts from somebody’s memory,and can also prompt somebody to certain actions. In the first case we shall call a sign asymbol, in the second a signal. The (nature of the) effect of the sign depends on contextand the actual mental situation of the reader. Van Dormolen, 1986, p.157.

RESUMEN

Usando la teoría de signos de Charles Sanders Peirce, este artículo introduce la nociónde riqueza matemática. La primera sección argumenta la relación intrínseca entre lasmatemáticas, los aprendices de matemáticas, y los signos matemáticos. La segunda,argumenta la relación triangular entre interpretación, objetivación, y generalización. Latercera, argumenta cómo el discurso matemático es un medio potente en la objetivaciónsemiótica. La cuarta sección argumenta cómo el discurso matemático en el salón declase, media el aumento del valor de la riqueza matemática del alumno, en formasincrónica y diacrónica, cuando él la invierte en la construcción de nuevos conceptos.La última sección discute cómo maestros, con diferentes perspectivas teóricas, influyenen la dirección del discurso matemático en el salón de clase y, en consecuencia, en elcrecimiento de la riqueza matemática de sus estudiantes.

PALABRAS CLAVE: Riqueza matemática, interpretación, relación con signos,la tríada interpretación-objetivación-generalización.

ABSTRACT

Using the Peircean semiotic perspective, the paper introduces the notion of mathematicalwealth. The first section argues the intrinsic relationship between mathematics, learnersof mathematics, and signs. The second argues that interpretation, objectification, andgeneralization are concomitant semiotic processes and that they constitute a semiotictriad. The third argues that communicating mathematically is a powerful means of semioticobjectification. The fourth section presents the notion of mathematical wealth, the learners’investment of that wealth, and the synchronic-diachronic growth of its value throughclassroom discourse. The last section discusses how teachers, with different theoreticalperspectives, influence the direction of classroom discourse and the growth of the learner’sinitial mathematical wealth.

225Relime, Número Especial, pp. 225-245.

Fecha de recepción: Febrero de 2006/ Fecha de aceptación: Mayo de 20061 Department of Mathematics and Statistics. University of North Carolina at Charlotte.

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KEY WORDS: Mathematical Wealth, Interpretation, Relationships with Signs,The Triad Intepretation-Objectification-Generalization.

RESUMO

Usando a teoria de signos de Charles Sanders Peirce, este artigo introduz a noção deriqueza matemática. A primeira secção argumenta a relação intrínseca entre amatemática, os aprendizes de matemáticas, e os signos matemáticos. A segunda,argumenta a relação triangular entre interpretação, objetivação e generalização. A terceira,argumenta como o discurso matemático é um potente meio na objetivação semiótica. Aquarta seção argumenta como o discurso matemático na sala de aula adequar o aumentodo valor da riqueza matemática do aluno, em forma sincrônica e diacrônica, quando eleinverte a construção de novos conceitos. A última seção discute como maestros, comdiferentes perspectivas teóricas, influem na direção do discurso matemático na sala deaula e, conseqüentemente, no crescimento da riqueza matemática de sus estudantes.

PALAVRAS CHAVES: Riqueza matemática, interpretação, relação com signos,a tríade interpretação-objetivação-generalização.

RÉSUMÉ

En utilisant la perspective sémiotique peircienne, cet article introduit la notion de richessemathématique. La première section soutient qu’il y a une relation intrinsèque entre lesmathématiques, les apprenants des mathématiques et les signes. La deuxième sectionsoutient que l’interprétation, l’objectivation et la généralisation sont des processussémiotiques concomitants et qu’ils constituent une triade sémiotique. La troisième sectionsoutient que la communication mathématique est un puissant moyen sémiotiqued’objectivation. La quatrième section présente la notion de richesse mathématique,l’investissement de cette richesse par les apprenants et la croissance synchronique etdiachronique de sa valeur à travers le discours de la salle de classe. La dernière sectiondiscute de la façon dont les enseignantes et enseignants, avec des perspectivesthéoriques différentes, agissent sur l’orientation de la discussion dans la salle de classeet sur l’enrichissement de la pensée mathématique initiale des apprenants.

MOTS CLÉS: Richesse mathématique, interprétation, relation avec des signes,la triade interprétation-objectivation-généralisation.

Mathematics and its IntrisicRelationship with Signs

Since ancient times, philosophers andmathematicians alike have beenconcerned with the definition ofmathematics as a scientific endeavor and

as a way of thinking. These definitionshave evolved both according to the stateof the field at a particular point in time andaccording to different philosophical

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perspectives. Davis and Hersh, assert that“each generation and each thoughtfulmathematician within a generationformulates a definition according to hislights” (1981, p. 8). To define mathematicsis as difficult as to define signs. It is noteasy to define either one withoutmentioning the other, as it is not easy todefine them in a paragraph and even lessin a couple of sentences. Mathematiciansmake use of and create mathematical signsto represent, “objectify”, or encode theircreations. On the other hand, learnersinterpret mathematical signs and theirrelationships both to decode the conceptualobjects of mathematics and to objectify (i.e.,encode) their own conceptualizations.

All kinds of signs and sign systems areubiquitous in our lives but so ismathematics. Given the fascinating andineludible dance between mathematics andsigns, it is not surprising that somemathematicians become semioticians.Peirce, for example, dedicated severalvolumes to analyze the relationshipbetween mathematical objects andmathematical signs (The New Elements ofMathematics, Vols. I, II, III, IV, 1976) as wellas several essays to discuss the essenceof mathematics (for example, the onepublished in Newman’s World ofMathematics, 1956). Peirce definesmathematics as the science that drawsnecessary conclusions and its propositionsas “fleshless and skeletal” requiring for theirinterpretation an extraordinary use ofabstraction. He also considers thatmathematical thought is successful onlywhen it can be generalized. Generalization,he says, is a necessary condition formathematical thinking.

Rotman (2000), inspired by Peirce’s theory,has dedicated a book to definemathematics as a sign. At the beginningof his book, he gives an overarching

definition of mathematics to conclude thatmathematics is essentially a symbolicpractice.

