- Email: [email protected]
Performing algebra: Emergent discourse in a fifth-grade classroom
JMB
ISSN 0364-0213. JOURNAL OF MATHEMATICAL BEHAVIOR, 16 (l), 3949 Copyright 0 1997 Ablex Publishing Corp. All rights of reproduction in any form reserved.
Performing Algebra: Emergent Discourse in a Fifth-grade Classroom BOB SPEISER CHUCK WALTER Brigham Young University
The children’s work we describe here grows from a teaching experiment, now in progress, at the Longview Elementary School in Murray, Utah, where the authors have been working for about four years. The goal of our experiment is to learn how algebraic thinking might begin in fourth- and fifth-grade classrooms. What we had planned as a preliminary trial for an experiment about children’s mathematics might best be viewed, we’ve come to realize, as drama. In this article, therefore, we treat mathematics as performing art, we regard the mathematics classroom as a theater, and we see learners of mathematics, and their teachers, as both authors and performers. In this way, we can raise new questions about what happens in a classroom, how we might describe it, and how we might support its action helpfully. For us, we need to emphasize, performance will be neither metaphor nor analogy-this article is not about how mathematical work resembles a performance, but rather about how mathematical work is performative, right down to the bottom. For three successive Friday mornings in May 1996, the two authors joined the children of a fifth-grade classroom, and their teacher, to produce what we discovered was a play. This play was unconventional in several ways. We had no formal script, and only tiny portions of the drama’s text were ever written down. Its subject, to a great extent, emerged only in part during our preliminary workshops with the acting company, on the first two Fridays, and its story, as we see it now, only became clear to us a full three months after its performance. That performance, on the final Friday, took place without rehearsal, began with an attempt to reconstruct a missing script fragment, and ended with an interruption, by some members of the company, who insisted on rethinking the whole story. We begin with background about children’s algebra, and then describe what happened in the three sessions. We are deeply grateful to Cindy Durante, the classroom teacher, for the welcome she extended, for the adventurous and very helpful insights she has shared with us, and for her own fine sense of classroom drama.’ Indeed, we often count ourselves among her students. Several children, then aged ten, emerged as full-fledged heroes in the classroom action, and play leading parts in the transcripts below. Among these, we wish especially to emphasize our debt to Amber, who, speaking for a group of dissidents, helped change fundamentally the way we look at classroom discourse.* Direct all correspondenceto: All correspondence should be directed to: Robert Speiser, Brigham Young University, Mathematics Department, Provo, UT 84602-6539.
39
40
SPEISER AND WALTER
Initial Views of Algebraic
Thinking
It seems accurate, but still potentially misleading, to say that algebra’s great power comes from leaving context temporarily behind, in order to work formally with symbols. It might be better to say that algebraic thinking, when it is appropriate, first reframes the original context, so as to designate certain features as important, and then makes these features more directly available through symbolic representation. For us, it seems especially important to investigate the emergence, in concrete problem situations, of particular decisions which focus attention on geometric or numerical patterns which may lend themselves, eventually, to expression by symbolic formulas and analysis by formal means. The initial choice to focus on specific patterns can be viewed as a first step in the construction of a theory.3 Further decisions, which might begin as quite vague hunches about how to represent the designated geometric or numerical patterns symbolically, can lead to at least potentially testable guesses about how to work with the emerging representations. Our hope is that a given representation, when it helps us understand an underlying process, will fold us back into the problem situation which gave rise to it, back to given facts. In this sense, algebraic thinking, like symbolic thinking generally, reframes its context in order to return to it. Whether an initial representation leads to symbolic thinking right away or not, the representation, to an extent, takes on a life of its own. In algebra, indeed, this life, often, seems like symbolic play. Played well, however, algebra is neither blind nor arbitrary. A representation succeeds precisely when its reframing of the problem context offers useful insight. To clarify and justify means to interpret, to reshape, to construct meaning through connections, and to advocate: in short, to perform. In precisely this sense, algebra is performative, and, through performance, enters, changes, and is changed by the discourse which surrounds it. Consider the parallel between the way symbolic thinking helps to shape, not just describe, the mathematical discussion which engenders it, and the way a research narrative, like this one, through its choice of what to tell and how to tell it, can help to shape what happens in the classrooms it describes. This parallel runs deep, and urges us to read symbolic algebra, each time it appears, in terms of wider human narratives.
Cognition
in the Context of Performance
Our view of mathematics as performance owes much to the anthropologist Victor Turner’s hermeneutic view of social drama (Turner, 1974, 1982, 1986). Attempts to build new meaning, to enact, to understand, emerge from powerful, unusual, not everyday, events. In Turner’s words, “These experiences that erupt from or disrupt routinized, repetitive behavior begin with shocks of pain and pleasure. Such shocks are evocative: They summon up precedents and likenesses from the conscious or unconscious past-for the unusual has its traditions as well as the usual.. . . What happens next is an anxious need to find meaning in what has disconcerted us, whether by pain or pleasure, and converted mere experience into UR experience. All this when we try to put past and present together.“4 Experience remains private, unless represented, sifted, shared. The shock of powerful experience, which evokes emotion and desire, the impact of what Dilthey calls value, drives cognition, through the need for social sharing. “In Dilthey’s view,” Turner contin-
THE BEGINNINGS
OF ALGEBRAIC
THINKING
41
ues, “experience urges toward expression, or communication, with others. We are social beings, and we want to tell what we have learned from experience.... The hard-won meanings should be said, painted, danced, dramatized, put into circulation.“5 For Turner, as for Dilthey, representation builds fundamentally toward social expression, toward performance, as our way of knowing and of being known. Indeed, our only access to others’ experiences is through interpreting, constructing meaning, in performances we share. This article is about the location and the function of algebraic thinking in the social process of the classroom. Like Turner, we see cognition as performative, and cognitive growth as something which emerges socially, through particular performances. As we planned our teaching experiment last winter, it became clear, in order to produce important growth, that the tasks we offered to the children, right from the start, would need to represent a very sharp departure from the children’s usual routine.
Two Workshop
Sessions
Last May,6 to begin our drama, we tried two variations on a task originally suggested by Bob Davis, who learned it in the 1960s from David Page. This task, which Davis and colleagues developed within classroom settings (with students in grades four and five in Weston, CT), is described in Davis (1980). In our fifth-grade setting, where the children have had only slight exposure to algebraic symbolism, our tasks change character: they produce situations which, at least potentially, can evoke that algebra, rather than simply appeal to it. The description which follows closely parallels our field notes, and concentrates, as did those notes, on cognitive discoveries. In the background, however, Cindy, both authors, and the children implicitly assume, and then test, dramatic roles. Inviting the Longview fifth-graders to explore our tasks elicited intense, sustained activity, predominantly modelbuilding. Each of the two first classroom sessions lasted for two hours. In the life of a performing company, these sessions would be called workshops, in which actors and actresses construct roles and situations, try interpretations with each other, and prepare, in some detail, the parts which they will later play. The first session centered, explicitly, on a very concrete problem. Start, for example, with one yellow Cuisenaire rod. Its volume is 5 cubic cm, while its surface area is 22 square cm. Place a second yellow rod directly alongside the first, as shown in Figure 1. The volume is now 10 cubic cm, while the surface area is 34 square cm. We continue stacking rods, one at a time, in the same way, to build progressively larger figures. The central challenge is to predict what will happen to the volume and surface area as the stack continues to grow, and then to find a way to write or say it. In this first trial, the children worked easily, and perhaps routinely, with the volume. Their work on surface area, however, produced intense engagement, much discussion, and surprising presentations.
