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Making sense of fraction quotients, one cup at a time
Journal of Mathematical Behavior 38 (2015) 1–8
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The Journal of Mathematical Behavior journal homepage: www.elsevier.com/locate/jmathb
Making sense of fraction quotients, one cup at a time Bob Speiser a,∗ , Chuck Walter b a b
3321 East Chaundra Avenue, Salt Lake City, UT 84124, USA PO Box 364, Levan, UT 84639, USA
a r t i c l e
i n f o
Keywords: Fractions Products Models Representations Elementary teachers
a b s t r a c t On the surface, we discuss a concrete case, in which a group of learners made sense of fraction division, beginning with a specific, concrete problem that demanded fresh insight. To meet its challenge, several education undergraduates, joined later by the authors, built mutable, evolving models to support their thinking. More fundamentally, we discuss those models in some depth, as foundations for a theoretical analysis of emergent sense and meaning. The initial problem begins, in effect, as a test case to investigate, but soon, with further understanding, it emerges as a special case, to support or represent an insight that will hold in general. The kind of knowledge to be built, therefore must shift. Hence, for us, the mathematics we discuss is mathematics in the making, anchored to evolving models, arguments, and explanations. © 2014 Elsevier Inc. All rights reserved.
1. Introduction To begin we treat a concrete case, in which several education undergraduates, for the first time, made sense of fraction division. We gave them a specific, concrete problem that required fresh insight. To meet its challenge, these learners, joined later by the authors, built mutable, evolving models to support their thinking. We discuss these models in some depth, as foundations for analysis of emergent sense and meaning. Our analysis will build directly on two recent investigations. The first (Speiser, Walter, & Sullivan, 2007), from a theoretical perspective, centers on how specific case investigations can underpin emerging generality. The second (Speiser & Walter, 2011) explores whole number products, in part to illustrate how learners’ models, understood simply as things that people build, discuss, and modify, support emerging proofs and explanations. “For us a model is a thing – no more, no less – a tool, designed, built, or imagined to help make sense of something that we seek to understand. Because whatever sense we make is our construction, the result of actions taken and considered over time, it follows that the models we discuss should be viewed as objects of reflection and design, hence as contingent, temporary, mutable, available for reconsideration, reconstruction, or rejection. . . Our analysis will emphasize what models can help people do.” (p. 271) Often we imagine creativity as the work of a special, isolated individual. Sometimes, certainly, it is. But here, instead, we explore how several people, in collaboration, address a challenge that impels them to extend and reshape what they know, can do, and understand (Speiser, Walter, & Maher, 2003). For this purpose we selected a proverbial tough nut to crack: fraction division, where mute submission to official rules might seem especially entrenched.
∗ Corresponding author. Tel.: +1 801 278 0326. E-mail addresses: [email protected] (B. Speiser), [email protected] (C. Walter). http://dx.doi.org/10.1016/j.jmathb.2014.12.001 0732-3123/© 2014 Elsevier Inc. All rights reserved.
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Fig. 1. A model for the operator product (2/3) × (4/5).
The shift from mimicking official practices to more thoughtful personal engagement (Speiser, Walter, & Lewis, 2004; Speiser et al., 2007) entails a change in how we understand ourselves as thinkers and as social actors. Building arguments from scratch can be a key experience, not just for oneself but perhaps especially in group collaboration. In prior work (Speiser, Walter, & Glaze, 2005) we documented how one learner, working with peers, first learned that mathematical ideas can offer helpful ways of thinking, rather than just templates to reproduce, fill in, and manipulate. As insights emerge, the models we discuss will undergo important shifts. Such shifts might highlight insights, drawn from earlier experience, for further exploration and development. Hence, as we have emphasized (Speiser & Walter, 2011), a model in the kind of practice we discuss should be viewed, right from the outset, as a temporary sketch, available to reconsider, redesign, reject, or reinterpret. Starting from a single, concrete problem, the models we discuss reflect more general concerns. The initial problem begins, in effect, simply as a test case (Speiser et al., 2007) to investigate. But soon, with further understanding, it emerges as a special case (loc. cit.), to support or represent an insight that will hold in general. The kind of knowledge to be built, therefore, has shifted. Hence, for us especially, the mathematics we discuss is mathematics in the making (Latour, 1987; Speiser & Walter, 2000), anchored to evolving models. The work presented here took shape in an extended conversation, beginning in a college mathematics class for future elementary teachers. That conversation would continue, once the class had ended, through the writing that we now complete.