Mathematics is many things; thescience of number and space; thestudy of pattern; an indispensable toolof technology and commerce; themethodological bedrock of thephysical sciences; an endless sourceof recreational mind games; theancient pursuit of absolute truth; aparadigm of logical reasoning; themost abstract of intellectualdisciplines. In all of these and as acondition for their possibility,mathematics involves the creation ofimaginary worlds that are intimatelyconnected to, brought into being by,notated by, and controlled through theagency of specialized signs. One cansay, therefore, that mathematics isessentially a symbolic practice restingon a vast and never-finishedlanguage—a perfectly correct butmisleading description, since bycommon usage and etymology“language” is identified with speech,whereas one doesn’t speakmathematics but writes it. (2000, p.ix, emphasis added).

But where does this symbolic practicecome from? Is mathematics, as anexpression of the symbolic behavior of thehuman species, a part of all cultures?Davis and Hersh (1981) argue thatmathematics is in books, in taped lectures,in computer memories, in printed circuits,in mathematical machines, in thearrangement of the stones at Stonehenge,etc., but first and foremost, they say, it mustexist first in people’s minds. Theyacknowledge that there is hardly a culture,however primitive, which does not exhibitsome rudimentary kind of mathematics.There seems to be a common agreement

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among White (1956), Wilder (1973), Bishop(1988), and Radford (2006a) for whommathematics is essentially a culturalsymbolic practice that encapsulates theprogressive accumulation of constructions,abstractions, generalizations, andsymbolization of the human species.Progress, White contends, would have notbeen possible if it were not for the humanability to give ideas an overt expressionthrough the use of different kinds of signs(or what he calls the human symbolicbehavior). He asserts that humancommunication, as the most important andgeneral of all symbolic behaviors, facilitatesnew combinations and syntheses of ideasthat are passed from one individual toanother and from one generation to thenext. White also stresses that mathematicslike language, institutions, tools, the arts,etc. is a cultural expression in the streamof the total culture. In fact, he argues thatmathematics is a synthesizing culturalprocess in which concepts react uponconcepts and ideas mix and fuse to formnew syntheses. For White, culture is thelocus of mathematical reality:

Mathematical truths exist in thecultural tradition in which theindividual is born and so they enterhis mind from the outside. But apartfrom cultural tradition, mathematicalconcepts have neither existence normeaning, and of course, culturaltradition has no existence apart fromthe human species. Mathematicalrealities thus have an existenceindependent of the individual mind,but are wholly dependent upon themind of the species. (1956, pp.2350-2351, emphasis added)

If mathematics is a symbolic practice, thenthe understanding of the nature of signsystems (i.e. the networking of signs oversigns to create new sign-references

according to a particular syntax, grammar,and semantics) is important for theteaching and learning of mathematics.Given that individuals, by nature, possesssymbolic behavior and mathematics is asymbolic practice, then why do somestudents come to dislike mathematics asa subject and very soon fall behind? Ingeneral, semiotics theories give us aframework to understand the mathematicaland the non-mathematical behavior of ourstudents. Among different theoreticalperspectives on semiotics, Peirce’s theoryof signs helps us to understand how wecome to construct symbolic relationshipsbased on associative iconic and indexicalones. A relation is iconic when it makesreference to the similarity between signand object; it is indexical when it makesreference to some physical or temporalconnection between sign and object; andit is symbolic when it makes reference tosome formal or merely agreed upon linkbetween sign and object, irrespective ofthe physical characteristics of either signor object.

Representation and interpretation are twoimportant aspects of Peirce’s theory. Hesees representation as the most essentialmental operation without which the notionof sign would make no sense (Peirce,1903) and considers that the mind comesto associate ideas by means of referentialrelations between the characteristics ofsign-tokens and those of the objects theycome to represent. As for interpretation,he considers that without the interpretationof signs, communicating with the self andwith others becomes an impossible task(Peirce, CP vols. 2 and 4, 1974). That is,without being interpreted, a sign as a signdoes not exist. What exists is a thing orevent with the potential of being interpretedand with the potential of becoming a sign.Metaphorically speaking, a sign is like aswitch; it becomes relevant and its

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existence becomes apparent only if it isturned on-and-off, otherwise, the switch isjust a thing with the potential to become aswitch. Likewise, a sign-token becomes asign only when its relationship to an objector event is turned on in the flow of attentionof the interpreting mind. That cognitiverelationship between the sign-token andthe interpreting mind is essential in Peirce’ssemiotic theory; in fact, it is whatdistinguishes his theory from other theoriesof signs. He crystallizes this interpretingrelation between the sing-token and theindividual as being the interpretant of thesign. This interpretant has the potential togenerate a new sign at a higher level ofinterpretation and generalization. At thishigher level, the new sign could, in turn,generate other iconic, indexical, orsymbolic relationships with respect to theobject of the sign. However, while theindividual generates new interpretants, theobject represented by the sign undergoesa transformation in the mind of theindividual who is interpreting. That is, theobject of the sign appears to be filtered bythe continuous interpretations of thelearner. In summary, Peirce considers theexistence of the sign emerging both fromthe learner ’s intellectual labor toconceptualize the object of the sign andfrom the construction of this object in thelearner’s mind as a result of his intentionalacts of interpretation.

A sign stands for something to theidea that it produces or modifies. Or,it is a vehicle conveying into themind something from without. Thatfor which it stands is called its object;that which it conveys, its meaning;and the idea to which it gives rise,its interpretant. (CP 1.339; emphasisadded)

By a Sign I mean anything whatever,real or fictile which is capable of a

sensible form, is applicable tosomething other than itself…andthat is capable of being interpretedin another sign which I call itsInterpretant as to communicatesomething that may have not beenpreviously known about its Object.There is thus a triadic relationbetween any Sign, and Object, andan Interpretant. (MS 654. 7)(Quoted in Pamentier, 1985;emphasis added).

Peircean semiotics helps to understandand explain many aspects of thecomplexity of the teaching and learning ofmathematics. For example, teachers’ andlearners’ expressions of theirinterpretations of mathematical signs bymeans of writing, reading, speaking, orgesturing; the interrelationship of themultiple representations of a conceptwithout confounding the concept with anyof its representations; and the dependencyof mathematical notation on interpretation,cultural context, and historical convention.In trying to understand the semiotic natureof the teaching and the learning ofmathematics, the above list about thesemiotic aspects of the teaching-learningactivity is anything but complete.

