Chapter

Chapter

12

Lifting the Labels: A Cautionary Story about Stories We Tell about Mathematics Students Miriam Ben-Yehuda Ilana Lavy Liora Linchevski Anna Sfard Beit-Berl College Emek Yezreel Hebrew University of University of College Jerusalem Haifa

High ability or low ability, brilliant or slow, strong or weak, gifted or learning-disabled—all these words sound only too familiar to mathematics teachers. As teachers, we use them, and many similar ones, while talking, or even just thinking, about our students. Most of the time, we choose the descriptors spontaneously, without consulting our records. On certain occasions, we seek the help of professional diagnosticians. However it is done, tagging students with ability-related descriptors is part and parcel of a teacher’s practice. Many reasons can be offered to explain our strong urge for this kind of labeling. Above all, categorization facilitates pedagogical decisions. Labels, we feel, encapsulate a vital part of what the teacher needs to know about her students in order to design their learning trajectory, and, indeed, to choose the young persons’ mathematical futures. But is this belief justified? Can these common descriptors be trusted as the teacher’s decision-making guides? This article is about a study that taught us to be wary of the labels. Our research has shown that only too often, the concise descriptors conceal more than they reveal and that the removal of these verbal screens allows us to see not just more, but also differently. Acutely aware of the potential significance of this finding for the teacher’s everyday practice—after all, if you use means that do not fit the needs, you may bring more harm than help—we decided to dig deeper. We devised ways of looking at learning-teaching processes that allowed us to see beyond the usual labels. The story we are now going to tell can be read as an attempt to retrain the reader’s eye so that it can reach what becomes available when one’s gaze is no longer (mis-)led by the ability-related signposts.

The Study: Understanding Failure

Our chapter is based on a study (Ben-Yehuda et al., 2005) that was a part of a long-term research project, the overall aim of which has been to understand in depth how and why students fail in mathematics. This undertaking was triggered by our numerous encounters with children and young people who could not manage even the simplest arithmetical calculation; it has been propelled by our own helplessness in the face of the phenomenon. The following story of two 18-year-old high school students is just one of many similar tales we could tell. At the time we met them, Talli and Mira (these names are pseudonyms) were eleventh-graders in a special vocational school for students with long histories of maladjustment and distinct learning difficulties. Mira had been preparing herself for a secretarial job, and Talli expected to become a hairdresser. Both girls were described by their mathematics teacher as “extremely weak” in arithmetic. This common label notwithstanding, they were seen as differing in more than one way. The teacher told us that Talli, in spite of her problems, was a student “with genuine potential.” In This chapter is adapted from Ben-Yehuda, M., Lavy, I., Linchevski, L., & Sfard, A. (2005). Doing wrong with words: What bars students’ access to arithmetical discourses. Journal for Research in Mathematics Education, 36, 176–247.

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More Lessons Learned from Research: Volume 1

contrast, she described Mira as “the weakest student” in her class, who clearly did not have “much chance.” Any effort on our part to perform arithmetic with Mira would be “a waste of time,” she said. The teacher’s assessment seemed in tune with the girls’ appearance and demeanor. Mira wore provocative clothes and heavy makeup, and she behaved and spoke like a helpless child. In contrast, Talli’s stern look and plain dark clothes gave the impression of a no-nonsense, mature person who knew exactly what she wanted. Thus, we were not surprised by the teacher’s evaluation or, for that matter, by the very fact that she offered these unsolicited comments. A surprise came later, when our interviews with the two girls failed to confirm the picture painted by the teacher. In the rest of this chapter we tell three alternative stories about our interviewees and of their mathematical abilities. We begin with a summary of the accounts given to us by people around Mira and Talli, including the report provided by the girls themselves. In the end, we paint our own, strikingly different picture of the two girls’ mathematics, as it enfolded during our hours-long sessions with them and in the subsequent detailed analyses. We follow these stories with a discussion of some possible sources for this observed clash of narratives. We end with a hypothesis that the stories we tell as parents, teachers, or psychologists play a major role in creating the phenomena they are supposed to merely describe. This hypothesis, of course, has important implications for our story-telling practices.