1.1. Setting, concepts, guiding questions We begin with student thinking from a mathematics class1 for future elementary teachers. The students, in groups of five or six, addressed key concepts through extended task investigations, where they built and tested explanations for solutions they had found. Further, they studied videos of young learners, from the Rutgers–Kenilworth longitudinal study (Davis, Maher, & Martino, 1992; Maher and Martino, 1992, 1996a, 1996b; Maher, Powell, & Uptegrove, 2010) who built, presented, and debated mathematically acceptable arguments. Our students, like the learners in these videos, sought explanations and supporting tools, in their own style and language, that would convince them (and us) on the spot, but also could potentially inform their future classroom work. With our students, we agreed on a specific, formal view of multiplication (Speiser & Walter, 2011), that of operator products.2 The first factor of an operator product, the operator, tells how many copies of the second factor, the operand, should be combined. For example, in the product 3 × 6, the operator, 3, counts groups of six. Students in a prior class (around 1999) extended this idea to fractions. The work that we consider here will build from that extension. To be precise, consider 2/3 as an operator. To multiply by 2/3, our students would seek 2/3 of the given operand. They would find, for instance, (2/3) × (4/5), based on a model like that shown in Fig. 1. Here one might, as shown, outline the area for 4/5 first, next draw further lines for 2/3, and finally shade 2/3 of the outlined area for 4/5. One explanation, for example, might proceed as follows. We see 3 × 5 = 15 small rectangles in the large rectangle, each worth 1/15.3 Of these, 2 × 4 = 8 have been shaded, so the sought-for result must be 8/15. The reasoning presented here,
1 Mathematics Education 306, Brigham Young University, January 2009, taught by the first author (Speiser & Walter, 2000, 2011; Speiser et al., 2004, 2007). 2 This idea came from Robert B. Davis (1980), from his classic Postman Stories activity for signed integer arithmetic. We discussed this activity further, for the present classroom context, in Speiser and Walter (2011). 3 Indeed, the array in Fig. 1 has three columns and five rows.
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in a test case4 where it might have been discovered, will also work in general. Because it underpins the recipe for fraction products that many of us learned by rote, it illustrates the strength of operator products as a working concept. Two specific questions set the research here in motion. The first was mathematical: What might division look like, especially for learners, in an operator product context? With our students, we understood division as the search for a missing factor, when the product and one factor have been given. In an operator product, the known factor might be either the operator or the operand. Suppose first that we know the operand (written A) and the product (written B). We then seek a missing operator (written X) so that X × A = B. In other words, we ask how many A’s, combined, will equal B. Classically, this kind of problem was called quotitive5 division. Or, instead, suppose we know the operator (now written A) and the product (again written B). This time, we seek a missing operand (denoted X) so that A × X = B. Classically, this was known as partitive6 division. We began in class with a quotitive problem. In a concrete situation, what might this entail? To construct an X where X times A gives B, we might try to picture A and B in such a way that we could combine copies of A experimentally, if possible, to produce B. Thus we need not just a model (Speiser & Walter, 2011), but also the beginnings of an action plan, some sense of how the work might go. To make sense of such action plans, it helps to introduce a further central concept, that of number name. In models like the ones we will consider, symbols like A, B and X will stand for quantities that measure objects. (In practice, for example, they might measure lengths, areas, volumes, weights, velocities, or intervals of time.) Following Davis (1980), we will call such measures number names. Here, to find a missing operator X, we need to count how many copies of an object labeled A will produce an object labeled B. But note: to count copies of the object we have labeled A, we necessarily must rename A so that it counts as one. Hence, as a result, our given problem reads: If we rename the object we have labeled A so that its number name is one, what number name should the object we had labeled B now have? In class, our students built from very concrete cases, where the need for number names emerged directly from the models they considered. This motivates our second guiding question: To seek a missing factor in a concrete setting, what models did our students find productive, and, given their models, how did we respond? Our students sketched rough models right away and then refined them to arrive at presentations where the classical division rule emerged from how they reasoned. Based on their presentations, we authors sought to refine and then extend the reasoning in play. We did so by proposing further models, based directly on the ones our students built. Along the way, new questions came into our conversation. We will treat those questions later, once their motivations have emerged.