Brousseau, for example, contends thatmathematicians and teachers both performa “didactical practice” albeit of a differentnature. Mathematicians, he says, do notcommunicate their results in the form inwhich they create them; they re-organizethem, they give them the most generalpossible form; “they put knowledge into acommunicable, decontextualized,depersonalized, detemporalized form”(1997, p. 227). This means, that theyencode their creations using mathematicalsign systems or they create new signs ifnecessary. That is, they objectify orsymbolize their creations (i.e., knowledge

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objects) through spacio-temporal signs. Onthe other hand, the teacher undertakesactions in the opposite direction. She,herself, interprets mathematical meaningsembedded in spacio-temporal signs (sign-tokens), decodes conceptual objects, andlooks for learning situations that couldfacilitate the endowment of those sign-tokenswith mathematical meanings in the minds ofthe learners. Thus, mathematicians andteachers of mathematics have a necessaryinterpretative relationship with the signsystems of mathematics (i.e., semioticmathematical systems) because theycontinuously use them to encode, interpret,decode, and communicate the mathematicalmeanings of conceptual objects.

Teacher’s and Learner’sInterpretations and Objectifications

The interpretation of signs is important fortwo reasons. First, signs are not signs ifthey are not interpreted; being a signmeans being a sign of something tosomebody. Second, the meaning of a signis not only in the sign but also in the mindinterpreting that sign. Now the question is:Does a sign objectify? According toPeirce’s definition of signs, the answer isyes. A sign does objectify (i.e., It does maketangible) the object (conceptual or material)that it stands for. However, the sign notonly objectifies but it also communicates(to the interpreting mind) something thathas not been previously known about the

object. Thus, Peirce’s definition of signsimplies a continuous process ofinterpretation and as a consequence, aconcomitant process of gradualobjectification.

Radford (2006b), on the other hand,considers that to objectify is to make visibleand tangible something that could not beperceived before. He definesobjectification as “an active, creative,imaginative, and interpretative socialprocess of gradually becoming aware ofmathematical objects and their properties”.This definition is not in contradiction withPeirce’s definition of signs. Radford (2003)also defines means of objectification as“tools, signs of all sorts, and artifacts thatindividuals intentionally use in social-meaning-making processes to achieve astable form of awareness, to makeapparent their intentions, and to carry outtheir actions to attain the goal of theiractivities” (p. 41). This definition is also inharmony with Peirce’s definition ofinterpretant.

Since mathematical objects make theirpresence manifest only through signs andsign systems, how can teachers helplearners to enter into the world of thesesemiotic systems and break the code, soto speak, to “see” those objects bythemselves? Which mathematical objectsdo learners interpret from signs2? Or better,what “objects” do sign-tokens stand for inthe minds of learners and teachers? Would

Peirce gave several definitions of signs without contradicting previous definitions; instead he extended them. The

invariant in his definitions is the triadic nature of the sign. The variation is in the names he gave to the sign-vehicle/ sign-

token or material representation of the sign. First, he called sign the material representation of the sign, then sign-vehicle,

and then representamen. Some mathematics educators have favored the sign triad object-sign-interpretant, others, like

myself, have favored the sign triad object-representamen-interpretant because it does not use the word sign to indicate,

at the same time, the triad and a term in the triad. In this paper, I use the words representamen, representation, and sign-

token interchangeably. However, Peirce used the term representation in the general sense of being a necessary operation

of the human mind.

2

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learners and teacher ‘interpret’ the samemathematical objects (i.e., knowledgeobjects) from sign relations in mathematicalsign systems? Who objectifies what? Whatare the “products or effects” of teacher’s andlearners’ interpretations and objectifications?What are the teacher’s interpretations of thelearners’ interpretations? It appears thatteachers’ and learners’ interpretations andobjectifications go hand in hand in theteaching-learning activity. Because of thetriadic nature of the sign, there is a necessaryand concomitant relationship betweenobjectification and interpretation; there is nointerpretation without objectification and noobjectification without interpretation. Inaddition, these two processes are linked to athird concomitant process, the process ofgeneralization.

Mathematicians objectify their creationsinventing new mathematical signs orencoding them, using already establishedsigns and sign systems. Teacher andlearners re-create knowledge objects byinterpreting mathematical signs in a varietyof contexts; by doing so, they undergo theirown processes of objectification. Thereseems to be running, in parallel, threeprocesses of objectification: theobjectification of the teacher, theobjectification of the learners, and theteacher’s objectification of the learners’objectifications. This seems to be acumbersome play with words, although thisis at the heart of the interrelationship betweenteaching and learning. Obviously, teacherand learners objectify, but do they objectifythe same thing? Are these objectificationsisomorphic or at least do they resemble eachother? Is the teacher aware of theseprocesses of objectification? If so, then theteacher has the potential: (a) to question andvalidate her own interpretations andobjectifications; (b) to make hypothesesabout the learners’ objectifications; (c) toquestion the learners to validate her

hypothesis in order to guide their processesof interpretation and objectification; and (d)to differentiate between her interpretationsand objectifications and the learners’interpretations and objectifications.

When teachers and learners engage in theteaching-learning activity, who interpretsand what is interpreted is somewhatimplied, but it is nevertheless tacit, in theprocesses of objectification andinterpretation. Obviously, in one way oranother, teachers appear to play animportant role in the learners’ processes ofinterpretation and objectification. Brousseauappears to indicate these levels ofinterpretation. “The teacher’s work …consists of proposing a learning situation tothe learner in such a way that [the learner]produces her knowing as a personal answerto a question and uses it or modifies it in orderto satisfy the constraints of the milieu [whichis managed by necessary contextual andsymbolic relationships] and not just theteacher’s expectations” (1997, p. 228,emphasis added). Here, Brousseau pointsout the difference between learners’interpretations and teachers’ interpretationsand intentions. The question is whether ornot the teacher ’s intentions andinterpretations are realized in the students’interpretations and objectifications. In otherwords, do the teacher’s and the learners’interpretations and objectifications, at least,resemble each other?