I
I
Figure 1.
42
SPEISER AND WALTER
Within half an hour, at least three-fourths of the children clearly recognized, by tabulating their experimental data, that the surface area grows by 12 each time a rod is added to the stack. More precisely, the numerical pattern is to begin with 22 for the first rod and repeatedly add 12 each time a new rod joins the stack. But how might these children justify this pattern? The first day’s work did not seem especially conclusive, as many students focused mainly on the numbers in their tables, once they had them, rather than the rods from which their numbers had been drawn. Nonetheless, some students produced tables for several different lengths of rods, leading to strong conjectures about modeling more general behavior. As we listened to the students’ presentations to each other, we were to some extent guided by two lines of thinking which had emerged in our preliminary discussions with the teachers. These lines of thinking emerged distinctly among children in the second session, as two main schools of thought-interpretive positions-held by different children. The differences between these two positions, and their arithmetic consequences, emerge quite clearly once a second rod is placed beside the first. We shall denote by A the surface area of a stack of N yellow rods. The first interpretative position begins with two rods, first held apart, giving a total area of 44 square cm. Bringing the rods together will “absorb” two 1 cm x 5 cm faces, hence we must subtract 10 from 44, to obtain 34 square cm for the surface area of a two-rod stack. This approach, therefore, sees 22 added and then 10 taken way each time a rod is added, with a net gain of 12 for each new rod. For N yellow rods, this leads to a formula of the form A = 22 + 12(N-
1).
No child, however, chose to express the number pattern in this way, although all had had some exposure to symbolic variables. Instead, a proponent of the first interpretive position, explaining at the overhead, with both pens and rods available, would say something like, “If you have five rods, then you’ve started with one rod, and then put down four. So you take 4 times 12, and add that to the 22.” This school of thought, in effect, regards each change in surface area as an adjunction followed by an absorption, The second interpretive position tells a very different story. Its central point emerged among the children, somewhat awkwardly at first, in response to one investigator’s question to a child who seemed to have a different strategy from the approach described above, yet seemed to have some difficulty explaining it. The question was, “Okay, we know the number grows by 12 each time you put a new rod on. Can you possibly show me the 12?” The child thought for a moment and then ran her index finger along the shaded band, which goes all the way around the stack, shown in Figure 2, while counting off the 12 square centimeters which form its area. Several other children, who were often girls, described similar strategies, once the investigators realized what to look for. In effect, the second position breaks the surface of the lower rod, the first one in the stack, into two parts: a bottom box (with area 17 square cm) and a top (with area 5 square cm). Then, as each new rod is added, that new rod contributes only a band (a bottomless, topless box, with area 12 square cm) which extends the original bottom box, and is closed above by the original top, which has been raised one centimeter for this purpose. Figure 2 shows the shaded band, together with the bottom box below and the top above, unshaded.
THE BEGINNINGS OF ALGEBRAIC THINKING
43
Figure 2.
Seeing the “value added” as a band leads to the exactly same formula the first interpretive position has obtained, but through a very different way of looking at the rods: each rod here, as it joins the stack, is decomposed into component pieces, some of which are joined to parts of the earlier stack to form the new one. Each interpretive position, then, reframes the problem context by seeing, and narrating, changes in surface area, but in different ways. Each way, it seems, has its own value. From a higher mathematical viewpoint, indeed, each interpretive position isolates, and motivates, important geometric principles. These principles emerge to some extent in children’s discourse, in the form of presentations which make different kinds of sense. In the first position’s discourse, the “absorbed faces” are precisely those which cancel in a two-dimensional vector integral, over the surfaces which bound the solids we have joined.7 The second position’s discourse, in contrast, seems to decompose the outer surface of the rod stack into a collection of simpler pieces, each formed of squares and rectangles, much as some topologists might do.* In the first session, it seemed to us that many children had preferred to concentrate attention on the number patterns in their tables, without returning to the rods, to any great extent, to justify their work, unless we or their teacher broke the children’s flow with questions. Hence, planning for the second session, our discussion with the teachers focused mainly on how we might bring home the need for explanations. To stimulate discussion, by children themselves, of issues which we viewed as fundamental, we modified the task design, in order to direct attention to the ways in which the number patterns changed between the first task and the second. The variation we proposed, also due to David Page (Davis, 1980, pp. 271-272), was quite straightforward: instead of placing each successive rod directly beside its predecessor, we offset the new rod 1 cm, to one side, as shown in Figure 3.
Figure 3.
44
SPEISER AND WALTER
This time the surface area grows by 14 square cm each time a rod is added to the stack, and many children, indeed, began investigating why it does, as we had hoped, quite early in the session. Proponents of the first interpretive position seemed, at least at first, to give the clearest verbal presentations. One student, for example, noted that when bringing two rods together, now only 8 square cm are absorbed, so the effect of adding one more rod adds 22 first, and then subtracts 8. Other children took advantage of the work already done, without explicit reference to the theory which produced the earlier results. The plan here is to modify the earlier results (by adding two square cm for each new rod adjoined) to account for the offset. The two square cm are easily located visually, as two one-centimeter squares, one above and one below, which are uncovered by the 1 cm offset, and several children pointed to these squares in the course of conversations with us. This idea, which seems related to the second interpretive position and its more decompositional approach, led several children to conjecture, and then justify, procedures for finding the surface area for rods of any length, and for any given offset. Two children told us later that they were so excited by our second task that they could hardly talk. As we watched one boy, almost breathless, supported by a group of friends, attempt four times, and fail each time, to find words for a number pattern, we sensed how much our tasks, even so early in the process, seemed to build experiences which challenged these children’s linguistic and expressive powers deeply, and yet fascinated them.
Preparing a Performance In a discussion after the second workshop session, Cindy Durante suggested to the research team that there was yet more, much more, to the story than we knew so far. Through the intervening week, her students were returning to the rod tasks and their tables, talking with each other, sharing further discoveries. Hence Cindy proposed a full-class unpacking,’ where her children could show us, in full-scale performance, what we’d missed. We joined her the following Friday morning, this time as audience, while she led the class. Here are two brief student presentations from this third session, transcribed from our videotape of children working at the overhead.” As with our description of the workshop, we’ll stay close to our transcripts and our working notes, and then comment on them. The task is to explain the calculation of the surface area of a stack of five yellow rods, with one cm offsets, to illustrate the computational pattern for finding the surface area with any number of rods. Cindy invites one boy, Brody, to present a discovery which he had made some days before. He has, in Cindy’s words, a theory which, as such, deserves some careful testing. Does Brody’s theory fit the tabulated data? Cindy’s invitation to present, however, has caught Brody completely by surprise. He does not, at first, remember how his calculations went. Cindy responds to Brody’s uncertainty by encouraging him to persevere, and then, to back him up, invites two of Brody’s closest friends to help him. Gradually, quite hesitant but eager nonetheless, Brody guides his small group, after several tense asides, toward a consensus. At this point Brody writes the equation (4x14)+22=78 on the overhead, and then, again, runs out of words.
