Brian Rotmans's semiotic model of mathematics is applied to examine the nature of the students' problematic relationship with word-problems. It is shown that word-problems are examples of the formal discourse of mathematics, therefore excluding the "Person", as semiotic figure, from mathematical activity.The investigation concludes that a student mathematical performance may be linked with the problematic of cultural encounters that exist between the society of practitioners of mathematics and those who do not belong (yet or never) to this community.*
During my professional activity as a mathematics teacher I was always surprise with the way that "medium-ability" students almost immediately decide to give up from a mathematical task that did not require a different pattern for its solution than one handled well previously by the student. It was this observation that motivated my interest in knowing the "unconscious" facet of the mathematical performer: I wanted to know what students think about when they think that they do not know how to solve a mathematical problem.
2. Rotmans's semiotic model of mathematics
Rotman's semiotic model of mathematics (1993) presents a theoretical framework to examine the doing of present-day mathematics. This model was conceptualized with mathematicians in mind, and is developed through an elaborated analysis, wherein the features of mathematical discourse are essential.
This semiotic model of mathematics discerns three different figures or agencies - the Person, the Subject, and the Agent - that jointly process any mathematical activity. To better introduce each one of the semiotic agencies, as well as to explain their different functions, it is necessary to consider at first the major discursive features of mathematics.
According to Rotman, mathematicians have two distinctive dimensions of discourse: the formal mode, or "Code", and the informal mode, or "metaCode". The formal discursive mode of mathematics is related to the objective and rigorous aspects of mathematics. It is what one finds in mathematical written texts, and it consists of the symbolic notation used in mathematical texts as well as its precise rules. The utterances made in natural language which are mixed in the symbolic notation are also a part of the Code (Rotman.1993:69).
The informal discursive mode, or metaCode, concerns the different ways whereby the Code is contextualized and linked to historical, empirical, social, psychological, or cultural realities. In Rotman's words (1993) it consists of:" drawing illustrative figures and diagrams; giving motivations: supplying cognate ideas; rendering intuitions; guiding principles, and underlying stories; suggesting applications; fixing the intending interpretations of formal and notational systems; making extra-mathematical connections" (p:69-70).
Although the construction of mathematics embraces these two discursive dimensions, the metaCode is traditionally viewed as unrigorous and epiphenomenal when compared to the Code. Consequently, what is established by the Code (mathematical definitions, procedures, enunciations, problems, and demonstrations, as well as its discursive means) is what is promoted as mathematics, and considered as the "true" corpus of mathematical knowledge.
In considering that mathematics is done by humans to humans, Rotman examines the "recipients" of mathematical discourse, and sets up the three semiotic agencies. Briefly summarizing they arise as follows:
The Person - that operates within the metaCode, and consequently is embedded in psycho-social, and historic-cultural references. The Person has access to dreams, motivations, narratives, and speaks with the personal pronoun of natural language.
The Subject - an idealized person that is addressed by the Code and functions within it. The subject cognitively interpret signs, and mathematical texts, according to the rules and conventions of the Code.
The Agent - a machine, that mechanically manipulates the signs of the Code, without concerns of meaning and interpretations.
As Rotman points out, the Subject is the most "visible and palpable" (Rotman.1993:81) of the three semiotic agencies. It is the Subject that is addressed by mathematical texts. However, the Subject's activity is expropriated from the excluded yet proximal presence of the Person, and it impels the Agent to produce.
The processes that accomplish the changes from the Person to the Subject correspond to the obscurement of reference and corporeality, and the processes that accompany the metamorphoses whereby the Subject gives rise to the Agent mainly coincide with the vanishing of "meaning and sense" (Rotman.1993:92).
As Rotman (1993) argues, mathematical activity demands the simultaneous presence of the three semiotic agencies; only in the discursive dimension of the metaCode can mathematicians interpret the wholeness of a mathematical proof that is not contained in each one of its logical steps; and only the Person is allowed to search for the "idea behind the proof" that gave rise to such a proof (Rotman.1993:80). Ultimately this means that the metaCode is the place where debate and eventually creation of the Code takes place.