Stories Told by Those Around Mira and Talli and by the Girls Themselves

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We begin with “institutional” stories—that is, those told by the school staff, social workers, and psychologists. The teacher’s introductory remarks summarized above were corroborated by the detailed records we found in Mira’s and Talli’s school files. According to these documents, the first eighteen years of the girls’ lives brought more suffering than other people experience in their entire life spans. Their misfortunes included illnesses and death in the family, parental abuse, and sexual assaults. As a result, they had been frequently moved from one school to another, never spending more than a couple of years in one place. According to school files, there was much similarity between Mira’s and Talli’s learning capacities. Both girls scored around average on IQ tests and were said to have a “deficiency in acquired knowledge” along with “much unrealized potential.” But there was also a difference: whereas Mira was said to have “slight learning disabilities (LD),” there was no mention of LD in Talli’s files. In addition, school records forcefully confirmed the teacher’s introductory remarks on Mira’s and Talli’s mathematical competence, or rather the lack thereof. Various tests and assessments invariably pointed to serious shortcomings in both girls’ arithmetical skills, except that Mira was said to be among the “weakest” in her different classes, whereas Talli was described as “strong in comparison to the school average” and as having a “good command over the four basic arithmetical operations.” Not surprisingly, the girls’ stories about themselves, compiled from the spontaneous comments they volunteered while grappling with arithmetical tasks and from their responses to our direct questions, proved close in their messages to the institutional narratives. As the girls commented on their own activities only when experiencing difficulty, their self-references were mostly negative. Mira’s self-descriptions, in particular, stood out in their self-denigrating, belittling tone. Some of her self-referential remarks depicted her current situation (“I don’t know” or “I don’t know how to do it”), but most of them expressed her more general opinion about herself: “I can’t write,” and “My brain is so slow.” These and similar utterances spoke about the girl’s permanent “disabilities.” Notably, this kind of message was absent from Talli’s self-referential comments, which tended to explain her poor performance with a local or temporary insufficiency of her knowledge or skills (“I am not good at the multiplication [table],” and “I am confused”). Thus, whereas Mira’s self-referential remarks demonstrated low self-esteem and presented the girl’s faults as permanent, Talli’s message about herself suggested a temporary shortcoming.

Lifting the Labels: A Cautionary Story about Stories We Tell about Mathematics Students

These messages were reinforced by the autobiographical information provided by the girls in the brief informal conversation that preceded the interview. On this occasion, Mira told us that her problems with mathematics began in the fourth grade, where she encountered the difficulty with multiplying. Since then, she felt unable to deal with mathematics: “When there are many numbers together, I get confused.” She expressed her conviction that she would never be able to memorize the multiplication table. She seemed to be telling us that she was not ‘the kind of person’ who could ever cope with mathematics. Talli’s story was, once again, almost the exact opposite of Mira’s. The first thing Talli told us was that she “never had problems with learning in school,” she “always invested great effort into mathematics,” and she was “crazy about mathematics.” She also stressed repeatedly that she wanted to graduate properly, with matriculation examinations, including mathematics. A clear, consistent image seemed to transpire from all these stories: both Mira and Talli were considered as mathematically weak, and perhaps even extremely so, but Talli, with her purported skills, potential, and motivation, was believed to have much brighter prospects than Mira.

Our Story: Talking about Numbers— Rituals or Explorations?

To construct our own story, we had to look at Mira’s and Talli’s mathematical performances closely and with an unprejudiced eye. Excerpts 1 and 2, in which the girls were trying to calculate 7 • 16, is a representative sample of the data we collected with this aim in mind. (Note that the interviews were in Hebrew; the excerpts below are the authors’ translations.) Excerpt 1: Mira calculates 7 • 16

Speaker

What was said

1. Interviewer:

Do you know how much 6 times 7 is?

2. Mira:

No.

3. Interviewer:

And if I asked you to figure it out, what would you do?

4. Mira:

I would use my fingers. Would count seven times.

5. Interviewer:

Show us.

What was done

………….. 6. Mira:

No. 70. Now 7 multiplied by 6 […………………………]

7 • 16

7. Interviewer:

Speak up, please.

Draws 6 rows of 7 strokes: ||||||| ||||||| etc.

7. Mira:

(a) No problem. Counts the strokes (b) 7 multiplied by 6 is 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 (c) just a second, 1, 2, 3, 4, 5, 6, 7 [......] 12. I do 14, 15 ... (d) [counts aloud up to 42]. Writes: 42 (e) 42 and 70 is [......], 4 and 7 is [......] 112 + 70 ____ 112

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More Lessons Learned from Research: Volume 1 Excerpt 2: Talli calculates 7 • 16

Speaker

What was said

What was done

1. Interviewer:

Do it please.

2. Talli:

I am not good at multiplication.

3. Interviewer:

Please try.