1.2. Background Our analysis will build on prior work that emphasizes the development, even by quite young learners, of convincing explanations. We focus, here especially, on logical necessity. As mathematicians with research experience, we see reasoning and proof as central to what makes mathematics mathematical. Hence we treat problem-centered exploration, communication, and debate as necessary practices through which learners, with support as needed (Freudenthal, 1991), re-invent the mathematics they will share. Our work reflects some key ideas contributed by the late Robert B. Davis. His approach to fraction arithmetic, division included, appears first, but briefly, in his introduction to Discovery in Mathematics (second ed., 1980). Here Davis used Cuisenaire rods strategically as raw materials for models, and emphasized the fundamental role of number names. “The main point. . . is to give students abundant experience with the relation between reality and mathematical descriptions of reality. . . About 20 years ago, I was introduced to Cuisenaire rods, for which I am eternally grateful. The first use of rods that I found myself making involved fractions. There is abundant evidence that most students do not ever learn to deal confidently, easily, and correctly with fractions (1980, Introduction, section X).” In his teaching, Davis encouraged learners to justify their findings, based on known properties of rods. His emphasis on grounded student explanation, and his personal example, inform the work presented here. Later work by Davis, and by Maher and her collaborators, discuss in detail how students, in collaboration, developed arguments that qualify as mathematical, albeit quite informal, proofs (Davis et al., 1992; Maher and Martino, 1992, 1996a, 1996b; Maher et al., 2010; Maher & Speiser, 1997). These papers emphasize how helpful student-generated (rather than instructed,) models are for building and refining arguments. That elementary school children, as early as grade 4, could grasp, not simply proofs, but also the idea of proof, directly motivates the work presented here. Further research on representational competence, by diSessa and his group (diSessa, 2000; diSessa & Sherin, 2000; Sherin, 2000) at the middle school level, also with learner-generated models, strongly suggests that one can hardly reason scientifically, even in a very concrete setting, without building iconic or symbolic presentations to develop and support
4 Davis suggested a model like this to the first author, then teaching elementary school, in 1970, but (later conversation, around 1996, at Rutgers) he told both authors that he had not thought of it in terms of operator products. 5 If, for example, A = 3, we want to find how many 3’s add up to B, so we are looking at a quotient in the usual sense. 6 To motivate the name, suppose A is 3. Then we would want to split B into three equal parts.
