Literate mathematical discourse:

What it is and why should we care?

Anna Sfard Michael Cole

The University of Haifa University of California Haifa, Israel San Diego, USA

Theses days, whenever mathematics is mentioned, one thing we can be sure to hear are remarks about how inefficient the current school teaching is in making people mathematically literate. Let us open with an example that speaks to this concern.

At the time we met her, Maria was learning in eleventh grade of a special vocational school for students with long histories of ill-adjustment and of low achievement and distinct learning difficulties. The girl expected to become a hairdresser, and was described by the teacher as being ‘extremely weak’ in mathematics. While interviewing her later, we had an ample opportunity to see that, indeed, the state of her arithmetic was well below what one would expect from a 17 year old. Even a simple multiplication of whole numbers seemed to exceed her computational abilities.

When asked to tell us the history of her mathematics learning, Maria asserted that as a young child she did not experience any difficulty with simple arithmetic. Difficulties began some time later:

“In the fourth grade, when we started to multiply… I lost the way… I thought it was not for me.. I did want to know how to do it… Sometimes I can do things and succeed…. But when I have to think hard, I give up.. multiplication table… No use in trying to remember. It is so confusing.”

The interviewer followed with a question “How much is 7?16?” and when subsequently the girl experienced difficulty with multiplying 6 by 7, the following exchange took place:

Interviewer:	Do you know how much is six times seven? [6?7]
Maria:	No
Interviewer:	And if I asked you to find out, what would you do?
Maria:	I would use my fingers. Would count seven times..
Interviewer:	Show us.
Maria:	No.
Interviewer:	Please do.
Maria:	No. I do it silently, so that people won’t see.

In the face of this and many similar examples we could bring from our later interview with Maria, it does not sound surprising that the girl was described by her teacher as being “extremely weak” and “without any potential” in mathematics. Yet, many of us would certainly protest against this description and say that the problem is not with the girl, but rather with her story. Another issue in point is the quality of school learning. If Maria was unable to cope with simple computational tasks it was not in spite of school learning, but rather because of it. For a girl coming from a broken working class family, this school learning seems to have little to do with anything that is of real importance in her difficult life, the critics are likely to say.

In the view of examples like this, it seems understandable that more and more voices can be heard questioning the wisdom of the traditional school emphasis on what is known as “formal” or “decontextualized” mathematics, as the subject is taught in school (Sfard, 2002a). This stance finds a source of support in research concentrating on the use of mathematics in real life situations, with the latter term being the antonym of the school setting. Time and again, cross-cultural studies conducted in societies where schooling is rare, poor, or absent, and the cross-situational investigations of mathematical usage in highly educated populations (Cole et al. 1971, Scribner & Cole 1981; Cole 1996, Scribner 1983/1997, Lave 1988, Hoyles & Noss 2002), have been showing that the mathematics used in real life is usually not the one that has been taught in school, and that the development of those mathematical skills which are of everyday relevance have little to do with school training. In addition, more often than not, the school and everyday learning differ substantially in quality and effectiveness. When children or adults are faced with school-like mathematical tasks (whether in school, at work, or at home), their performance is often far from perfect: they struggle, express confusion, and err repeatedly before arriving at a solution, if any. By contrast, the learning that occurs naturally through routine everyday activities leads to performance that is relatively smooth and error free. Based upon such evidence, policy makers, with the authors of the NCTM (2000) influential policy statement Principles and Standards for School Mathematics among them, stress the importance of “contextualization”, whereas other writers propose turning schools into sites of “cognitive apprenticeship”, the activity designed in such a way as to imitate spontaneous everyday learning (Brown et al., 1989). Even if no such claim is made explicitly, this shift in emphasis may be read as an attempt to undermine the importance of the formal, symbolic mathematics.

In this paper, we take a different position. While it is not our intention to defend poor teaching which produces useless procedural knowledge, we do want to argue that the ability to deal with abstraction and symbolism with rigor should be viewed as a vital part of mathematical literacy in modern societies and should by no means be barred from schools and made the privileged domain of elite experts. We will now take time to argue that the very studies which gave rise to arguments against formalized mathematics in school may, in fact, become a good source of evidence in its favor. Later, we will analyze the reasons for the students’ frequent failure to become literate in the ways we advocate. The understandings that we have to offer are not yet a solution to the problem, but we hope it may be helpful in finding one.