4. Talli:

6 multiplied by 7 is 36. Okay? I am asking you. [laughs].

5. Interviewer:

It is not. Can you correct it by yourself?

6. Talli:

Ah?

7. Interviewer:

7 multiplied by 6 is not 36. If I tell you that 6 multiplied by 6 is 36, what will be the answer for 7 multiplied by 6?

Talli remembers numbers from the multiplication table but doesn’t recall exactly which result belongs to which pair of multipliers.

8. Talli:

It is 42.

[6 • 7] = 42

9. Interviewer:

How did you know?

10. Talli:

Guess. [laughs]

11. Interviewer:

6 multiplied by 7 is 42. Ok.? What do you do next?

12. Talli:

I take it down. [mumbling] 1 multiplied by 7, it makes 7. So it makes 742.

13. Interviewer:

How much?

14. Talli:

No . . . I was confused . . . No, 12. No, 122.

15. Interviewer:

Now try to write down what have you said.

16. Talli:

[mumbling and writing the exercise]

6 • 7 = 36?

Seems like trying to perform the algorithm for long multiplication, but in the end simply puts results of multiplying two digits of 16 by 7, one next to other.

Writes:

4 • 16 7 --122 (looks like she added the 4, 1 and 7 to get 12)

At the first sight, not much can be said about Mira’s and Talli’s performance except that it was “very poor” as for eighteen-year-olds (and this seems true even if one of the girls did succeed, in the end, to give the correct answer.) And yet, as explained before, we were determined to look beyond such labels, and for this, we had to design our own analytic lens. The point of departure for our tool-building efforts was the realization that mathematics can be seen as an activity of telling stories about mathematical objects. In arithmetic, the objects we talk about are called numbers.

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Even the simple equality, such as 7 • 16 = 112, can be unpacked as a brief narrative about three numbers: “If you multiply 7 and 16, you get 112.” Similar observation applies to any other school subject. In biology, for example, one tells stories about “biological objects” called plants and animals, and in physics one tells stories about physical things. But there is also an important difference between mathematics and all the other sciences: While most objects in biology or physics are material and accessible to our senses, even if only with the help of special instruments, the objects of mathematics, such as numbers, functions, and sets, are not. We say that these objects are abstract, and this means that they cannot be seen, smelled, heard or touched. “And what about symbols?” somebody may object; “Aren’t they material and visible?” In fact, the mathematical symbols we discuss are considered to be merely the material avatars of mathematical objects. We stress the role of symbols as merely proxies of the “real things” by calling them representations. Thus, the symbols 7 • 16 and 112 are said to be representations of the same number, as are, for that matter, any of the following: the expression 96 + 16, the binary numeral 1 110 0002, eleven bundles of ten sticks together with two separate ones, and the Roman numeral CXII. The number “represented” by all these symbols, rather than being identifiable with any of them, is to be understood as an intangible entity, existing independently of the representations. The impalpability of all “things” mathematical makes mathematical storytelling particularly difficult. It may well be this lack of the direct accessibility of mathematical objects that constitutes the main challenge in teaching and learning of mathematics. If you wish to teach children, say, how to move around a new school building, the first thing to do would be to take them for a tour or at least to show them schematic pictures of the place. Once again, somebody may object, saying that there is an alternative: The newcomers could also get to specific places—to the gymnasium or to the mathematics classroom—by following explicit, step-by-step instructions. In the long run, however, this kind of behavior would clearly be much less effective. Just think about the predicament of the instructions-follower whose memory fails or who needs to get to a place for which she has no memorized instructions. It seems therefore, that in the mathematics classroom, exploration is the possibility to opt for. But what are you going to do when you have a mathematical object instead of a building? How do you take the student “for a tour around” a number, a set, or a function? The challenge of telling stories about invisible mathematical objects seems so formidable that many students, and even some teachers, compromise by using the memorize-symbolic-manipulations type of “mathematizing,” a method that leaves the students at the mercy of memorized stepby-step instructions. Indeed, many of our university students recall their mathematical experience as that of “parroting” the teacher and “pairing formulas with exercises.” For them, mathematics will always only be a collection of rituals. It is our deep conviction that this is not the kind of learning we should be aiming at as teachers. In spite of its difficulty, we would opt for explorative mathematizing. Rather than viewing mathematics as a bunch of memorized prescriptions to follow with the sole aim of adhering to some inexplicable social norms, we would like the students to be masters of their mathematical activity—that is, to be able to derive new facts from those already known and to decide when and how to use mathematics for their own needs. For this to happen, they need to graduate with a pretty good sense of mathematical objects and with the ability to use these objects as a source and ultimate touchstone of their mathematical stories. Only when students have populated their mathematical universe with abstract objects, and they no longer have the need for an “expert opinion” to be sure of their decisions, can we feel that our job as teachers has been properly done. At this point, somebody may protest and claim that numerical calculations, when performed as a part of everyday activities, do not feel abstract at all. It may well be, one could say, that the pedagogical problem described above would simply disappear if we always taught arithmetic in real-life context, and if, rather than speaking about “bare” numbers, we used words such as three or two thirds to count and measure familiar objects. We beg to differ. We believe that, in restricting