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one’s reasoning. In all this prior work, the learners start by building models they can structure. Then they reason, based on structures they have built (Speiser & Walter, 2011). 1.3. Overview Our exposition goes as follows. First we pose the motivating problem that we gave our students: a concrete situation that demands quotitive division by 3/4. Our students, without teacher input, agreed on a first model, and with it fully solved the problem. Then, based on that model, we proposed two further models. We designed the second model to highlight our students’ central observation, in effect a shift of number names. Heuristically, our model localizes the initial task. Then, in a final model, in effect, we reassembled the initial situation. There, perhaps surprisingly, we show how this very model can directly underpin a full solution of the corresponding partitive problem. We owe special thanks to our late friend and mentor, Robert B. Davis, and to Carolyn A. Maher, with whom we have collaborated and exchanged ideas for more than 20 years. Ricardo Nemirovsky, Chris Rasmussen, and Andrea diSessa also shared important insights. Our students, finally deserve a different kind of recognition. They began as authors, too, first of the mathematics they produced with so much interest and enthusiasm, but also of themselves, as thinkers and learners, often in ways that have surprised us all. The analysis presented here, in a sense, began in 1971, when Davis challenged the first author to propose a concrete situation in which fraction quotients might become accessible to student exploration. There the challenge rested until both authors undertook the work presented here. 2. Task7 You’re working in the kitchen, pressed for time. You need to transfer 5 cups of flour from a storage jar into a mixing bowl, right now! You look around in desperation for your one-cup measuring cup, and then you find it, smeared with butter, floating in the sink. Beneath the counter, you find a clean 3/4-cup measuring cup. How can you use it to transfer the 5 cups of flour? This task invites students to invent division by 3/4 in a concrete, familiar setting. In that setting there are strong conventions about how to work with realistic quantities of flour. Measuring cups customarily come marked vertically in fractions of a cup. And ingredients, once measured, are set out in containers horizontally, ready for sequential use. Such conventions, in effect, suggest a model (via cups), and a concrete action plan for moving flour (one given measure at a time). Our everyday language, when we speak of quantities, often reflects this vertical/horizontal interplay. Speaking metaphorically, we refer to larger quantities as “higher”. Further, we tend to think of objects to be counted as arranged in horizontal rows, where we “go on” from one specific item to the next. Such metaphors, deeply embedded psychologically, anchor how ˜ we work mentally with many quantities in daily life (Lakoff & Nunez, 2000). Thus, to model operator products, the operators, like the number 5 above, might most naturally be pictured horizontally, while operands, like each cup above, might, in contrast, be structured vertically. Each model we discuss below will make consistent use of these conventions. 3. First example This sequence of two models was proposed by several of our students, preservice elementary teachers, in 2008. In Fig. 2 we see two models. The first begins with five “official” cups, the quantity we need to move. These are shown in the top line, numbered 1 through 5. So far, we are picturing a solution to the problem, but not (so far) the tool we have, a 3/4 cup measure, the “working cup” we need to use to move8 the flour. To operate with working cups, this initial model will be elaborated first and then replaced. In the second line, we see the same five official cups, but now each cup has been divided into quarter-cups. In the third line, those quarters have been grouped, as far as possible, in threes, with symbolic labels “x” and “o”. These labels prepare for a complete reshaping, a second model, shown at the bottom of Fig. 2. There, each labeled group now constitutes a working cup. These cups, with their contents, form the second model. Each working cup is full except the last, which holds two quarter-cups, each marked “x”. Counting, we therefore find 6 and 2/3 working cups. We can move the five official cups of flour with these. Let us now reflect on the arithmetic these models anchor. From the initial five official cups, we find 5 × 4 = 20 quarter cups. Dividing this 20 by 3, we obtain the quotient 6 and the remainder 2. In effect, we have found 5 × (4/3). But note, we did not use 4/3 directly in the process. Instead, entirely in the context of whole number arithmetic, we multiplied by 4 and then divided the result by 3. Nonetheless, to make sense of the remainder as a count of thirds (not fourths) we need to fold back to the concrete problem. There, as in the second model, we need to count working (rather than official) cups.
7 The task dates from January 2003. (In class we called it Kitchen Confidential. No connection to the chef’s memoir, Bourdain, 2000, of that title.) We used it every year thereafter. 8 We do not build models like this physically in kitchens. There, we would need to operate with just one working cup. In the picture, though, our students first imagined five official cups, then seven working ones. In life, the operator works sequentially in time. In the model, though, it plays out horizontally in space.
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Fig. 2. Two models by our students.