3. Analysis of word- problems' texts
As I have mentioned before it was the "unconscious" aspects of the mathematical performer that interested me mostly. Thus, I started to meet personally with students and asked them to make some assignments from mathematical text-books that they were familiar with. By this way, during my mathematical encounter with students, several tasks were done without mathematical obstacles. Despite this sucess, some word-problems appeared that provoked failure. Two of the word-problems that students firstly, did not know how to solve, or were not sure about their solution, are enunciated by the following texts:
TEXT 1 - "Mr Joaquim has a rectangular land with 50m by 20m. He wants to put four rows of barbed wire in the fence. How many rolls of 50m of barbed wire each does he need to purchase?"( Neves & Monteiro, 1996:117).
TEXT 2 - "Aunt Helen is making a 90-in-wide by 105-in-long bedspread. She wants to add fringe to the two long sides and one short side of the bedspread. How many yards of fringe should she purchase?" (Burton, Hopkins, Johnson, Kaplan, Kennedy & Schultz, 1992:411).
The above problems, collected from two different contemporary mathematics text-books, present similar discursive characteristics. For example, an overall configuration emerges from the texts above; it is possible to break each of them into two distinct sections. In the first section a situation is introduced and described. The second section poses a question or questions, regarding the situation previously described.
The textual organization in each one of the above texts, because they clearly separate off the description of a situation from the posing of a question about the situation previously described, suggests that the above texts have the same conceptual characteristics, and two clear distinct moments in that conception.
The language pattern of the above problems' texts is similar. The first sections mainly use the indicative, in its declarative form. Thus, the language is used mainly to convey information, and each time that new information - new data conditions for the problem - are presented, it is "signaled syntactically and lexically". Thus, no redundancy exists in the text, as each sentence with its new information is "explicitly marked". The language pattern in the second sections is the interrogative. Therefore language is used predominantly to request information.
The systematic choice of the indicative either in the declarative form or in the interrogative form, brings on to the reader, as well as to the writer, specific roles. As Rotman (1988:7) noticed by citing (Berry 1975:166), "The speaker of a clause which has selected the indicative plus declarative has selected for himself the role of informant and for his hearer the role of informed". Because, practically, no alternative verbal mode is used in all these texts, nothing that the reader (or the writer) might know already about the situation which is being presented is evoked or recognized from the writer to the reader as relevant to the text.
Moreover, the use of the indicative is not a strange occasion in word-problems. Rather it is in accordance with mathematical more general discursive features. As Rotman (1988) points out "For mathematics, the indicative governs all those questions, assumptions, and statements of information - assertions, propositions, posits, theorems, hypotheses, axioms, conjectures, and problems - which either ask for, grant, or deliver some piece of mathematical content" (p.7).
The information in each one of the texts is factual, punctual and quantitative and they all display socially isolated contexts, with undefined or unknown actors. In regard to time information the verbal mode can be changed without creating constraints to the information presented in the text. Thus, the texts' spacetime references for the actions that are being described, if there, are useless and rhetorical. They just appear to help to set up scenarios where unfamiliar actors mimic actions.
In addition, because the information offered by the writer is no more or less than what the reader will need to answer the question (the which is itself posed by the writer), the function of what the writer mentions in the first section of the text is only discovered in its second section after the posing of a question. And, since the answer of such a question is totally dependent upon the previously presented information, the inflection introduced on the text by the question remits the reader back to already described situation, establishing, hence, a closed cyclical type of interaction between the reader and the text that only will end when the question is totally fulfilled. Moreover, such a question is only correctly responded to through the performance of mathematical procedures. Hence, although no allusion is made to mathematics, or mathematical procedures, they are present in a pre-assumed unmentioned context that the text actually demands. Consequently, the contrast reached by the breakdown between given information (first section) and requested information (second section) is more a rethorical distinction than a discursive one, this is, it is not a different topic that is introduced in the second section, but rather the exhortation of a particular look (a mathematical look) into the text's first section.