The teacher may design learning situationsto induce learners’ construction ofmathematical objects and relationshipsamong those objects; or the teacher maydesign learning situations in which themathematical object is directly delivered asif it were a cultural artifact ready to be“seen” and memorized by the learners,while saving them the cost of their ownabstractions and generalizations. In thelatter case, the learners could be

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objectifying only the iconic or indexicalaspects of the mathematical signs withoutcapturing the symbolic aspects of thosesigns and their symbolic relations with othersigns. In the former case, the learnerscapture both the symbolic aspects of thesigns and their symbolic relations with othersigns. This means that the learner is ableto unfold those signs to “see” not only thesymbolic aspects but also the indexical andiconic aspects embedded in them. Thus,learners and teacher could be interpretingdifferent aspects of the mathematical signs(iconic, indexical, or symbolic) and, inconsequence, interpreting the nature ofmathematical objects from different levelsof generalization and abstraction.

But what is the nature of the mathematicalobjects? How many types of objects couldbe interpreted from mathematical signs?Duval (2006) calls our attention to differenttypes of objects:

(1) Objects as knowledge objects whenattention is focused on the invariant ofa set of phenomena or on the invariantof some multiplicity of possiblerepresentations. Mathematical objectslike numbers, functions, vectors, etc. areall knowledge objects.

(2) Objects as transient phenomenologicalobjects when the focus of attention ison this or that particular aspect of therepresentation given (e.g., shape,position, size, succession, etc.).

(3) Objects as concrete objects when thefocus of attention is only on theirperceptual organization.

Thus given a sign-token (i.e., arepresentamen or a representation), onecould interpret at face value a concrete objectif one focuses strictly on the material aspectsof this semiotic means of objectification

without constructing relationships with otherrepresentations. One could also interpret aphenomenological object if one goes beyondpure perception and focuses on aspects ofthose representations in space and time. Orone could also interpret a knowledge objectif one focuses on the invariant relations in arepresentation or among representations.For example, Duval (2006) considers that thealgebraic equation of a line and its graphcould be seen as phenomenological objectswhen one focuses on the material aspectsof these representations (i.e., iconic andiconic-indexical aspects of the sign-tokensor representations); they could be knowledgeobjects if one focuses on the invariance ofthese representations (i.e., symbolicaspects). Once one is able to interpret andto objectify knowledge objects, one shouldbe able to unfold the phenomenological (i.e.,iconic, iconic-indexical) and material (i.e.,iconic) aspects of those objects. However,if one objectifies only phenomenological andconcrete objects, one would not necessarilycome up with the symbolic aspects of theircorresponding knowledge objects.

In a nutshell, Duval’s characterization of‘objects’ points out the semiotic stumblingblocks of the teaching and learning ofmathematics. In this characterization, themanifestation of a knowledge objectdepends not only on its representation butalso on the human agency of theinterpreter, user, producer, or re-producerof that object. Objects could be either theinterpretation of pure symbolic relationsembedded in the sign-tokens orrepresentations (i.e., knowledge objects orpure signifieds); or they could be purematerial tokens with no signifieds (i.e.,concrete objects or concrete things); orthey could be materially based tokensinterpreted in time and space (i.e.,phenomenological objects). The best casewould be when the knowledge object isobjectif ied in space and time with

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structured signifieds and with the potentialof being used again in private and inter-subjective conceptual spaces; and, viceversa, when mathematical knowledgeobjects are decoded from the material sign-tokens or representations without escapingtheir extension in space and theirsuccession in time.

As teachers and learners engage in theteaching-learning activity, which objects arethe teacher referring to and which objects arethe learners interpreting, objectifying, andworking with? In the best of all scenarios,teacher and learners could interpret, from thesame sign-token or representation, the sameknowledge object. However, sometimeslearners might only be interpreting concreteobjects (i.e., concrete marks) orphenomenological objects missing, in theprocess, the knowledge object; meanwhilethe teacher might be interpreting that learnersare interpreting knowledge objects. Thissituation would clearly mark a conceptualrupture between teacher and learners.Therefore, interpreting in the classroom isa process that unfolds at three levels: (1)the level of those who send an intentionalmessage (the teacher or the students); (2)the level of those who receive and interpretthe message (the learners or the teacher);and (3) the level of the sender’s interpretationof the receiver’s interpretation. Thus, in theteaching-learning activity, the interpretationprocess is not only a continuous process ofobjectification but it is also a relativeprocess (relative not only to teachers andlearners but also relative to their priorknowledge, not to mention their beliefsabout knowledge and knowing).

Communicating Mathematically as aMeans of Objectification

Communication in the mathematicsclassroom was viewed as depending

exclusively on language (syntax andgrammar), the active and passive lexiconof the participants, and the nature of thecontent of the message (Austin andHowson, 1979). Now, we have becomeaware that communication depends notonly on natural language but also on thespecific sublanguages of different fields ofstudy, on linguistic and non-linguisticsemiotic systems, and on a variety of socialand cultural contexts in which the contentof the message is embedded (Halliday,1978; Habermas, 1984; Bruner, 1986;Vygotsky, 1987; Steinbring et al. 1998).Communication is also influenced by thebehavioral dispositions and expectationsof the participants as well as by theirintersubjective relations of power(Bourdieu, 1991). Thus, perspectives oncommunication, in general, appear to havegained in complexity rather than insimplicity. Hence, perspectives oncommunication in the mathematicsclassroom have changed. Thiscommunication depends on naturallanguage, mathematical sublanguage, andmathematical sign systems that mediateteacher’s and learners’ interpretations ofmathematical objects.

Rotman (2000) points out a special featureof mathematical communication. Hecontends that in order to communicatemathematically, we essentially write. Hecontends that writing plays not only adescriptive but also a creative role inmathematical practices. He asserts thatthose things that are described (thoughts,signifieds, and notions) and the means bywhich they are described (scribbles) makeup each other in a reciprocal manner.Mathematicians, as producers ofmathematics, Rotman says, think theirscribbles and scribble their thinking.Therefore, one is induced to think thatlearners of mathematics should do thesame in order to produce and increase their

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personal ‘mathematical wealth’ as aproduct of their own mathematical labor.Such wealth does not accumulate all atonce, but rather, it accumulates graduallyin a synchronic as well as in a diachronicmanner. We will enter the discussion ofmathematical wealth and its synchronic-diachronic formation in the next section.