1. What is mathematical literacy?

In this paper, we adopt the definition of literacy formulated by James Gee, one that presents literacy as the ability to use secondary discourses. This latter type of discourse is defined by opposition to discourses with which people grow up (Gee, 1991). Unlike spontaneously acquired everyday discourses, secondary discourses require deliberate teaching.

Such discursive framing seems particularly well suited to the communicational approach to cognition, adopted by the authors of the present text. Within the communicational framework, mathematics is seen as a special type of discourse. A discourse counts as “mathematical” if it deals with mathematical objects, such as quantities and shapes. Since, however, there is more than one way of communicating about quantities and shapes, this definition needs elaboration. To illustrate how diverse mathematical communication can be, let us look at just one example.

Sylvia Scribner (1983/1997), an American investigator, was watching dairy warehouse workers who were loading trucks with dairy products in amounts specified in customers’ orders. In each case, the preloaders task was to compose a given amount of milk by putting together a number of containers of different sizes. For comparison, some 9th grade students and a few office workers were asked to perform the same tasks. The ways the workers and the “novices” (as the students and the clerks were dubbed) performed the tasks were meticulously recorded and analyzed. Many differences between the mathematical discourses of the two populations can be seen from Scribner’s report, but the one that deserves particular attention in our present context transpires from the following observations:

Students, “were .. single-algorithm problem-solvers.. Even when [they] selected an optimal strategy, they… relied heavily on numerical solutions and counting operations [like those performed in school]”

“In contrast, the [warehouse workers] often appeared to shortcut the arithmetic and worked directly from the visual display.” (p. 362)

The most salient difference between the two discourses on quantities observed in the study is that they were mediated by different visual/symbolic tools. Let us dwell for a moment on this point. Every discourse is about something, and if the discourse is to go on, this “something” must be either actually visible or imagined. This is what we mean while speaking of the visual/symbolic mediation of discourses. In Scribner’s study, the novices preceded the actual implementation of every order with a purely discursive act of numerical calculations, one that was most probably mediated by written, or just imagined, numerical symbols they had learned in school.[1] In contrast, the preloaders performed their manipulations directly on containers of different shapes and sizes. One of them testified to this explicitly: “I don’t never [!] count when I’m making the order, I do it visual, a visual thinking, you know” (p. 362). Thus, the quantitative discourse (and thus thinking) of the experienced workers was visually mediated but readily available, familiar concrete objects, rather than by symbolic artifacts. Similar phenomena were observed in a long series of other studies, such as those on Kpelle rice sellers (Cole, 1996), on Vai tailors (Lave & Wenger, 1991), on Brazilain street vendors (Nunes et al., 1993), to name but a few. In all these studies, the discourses of experts were invariably mediated by concrete familiar objects, at least partially. As was recently shown by a British team (Hoyles & Noss 2002), this is true also of such highly qualified workers as nurses.

If so, we should probably speak about mathematical discourses, in the plural. In particular, it may be useful to distinguish between everyday, or colloquial, mathematical discourses[2], such as those observed in dairy warehouse workers, Kpelle rice sellers or Brazilain street vendors, and literate mathematical discourse, which is the objective of school learning and which was employed by the students and office workers who participated in Scribner’s studies.[3] This latter type of discourse does not develop spontaneously and may thus be regarded as the mathematical instantiation of Gee’s general idea of secondary discourse. The main property that sets this deliberately taught discourse apart from the discourses of experienced workers is its being visually mediated by symbolic artifacts created specially for the sake of communication about quantities. This also means that unlike the workers, a mathematically literate person is capable of helping herself using written records, where the specialized symbols become fully visible and acquire permanence (in Jerome Bruner’s terms, such person is “amplifying her cognitive capacities” (Bruner 1966))[4]. Two additional dimensions along which mathematical discourses can be distinguished from each other are their special use of words and by the distinct discursive routines with which the participants implement well defined types of tasks.[5]

Although everyday and literate mathematical discourses differ in all these respects, the difference in the type of mediation can probably be used as its single most important property. Indeed, symbolic mediation is the most salient characteristic that sets literate discourses apart from any other. Since symbols have both oral and graphical forms, this special type of discourse is routinely implemented as a mixture of pronounced (or just silently thought) and written utterances and, as will be argued below, this property is the one that gives it its special strength and makes it potentially effective in solving a wide range of problems. The other features of literate mathematical discourses mentioned above – the distinctive use of words and their unique routines – derive from, and build on, the symbolic and recordable nature of the discourse.