In conclusion, the
above analysis shows that the texts of word-problems are written in such
a way that they deviate the reader from the requisite of direct experiencing
of what they present. The texts, because they make use of squeezed, simulated
real life based situations with obscure spacio-temporal references, pretend
that the reader's experience is already insert in them. In so doing, word-problems
induce the mathematical performer to eschew the empirical situation, but
to imagine that he/she is acting in it. Moreover, the textual pattern of
the above texts is in accordance with the general features of the formal
discursive mode of mathematics, in the sense that, their indexality, if
there, is fictitious, and they exhort no more or less than the procedures
that are required in order to answer mathematically their questions. Therefore,
mathematical word-problems only make sense if put within the discursive
context where they belong and are embedded. Each text is a microworld that
has no existence other than in the mathematical language of objectivity.
They can be taken as objects of mathematical discourse.
4. Facing exclusion: the student apprentice as Person
I am going to present, now, the interaction established between the above word-problems and the students who gave up to solve them.
In regard to TEXT 1, the student, a 10 years old 5th grader, prompted calculated the perimeter of the land and stoped after this. Since she did nothing more, I started to ask her what was happening, and, after a while, I knew that all the situation described in the problem was, for her, a completly virtual situation. This is, if the student was not aware that one may purchase barbed wire on rolls, on the other hand, she was trying to figure out a situation that required a fance of barbed wire. After we talked for a while about some situations that eventually would require fances of barbed wire, as the Zoo, for example, and after we both remember materials that are usually sold in rolls, the student did the problem without mathematical difficulties.
In regard to the problem
stated by TEXT 2, the student, a 14 years olde 8th grader, told me that
she was not capable of doing the problem. This is her first attempt to
explain me the reasons.
They say that she wants to put fringe in... How much did she?....
I do not understand. They do not say...They are not giving any indication...
I asked her: What do they not say? What do you think is missing?
And the student replied:
She wants to do a bedspread, and she wants to put a fringe on both
the long side and on one wide.
How many yards of string does she need to buy to do...
It is not coming to my head.
Here, the student looked at me, kept silence for a few seconds and did the problem. I kept asking her what was the missing information in the problem's enunciation that she was looking for at the beginning. She said:
They could have told that in order to do a yard, she would need to buy a certain amount of string
Like that, it would be easier.
Because then, I would know how much was necessary for a longside, and then for all the bedspread.
I think that they did not give enough information about how much string
would she need to do a yard of fringe.
The particular situation presented in this problem - the making of a bedspread - is common among this student socio-cultural community, thus, she was connecting the above problem's information with her experience. However, in the student's community, a bedspread is usually laced, and it includes the lacing of the fringe as well. As the student's commentaries revealed, she wanted to know the total amount of string necessary to do the all fringe, which firstly required that she knew how much string would be necessary to do a yard of fringe. Therefore, the student was looking for missing information which obviously was not there. The data condition given in the problem was neither enough to solve mathematically the reality that she wanted described in the problem, nor to figure out what could be the solution for the correspondent problem's question.
As the analysis of word-problems' texts had shown above, and the interaction between students and mathematical word-problems ilucidate, firstly, the mathematical problem-solver is directed to a hypothetical situation, a scenario, and secondly, she/he is commanded to act within this scenario. The correct performance expected from the problem solver is only possible through the exhibition of mathematical procedures. In this context, implicated with the mathematical reasoning that will lead to the construction of the different steps required for the problem's solution is the semiotic agency of the "Subject", (the idealized suject addressed by mathematical word-problems), who will make sense of the text by reading it according to the conventions of the mathematical Code, and who will launch the process that makes possible the performance of mathematical procedures. Such procedures are subsequently worked out by the semiotic "Agent" figure, that manipulates signs according to the exact rules of the Code already far removed from concerns of meaning and interpretation.
In all the processes of problem-solving, the "Person" is never invoked to act. It is, rather, excluded in the doing of the mathematical activity proposed by word-problems, since the discursive dimension in which the Person operates, the metaCode, is not engaged by word-problems, written as they are in the formal discursive mode of mathematics.