THEBEGINNINGSOFALGEBRAICTHINKING
45
Speaking for the group, Derek, at Brody’s side, jumps forward to explain. Next to Brody’s equation, a stack of five rods, offset carefully, shows clearly on the screen. Derek:
Cindy: D: c: D:
The first one’s 22. So, then if you put them down, it adds 14 every time, so if you were going to do five rods, then it would be, it would increase by fourteen four times. So you’d do four times fourteen... Four times fourteen. ...and then you’d add... You’d add. 1.. the twenty-two, and then you’d get the answer. You’d add 22. You do these increases first, and then you add what it’s equal to.
At this moment, another student, Amber, at the far side of the room, stands up to challenge both the form of Brody’s equation and the content of Derek’s explanation. The multiplier in the first term, she suggests, should be the number of rods, namely five, not four. To focus her objection, Amber proposes a modification of Brody’s formula, hence of its underlying numerical process. Further, she supports her thinking with a concrete argument, based upon, but modifying, Derek’s. Amber:
They had five rods, and they did four times fourteen, plus 22. They said, that if you take this rod [pointing to the first rod] away, it comes up in the 22. But I, urn, if you add, uh, I didn’t get that. So I just thought, if you add the extra rod, what would that be? And just take that from the 22.
Amber does not question Brody’s numerical Instead, on conceptual grounds, she takes exception been expressed.” Here, she walks across the room, at the overhead. Having outlined her basic strategy, Amber:
result; indeed, she tacitly applies it. to the form in which Brody’s result has and takes her place, with two friends, Amber now presents details.
So, urn? [Her partner, Kimberley, whispers something.] Okay, we had five rods. [She writes “(5 x 14)” on the overhead, just below Brody’s formula.] You had five rods. And then what you added, and you have to subtract from the 22, this extra rod, here, so you subtract. I don’t know what we subtracted, but we got 8. [Here she completes the expression “(5 x 14) + 8”, exactly parallel to Brody’s.]
Amber’s work here looks to us like mature, unvarnished algebra, but with the number 5 as a specific stand-in for a variable.12 In effect, she takes Brody’s expression, 14(/V1) + 22, and rewrites it as 14N + 8, subtracting 14 from 22 to obtain her constant term. She does not reconstruct the final step in detail, which suggests that her attention focuses instead upon her underlying plan, which she explains quite clearly. We see two levels of reframing here, now that algebraic thinking has emerged. At the first level, examining a number pattern through the optic of a particular representation of the rod stack emerged slowly as children discussed the second task, often to reconcile contrasting calculation strategies. But further, as in Amber’s work above, patterns ofcalcuZution also became objects to reframe. At this second level of reframing, algebraic thinking appears clearly, but, as Amber shows, only through a further reference to the problem con-
46
SPEISER AND WALTER
text. Amber’s work, indeed, might not have come to light, until she disagreed with Brody on the grounds that his formula did not show clearly enough the role of five rods in the calculation of the area. The performance here began with a theory, in this case a conjectured model to be tested. In effect, does 14(N - 1) + 22 describe the given data? With our camera running, a group of nervous boys explained the fit: with just a few rods, their model clearly accords with experimental data, and each time a new rod is added, the area must, just as they claim, grow by fourteen. As they go along, although they build their scene from partly remembered fragments, they get their story right. If the play is about conjecturing and testing models, then these boys have played successfully, building suspense as well as interest, against heavy odds. In this context, Amber’s challenge seems especially remarkable. At first glance, however, Amber simply seems to break the flow. Watching the class, indeed, neither we nor Cindy grasped the full import of Amber’s objection. Amber, too, spoke for a group, but one that formed, as if by itself, unbidden. Was their message heard? After the session ended, Amber, as well as others in her group, indeed, would walk away unsatisfied. Only three months later, as the two authors checked their video transcript for the third time, did we realize the fundamental strength of Amber’s argument, and hence the deeper import of the confrontation we had witnessed. Amber’s concern was not with data-concordance with the data, she might argue, is the easy part-but with sense and meaning. Because she has proposed a radical critique, she will now need to break the flow. Against the boys’ stock story, built with their teacher’s support, Amber will need to tell a counterstory.13 Her counterstory sees the mathematics, and the surrounding action in her classroom, in new ways.
Dramatic Discourse To understand dramatic action, we might begin by looking closely at the stories which the actors tell. To understand a story, we might begin by asking, for example, what the story suys. But we also. often, need to step outside that story, in order to examine closely what that story does. A stock story might help one group of tellers to exert power through a dominant discourse, while a counterstory (Delgado, 1989) might help a different group of tellers to resist that power, by constructing an emergent discourse. Dramatic conflict then, can take place between discourses, not just between the characters who represent them. According to what seems to be a dominant discourse among many educators, for example, girls often have trouble learning algebra, beginning at around age twelve. Against this dominant discourse, Amber’s confrontation with three boys takes on special meaning. Let’s explore the possibility that Amber and her coworkers, through a counterstory, might be building an emergent discourse in their classroom. How, specifically, might this view help shape our reading of Amber’s dissent? As spokesperson, Amber represents a group, which includes but may not be limited to the girls who stand beside her, and this group moves into prominence because, collectively, the discourse is theirs, and Amber therefore speaks, at least in part, for them. Further, this group’s critique of Brody’s formula first outlines, along the way, how its members read the dominant discourse, which seems to be about checking formulas to fit “the facts,” and then stopping once some kind of fit has
THE BEGINNINGS OF ALGEBRAIC THINKING
47
been established. Hence we may identify Brody’s “theory” as supporting, at least relative to this group of girls, a stock story. The counterstory, judging by Amber’s careful emphasis, seems to be that formulas, themselves, tell stories about data, and therefore that we should examine closely what kinds of sense these stories make. Furthermore, as she argues by example-inventing algebra to make her point-that appropriately chosen transformations of a given formula can help us to make better sense. For Amber’s group, Brody’s formula, even though it fit the facts, had failed in an important way to tell the story offive rods. To make it do so, Amber proposes to alter Brody’s formula in a coherent way, based firmly on the meaning of that formula (especially the observation that the area must grow by 12 each time we add another rod), and upon the structure of the calculation which it represents (which allows her to revise the constant term). For us, the discourse which Amber leads presents important pieces of the algebra we know, with an assurance which continues to surprise us. This reframing of Amber’s dissent accomplishes, we feel, two central purposes. First, it connects what we observe to widely shared concerns about students at risk, and how we might respond most helpfully to the ideas they advocate. That Amber’s insight, and the discourse which it helped to constitute, did not-at least so far-come to play a strong part in the class discussion, therefore seems especially worth noting. Second, our new reading of Amber makes sense of far more detail on our tapes and transcripts than simply what this one child did and said. Hence we build a more coherent, and perhaps more compelling, picture of what happened in the surrounding classroom, just as algebraic thinking broke the surface. It helps, certainly, to emphasize how quietly an important new idea might be expressed, and hence how easily that new idea, together with the larger discourse it could help initiate, might pass unnoticed.