After the above conceptual clarification, we can say that being mathematically literate means to be a skillful and proactive participant of literate mathematical discourse. The term proactive means that the mathematically literate person has a general disposition toward using the literate mathematical discourse[6] in a broad range of situations, including situations much different from the one in which this discourse was originally learned.

2. What difference does literacy make?

In Scribner’s study, those who perform the task on an everyday basis, the warehouse workers, were found working in their own extremely effective ways. Their proficiency in applying these methods exceeded that of the novices who had no other means of dealing with the situation than their literate mathematical discourse. Indeed, Scribner timed her subjects’ performance and found that for the novices, the average time on task was more than twice as long as for the workers. Other cross-situational studies have shown that people with a mastery of specialized everyday mathematical techniques are usually not only quicker, but also more accurate in solving real-life problems (see e.g. Hoyles & Noss, 2002; Lave, 1988). Hence, on the face of it, literate mathematical discourse fostered in schools has no advantage over the self-made mathematical discourses, developed spontaneously through repetitive practice.

It is now time to ask, therefore, what we gain in exchange for the considerable effort that has to be invested in becoming mathematically literate. Based on the dominant interpretations of the cross-cultural, cross-situational studies, not much. But let’s take a new look at this research. Scribner’s study, along with many others, has shown that even educated people use everyday mathematical discourses rather than literate mathematical discourse in their daily lives. And yet, we have no grounds for interpreting this result as showing the redundancy of the literate mathematical discourse. The fact that diary workers and supermarket buyers use unique forms of communication and are more skilled than school students in performing certain tasks may be accounted for by the fully understandable human tendency for minimization of effort (cf. Scribner 1983/1997)[7] and does not mean that the forms of mathematical communication learned at school could not be transferred to the warehouse. On the contrary. Although what caught our eye in Scribner’s study was the fact that the office workers and students were less skilled than the warehouse workers, this story has another moral as well. It is thanks to their literate mathematical discourse that the novices could manage at all. Since this discourse comes with its own mediators, it does not depend as heavily on the situation for its effectiveness. Rather, it is the specialized discourses of skilled workers that are less likely to be easily accommodable to new settings. Indeed, the colloquial discourses are tied to the situations in which they developed through the situation-specific perceptual mediators. While mathematically literate people may sometimes be less proficient than the expert users of specialized discourses, the odds are that (if they are properly taught) they would be able to deal with incomparably wider range of situations than those whose discursive proficiency depends on situation-specific mediational means. Such generality and wide applicability are thus the main advantage of literate mathematical discourse over specialized everyday discourses.

Let us address yet another aspect of symbolism which is of principal importance when it comes to understanding the special power of this literate discourse. Symbolic records are exceptionally concise and render mathematical discourse permanent and manipulable in a way that the spoken discourses could never be. These two features, permanence and conciseness, are highly consequential and they become a basis both for the growth of literate mathematical discourse and for a further increase in its generality. Once rendered permanent, the discourse may now become an object of reflection, that is, turn into the objects of another discourse. When the mathematical discourse turns on itself, its own patterns turn into an object of study and become a basis for new abstractions and generalizations. The more abstract the literate mathematical discourse, the greater its potential as a widely generalizable problem-solving tool.

The main features and relative advantages of colloquial and literate mathematical discourses are summarized in the table below. The principal conclusion of the above discussion is that the compensation for the relative complexity of literate mathematical discourse is its generality. Literate discourses are general-purpose, whereas colloquial mathematical discourses are specialized and highly limited in their applicability. A mathematically literate person can thus be thought of as endowed with a toolkit that allows her to deal successfully with a wide spectrum of situations, including those she had never encountered before. And yet, as we all know only too well, possessing the tools is not tantamount to the ability to use them whenever appropriate. The reminder of this paper is devoted to the question why literate discourses seem so far removed from “real life” and whether the students’ ability to actualize the practical potential of the literate mathematical discourse can be fostered.