It appears that communicatingmathematically is first and foremost an actof writing in the form of equations,diagrams, and graphs, supported all alongby the specialized sublanguage ofmathematics (mathematical dictionariesare a living proof that a mathematicalsublanguage exists). We also need toconsider that writing is not an isolated act.Acts of writing are concomitant with acts ofreading, listening, interpreting, thinking, andspeaking. All these acts intervene insemiotic processes of objectificationresulting from personal processes ofinterpretation by means of contextualizationand de-contextualization, concretizationand generalization. That is, communicatingmathematically depends on the synergy ofthe processes of interpretation,objectification, and generalization.

Gay (1980), Rossi-Landi (1980), andDeacon (1997) argue that any semiologicalsystem only has a finite lexicon but itssemantics accounts for an unlimited seriesof acceptable combinations and that someof these combinations propose originalways of describing linguistic andextralinguistic reality. By the same token,the semiotic system of mathematics has afinite number of tokens and a finite set ofaxioms, theorems, and definitions (Ernest,2006). When these elements arecombined, they account for a large numberof acceptable combinations that describeor justify, create or interpret, prove or verify,produce or decode already culturallystructured mathematical objects. In

discovering, constructing, apprehending,reproducing, or creating mathematicalobjects, reading and writing, listening andspeaking become essential means forproducing and interpreting combinations ofreferential relations (whether iconic,indexical, or symbolic) in a space that isboth visible and intersubjective.

Vygotsky (1987) contends that in anynatural language the writing and speakingacts are of different nature. Writing, hesays, is a monological activity in whichcontext is mental rather than physical andtherefore it does not benefit from theimmediate stimulation of others. Thismakes writing a demanding mental activitythat requires not only the syntax andgrammar of the language in use, but alsothe conceptual objects (i.e., knowledgeobjects) to be encoded or decoded usingparticular signs or combination of signs. Incontrast, Vygostsky argues that oraldialogue is characterized by the dynamicsof turn-taking determining the direction ofspeech: in oral dialogue, questions lead toanswers and puzzlements lead toexplanations. Written speech, instead, isnot triggered by immediate responses asin oral dialogue. In writing, the unfoldingof an argument is based much more onthe personal and private labor of theindividual. What Vygotsky argues aboutwritten and oral speech in the context oflanguage can be transferred to the contextof mathematical communication inside andoutside of the classroom. It is one thing toclarify one’s mathematical ideas whendebating them and another to producethem as the result of one’s own isolatedmental labor and personal reflection. Bothtypes of communication are commonlyused among mathematicians (Rotman,2000). In the last decades, oral and writtenmodes of interacting in the classroom havebeen accepted as appropriate ways ofcommunicating mathematically in the

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classroom (National Council of Teachersof Mathematics, Standards, 2000).

Rotman (2000) also considers that writingand thinking are interconnected and co-terminous, co-creative, and co-significant.There is no doubt that for professionalmathematicians who are in the businessof producing mathematical knowledge thisshould be the case. But are writing andthinking always interconnected, co-creative, and co-significant activities for thelearners? Or are the learners using writingto take into account only the perceptual levelof mathematical signs (i.e., sign-tokens orconcrete objects) to automatically performalgorithmic computations in order to surviveacademically? Do multiple-choice examsinterfere with the development of the learners’thinking-writing capacity? Do teachers makelearners aware that reading, writing, listening,and speaking are effective means ofobjectifying mathematical knowledgeobjects? Do teachers make learners awarethat communicating mathematically is alsoconstituted by justifying in terms ofexplanation, verifications, making validarguments, and constructing proofs?

To communicate mathematically in theclassroom, the teacher has: (a) to flexiblymove within and between different semioticsystems (e.g., ordinary language,mathematical sub-language, mathematicalnotations, diagrams, graphs, gestures, etc.)(Duval 2006); (b) to refer to mathematicalobjects that are other than visible andconcrete (e.g., patterns, variance, andinvariance across concepts) (see forexample, Radford, 2003); (c) to address thelearners in ways that are supposed to bemeaningful to them (see for example,Ongstad, 2006); and (d) to express(verbally and nonverbally) the encodingand decoding of mathematical objects(Ongstad, 2006). Thus communicatingmathematically between teacher and

learners also requires the triad referring-addressing-expressing within and betweenseveral semiotic systems.

Interpreting mathematical signs is, inessence, a dynamic process of objectificationin which the individual gradually becomesaware of knowledge objects represented inverbal, algebraic, visual, and sometimesimaginary representations (Davis and Hersh,1981) and these representations have theirown inherent properties. Becoming awareof knowledge objects through a variety ofrepresentations is in itself a demandingintellectual labor because of thecharacteristics of different representations.Skemp (1987), for example, points outdifferences between visual and verbal/algebraic representations: (1) Visualrepresentations, such as diagrams,manifest a more individual and analog typeof thinking; in contrast, verbal/algebraicrepresentations manifest a more socializedtype of thinking. (2) Visual representationstend to be integrative or synthetic; incontrast, verbal/algebraic representationsare analytical and show detail. (3) Visualrepresentations tend to be simultaneous; incontrast, verbal/algebraic representationstend to be sequential. (4) Visualrepresentations tend to be intuitive; incontrast, verbal/algebraic representationstend to be logical. All these tacitdifferentiations are part and parcel of the tacitknowledge underpinning the classroommathematical discourse and they maycreate difficulties for some learners(Presmeg, 1997). Yet another source oftacit knowledge in the classroom discourseis the variety of speech genres inmathematical discourse, for example,debating, arguing, justifying, and proving(Seeger, 1998). For Rotman, persuading,convincing, showing, and demonstratingare discursive activities with the purposeof achieving intersubjective agreement,generalization, and semiotic objectification.

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This kind of tacit knowledge is not evenremotely considered to be a part of theinstitutionalized school curriculum andmany teachers are not even aware of it.The lack of explicitness of the tacitknowledge (expected to be understood bythe learners) contributes to their abrupt andfoggy entrance into the territory of themathematical world, where those who willsuccessfully accumulate ‘mathematicalwealth’ are the ones who have the capacityof making explicit for themselves the tacitunderpinnings of mathematical discourseand the triadic nature of the process ofconceptualization (interpretation,objectification, and generalization).