Emerging
Views of What School Mathematics
Might Become
As working mathematicians, we have emphasized connections, still hidden to many teachers, leading from school mathematics to the mathematics which we practice. We see algebra, in particular, not simply as a body of notations and techniques adhering to time-worn examples, but rather as a way of working, of perjiorming, in the world. In Cindy Durante’s classroom, algebra, we now think, supported an emergent discourse, constituted by a team of girls with Amber in the lead, counterposed against a dominant discourse, built here by several boys with strong teacher support. This dominant discourse takes data as given and then stresses numerical agreement between conjectured computations (denoted “Brody’s theory” here) and data. Amber and her friends, however, tell a very different story. This second story, as we read it, emphasizes why, given the increase of 14 square cm with each rod, the computation, once Amber and her friends reframed it, could explain and justify the duta. The second story wasn’t heard that day, however, outside the group where it beganI How does this algebraic way of working work? Nemirovsky (1994) distinguishes between symbol systems, which embody ways of working in the world through rules, and symbol use, “embedded in personal intentions, in specific histories, and in the qualities of a situation.” Amber and her friends provide a glimpse which helps to clarify the ways in which symbolic thinking might emerge from children’s direct, personal engagement. What mathematics means, specifically, in given situations, depends precisely on its human context. Here Nemirovsky quotes Bakhtin: “The sentence, as a unit of language, like the word,
48
SPEISER AND WALTER
has no author.” An equation, to convey meaning, needs an author, and we seek to understand her, just at the decisive moment when she chooses what and how to write. But here we need to go much further. Symbolic texts, as we have seen, need actors, not just authors. Nemirovsky, evoking Bakhtin’s ghost, who offers him a sword, attempts to give the text back to its author, and so, by implication, to return the author’s words to social contexts. This return can only be accomplished, we may now assert, when personal experience, communicated, shared and worked through socially, has been effectively performed. Like Derek and Amber, we and the participating teachers seek to learn, through personal intent, within specific histories, ideas and processes which, potentially, can shape our lives. Some of these ideas and processes are deeply mathematical. Some have more to do with how to build environments and work in them. Some map potential common ground, perhaps surprising, shared by researchers like us, and by colleagues, starting at age ten, who work in schools. Precisely how we choose to stage this cognitive and social action, as we have sought to show, will matter deeply.
Notes 1. Parallel to the first two sessions in Cindy’s room, we explored the tasks in a second fifthgrade room, led by Gary Mimer. Both Cindy and Gary helped us plan all three Friday sessions, exploring each task personally, and offering many suggestions along the way. In the work below, we sometimes refer to discussions with both Cindy and Gary, which took place in the preparations for each classroom session. Our debt to Gary, who shares longstanding interests in literature and anthropology, is also very large. He and Cindy helped to teach us about Longview as a special human culture, joined by several other teachers, most especially Karen Cristopulos, Robin Griego, Pam Aoyagi, Jeff Nalwalker, and their Principal, Dr. Marilyn Prettyman. 2. A series of informal discussions of this classroom drama with colleagues at the University of Wyoming, in September 1996, also helped to clarify our thinking, as we moved from fairly focused hunches toward interpretation. We are especially grateful to Karen Bartsch (psychology), Susan Frye, Mark Booth and Bob Torry (English), Janice Harris (women’s studies), Audrey Shalinsky (anthropology), John Dorst and Eric Sandeen (American Studies), and Lynn Ipifia (mathematics). The first author, as well, extends his deepest thanks to Elizabeth Hacker, then ten years old, also of Laramie, who emphasized the dramatic power of conflicts, intuitions, dreams and metaphors in children’s thought. 3. We believe that theory-building centers all cognition. (See Baillargeon, 1993; Bartsch & Wellman, 1995; Carey, 1985; Spelke, 199 1.) This view, sometimes called the “theory theory” grows from subtle and often ingenious experiments, often with quite young children. Theories, in particular, may drive perception, right from birth. 4. Turner (1986) pp. 35-36. 5. Turner (1986), p. 37. 6. In 1996. 7. McCallum, Hughes-Hallet (1995) Chapter 19. 8. One of the authors was reminded strikingly of the standard Morse decomposition of a sphere. (See Milnor, 1963, pp. 1-2, 14, for illustrations on a torus.) At a deeper level, which undergraduates, unfortunately, seldom see, the analysis and the topology will tend to merge. (See Fulton, 1995.)
THE BEGINNINGS
OF ALGEBRAIC THINKING
49
9. The term “unpacking” goes back to a workshop which Speiser led at Longview for two weeks in June, 1994. It refers to detailed presentation and analysis, by the whole group, of a significant exploratory finding, in order to make that finding available for connections later. 10. We checked the transcript three times. Only on the third pass did we begin to build consistent versions even of the children’s words. 11. In this, she differs from the direction taken by her teacher, who invited students to test the numerical process underlying Brody’s formula with different numbers of rods, to check agreement with their tables. 12. Diophantus, for example, wrote this way. 13. Delgado (1989). 14. In the meantime, the Longview term had ended, and tape editing equipment arrived, months late, needing repair. To make good use of our time, we began to read the anthropology of social drama, focused on a tape of undergraduate education students as they built a number system (Speiser & Walter, 1996), and began to sharpen our analysis. When we returned to the Longview tapes in August, 1996, our viewpoint now included narrative as well as cognitive construction, and hence we heard the Longview tapes with different ears.
REFERENCES Carey, Susan (1985). Conceptual change in childhood. Cambridge, MA: MIT Press. Bartsch, Karen, & Wellman, Henry M. (1995). Children talk about the mind. New York: Oxford University Press. Baillargeon, Rede (1993). The object concept revisited: New directions in the investigtion of infants’ physical knowledge. In C. E. Granrud (Ed.), Visual perception and cognition in infancy (pp. 265-3 15). Hillsdale, NJ: Erlbaum. Delgado, Richard (1989). Storytelling for oppositionists and others: A plea for narrative. Michigan Law Review, 87,241 l-2441. Davis, Robert B. (1980). Discovery in mathematics: A text for teachers. White Plains, NY: Cuisenaire. Fulton, William (1995). Algebraic topology. A first course. New York: Springer-Verlag. Milnor, John W. (1963) Morse theory. Princeton, NJ: Princeton University Press. McCallum, William G., Hughes-Hallet, Deborah et al. (1995). Multivariable calculus: Preliminary edition. New York: Wiley. Nemirovsky, Ricardo (1994). On ways of symbolizing: The case of Laura and the velocity sign. The Journal of Mathematical Behavior, 13, 389-422. Speiser, Robert, & Walter, Chuck (1996). Five women build a number system. Manuscript in preparation. Spelke, Elizabeth S. (1991). Physical knowledge in infancy. In S. Carey & R. Gelman (Eds.), The epigenesis of mind: Essays on biology and cognition (pp. 133-169). Hillsdale, NJ: Erlbaum. Turner, Victor W. (1974). Dramas, fields, and metaphors: Symbolic action in human society. Ithaca, NY: Cornell University Press. Turner, Victor W. (1982). From ritual to theater: The human seriousness ofplay. New York: PAJ Publications. Turner, Victor W. (1986). Dewey, Dilthey, and drama: An essay in the anthropology of experience. In V. W. Turner & E. M. Bruner (Eds.) The anthropology ofexperience (pp. 33-44). Urbana, IL: University of Illinois Press.