Hidden, therefore, in these pictures is a fundamental shift of reference for the number one: This number one began, in Model 1, by counting an official cup. But then, in Model 2, it needs to count a working cup. In the process, 1/4 of an official cup, therefore, needs to be understood as 1/3 of a working one. At the outset, therefore, the official cup had one as number name, while the working cup had number name 3/4. In contrast, to solve the given problem, we need to find how many working cups we would need to measure out. Hence the working cup, not the official one, must count as a one.9 But note: when we rename that working cup, the official cup, by implication, must also be renamed. Its name (implicit here) becomes 4/3. This further shift of names will motivate the next example. 4. Second example We now compare official and working cups directly, as specific objects that can carry different number names. At the top of Fig. 3, is an official cup with one as number name. In it, three-fourths have been shaded. These three fourths form a working cup. Beneath it, the same working cup, the same experiential object, now has one as number name, while the official cup, as a result, has number name 4/3. More concretely, in relation to the last example, this model highlights what the labels “x” and “o” accomplished there: to shift attention from the five official cups we had to break apart, to the newly constituted groups of three component pieces that we needed for the working cups. In our view there’s just one model here, drawn twice: a rectangle, split vertically into four equal pieces, with three shaded. The shift in number names from top to bottom could just as naturally have gone from bottom to top. In this we way we can understand, in concrete terms, the reciprocal relation between 3/4 and 4/3. Or just as naturally, that 3/4 of 4/3, or 4/3 of 3/4, must equal 1. Which piece of a given model, therefore, ought to count as one? That depends, our students emphasized, on what you need to count. First they counted five official cups. Then they found 6 and 2/3 groups of 3, where each such group counted as one. Each time, in other words, there is something new we need to count, we should reassign the use of “one”, and track the resulting further name changes explicitly, supported by models that we have designed to help us. To summarize, the model shown in Fig. 3 has localized the students’ first solution. Once we know how to move just one official cup, then, iterating, we can easily move five. The full solution, in working cups, must therefore be the operator product 5 × (4/3), which readily unpacks to 20/3. Based on Fig. 3, we propose a model for a full solution next.
9 As Davis emphasized repeatedly (personal communication) it is irrelevant, and frequently confusing, to refer to “units” when discussing fractions. Instead, we simply note, as many of our students did, that when we need to count a working cup, it has to count as one.
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Fig. 3. An official cup and a working cup, compared.
Fig. 4. Five copies.
5. Third example Fig. 4 shows five copies of the local model arranged side by side, with number names as in the lower half of Fig. 3. On the one hand, by construction, the full rectangle presents five times 4/3. On the other hand, because the shaded area, viewed as a single object, has number name 5, the large rectangle shows 4/3 of 5. Now let us go back to the kitchen. Writing 5 × (4/3) as 6 and 2/3, we can solve the concrete problem by moving 6 and 2/3 working cups of flour. In other words, we just found how many times 3/4 goes into 5. Our model therefore anchors a proof that, to obtain a correct numerical solution, we can invert 3/4 and then multiply by 5. More generally, recall our earlier discussion of division as a search for missing factors. The problem we just solved was quotitive: we looked for a missing operator, given 3/4 as operand and 5 as product. In a partitive problem, however, we would look for a missing operand instead. In the present case, we would try to find a number (call it A) such that 3/4 of A gives 5. The diagram above, perhaps surprisingly, displays exactly such a number. Indeed, we see the shaded area both as 5 and as 3/4 of the full rectangle, whose number name is 5 × (4/3), by construction. To sum up, not only does the same rule (invert and multiply) solve both the quotitive and the partitive division problems, but the same model (in particular with number names unchanged) can underpin a common explanation. 6. Discussion The work described above exemplifies what we have called a method (Speiser et al., 2004) rather than a rote procedure (like the recipe “invert and multiply”) while it justifies, in several ways, the latter as a working rule. One hallmark of a method is a quite specific kind of mutability, not just of models, but also of the language and procedure details that the models anchor. Here, for each model we considered, its mutability reflects emerging understandings that can hold in general. Remember that the students’ models in the first example helped them solve the given problem just as well as the reshaped, reconsidered models of the following examples. Those first models, as we understand them now, illustrate a theme or insight shared by every model we considered: the need to change the name of an official cup, based on a new name for the working cup. What we have called a method demonstrates its presence through just such a range of valid, socially accepted variations on a common theme or insight. Our examples also clarify important changes in the kind of knowledge that emerging models can support. Specifically, we see a test case (the concrete situation of the first example) reshaped in stages to become a special case of a more general,
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Fig. 5. Same cups, new number names.