To summarize, the emergence ofmathematical objects and their meaningsare in no way independent from intentionalacts of interpretation and objectificationmediated by reading and writing, speakingand listening. These acts are essential inthe gradual mathematical growth of themathematical wealth of the learners.Communicating mathematically in terms ofreading, writing, listening, and debatingshould be considered means ofinterpretation and objectification. Hence,knowledge of semiotics appears to be anecessary conceptual tool in the classroom,not only for theoretical and explanatorypurposes but also for pragmatic ones.

Communicating Mathematically andMathematical ‘Wealth’

We would like to consider mathematicalwealth as a metaphor to refer to the learner’scontinuous accumulation of mathematicalknowledge as the product of his intellectuallabor in an intra-subjective or inter-subjectivespace. This mathematical wealth is personal,although socially and culturally constituted,in addition to continuously being in themaking.

As learners initiate and continue theirjourney in a mathematical world (which isplanned by the institutionalized curriculumand/or by the learners’ own interests), theycontinuously invest their existingmathematical wealth in order to increaseits value. This investment is a continuousprocess of evolution, development, andtransformation of the learner’s referentialrelations using signs of iconic, indexical,and symbolic nature. Sign-tokens are notinherently icons, indices, or symbols; theyare so only if interpreted in that way. Thelearner’s interpretation of the referentialrelations of signs is manifested in his verbaland written responses. Say for example,that a learner is capable of keeping inmemory the expression “positive timespositive is positive and negative timesnegative is positive”(*). What is themeaning of this expression for a learner atdifferent phases of his mathematicaljourney? Does it change? Does it remainthe same?

It could be that he has memorized thisexpression as we memorize prayers whenwe are little; they just stick in our mindsand we regurgitate them, even if we do notknow what they mean. It could be that thelearner interprets that expression asfollows: “I remember that with a ‘-’and a ‘-’ I can make a ‘+’’; and with a ‘+’ and a ‘+’ Ican only make a ‘+’”. In these cases, thelearner has only an iconic relationship withthe expression (*). The learner is trying tomake sense by focusing on the physicalresemblances of the sign-tokens. Wouldhe be able to ascend from the level ofhaving an iconic relation with theexpression (*) to the level of having anindexical relation with it? If the learnersays, for example, “I know that 2 times 3is 6 and -2 times -3 is 6”, then the learnerhas an iconic-indexical relation with theexpression (*) because he has a particularcase that, in a way, indicates the possibility

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of the generality of this statement.However, when the learner comes totransform the above expression into anexpression like “xy>0 only in cases whenx>0 and y>0 or when x<0 and y<0” or to re-cognize that “-x” could be positive ornegative depending on the value of x; thenthe learner has a symbolic relation with theexpression (*). In the latter, the learner hascome to enrich the meaning of theexpression (*) as he works with variablequantities in the context of algebra.

In fact, as the learner comes to develop asymbolic relationship with this expression,or the expression (*) becomes symbolic forthe learner, he will also come to have aniconic and iconic-indexical relationship withit. This is to say that once a learner has asymbolic relation with a sign, he would beable to unfold it into iconic and iconic-indexical relations whenever necessary.But the other way around is not necessarilytrue. A learner, who has an iconic or aniconic-indexical relationship with a sign-token (in this case the expression (*)) maynot necessarily have a symbolicrelationship with it (i.e., the sign-token doesnot yet stand for a knowledge object in themind of the learner). What does this meanin terms of objects? A learner who hasconstructed either a concrete or aphenomenological object may very wellhave not yet constructed a knowledgeobject. However, if the learner hasconstructed a knowledge object, one cansafely infer that he also has constructedthe corresponding concrete andphenomenological objects (i.e., the learnercould be able to deconstruct the knowledgeobject into phenomenological and concreteobjects).

When a learner repeats the expression“positive times positive is positive andnegative times negative is positive”, itmeans that he could have an iconic, an

iconic-indexical, or a symbolic relationshipwith the expression. What is therelationship that the learner hasconstructed? This is not evident until thelearner has the opportunity to use it indifferent contextual situations. How doesthe teacher, who is in charge of guidingthe learner, interpret the kind of relationshipthat the learner has with the expression?The teacher could have a symbolicrelationship with the expression (*) andthink that the learner also has a symbolicrelationship with it. In addition, if theteacher considers that any sign-token orrepresentation is inherently symbolic,independently of the learner ’sinterpretation, she would firmly believe thatthe learner could have only a symbolicrelationship with it. Henceforth, the teacherwill not change her interpretation of thelearner’s interpretation, and this mightrupture the semantic l ink in thecommunication between the teacher andthe learner. The teacher’s expectationswould run at a level higher than the currentlevel of the learner’s possibilities. Thiscould prompt the teacher to misjudge thecapabilities of the learner and to give upon the learner instead of creating newlearning situations to induce theconstruction of the learner’s symbolicrelationship with sign-tokens (in this casethe expression (*)). The worst case wouldbe when the learner stops increasing thevalue of his initial mathematical wealth andsoon falls behind others and with feelingsof not having any intellectual capacity formathematics.

The teacher needs to understand that theexpression (*) or any other sign could haveiconic, iconic-indexical, or iconic-indexical-symbolic meanings for the learner atdifferent points of his mathematical journey.That is, the teacher should be aware thatwhat one routinely calls “symbols” arenothing else than sign-tokens that can be

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interpreted at different levels ofgeneralization. The teacher who comesto understand what is symbolic and forwhom, what is iconic-indexical and forwhom, what is iconic and for whom,should also come to see her teachingdeeply rooted in her own learning ofmathematics and in her learning of herstudents’ learning.

A teacher unaware of hers and the learners’possible iconic, indexical, and symbolicrelationships with signs has no grounds formaking hypotheses about the learners’interpretations. Then, the teacher will onlyinterpret her own interpretations but notthose of the learners. That is, the teachercomes to collapse the three levels ofinterpretation (her own interpretation, thelearners’ interpretation, and herinterpretation of the learner’s interpretation)making it only one muddled level that barelyreflects the cognitive reality of thoseinvolved in the teaching-learning activity. Indoing so, the teacher loses cognitivecontact with the learner and thus theopportunity to support his personalprocesses of re-organization andtransformation of his prior knowledge. It isnot surprising, then, that Bauersfeld (1998)noticed that learners are alone in makingtheir own interpretations and that there isa difference between “the matter taught”and “the matter learned”. In our framework,this would translate as the existence of adifference between the matter interpretedby the teacher, the matter taught by theteacher, and the matter interpreted by thelearners.