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ourselves to the tangible and the familiar, we would lose something important. If real-life activities with numbers do not feel abstract, it is because in the midst of our daily affairs we do not tell stories about numbers as such; rather, we use numbers to tell stories about the world around us. Indeed, we say things such as “three children met another five children and there were eight of them altogether”, and not just “three plus five is eight.” In the colloquial talk, numbers serve as descriptors of objects, not as objects in their own right. This is what makes everyday arithmetic more accessible, but this is also what makes it less powerful. The school-type abstract computations, which for the sake of distinction we will now call literate, are universal and can involve numbers of any size. These two advantages are lost when number words are used as descriptors of objects such as groups of people or coins and banknotes. Our everyday number-related stories, and even our very ability to tell them, are highly dependent on the physical objects that are being described. The fact that we can easily implement a complex payment does not mean we can equally effortlessly calculate distances or amounts of food, and this is true even if, theoretically, all these calculations involve the same numbers and identical operations. It is thus clear that we cannot give up the literate storytelling. It is also clear, however, that while learning abstract arithmetic, one may use the colloquial talk as a stepping- stone for the literate. Based on this understanding of what mathematical literate activity is all about, we decided to present Mira and Talli with a series of school-like oral and written tasks requiring calculations, estimations, and comparisons. Later, after carefully transcribing the videotaped interviews, we scrutinized the girls’ performance for signs of, first, “objectification” and second, of genuine exploratory activity. There was also an independent third question: with the help of some buy-andsell tasks implemented with real money, we tested Mira’s and Talli’s colloquial arithmetic, asking how it was related to the girls’ literate numerical skills and understandings. Following is a report on what we actually did and what we found.

Question 1: Have the interviewees objectified number?

More specifically, have the girls used two, seven, or sixteen as if these words designated independent entities, for whom the words themselves, as well as the corresponding symbols 2, 7, or 16, were mere representations? The alternative would be for the interviewees to use number-words as if these words referred to the symbols themselves, possibly to different symbols in different context. Our answer to question 1 was that the former option applied to Mira’s talk, whereas the latter one matched the case of Talli’s. In other words, there was a degree of objectification in Mira’s numerical talk, whereas Talli’s stories were exclusively about manipulating symbols. To show this, let us examine some representative interview excerpts. Excerpt 3: Mira calculates 86 + 37 (orally)

First try: “[…] six and seven. […] It makes eleven. Oh well, it can’t be […] … I mixed it up. Thirteen … And then I do eight and three […]. It makes […] eleven.”

Second try: “Eighty and thirty […] … is hmm […] one hundred […] and ten, one hundred and ten […] six and seven is thirteen […] twenty three. Twenty three and one hundred […] hundred twenty three.”

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Lifting the Labels: A Cautionary Story about Stories We Tell about Mathematics Students Excerpt 4: Talli calculates 86 + 37 (orally) “I put the thirty-seven down, under the eighty-six; I put the three down and put the one over the three of the thirty-seven. I added and it made four; and four and eight makes twelve. Hundred twenty-three.”