Nonetheless, the accounts of students' performance in mathematical word-problems' contexts shows that they were involving in their efforts considerations that are not allowed either to the Subject or to the Agent. By doing so, they were trying to produce mathematics by acting with the characteristics of the "Person" semiotic agency, as well. As Rotman says,
only on the basis "it is like me" is the Subject in a position to be persuaded that what happens to the Agent in its imagined world mimics what would happen to the Subject in the actual or projected world. But it is this recognition, this judgment or affirmation of sufficient similitude, that the Subject cannot articulate, since to do so would require access to an indexical self-description necessarily denied to any user of the Code (1993:78)
It was this kind of authentication, this kind of "it is like me" that students were seeking when trying to figure out possible relationships between the situation that they were encountering in word-problem's text and their own realities. Before trying to discover such a similarity, these students did not allow themselves to be convinced by the proposal of the mathematical word-problem, and consequently they inhibited their passage to the scenarios where the Subject and Agent act in isolation from the Person.
As apprentices of mathematics, students were looking for an "underlying story", some "idea behind the problem" that would acknowledge the mathematical problem, and subsequently empower the imagining of what it proposes, and the acting out of what it demands. Thus, in the particular cases of the above problems, the presence of the Person represent meaningful and necessary mathematical reasoning.
In conclusion, the data show that even mathematical word-problems are a locus of subjectivity. While interpreting the above problems, students invoke the context in which they were designed. By doing so, they are consistently linking their mathematical performance with other contexts, and acting as persons embedded in their socio-cultural realities. However, this subjectivity is neither compatible nor meaningful within the contexts proposed by the mathematical problems, and generates conflicts for they as mathematical performers. In addition, what the performer beforehand "unconsciously knows" is that, in mathematical activities, his acting as a subjective, culturally and socially embedded actor is avoided.
The mathematical community already shares, indeed is complicit in, practices that denote the metaCode to a place of subalternity, or even denies its existence, in relation to the Code; the student apprentice, on the other hand, is not yet a member of this community, (and most likely will never be), and is still in the process of being enculturated in its practices. Being so, the student is still developing the "habitus" that will "produce the practice" (Bourdieu 1977:78) of denying the invocations incited by the ambivalent nature of word-problems. Consequently, the unconscious aspect of the practice of abandoning attempts to solve a mathematical word-problem can be interpreted as a demonstration of the implicit knowledge that the student apprentice arrives at: that the revealing of subjectivity in mathematical contexts will never be taken seriously, and even collides with present day dominant culture of mathematics.
Thus, contrary to a person who is already fluent in mathematical discourse, and as such, has already acquired the "habitus" that permits prompt Subject's incarnation in order to enact the scenarios of mathematical word-problems exactly as they demand, the student apprentice as a mathematical problem solver still allows subjective realities to affect their performance. After all, students apprentices are only starting the mathematical training of "forgetting" about themselves, and their surrounding cultural and social realities.
5. Final considerations
What emerges from the present investigation is that a student's mathematical performance can be linked with the problematic of cultural encounters, in its discoursive dimension, that exist between the society of practitioners of mathematics and those who do not belong (yet or never) to this community. The malfunctions of the mathematical problem solver can be interpreted as a broader type of cultural tension that arises when she/he is encountering a discourse that excludes the possibility of her/his participation as an embodied Subject, a Person. Mathematical discourse - and more concretely word problems - alienate the possibility of the student's drawing on her/his own emergent local realities, indeed, own culture, when she/he is performing her/his role as a mathematical problem solver. Feelings of strangeness toward mathematics can be linked with the requirement of the mathematical discourse, that social and cultural references and emotions must be deleted in enacting the imperative orders and the transcultural attributes that characterize the discursive features of the Code. After all, it is the Code's primordial position within mathematics that forces the student into a position of outsider in relation to mathematics, by not tolerating in its discursive characteristics the student's self expression and socio-cultural embodiment.8
Future research should focus on further development of theoretical frameworks that can elucidate the influences of mathematical discourse and perhaps other forms of scientific discourse on student apprentices. This work will have to be accompanied by the development of methods for gathering empirical evidence for demonstrating how mathematical activities are locus of practitioner's subjectivity, as well as how such subjectivity is related to mathematical performance.
8 As Restivo, S. (1990)has
also said, refering to a broader context "technical talk about mathematics
(as a system of formal relations among symbols, for example) is not sufficient
for a complete understanding of mathematics. Social talk, in contrast to
merely technical talk or "spiritualizing" the technical realm, connects
mathematics to human experiences, goals, and values" (p.120-121).
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