ISSN 0364-0213. JOURNAL OF MATHEMATICAL BEHAVIOR, 16 (l), 3949 Copyright 0 1997 Ablex Publishing Corp. All rights of reproduction in any form reserved.
Performing Algebra: Emergent Discourse in a Fifth-grade Classroom BOB SPEISER CHUCK WALTER Brigham Young University
The children’s work we describe here grows from a teaching experiment, now in progress, at the Longview Elementary School in Murray, Utah, where the authors have been working for about four years. The goal of our experiment is to learn how algebraic thinking might begin in fourth- and fifth-grade classrooms. What we had planned as a preliminary trial for an experiment about children’s mathematics might best be viewed, we’ve come to realize, as drama. In this article, therefore, we treat mathematics as performing art, we regard the mathematics classroom as a theater, and we see learners of mathematics, and their teachers, as both authors and performers. In this way, we can raise new questions about what happens in a classroom, how we might describe it, and how we might support its action helpfully. For us, we need to emphasize, performance will be neither metaphor nor analogy-this article is not about how mathematical work resembles a performance, but rather about how mathematical work is performative, right down to the bottom. For three successive Friday mornings in May 1996, the two authors joined the children of a fifth-grade classroom, and their teacher, to produce what we discovered was a play. This play was unconventional in several ways. We had no formal script, and only tiny portions of the drama’s text were ever written down. Its subject, to a great extent, emerged only in part during our preliminary workshops with the acting company, on the first two Fridays, and its story, as we see it now, only became clear to us a full three months after its performance. That performance, on the final Friday, took place without rehearsal, began with an attempt to reconstruct a missing script fragment, and ended with an interruption, by some members of the company, who insisted on rethinking the whole story. We begin with background about children’s algebra, and then describe what happened in the three sessions. We are deeply grateful to Cindy Durante, the classroom teacher, for the welcome she extended, for the adventurous and very helpful insights she has shared with us, and for her own fine sense of classroom drama.’ Indeed, we often count ourselves among her students. Several children, then aged ten, emerged as full-fledged heroes in the classroom action, and play leading parts in the transcripts below. Among these, we wish especially to emphasize our debt to Amber, who, speaking for a group of dissidents, helped change fundamentally the way we look at classroom discourse.* Direct all correspondenceto: All correspondence should be directed to: Robert Speiser, Brigham Young University, Mathematics Department, Provo, UT 84602-6539.
39
40
SPEISER AND WALTER
Initial Views of Algebraic
Thinking
It seems accurate, but still potentially misleading, to say that algebra’s great power comes from leaving context temporarily behind, in order to work formally with symbols. It might be better to say that algebraic thinking, when it is appropriate, first reframes the original context, so as to designate certain features as important, and then makes these features more directly available through symbolic representation. For us, it seems especially important to investigate the emergence, in concrete problem situations, of particular decisions which focus attention on geometric or numerical patterns which may lend themselves, eventually, to expression by symbolic formulas and analysis by formal means. The initial choice to focus on specific patterns can be viewed as a first step in the construction of a theory.3 Further decisions, which might begin as quite vague hunches about how to represent the designated geometric or numerical patterns symbolically, can lead to at least potentially testable guesses about how to work with the emerging representations. Our hope is that a given representation, when it helps us understand an underlying process, will fold us back into the problem situation which gave rise to it, back to given facts. In this sense, algebraic thinking, like symbolic thinking generally, reframes its context in order to return to it. Whether an initial representation leads to symbolic thinking right away or not, the representation, to an extent, takes on a life of its own. In algebra, indeed, this life, often, seems like symbolic play. Played well, however, algebra is neither blind nor arbitrary. A representation succeeds precisely when its reframing of the problem context offers useful insight. To clarify and justify means to interpret, to reshape, to construct meaning through connections, and to advocate: in short, to perform. In precisely this sense, algebra is performative, and, through performance, enters, changes, and is changed by the discourse which surrounds it. Consider the parallel between the way symbolic thinking helps to shape, not just describe, the mathematical discussion which engenders it, and the way a research narrative, like this one, through its choice of what to tell and how to tell it, can help to shape what happens in the classrooms it describes. This parallel runs deep, and urges us to read symbolic algebra, each time it appears, in terms of wider human narratives.
Cognition
in the Context of Performance
Our view of mathematics as performance owes much to the anthropologist Victor Turner’s hermeneutic view of social drama (Turner, 1974, 1982, 1986). Attempts to build new meaning, to enact, to understand, emerge from powerful, unusual, not everyday, events. In Turner’s words, “These experiences that erupt from or disrupt routinized, repetitive behavior begin with shocks of pain and pleasure. Such shocks are evocative: They summon up precedents and likenesses from the conscious or unconscious past-for the unusual has its traditions as well as the usual.. . . What happens next is an anxious need to find meaning in what has disconcerted us, whether by pain or pleasure, and converted mere experience into UR experience. All this when we try to put past and present together.“4 Experience remains private, unless represented, sifted, shared. The shock of powerful experience, which evokes emotion and desire, the impact of what Dilthey calls value, drives cognition, through the need for social sharing. “In Dilthey’s view,” Turner contin-
THE BEGINNINGS
OF ALGEBRAIC
THINKING
41
ues, “experience urges toward expression, or communication, with others. We are social beings, and we want to tell what we have learned from experience.... The hard-won meanings should be said, painted, danced, dramatized, put into circulation.“5 For Turner, as for Dilthey, representation builds fundamentally toward social expression, toward performance, as our way of knowing and of being known. Indeed, our only access to others’ experiences is through interpreting, constructing meaning, in performances we share. This article is about the location and the function of algebraic thinking in the social process of the classroom. Like Turner, we see cognition as performative, and cognitive growth as something which emerges socially, through particular performances. As we planned our teaching experiment last winter, it became clear, in order to produce important growth, that the tasks we offered to the children, right from the start, would need to represent a very sharp departure from the children’s usual routine.
Two Workshop
Sessions
Last May,6 to begin our drama, we tried two variations on a task originally suggested by Bob Davis, who learned it in the 1960s from David Page. This task, which Davis and colleagues developed within classroom settings (with students in grades four and five in Weston, CT), is described in Davis (1980). In our fifth-grade setting, where the children have had only slight exposure to algebraic symbolism, our tasks change character: they produce situations which, at least potentially, can evoke that algebra, rather than simply appeal to it. The description which follows closely parallels our field notes, and concentrates, as did those notes, on cognitive discoveries. In the background, however, Cindy, both authors, and the children implicitly assume, and then test, dramatic roles. Inviting the Longview fifth-graders to explore our tasks elicited intense, sustained activity, predominantly modelbuilding. Each of the two first classroom sessions lasted for two hours. In the life of a performing company, these sessions would be called workshops, in which actors and actresses construct roles and situations, try interpretations with each other, and prepare, in some detail, the parts which they will later play. The first session centered, explicitly, on a very concrete problem. Start, for example, with one yellow Cuisenaire rod. Its volume is 5 cubic cm, while its surface area is 22 square cm. Place a second yellow rod directly alongside the first, as shown in Figure 1. The volume is now 10 cubic cm, while the surface area is 34 square cm. We continue stacking rods, one at a time, in the same way, to build progressively larger figures. The central challenge is to predict what will happen to the volume and surface area as the stack continues to grow, and then to find a way to write or say it. In this first trial, the children worked easily, and perhaps routinely, with the volume. Their work on surface area, however, produced intense engagement, much discussion, and surprising presentations.