There is no need for sophisticated tools to notice a considerable difference between Mira’s and Talli’s ways of dealing with the numbers. In her second try, Mira sounded as if she was telling a story of modular objects combining one with another in different ways, almost like Lego pieces. This fact stands out with a particular force in the direct comparison with Talli’s exclusive focus on her own symbolic activity. Let us try to pinpoint those specific properties of our interviewees’ talk that gave raise to these impressions. First, we considered those features of the interviewees’ talk that indicated for us that Mira, unlike Talli, was dealing with objects other than the symbols she was looking at. These features included reading the same digits in several different ways and having recourse to a number of visual mediators—symbols, drawings, and “manipulatives.” Thus, while naming numbers, Mira would often utter several different words in reaction to one particular digit. Look at her two attempts to calculate 86 + 37 and note the number words we have emphasized in the two parts of this excerpt. In the first try, the digits 8 and 3 were read as eight and three, respectively, and in the second round they gave rise to words eighty and thirty. If Mira was reading symbols such as 8 and 3 in different ways in different contexts, she could not possibly be using number words as referring to the symbols as such. This impression gets even stronger when we contrast Mira’s talk with Talli’s. Not only did the other girl stick to just one way of reading each of the digits, she also used number words as if they designated physical entities occupying a certain position relatively to one another: I put the 37 down, under the 86 or I put the 703 above the 245. It was clear that these words referred to the symbols rather than to anything they could possibly stand for. Written symbols are the most common type of visual means (mediators) we use in literate arithmetic. In excerpts 1 and 2, both Mira and Talli were clearly mediating their computations with numerical symbols, even if the latter were only imagined. And yet, the girls differed significantly in the manner in which they employed the symbols: Talli made a clear effort to scan the symbolic strings and replace them by other symbols in a unique, algorithmically prescribed way. This manner of attending to the numerical symbol may be called syntactic, as it does not require more than knowing the names and order of the digits along with the rules for replacing digits with other digits. What made Mira’s uses of symbols distinct was the already mentioned fact that she frequently uttered several different words for the same symbol, and this made her utterances sound as though they were referring to objects other than the symbols themselves. For this reason, her use of symbols can be described, once again, as objectified. Excerpt 1 brings evidence that Mira was adept also in two additional visual modes: in the use of concrete manipulatives (see her remark about fingers in line 4 of excerpt 1) and of drawings (in line 7, same excerpt). This excerpt also shows that she was quite skillful in combining the different mediational modes together and in making transitions from one to another. None of these was ever observed in the case of Talli, who never renounced the symbolic syntactic calculation. We may conclude saying that for Mira, number-words, drawings, and manipulatives were different representations of something else—the abstract entity called number, whereas for Talli literate arithmetic was exclusively about symbols (or, at least, this was the only kind of approach she was prepared to show us). Certain additional characteristics of the interviewees’ talk indicated that for Mira, numbers existed independently of any human action, whereas for Talli they were transient entities coming into

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being only in the course of her own activities. To see this, note that Mira spoke about the numerical operations such as addition or multiplication as if they were the numbers’ own acts of combining with one another. In her impersonal sentences such as “Eighty and thirty is one hundred” or “six and seven is thirteen,” there was no reference to human intervention. In fact, these utterances did not even sound as though they were referring to processes. Rather, they seemed to describe the structure of numbers: there were no verbs here, except “is,” and there was no action-performing agent. This impersonal description of the structure of composite numbers sounded as if Mira had constructed them the way one constructs a description of a building simply by exploring. All this stands in even fuller relief when compared to Talli’s talk, in which the operations were presented with personalized utterances such as “I added the 6 to 7”—that is, with verbs and in the first person. Once again, Mira was telling a story of self-sustained objects “doing their own thing,” whereas Talli was reporting on her own actions with symbols.

Question 2: Was Mira’s and Talli’s literate arithmetic ritualized or explorative?