emerging understanding (Speiser et al., 2007). This reshaping began locally, in the second example, where one model isolates the core of a more global understanding, to be developed further in the third example. When a test case evolves into a special case, as we saw here, its model can return to its initial status as a rough sketch, potentially for later use, if helpful, to address new situations. In particular, its mutability returns, but now to underpin a newly constituted understanding. We see such mutability as central to what has been called meta-representational competence (diSessa, 2000; diSessa & Sherin, 2000; Sherin, 2000), the ability to design and purposefully modify specific models in the course of an investigation. To illustrate, let us look again at partitive division, where this time we know the operator and the product, and look for a missing operand. Suppose again the operator is 3/4, but this time that the product is 5/7. We need to find an unknown number such that 3/4 of that number gives 5/7. Our rough sketch of the official cup, with working cup shaded inside, now comes directly into play, but with new number names, as shown in Fig. 5. We can now find directly, with the local model shown in Fig. 3, that the missing operand must be 4/3 of 5/7. But reflect: Is this example just a test case for us now, or has it emerged, based on further work that we have done, as a special case of something we sensed tacitly, through work already done, and now can make explicit? Suppose the latter. If so, the key transition, from test case to special case, must be more than just a pattern in a narrative of personal development. Something has shifted, not simply in our understanding, but perhaps also in the way that we embody mathematics through our action. More explicitly, consider how Robert Bringhurst (2011), thinking for the moment as an anthropologist, views the Haida artists who transmit their culture’s constituting narratives through the lively, personal, surprising ways that they convey them. “I would be happy to have Claude Lévi-Strauss’ gift for seeing narrative patterns too—since I suppose, as he did, that stories often tell us more than their tellers ever know. . . I want to know, as he did, how myths think themselves in people, but I also want to know how people think themselves in myths. These are two quite different modes of thinking. Only one of them is, strictly speaking, human, though both are in the broad sense humanistic, and both take place in human hearts and heads.” (p. 15) In a similar way, we might suggest, with Levi-Strauss, that mathematics thinks itself in people, while our work here goes further, as with Bringhurst, to recognize, and indeed emphasize, how people can think themselves in mathematics. The work reported here began when students in our classroom thought themselves into a kitchen, and then followed how that kitchen, in effect, began to think itself through them, and, later, through several of us. Because the thought was mathematical, so was the sense that each of us could make. Because we made that sense as humans, with important human limitations, we began with models that we first sketched roughly and, in an extended conversation, modified. That we ended with rough sketches therefore seems appropriate as well as necessary: through their mutability and roughness we have thought ourselves in mathematics. References Bourdain, A. (2000). Kitchen confidential. New York: Harper Perennial. Bringhurst, R. (2011). A story as sharp as a knife. The classical Haida mythtellers and their world. Vancouver/Toronto/Berkeley: Douglas & McIntyre. Davis, R. B. (1980). Discovery in mathematics: A text for teachers. White Plains, NY: Cuisenaire. Davis, R. B., Maher, C. A., & Martino, A. M. (1992). Using videotapes to study the construction of mathematical knowledge by individual children working in groups. Journal of Science Education and Technology, 1(3), 177–189. diSessa, A. A. (2000). Changing minds: Computers, learning, and literacy. Cambridge, MA: MIT Press. diSessa, A. A., & Sherin, B. L. (2000). Meta-representation: An introduction. Journal of Mathematical Behavior, 19, 385–398. Freudenthal, H. (1991). Revisiting mathematics education: China lectures. New York: Springer. ˜ R. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books. Lakoff, G., & Nunez, Latour, B. (1987). Science in action. Cambridge, MA: Harvard University Press. Maher, C. A., & Martino, A. M. (1992). Teachers building on students’ thinking. Arithmetic Teacher, 39(7), 32–37.