At any given moment, learners start with aparticular set of mathematicalconceptualizations to be transformed andre-organized. This initial set of conceptualelements, with whatever mathematicalvalue (iconic, indexical, or symbolic) , iswhat we would like to call the initial

mathematical wealth. This wealth, ifinvested in well designed learningsituations using a variety of contexts, willallow the learner to embed iconicrelationships into iconic- indexicalrelationships and to embed iconic-indexical relationships into symbolicones. By doing so, the learner will cometo construct mathematical patterns (atdifferent levels of generalization), andregulated combinations of mathematicalsigns according to the structure of themathematical sign systems he is workingwith at that moment. For example,learners’ generalization, in the naturalnumbers, that multiplication makes biggerand division makes smaller, has to be re-conceptualized or re-organized when theystart working with decimals. Later on,multiplication needs to be generalized asan operation with particular properties.And even later, division needs to be re-cognized and re-organized as a particularcase of multiplication. That is, the learner’srelationship with multiplication and itsresults needs to be transcended andattention needs to be focused on thenature of the operation itself, leavingimplicit the indexicality of particularresults as well as the iconicity of the sign-tokens “times” or “x” (like in 4 times 2 or4x2) used for multiplication in gradeschool. That is, multiplication, in the longrun, should become a symbolic operationin the mind of the learner and not only themere memorization of multiplication factsand the multiplication algorithm.

Hence, the nature of the investment of thelearner’s mathematical wealth resides inhis capacity to produce new levels ofinterpretations and concomitantly newobjects (concrete, phenomenological, andknowledge objects) at different levels ofgenerality (iconic, indexical, or symbolic).This kind of investment increases thelearner’s mathematical wealth and goes

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beyond the manipulation of “sign-tokens 3”.That is, the value of the investmentincreases as the learner’s interpretation ofsigns ascends from iconic, to iconic-indexical, to iconic-indexical-symbolicalong his recursive and continuouspersonal processes of interpretation,transformation, and re-organization.Moreover, what becomes symbolic at aparticular point in time in the learner’sconceptual evolution could become theiconic or iconic-indexical root of a newsymbolic sign at a higher level ofinterpretation. For example, our middleschool knowledge of the real numbers withthe operations of addition and multiplicationbecomes the root for interpreting, later on,the field structure of real numbers (i.e., theset of real numbers with the operations ofaddition and multiplication constitutes anadditive group and a multiplicative grouprespectively and also the operation ofmultiplication distributives over theoperation of addition).

In summary, learners who becomemathematically wealthy are those who,along the way, are able to interpretknowledge objects from concrete sign-tokens and, in the process, are able totranscend their phenomenological aspects(i.e., iconic-indexical) and ascend tosymbolic relationships with them throughcontinuous acts of interpretation,objectification, and generalization. Nomatter through what lens one sees teachingand learning (i.e., learners discover,construct, or apprehend mathematicalconcepts), this triadic intellectual process(interpretation-objectification-generalization)is in reality a continuous recursivesynchronic-diachronic process in theirintellectual lives. This process is not only

It is worthwhile to notice that the expression manipulation of symbols becomes an oxymoron in Peirce’s theory

of signs and it could be replaced by the expression manipulation of sign-tokens.

2

synchronic. It would be impossible for thelearner to appreciate, all at once, currentand potential meanings embedded incontextual interpretations of mathematicalsigns. Only when the learners havetraveled the mathematical landscape forsome time, they are able to “see” deepermeanings in mathematical signs as theyinterpret them in new contexts and in newrelationships with other signs. Hence, theprocess is also diachronic. In thediachronicity of the process, the learnercomes to understand the meaning potentialof different signs.

Continuity and recurrence (i.e., going backin thought to consider something againunder a new light) is the essence of thissynchronic-diachronic process. Continuityand recursion allow learners (1) to carryon with their personal histories ofconceptual development and evolution and(2) to transcend conceptual experiencesin particular contexts through theobservation of invariance and regularitiesas they see those experiences from newperspectives. That is, the sequentialnature of the synchronic-diachronicprocess upholds all personal acts ofinterpretation, objectif ication, andgeneralization as well as of self-persuasion. Essentially, this is a mediatedand a dialectical process between learner’sknowing and knowledge in the permanentpresence of the continuous flow of time,not only synchronically (in the short livedpresent) but also diachronically (acrosspast, present, and future). As learnerstravel through the world of schoolmathematics, they construct and interpretfor themselves a network of mathematicalconceptualizations that is continuously re-organized through mathematical discourse

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(reading, writing, speaking, and listening)and de-contextualized through abstractionand generalization. As the learners’ networksof mathematical conceptualizations becomeincreasingly re-organized and transformedover time, the earlier value of theirmathematical wealth also increases.

Where do Learners Build up andConsolidate their Mathematical

Wealth?

As learners travel through a particularterritory of the mathematical world (e.g., theinstitutionalized school curriculum) theybecome mathematically wealthier becausethey become better acquainted with the insand outs of the territory (i.e., they are ableto produce symbolic interpretations ofsigns, or they relate to signs iconically andindexically but in a systematic manner).Others have a bird’s eye view of the territory(i.e., they are able to produce only isolatediconic, or indexical interpretations of signsor they relate to signs iconically orindexically but in an unsystematic manner)and soon forget they have seen thelandscape because they have made nogeneralizations. Still others are able tofinish their journey traveling on automaticmode (i.e., using calculators andmemorized manipulations) to establish theirown peculiar relationships with themathematical code or mathematicalsemiotic systems. Henceforth, they areable to produce, at best, only iconicinterpretations from signs that soon will beforgotten.