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The terms exploration and ritual signify two types of routines, with this latter word signifying activities that follow a well-defined course of action and repeat themselves in similar situations. The first, defining difference between rituals and explorations is in their goals: explorations aim at producing new stories about mathematical objects, whereas rituals are routine activities undertaken for the sake of solidarity with other people. Our analyses of the transcripts led us to the conclusion that for Mira, numerical calculations were explorations, whereas for Talli they were rituals. Here are some explanations about what it was in the two girls’ talk that inspired this interpretation. First, we note that the same routine procedure (e.g., calculation) may be performed by one person as an exploration and by another as a ritual. Although the difference is subtle, there are several signs that allow us to tell one from another. Consider our earlier metaphor of new mathematical objects as unfamiliar buildings. As with the building, a person can try to find her way with numbers by “walking around” and exploring their structure; alternatively, she may satisfy herself with a bunch of memorized trajectories, along which she moves her legs according to prescriptions learned from other people. It seems clear that objectification—having a free access to the building—is the first condition for being capable of explorations. A person who deals merely with signs on paper is like one who is not even aware of her being in a new building. Unlike the explorer who, when necessary, derives new routes from the mental map she built in her explorations, the sign-manipulator will always remain dependent on memorized prescriptions. As Mira’s talk was more objectified than Talli’s, it is not surprising that her calculations had several other characteristics of explorations: they were more flexible, easier to correct, more conducive to justification, and more widely applicable. Let us say more about each of these features. One’s calculations count as flexible if the performer is capable of producing the same result in more than one way. In Mira’s case, the flexibility found its expression, among others, in the multimodality of her calculations (see excerpt 1 for Mira’s uses of symbolic, iconic, and concrete mediators to calculate 7 • 16). This feature indicated the girl’s awareness of the equivalence of differently performed calculations. Another manifestation of flexibility was Mira’s tendency to make different uses of symbolic mediators in the same calculation (see in excerpt 3 the two attempts to calculate 37 + 86 in which she uses symbols in two different ways, syntactically and in the objectified way). In contrast, Talli’s performance was extremely rigid, and this rigidity found its expression not only in her already mentioned automatic choice of the syntactical mode of action, but also in her firm refusal to follow the interviewee’s suggestions to try other modalities. The flexibility is tightly related to the interlocutor’s ability to correct her own performance. Both Mira and Talli erred quite frequently, and equally often they had a visible difficulty with continuing their calculations. Still, the girls differed considerably in the way they dealt with obstacles. In excerpts 1 and 3 we saw Mira retracing her moves, switching modalities and changing procedures whenever unable to follow the path she had chosen. We have not observed Talli in this kind of


action in excerpt 2, where she experienced a distinct difficulty, or in any other part of the interview for that matter. We take Talli’s inability to correct herself as yet another symptom of ritualization of her actions. Indeed, in the ritual, the exact performance according to rigid guidelines is the end in itself. Rituals, therefore, are not conducive to changes, and thus to local corrections. When a mistake is made, the whole procedure has to be repeated from the beginning. In this situation, it is not surprising that Talli, unlike Mira, was incapable of either evaluating her results or justifying them by presenting an alternative way of performing the same calculation. In judging the acceptability of her calculations she was thus dependent on somebody else’s opinion (and this is, indeed, why we used here the word acceptability rather than correctness). Mira, in contrast, as unskillful as she was in her literate calculations, tried to be the master of her own arithmetical activity, able to perform and judge her calculations by herself, without help from other people. The last question to ask when trying to distinguish between explorations and rituals regards one’s ability to apply a given routine in multiple contexts—to match them with a variety of tasks. We will attend to this question while reporting on aspects of Mira’s and Talli’s colloquial arithmetic and trying to assess the two girls’ ability to use their everyday and literate skills interchangeably.

Question 3: Could Mira’s and Talli’s colloquial numerical skills serve as a stepping-stone for their literate arithmetic?

It will probably surprise no one if we say that our interviewees’ clumsiness with abstract numerical calculations notwithstanding, both of them showed much agility in dealing with money. It may also sound too predictable if we add that the girls did not seem to recognize the connections between money tasks and abstract numerical operations. However common, these phenomena still tend to be regarded as rather puzzling. For instance, Terzinha Nunes, Analucia Schlieman, and David Carraher, the researchers who did pioneering work on what they called “street mathematics,” wondered: “How is it possible that children capable of solving a computational problem in the natural situation will fail to solve the same problem when it is taken out of its context?” (Nunes et al., 1993, p. 23; emphasis added). A close scrutiny of the mechanisms of literate and colloquial arithmetic, however, makes the phenomenon more understandable. It shows that from the child’s point of view, there may be no justification to regard the money task and the paper-and-pen calculation as in any way “the same.” When considered in all their detail, these two types of activity reveal themselves as differing in every possible respect: in their goals, in the objects that are being dealt with, and in the ways they are performed. To begin with, the objects that are being talked about in money transactions are coins and banknotes, not numbers. Indeed, consider the following excerpt. Excerpt 5: Mira pays for 3 cookies that cost 75 agoras each. Just a minute, 75 and 75 is 70 and 70 […] it’s one shekel and 40 agoras, one shekel and 50, here [handing over the coins of 1 and of 1/2 IS one after another] one shekel and 50 and 75 agoras I need to give you. [passes another 1/2 IS coin and to coins of 10 agoras] […] One shekel fifty, add seventy […] … I cheated you. I am not smart [laughter] [mumbling] I forgot the 5 agoras [passes a coin of 5 agoras]. Note: Agora is the smallest monetary unit in Israel. There are 100 agoras in 1 shekel (IS).