Characteristic	Colloquial (Primary) mathematical discourse	Literate (Secondary) mathematical discourse
Development	Spontaneously	Through reflection, that is, at meta-level with respect to the primary
Visual mediation	predominantly physical	predominantly symbolic
Durability	Transient	lasting
Applicability	Restricted (the discourse is highly situated)	Universal

3. What is the problem and what can be done about it?

So far, we have been talking about literate mathematical discourse in a somewhat “agentless” manner, as if it this discourse existed independently of people who make it happen. It is therefore time to return to our definition of mathematical literacy, which draws the link between the discourse and its participants. Let us remember that with the help of the concept of literate mathematical discourse we formulated two conditions for individual mathematical literacy: To be regarded as mathematically literate, a person has to know both the “how” and the “when” of this discourse, that is, has to be able to use this discourse not only when the initiative comes from others, but also on her own accord, in any situation in which this discourse can be helpful. Of course, these two abilities – the command of the literate discourse and the ability to use it, although presented as separate, are dialectically interrelated. And yet, as we will now be arguing, development of each of them may be hindered by its own special type of deep-seated circularity.

Let us begin with the circularity that is inherent in the activity of matching discourses with situations. Perhaps the most intriguing question that follows from findings of cross-situational studies is how people decide in what way to communicate about settings and problems they encounter. The nature of the perceptual mediation available in a given situation is certainly among the main factors. Salient mediational possibilities make us turn to certain specific discourses even before getting well acquainted with the new situation. After all, since thinking is communicating, such spontaneous discursive framing is indispensable if we are to think about new situations at all. On the other hand, since we are thinking as long as we are not asleep, we are already participating in a certain discourse when stepping into a new situation for the first time. This default discourse regulates our sensibility to the types of mediation available in the given environment. Moreover, whatever discourse it is, it comes endowed with routines that push us even further into a particular way of looking and noticing. It seems, therefore, that the spontaneous coupling of discourses with situations is a matter of a delicate dialectic: We choose discourses according to what we see, but we see only what is made visible by the discourse we choose. If so, the odds are that the familiar discourses to which we turn habitually would often bar access to less familiar ways of communication, with the literate mathematical talk among them.

This means that if schools are to be successful in promoting students’ flexibility in using the literate discourse, a special thought must be given to possible ways of lowering these habitual barriers. And yet, not much seems to be done in this respect. Traditionally, high grades are rewarded to those students who show ability to participate in mathematical discourse as, and when, dictated by the teacher. These students do not have to develop the ability to use the discourse whenever appropriate. In other words, school learning would often lead to literate mathematical discourse that remains encapsulated and stand-alone rather than playing the subsuming role in the overall repertoire of mathematical discourses[8].

Let us now turn to the other circularity, the one that hinders the successful participation in literate mathematical discourse even when it takes place in the setting that imposes its use – at school. This time we are not talking about “inherent” circularity of the learning process, but rather about one that is man-made and could, perhaps, be avoided altogether if we only cared enough. Our present concern is with the self-reproducing nature of failure.

To illustrate this point, let us return to Maria, whom we met in the beginning of this talk. While interviewing the girl we noticed that her answers were densely interspersed with self-referential remarks. More often than not, these spontaneous comments expressed the girl’s opinions about herself as actual or potential participant in the arithmetic discourse. Maria’s self-references were mostly pejorative and self-denigrating. Some of them evaluated the girl’s current situation (e.g. “I don’t know” or “I don’t know how to do it”), whereas some others express her more general opinion about herself (“I can’t write”, “Even if I write, I won’t be able to”, “My brain is so slow”). The latter remarks alluded to a permanent disability: they implied that girl’s present failures were not an isolated event but rather a result of her generally low potential (see the use of the words “can’t”, “am not able to” and “my brain”). Maria’s talk conveyed lack of self-esteem and the girl seemed to be trying to lower the interviewers’ expectations as to her future performance. In addition, the girl’s faults were presented as permanent. There was, therefore, a sense of hopelessness in what Maria was telling us about herself: If failures are conceptualized as a result of natural givens, there is no reason to expect that the situation can be changed.