I
I
Figure 1.
42
SPEISER AND WALTER
Within half an hour, at least three-fourths of the children clearly recognized, by tabulating their experimental data, that the surface area grows by 12 each time a rod is added to the stack. More precisely, the numerical pattern is to begin with 22 for the first rod and repeatedly add 12 each time a new rod joins the stack. But how might these children justify this pattern? The first day’s work did not seem especially conclusive, as many students focused mainly on the numbers in their tables, once they had them, rather than the rods from which their numbers had been drawn. Nonetheless, some students produced tables for several different lengths of rods, leading to strong conjectures about modeling more general behavior. As we listened to the students’ presentations to each other, we were to some extent guided by two lines of thinking which had emerged in our preliminary discussions with the teachers. These lines of thinking emerged distinctly among children in the second session, as two main schools of thought-interpretive positions-held by different children. The differences between these two positions, and their arithmetic consequences, emerge quite clearly once a second rod is placed beside the first. We shall denote by A the surface area of a stack of N yellow rods. The first interpretative position begins with two rods, first held apart, giving a total area of 44 square cm. Bringing the rods together will “absorb” two 1 cm x 5 cm faces, hence we must subtract 10 from 44, to obtain 34 square cm for the surface area of a two-rod stack. This approach, therefore, sees 22 added and then 10 taken way each time a rod is added, with a net gain of 12 for each new rod. For N yellow rods, this leads to a formula of the form A = 22 + 12(N-
1).
No child, however, chose to express the number pattern in this way, although all had had some exposure to symbolic variables. Instead, a proponent of the first interpretive position, explaining at the overhead, with both pens and rods available, would say something like, “If you have five rods, then you’ve started with one rod, and then put down four. So you take 4 times 12, and add that to the 22.” This school of thought, in effect, regards each change in surface area as an adjunction followed by an absorption, The second interpretive position tells a very different story. Its central point emerged among the children, somewhat awkwardly at first, in response to one investigator’s question to a child who seemed to have a different strategy from the approach described above, yet seemed to have some difficulty explaining it. The question was, “Okay, we know the number grows by 12 each time you put a new rod on. Can you possibly show me the 12?” The child thought for a moment and then ran her index finger along the shaded band, which goes all the way around the stack, shown in Figure 2, while counting off the 12 square centimeters which form its area. Several other children, who were often girls, described similar strategies, once the investigators realized what to look for. In effect, the second position breaks the surface of the lower rod, the first one in the stack, into two parts: a bottom box (with area 17 square cm) and a top (with area 5 square cm). Then, as each new rod is added, that new rod contributes only a band (a bottomless, topless box, with area 12 square cm) which extends the original bottom box, and is closed above by the original top, which has been raised one centimeter for this purpose. Figure 2 shows the shaded band, together with the bottom box below and the top above, unshaded.
THE BEGINNINGS OF ALGEBRAIC THINKING
43
Figure 2.
Seeing the “value added” as a band leads to the exactly same formula the first interpretive position has obtained, but through a very different way of looking at the rods: each rod here, as it joins the stack, is decomposed into component pieces, some of which are joined to parts of the earlier stack to form the new one. Each interpretive position, then, reframes the problem context by seeing, and narrating, changes in surface area, but in different ways. Each way, it seems, has its own value. From a higher mathematical viewpoint, indeed, each interpretive position isolates, and motivates, important geometric principles. These principles emerge to some extent in children’s discourse, in the form of presentations which make different kinds of sense. In the first position’s discourse, the “absorbed faces” are precisely those which cancel in a two-dimensional vector integral, over the surfaces which bound the solids we have joined.7 The second position’s discourse, in contrast, seems to decompose the outer surface of the rod stack into a collection of simpler pieces, each formed of squares and rectangles, much as some topologists might do.* In the first session, it seemed to us that many children had preferred to concentrate attention on the number patterns in their tables, without returning to the rods, to any great extent, to justify their work, unless we or their teacher broke the children’s flow with questions. Hence, planning for the second session, our discussion with the teachers focused mainly on how we might bring home the need for explanations. To stimulate discussion, by children themselves, of issues which we viewed as fundamental, we modified the task design, in order to direct attention to the ways in which the number patterns changed between the first task and the second. The variation we proposed, also due to David Page (Davis, 1980, pp. 271-272), was quite straightforward: instead of placing each successive rod directly beside its predecessor, we offset the new rod 1 cm, to one side, as shown in Figure 3.
Figure 3.
44
SPEISER AND WALTER
This time the surface area grows by 14 square cm each time a rod is added to the stack, and many children, indeed, began investigating why it does, as we had hoped, quite early in the session. Proponents of the first interpretive position seemed, at least at first, to give the clearest verbal presentations. One student, for example, noted that when bringing two rods together, now only 8 square cm are absorbed, so the effect of adding one more rod adds 22 first, and then subtracts 8. Other children took advantage of the work already done, without explicit reference to the theory which produced the earlier results. The plan here is to modify the earlier results (by adding two square cm for each new rod adjoined) to account for the offset. The two square cm are easily located visually, as two one-centimeter squares, one above and one below, which are uncovered by the 1 cm offset, and several children pointed to these squares in the course of conversations with us. This idea, which seems related to the second interpretive position and its more decompositional approach, led several children to conjecture, and then justify, procedures for finding the surface area for rods of any length, and for any given offset. Two children told us later that they were so excited by our second task that they could hardly talk. As we watched one boy, almost breathless, supported by a group of friends, attempt four times, and fail each time, to find words for a number pattern, we sensed how much our tasks, even so early in the process, seemed to build experiences which challenged these children’s linguistic and expressive powers deeply, and yet fascinated them.
Preparing a Performance In a discussion after the second workshop session, Cindy Durante suggested to the research team that there was yet more, much more, to the story than we knew so far. Through the intervening week, her students were returning to the rod tasks and their tables, talking with each other, sharing further discoveries. Hence Cindy proposed a full-class unpacking,’ where her children could show us, in full-scale performance, what we’d missed. We joined her the following Friday morning, this time as audience, while she led the class. Here are two brief student presentations from this third session, transcribed from our videotape of children working at the overhead.” As with our description of the workshop, we’ll stay close to our transcripts and our working notes, and then comment on them. The task is to explain the calculation of the surface area of a stack of five yellow rods, with one cm offsets, to illustrate the computational pattern for finding the surface area with any number of rods. Cindy invites one boy, Brody, to present a discovery which he had made some days before. He has, in Cindy’s words, a theory which, as such, deserves some careful testing. Does Brody’s theory fit the tabulated data? Cindy’s invitation to present, however, has caught Brody completely by surprise. He does not, at first, remember how his calculations went. Cindy responds to Brody’s uncertainty by encouraging him to persevere, and then, to back him up, invites two of Brody’s closest friends to help him. Gradually, quite hesitant but eager nonetheless, Brody guides his small group, after several tense asides, toward a consensus. At this point Brody writes the equation (4x14)+22=78 on the overhead, and then, again, runs out of words.