Here, Mira’s actions with money do not sound as numerical calculations. For one thing, there is no explicit mention of numerical operations except, perhaps, addition (if we read such expressions as one shekel and 50 and 75 agoras as expressing addition). Multiplication is not used even though the task seems to be asking for one. Above all, there is no announcement of the result. The performance of the transaction is a sequence of physical actions with money, accompanied by a very sparse commentary. Here, Mira is constructing the required payment by giving the recipient

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one coin after another, until she feels that the three payments of 75 agoras have been completed. She never inquires about the sum she actually gave away. Clearly, she views paying as a practical act in which the only thing that matters is to ensure that everybody is satisfied with his or her share. This and similar episodes convinced us that for our interviewees, money transactions were neither explorations nor rituals, but rather deeds—actions undertaken for the sake of a specific change in the state of things around them. More generally, banknotes and coins, not numbers, were, indeed, what the interviewees seemed to be talking about all along. This fact became particularly obvious in statements such as In 1 shekel there are 10 agoras (Talli) and In 2 shekels there are 20 agoras (Mira), both of which are correct only if the word agora is interpreted as referring to the coin of 10 agoras, not to a single agora (it must be stressed that there is no such thing as 1 agora coin). What we saw in these examples was true throughout the interviews. Mira and Talli tended to keep the literate and colloquial arithmetic separate. Symbolic calculations were almost completely absent from both girls’ implementations of money transactions. The reverse was true as well: Colloquial procedures rarely appeared in the execution of school-like tasks. To be sure, Mira occasionally did have recourse to colloquial arithmetic in paper-and-pencil calculations (recall her use of fingers and drawings in multiplications), and yet, she was uneasy about this fact and tried to conceal it (see line 6 of excerpt 1, where she refuses to show how she helps herself with fingers). Obviously, she observed the norm, according to which the everyday arithmetic should be kept out of school. If she ever agreed to use it in our interviews, it was only because of our insistent encouragement. Talli was even more extreme in her refusal to do so, and we cannot tell whether her resistance was a mere result of the strict adherence to the norm or was caused by the girl’s outright inability to relate the literate and the colloquial arithmetic in any way. We conclude that our interviewees’ colloquial arithmetic, although quite impressive at times, was not contributing to their literate skills and understandings, and that this fact was only partially the result of the girls’ inability to see the vital connections between the two. Another contributing factor was the fact that Mira and Talli evidently regarded their colloquial skills as inappropriate for school use, and thus also for our school-like interviews.

Summary of Our Study

Although both our interviewees had distinct difficulty even with relatively simple numerical tasks, our close inspection has shown that Mira’s arithmetic was, in many ways, superior to Talli’s. Indeed, Mira’s numerical talk was more objectified, and her calculations, unlike Talli’s ritualized performance, had all the characteristics of explorations. As a result, while encountering an obstacle, Mira was able to find a proper “detour” with the help of the mental map she built in her explorations. At times like this, Talli had no choice but to rely on her manifestly imperfect memory. All these findings may appear surprising if you recall the numerous stories we heard about Mira’s mathematical “disabilities” versus Talli’s mathematical “potential.” We will end this article with some conjectures about the source of this paradox, its practical implications, and its possible solutions.

Discussion and Conclusions: Lessons about Learning and Teaching Mathematics

While going through our data we could not stop wondering why other people’s evaluations of Mira’s and Talli’s arithmetic was so different from our own. How did it happen, we wondered, that a story that we considered as misled and misleading was shared by so many people, including Mira and Talli themselves? Upon reflection, we understood that the similarity between the institutional account and the girls’ own descriptions could be the key to the whole puzzle. It highlighted the fact that narratives co-constitute one another. We conjectured that, over the years, the third- and first-person stories about the girls fed into each other, gradually unifying and reinforcing their message with the help