In view of this, as well as of our own first impression and the teacher initial comment, it came to us as a surprise that while watching closely and in the finest details the way Maria coped with arithmetical questions we noticed that her literate computational skills, although not fully developed owing to a simple lack of sufficient practice, were impressively diverse and rich in possibilities. Our analysis has shown considerable flexibility in her use of symbolic mediators, her clear ability to go from one mediator to another whenever appropriate, her tendency for self-control and for correcting mistakes, and her good sense of literate interpretation of such task as calculation, estimation, explanation, or justification. So, if Maria failed in all these tasks, it was not because of not knowing of what or how to do, but because of not being sufficiently proficient in this doing. If we were to use Maria’s teacher’s language, we would say that the girl certainly had “much potential”.

The question which now cries out to be asked is why this potential was not sufficiently realized. Watching the girl, we had a sense of her being entangled in a vicious circle of low expectations and poor performance, with each of the two feeding back into the other. Indeed, it is reasonable to assume that a person’s opinion about herself is reflexively related to the quality of her participation: the better, smoother the participation, the stronger interlocutor’s self-confidence, and vice versa. This is particularly true about participation which is highly valorized by the society[9], and the participation in literate mathematical discourse is certainly a case in point.

The next question to ask is where this vicious circle has been put in motion, in the first place. To find the answer, let us have another look at Maria’s history as a mathematics student. The girl told us that already in elementary school she had a hard time with mathematics, and that this led to an aversion toward the subject: “I didn’t succeed [in math], so I didn’t like [it]”. She also complained about her inability to deal with mathematics: “When there are many numbers together, I am confused”. The pressure she used to feel was often difficult to bear and more often than not, the girl would feel the urge to escape and to abandon the task altogether. This is what usually happened, for example, whenever the teacher offered her help:

Many numbers at once give me a headache.... I can’t… [The teacher] tries to explain. Say, there is an exercise. She asks me to do it and then tries to explain. “Now, let’s take an example, and then another example..” And I already feel confused. And then suddenly she shows me several different ways and this is already too much.. “Well, let me go, I don’t want to hear anymore; it’s a waste of time and I am unable to keep grappling with this”.

It seems that it was Maria’s early unsuccessful experiences with mathematics and the teacher’s help that created, and subsequently fed, her low opinion about her abilities. Eventually, her view of herself became self-defeating. This scenario seems even more plausible if we consider Maria’s further testimony, according to which she was not generally inclined to give up in the face of intellectually challenging tasks. On the contrary, she would sometimes find quitting difficult:

If this is something that I know, I have no problem spending time on it. If I know I am going to succeed, I will sit for who knows how long…. Take the Bulgarian [Hungarian] cube, tasks with a computer, puzzles about coins, and the likes. I would sit and try. It may take me a week or two, but I won’t leave it. I have to go on.

It seems, therefore, that Maria’s lack of self-confidence was not a general trait of all her discursive activities related to problem solving, but was rather subject matter-dependent. This justifies raising the conjecture that the girl’s vision of her arithmetic ‘potential’ was shaped by the manner in which her former lack of success was handled. The way the girl reacted to the teacher’s interventions lends support to this hypothesis.

To sum up, and not at all surprisingly, when it comes to the poor results in developing mathematical literacy, the school is found to be the principal culprit. As obvious as this fact seems to appear, we feel we must stress it again. In particular, it is important to note that the guilt is not in the literate discourse itself. If anything, for a girl like Maria, the abstract, detached nature of this discourse was a challenge which she was willing, and probably perfectly able, to face. If she now disliked literate mathematics and was unsuccessful in speaking it, it was because her experience with mathematics at school, when cast against the manner in which mathematical literacy is used in her society, made her feel an unworthy person, doomed for exclusion and for life on a margin.

There is not enough room in this talk to elaborate on the all-important question of how to teach so as to break out of the dangerous circularities and to maximize student’s chances for full-fledged participation in literate mathematical discourse. One thing that has to be stressed is that if we are to be successful in this project, and if the literate discourses are to be promoted without risking a harm to students’ views of themselves, changes are necessary not only in schools, but also beyond them. One such change would be discontinuing the practice of using mathematics as a tool for measuring “human potential”.

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