THEBEGINNINGSOFALGEBRAICTHINKING
45
Speaking for the group, Derek, at Brody’s side, jumps forward to explain. Next to Brody’s equation, a stack of five rods, offset carefully, shows clearly on the screen. Derek:
Cindy: D: c: D:
The first one’s 22. So, then if you put them down, it adds 14 every time, so if you were going to do five rods, then it would be, it would increase by fourteen four times. So you’d do four times fourteen... Four times fourteen. ...and then you’d add... You’d add. 1.. the twenty-two, and then you’d get the answer. You’d add 22. You do these increases first, and then you add what it’s equal to.
At this moment, another student, Amber, at the far side of the room, stands up to challenge both the form of Brody’s equation and the content of Derek’s explanation. The multiplier in the first term, she suggests, should be the number of rods, namely five, not four. To focus her objection, Amber proposes a modification of Brody’s formula, hence of its underlying numerical process. Further, she supports her thinking with a concrete argument, based upon, but modifying, Derek’s. Amber:
They had five rods, and they did four times fourteen, plus 22. They said, that if you take this rod [pointing to the first rod] away, it comes up in the 22. But I, urn, if you add, uh, I didn’t get that. So I just thought, if you add the extra rod, what would that be? And just take that from the 22.
Amber does not question Brody’s numerical Instead, on conceptual grounds, she takes exception been expressed.” Here, she walks across the room, at the overhead. Having outlined her basic strategy, Amber:
result; indeed, she tacitly applies it. to the form in which Brody’s result has and takes her place, with two friends, Amber now presents details.
So, urn? [Her partner, Kimberley, whispers something.] Okay, we had five rods. [She writes “(5 x 14)” on the overhead, just below Brody’s formula.] You had five rods. And then what you added, and you have to subtract from the 22, this extra rod, here, so you subtract. I don’t know what we subtracted, but we got 8. [Here she completes the expression “(5 x 14) + 8”, exactly parallel to Brody’s.]
Amber’s work here looks to us like mature, unvarnished algebra, but with the number 5 as a specific stand-in for a variable.12 In effect, she takes Brody’s expression, 14(/V1) + 22, and rewrites it as 14N + 8, subtracting 14 from 22 to obtain her constant term. She does not reconstruct the final step in detail, which suggests that her attention focuses instead upon her underlying plan, which she explains quite clearly. We see two levels of reframing here, now that algebraic thinking has emerged. At the first level, examining a number pattern through the optic of a particular representation of the rod stack emerged slowly as children discussed the second task, often to reconcile contrasting calculation strategies. But further, as in Amber’s work above, patterns ofcalcuZution also became objects to reframe. At this second level of reframing, algebraic thinking appears clearly, but, as Amber shows, only through a further reference to the problem con-
46
SPEISER AND WALTER
text. Amber’s work, indeed, might not have come to light, until she disagreed with Brody on the grounds that his formula did not show clearly enough the role of five rods in the calculation of the area. The performance here began with a theory, in this case a conjectured model to be tested. In effect, does 14(N - 1) + 22 describe the given data? With our camera running, a group of nervous boys explained the fit: with just a few rods, their model clearly accords with experimental data, and each time a new rod is added, the area must, just as they claim, grow by fourteen. As they go along, although they build their scene from partly remembered fragments, they get their story right. If the play is about conjecturing and testing models, then these boys have played successfully, building suspense as well as interest, against heavy odds. In this context, Amber’s challenge seems especially remarkable. At first glance, however, Amber simply seems to break the flow. Watching the class, indeed, neither we nor Cindy grasped the full import of Amber’s objection. Amber, too, spoke for a group, but one that formed, as if by itself, unbidden. Was their message heard? After the session ended, Amber, as well as others in her group, indeed, would walk away unsatisfied. Only three months later, as the two authors checked their video transcript for the third time, did we realize the fundamental strength of Amber’s argument, and hence the deeper import of the confrontation we had witnessed. Amber’s concern was not with data-concordance with the data, she might argue, is the easy part-but with sense and meaning. Because she has proposed a radical critique, she will now need to break the flow. Against the boys’ stock story, built with their teacher’s support, Amber will need to tell a counterstory.13 Her counterstory sees the mathematics, and the surrounding action in her classroom, in new ways.
Dramatic Discourse To understand dramatic action, we might begin by looking closely at the stories which the actors tell. To understand a story, we might begin by asking, for example, what the story suys. But we also. often, need to step outside that story, in order to examine closely what that story does. A stock story might help one group of tellers to exert power through a dominant discourse, while a counterstory (Delgado, 1989) might help a different group of tellers to resist that power, by constructing an emergent discourse. Dramatic conflict then, can take place between discourses, not just between the characters who represent them. According to what seems to be a dominant discourse among many educators, for example, girls often have trouble learning algebra, beginning at around age twelve. Against this dominant discourse, Amber’s confrontation with three boys takes on special meaning. Let’s explore the possibility that Amber and her coworkers, through a counterstory, might be building an emergent discourse in their classroom. How, specifically, might this view help shape our reading of Amber’s dissent? As spokesperson, Amber represents a group, which includes but may not be limited to the girls who stand beside her, and this group moves into prominence because, collectively, the discourse is theirs, and Amber therefore speaks, at least in part, for them. Further, this group’s critique of Brody’s formula first outlines, along the way, how its members read the dominant discourse, which seems to be about checking formulas to fit “the facts,” and then stopping once some kind of fit has
THE BEGINNINGS OF ALGEBRAIC THINKING
47
been established. Hence we may identify Brody’s “theory” as supporting, at least relative to this group of girls, a stock story. The counterstory, judging by Amber’s careful emphasis, seems to be that formulas, themselves, tell stories about data, and therefore that we should examine closely what kinds of sense these stories make. Furthermore, as she argues by example-inventing algebra to make her point-that appropriately chosen transformations of a given formula can help us to make better sense. For Amber’s group, Brody’s formula, even though it fit the facts, had failed in an important way to tell the story offive rods. To make it do so, Amber proposes to alter Brody’s formula in a coherent way, based firmly on the meaning of that formula (especially the observation that the area must grow by 12 each time we add another rod), and upon the structure of the calculation which it represents (which allows her to revise the constant term). For us, the discourse which Amber leads presents important pieces of the algebra we know, with an assurance which continues to surprise us. This reframing of Amber’s dissent accomplishes, we feel, two central purposes. First, it connects what we observe to widely shared concerns about students at risk, and how we might respond most helpfully to the ideas they advocate. That Amber’s insight, and the discourse which it helped to constitute, did not-at least so far-come to play a strong part in the class discussion, therefore seems especially worth noting. Second, our new reading of Amber makes sense of far more detail on our tapes and transcripts than simply what this one child did and said. Hence we build a more coherent, and perhaps more compelling, picture of what happened in the surrounding classroom, just as algebraic thinking broke the surface. It helps, certainly, to emphasize how quietly an important new idea might be expressed, and hence how easily that new idea, together with the larger discourse it could help initiate, might pass unnoticed.