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of descriptors such as “weak student” and “learning disabled.” When translated from the third- to the first-person accounts, these narratives became also powerful shapers of their protagonists’ actions. In this way, the allegedly “objective” reports turned into self-fulfilling prophecies. We illustrate this latter claim with a hypothetical history of Mira’s mathematical underperformance. While what happened to Mira in the fourth grade—the sudden breach in her ability to keep pace with the requirements—is not unusual, the fact that she was unable to overcome the difficulty raises questions. We conjecture that this fact might have to do with the widespread belief that being skillful in literate mathematics is an evidence of a person’s “general smartness”—of her “being born for success.” Coming from where she did, Mira did not feel that way. Others— teachers and diagnosticians—might have shared this opinion and some consequential tracking would follow. Subsequently left in the exclusive company of children who were not any more versed in mathematics than herself, Mira was deprived of proper opportunities for overcoming her difficulty. Her continuing failure yielded the tag of “learning disabled,” thereby sealing the verdict of her not being “born for mathematics.” This hypothetical scenario brings into the full relief the dangers of ability-related labels and makes us aware of the responsibility that rests on the shoulder of their users. Indeed, our interviewees’ mathematical failure was a product of collective doing. In Mira’s and Talli’s case, everybody seemed involved: the parents, the teachers, social workers, the girls’ peers and the society at large. This, we believed, was true even if most of the parties involved had acted with the best of intentions. Our close analyses have shown a number of ways in which the society at large, and those around Mira and Talli in particular, contributed to the girls’ difficulty with numbers. Let us count some of these factors, while also asking ourselves what could be done to stop such an inadvertent wrongdoing. The most obvious reason for Mira’s and Talli’s persistent difficulties with school mathematics was the fact that the girls had too many other worries to be able to deal effectively with all the difficulties that occur naturally on one’s way toward mathematical abstraction. At the time we met them, the past maltreatments and abuses could not be undone. And yet, we could think of using Mira’s and Talli’s stories as a paradigmatic cases to learn from—as a cautionary tale that could help in fighting for other children’s better futures. Moreover, from these two narratives three other, more treatable, reasons for our interviewees’ problems with school mathematics could be deduced. The first among these disabling factors is the practice of labeling students. Descriptions such as “weak,” “low-potential,” and especially “learning disabled” misrepresent properties of one’s mathematical performance as properties of the performer herself. As such, these descriptors have a self-perpetuating effect. Indeed, the story once told cannot be untold. It also tends to spread uncontrollably, especially if it has been heard from the mouth of an authority: a teacher, psychologist, or educational consultant. Thus, if a child is said to be “learning disabled,” there is little she can do to prove otherwise. The unsuccessful attempts to change everybody’s minds will soon stop her from trying. This is how stories about students’ properties rather than their deeds become reality. Let us keep this in mind while choosing the words in which to talk about mathematics learners: While using a label such as learning disabled we may be creating the disability rather than merely describing it. Related to labeling is the fact that only too often, our pedagogical decisions are not s upported by proper assessment of the students’ needs. Our research has shown that the most common assessment tools, such as the school grades or diagnostic instruments used in preparing Mira’s and Talli’s school files, have too little differentiating power to be useful. These tools gloss over the inherent complexity of arithmetical activity and make individual differences among their participants practically invisible. Obviously, overlooking the differences largely diminishes the chances for an effective intervention. In this article we offered alternative, label-free way of listening to mathematics learners. By doing so, we hoped to promote the practice of customizing each student’s paths to success.

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For instance, our analyses yielded two separate sets of suggestions for Mira’s and Talli’s teachers. As Mira’s difficulties were clearly related to her inability to help herself with memorized numerical facts and algorithms, we believed that her performance could greatly benefit from the use of a handheld calculator accompanied by a systematic practice in all those areas we had identified as requiring improvement. The resulting success would minimize stress and, in the long run, would hopefully disprove her debilitating story about herself. Helping Talli might be more difficult, if only because we could not be sure that we had seen the full arsenal of her arithmetical possibilities. The first thing to do would be to relax Talli’s vision of what counts as legitimate mathematical activity. The next step would be to try to connect her literate arithmetic with the colloquial. Talli could also benefit from more direct guidance in distinguishing between the features of concrete performances that must be preserved and those that require modification in transition to other situations. The last harmful practice highlighted by our study is creating and sustaining norms that prevent students from taking full advantage of their skills. According to one such norm, the use of colloquial arithmetic during mathematics lesson would seem as inappropriate as using slang while writing an essay in literature or history. Another harmful norm is the one that underlies the use of literate mathematics as a general predictor of success. Based on all these observations, the struggle against mathematical failure may begin with perfecting our tools for conceptualizing, analyzing, and assessing students’ mathematical activity. It should continue with an attempt to stop the practice of labeling and of using the resulting tags as a tool for measuring “human potential.”

References

Ben-Yehuda, M., Lavy, I., Linchevski, L., & Sfard, A. (2005). Doing wrong with words: What bars students’ access to arithmetical discourses. Journal for Research in Mathematics Education, 36, 176–247. Nunes, T., Schliemann, A., & Carraher, D. (1993). Street mathematics and school mathematics. New York: Cambridge University Press.

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