The learners’ mathematical wealth is builtin a socio-cognitive classroom environmentgrounded on collective mathematicaldiscourse as opposed to the unidirectionaldiscourse from the teacher to the students.The quality of this discourse and theteacher’s focus of attention on the learners’

mathematical arguments influence theways in which learners invest theirmathematical wealth and how theybecome mathematically wealthier. It is wellknown that teachers, who are in charge ofdirecting the classroom discourse, guidetheir practices according to conscious orunconscious theoretical perspectives onmathematics and the teaching ofmathematics and they focus their attentionon different aspects of classroomdiscourse. Sierpinska (1998) delineatesthe theoretical perspectives of teacherswithin three ample frameworks:constructivist, socio-cultural, andinteractionist theories. Constructivistperspectives focus primarily on thelearners’ actions and speech while theactions and speech of the teacher are seenas secondary; that is, the constructivistteacher focuses essentially on the learnersand their mathematics. Socio-cultural(i,e.,Vygotskian) perspectives focus on thesocial and historical character of humanexperience, the importance of intellectuallabor, the mediating role of signs as mentaltools, and the role of writing in the individual’sintellectual development; that is, the socio-cultural teacher focuses essentially onculture and mediated socio-cognitiverelations. Interactionist perspectives focuson language as a social practice; that is, theinteractionist teacher focuses essentially ondiscourse and intersubjectivity. Thebehaviorist perspective could be added tothose emphasized by Sierpinska. Thebehaviorist teacher focuses essentially onthe learners’ performance and pays littleattention, if any, to the learners’ ways ofthinking. Finally, eclectic teachers seem tointertwine one or more theoreticalframeworks according to the needs of thelearners and their personal goals asteachers.

In any classroom, one needs to be cautionsabout what could be considered successful

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classroom communication. Successfulclassroom discourse may not be anindication of successful mathematicalcommunication. Steinbring et al. (1998)contend that learners may be successfulin learning only the rituals of interaction withtheir teachers or the routine and stereotypedframes of communication (like the well-knowninitiation-response-evaluation and funnelingpatterns). This kind of communication, theyargue, leaves the learners speechlessmathematically although keeping theappearance of an exchange of mathematicalideas. Brousseau (1997), and Steinbring etal. (1988), among others, present us withclassical examples in which teachers,consciously or unconsciously, hurry up ormisguide learners’ processes ofinterpretation. Thus, communicatingmathematically is more than simple ritualisticmodes of speaking or the manipulation ofsign-tokens; it is based on a progressivefolding of meaningful interpretations passingfrom iconic, to iconic-indexical, to iconic-indexical-symbolic, and vice versa theunfolding of these relations in the oppositedirection. Or as Deacon (1997) puts it:“symbolic relationships are composed ofindexical relationships between sets ofindices, and indexical relationships arecomposed of iconic relationships betweensets of icons” (p. 75). That is, more complexforms of objectification emerge from simplerforms (i.e., simpler forms are transcended butremain embedded in more complex ones).

This is to say that the learner’s process ofmathematical interpretation is mediated bymathematical sign systems (icons, indexes,or symbols and their logical and operationalrelations) to constitute networks ofconceptualizations and strategies formeaning-making. Communicatingmathematically is, in fact, a continuoussemiotic process of interpretation,objectification, and generalization. Theconstruction of generalizations takes the

learner from simple iconic relations, toindexical relations, and then to symbolicrelation (i.e., folding of iconic relations intoindexical ones, and then embeddingindexical relations into symbolic ones) inorder to make new interpretations and newobjectifications that produce newgeneralizations. Moreover, deconstructinggeneralizations takes the learner in theopposite direction (i.e., unfolding ofsymbolic relations into iconic-indexicalones, and unfolding iconic-indexicalrelations into iconic ones) in order toexemplify, in particular cases, the skeletalinvariance arrived at in generalization. Bothdirections are necessary because,together, they manifest not only therecursive and progressive constructivepower of individual minds but also theymanifest the human and socio-culturalroots of mathematical thinking.

Concluding Remarks

Using a Peircean perspective on semiotics,this paper argues the notion ofmathematical wealth. The initial cognitivemathematical wealth of any learner beginsearly in life. In his years of schooling andwith the guidance of teachers, this initialwealth is progressively invested and itsvalue gradually increased. The processof investment is, in essence, a mediated-dialectical process of decoding a varietyof semiotic systems and, conversely, theencoding of thoughts and actions in thosesemiotic systems that intervene. Suchsystems could be of socio-cultural,pedagogical, or mathematical nature.

For mathematical wealth to increase invalue in the process of investment, thelearner has to decode not only themathematical code but also the tacit codeof socio-cognitive rules of engagement inthe classroom. A priori and implicitly, he is

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expected to understand, that reading andwriting, constructing and interpretingmathematical arguments, listening andspeaking, and justifying in the form ofexplanation, verification, and proof arenecessary activities for the learning ofmathematics. He also has to understandthat these activities can effectively mediatethe appropriation and construction ofmathematical meanings from mathematicalsigns and the encoding of his owninterpretations and meaning-makingprocesses back into mathematical signs.

The paper also argues three levels ofinterpretation in the classroom: (a) thelearner’s level of mathematical interpretation;(b) the teacher’s own level of mathematicalinterpretation; and (c) the teacher’s level ofinterpretation of the learners’ mathematicalinterpretations. It is also argued thatmathematical meanings are not only inherentin mathematical signs but also inherent in thelearner’s cognitive relationship with thosesigns. Such relationships could be of aniconic, indexical, or symbolic nature. Theserelationships are not necessarily

disconnected since an iconic relationshipcould ascend and become an indexicalrelationship, and the latter could ascendand become a symbolic relationship. Viceversa, a symbolic relationship could beunfolded into an indexical relationship, andthe indexical relationship could be unfoldedinto an iconic relationship. In fact, whenlearners manipulate sign-tokens, it issometimes necessary, for efficiency, tokeep symbolic relations implicit in one’smind. Keeping the ascending anddescending directions of relationships withsigns and sign systems allow learners tomove from the particular to the general andfrom the general to the particular. Thelearners’ relationships with mathematicalsigns and sign systems are the result ofmediated-dialectical processes betweenthe learner’s knowing and knowledge in thesynchronic and diachronic triadic processof interpretation, objectification, andgeneralization. The reader is referred toRadford (2003) and Sáenz-Ludlow (2003,2004, and 2006) for other instances oflearners’ processes of interpretation,objectification, and generalization.

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Adalira Sáenz-LudlowDepartment of Mathematics and StatisticsUniversity of North Carolina at CharlotteUSA

E-mail: sae@email.uncc.edu

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