Emerging
Views of What School Mathematics
Might Become
As working mathematicians, we have emphasized connections, still hidden to many teachers, leading from school mathematics to the mathematics which we practice. We see algebra, in particular, not simply as a body of notations and techniques adhering to time-worn examples, but rather as a way of working, of perjiorming, in the world. In Cindy Durante’s classroom, algebra, we now think, supported an emergent discourse, constituted by a team of girls with Amber in the lead, counterposed against a dominant discourse, built here by several boys with strong teacher support. This dominant discourse takes data as given and then stresses numerical agreement between conjectured computations (denoted “Brody’s theory” here) and data. Amber and her friends, however, tell a very different story. This second story, as we read it, emphasizes why, given the increase of 14 square cm with each rod, the computation, once Amber and her friends reframed it, could explain and justify the duta. The second story wasn’t heard that day, however, outside the group where it beganI How does this algebraic way of working work? Nemirovsky (1994) distinguishes between symbol systems, which embody ways of working in the world through rules, and symbol use, “embedded in personal intentions, in specific histories, and in the qualities of a situation.” Amber and her friends provide a glimpse which helps to clarify the ways in which symbolic thinking might emerge from children’s direct, personal engagement. What mathematics means, specifically, in given situations, depends precisely on its human context. Here Nemirovsky quotes Bakhtin: “The sentence, as a unit of language, like the word,
48
SPEISER AND WALTER
has no author.” An equation, to convey meaning, needs an author, and we seek to understand her, just at the decisive moment when she chooses what and how to write. But here we need to go much further. Symbolic texts, as we have seen, need actors, not just authors. Nemirovsky, evoking Bakhtin’s ghost, who offers him a sword, attempts to give the text back to its author, and so, by implication, to return the author’s words to social contexts. This return can only be accomplished, we may now assert, when personal experience, communicated, shared and worked through socially, has been effectively performed. Like Derek and Amber, we and the participating teachers seek to learn, through personal intent, within specific histories, ideas and processes which, potentially, can shape our lives. Some of these ideas and processes are deeply mathematical. Some have more to do with how to build environments and work in them. Some map potential common ground, perhaps surprising, shared by researchers like us, and by colleagues, starting at age ten, who work in schools. Precisely how we choose to stage this cognitive and social action, as we have sought to show, will matter deeply.
Notes 1. Parallel to the first two sessions in Cindy’s room, we explored the tasks in a second fifthgrade room, led by Gary Mimer. Both Cindy and Gary helped us plan all three Friday sessions, exploring each task personally, and offering many suggestions along the way. In the work below, we sometimes refer to discussions with both Cindy and Gary, which took place in the preparations for each classroom session. Our debt to Gary, who shares longstanding interests in literature and anthropology, is also very large. He and Cindy helped to teach us about Longview as a special human culture, joined by several other teachers, most especially Karen Cristopulos, Robin Griego, Pam Aoyagi, Jeff Nalwalker, and their Principal, Dr. Marilyn Prettyman. 2. A series of informal discussions of this classroom drama with colleagues at the University of Wyoming, in September 1996, also helped to clarify our thinking, as we moved from fairly focused hunches toward interpretation. We are especially grateful to Karen Bartsch (psychology), Susan Frye, Mark Booth and Bob Torry (English), Janice Harris (women’s studies), Audrey Shalinsky (anthropology), John Dorst and Eric Sandeen (American Studies), and Lynn Ipifia (mathematics). The first author, as well, extends his deepest thanks to Elizabeth Hacker, then ten years old, also of Laramie, who emphasized the dramatic power of conflicts, intuitions, dreams and metaphors in children’s thought. 3. We believe that theory-building centers all cognition. (See Baillargeon, 1993; Bartsch & Wellman, 1995; Carey, 1985; Spelke, 199 1.) This view, sometimes called the “theory theory” grows from subtle and often ingenious experiments, often with quite young children. Theories, in particular, may drive perception, right from birth. 4. Turner (1986) pp. 35-36. 5. Turner (1986), p. 37. 6. In 1996. 7. McCallum, Hughes-Hallet (1995) Chapter 19. 8. One of the authors was reminded strikingly of the standard Morse decomposition of a sphere. (See Milnor, 1963, pp. 1-2, 14, for illustrations on a torus.) At a deeper level, which undergraduates, unfortunately, seldom see, the analysis and the topology will tend to merge. (See Fulton, 1995.)
THE BEGINNINGS
OF ALGEBRAIC THINKING
49
9. The term “unpacking” goes back to a workshop which Speiser led at Longview for two weeks in June, 1994. It refers to detailed presentation and analysis, by the whole group, of a significant exploratory finding, in order to make that finding available for connections later. 10. We checked the transcript three times. Only on the third pass did we begin to build consistent versions even of the children’s words. 11. In this, she differs from the direction taken by her teacher, who invited students to test the numerical process underlying Brody’s formula with different numbers of rods, to check agreement with their tables. 12. Diophantus, for example, wrote this way. 13. Delgado (1989). 14. In the meantime, the Longview term had ended, and tape editing equipment arrived, months late, needing repair. To make good use of our time, we began to read the anthropology of social drama, focused on a tape of undergraduate education students as they built a number system (Speiser & Walter, 1996), and began to sharpen our analysis. When we returned to the Longview tapes in August, 1996, our viewpoint now included narrative as well as cognitive construction, and hence we heard the Longview tapes with different ears.
REFERENCES Carey, Susan (1985). Conceptual change in childhood. Cambridge, MA: MIT Press. Bartsch, Karen, & Wellman, Henry M. (1995). Children talk about the mind. New York: Oxford University Press. Baillargeon, Rede (1993). The object concept revisited: New directions in the investigtion of infants’ physical knowledge. In C. E. Granrud (Ed.), Visual perception and cognition in infancy (pp. 265-3 15). Hillsdale, NJ: Erlbaum. Delgado, Richard (1989). Storytelling for oppositionists and others: A plea for narrative. Michigan Law Review, 87,241 l-2441. Davis, Robert B. (1980). Discovery in mathematics: A text for teachers. White Plains, NY: Cuisenaire. Fulton, William (1995). Algebraic topology. A first course. New York: Springer-Verlag. Milnor, John W. (1963) Morse theory. Princeton, NJ: Princeton University Press. McCallum, William G., Hughes-Hallet, Deborah et al. (1995). Multivariable calculus: Preliminary edition. New York: Wiley. Nemirovsky, Ricardo (1994). On ways of symbolizing: The case of Laura and the velocity sign. The Journal of Mathematical Behavior, 13, 389-422. Speiser, Robert, & Walter, Chuck (1996). Five women build a number system. Manuscript in preparation. Spelke, Elizabeth S. (1991). Physical knowledge in infancy. In S. Carey & R. Gelman (Eds.), The epigenesis of mind: Essays on biology and cognition (pp. 133-169). Hillsdale, NJ: Erlbaum. Turner, Victor W. (1974). Dramas, fields, and metaphors: Symbolic action in human society. Ithaca, NY: Cornell University Press. Turner, Victor W. (1982). From ritual to theater: The human seriousness ofplay. New York: PAJ Publications. Turner, Victor W. (1986). Dewey, Dilthey, and drama: An essay in the anthropology of experience. In V. W. Turner & E. M. Bruner (Eds.) The anthropology ofexperience (pp. 33-44). Urbana, IL: University of Illinois Press.