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Maher, C. A., & Martino, A. M. (1996a). Young children invent methods of proof: The gang of four. In P. Nesher, P. Nesher, et al. (Eds.), Theories of mathematical learning (pp. 431–447). Hillsdale, NJ: Erlbaum. Maher, C. A., & Martino, A. M. (1996b). The development of the idea of mathematical proof: A 5-year case study. Journal for Research in Mathematics Education, 27(2), 194–214. Maher, C. A., & Speiser, B. (1997). How far can you go with block towers: Stephanie’s intellectual development. Journal of Mathematical Behavior, 16, 125–132. Maher, C. A., Powell, A. B., & Uptegrove, E. B. (2010). Combinatorics and reasoning: Representing, justifying and building isomorphisms. Dordrecht/Heidelberg/ London/New York: Springer. Sherin, B. L. (2000). How students invent representations of motion: A genetic account. Journal of Mathematical Behavior, 19, 399–441. Speiser, B., & Walter, C. (2000). Five women build a number system. Stamford, CT: Ablex. Speiser, B., & Walter, C. (2011). Models for products. Journal of Mathematical Behavior, 30, 271–290. Speiser, B., Walter, C., & Glaze, T. (2005). Getting at the mathematics: Sara’s journal. Educational Studies in Mathematics, 58(2), 189–207. Speiser, B., Walter, C., & Lewis, T. (2004). Talking through a method. For the Learning of Mathematics, 24(3), 40–45. Speiser, B., Walter, C., & Maher, C. A. (2003). Representing motion: An experiment in learning. Journal of Mathematical Behavior, 22, 1–35. Speiser, B., Walter, C., & Sullivan, C. (2007). From test cases to special cases: Four undergraduates unpack a formula for combinations. Journal of Mathematical Behavior, 26, 11–26.
Contents lists available at ScienceDirect
The Journal of Mathematical Behavior journal homepage: www.elsevier.com/locate/jmathb
Making sense of fraction quotients, one cup at a time Bob Speiser a,∗ , Chuck Walter b a b
3321 East Chaundra Avenue, Salt Lake City, UT 84124, USA PO Box 364, Levan, UT 84639, USA
a r t i c l e
i n f o
Keywords: Fractions Products Models Representations Elementary teachers
a b s t r a c t On the surface, we discuss a concrete case, in which a group of learners made sense of fraction division, beginning with a specific, concrete problem that demanded fresh insight. To meet its challenge, several education undergraduates, joined later by the authors, built mutable, evolving models to support their thinking. More fundamentally, we discuss those models in some depth, as foundations for a theoretical analysis of emergent sense and meaning. The initial problem begins, in effect, as a test case to investigate, but soon, with further understanding, it emerges as a special case, to support or represent an insight that will hold in general. The kind of knowledge to be built, therefore must shift. Hence, for us, the mathematics we discuss is mathematics in the making, anchored to evolving models, arguments, and explanations. © 2014 Elsevier Inc. All rights reserved.
1. Introduction To begin we treat a concrete case, in which several education undergraduates, for the first time, made sense of fraction division. We gave them a specific, concrete problem that required fresh insight. To meet its challenge, these learners, joined later by the authors, built mutable, evolving models to support their thinking. We discuss these models in some depth, as foundations for analysis of emergent sense and meaning. Our analysis will build directly on two recent investigations. The first (Speiser, Walter, & Sullivan, 2007), from a theoretical perspective, centers on how specific case investigations can underpin emerging generality. The second (Speiser & Walter, 2011) explores whole number products, in part to illustrate how learners’ models, understood simply as things that people build, discuss, and modify, support emerging proofs and explanations. “For us a model is a thing – no more, no less – a tool, designed, built, or imagined to help make sense of something that we seek to understand. Because whatever sense we make is our construction, the result of actions taken and considered over time, it follows that the models we discuss should be viewed as objects of reflection and design, hence as contingent, temporary, mutable, available for reconsideration, reconstruction, or rejection. . . Our analysis will emphasize what models can help people do.” (p. 271) Often we imagine creativity as the work of a special, isolated individual. Sometimes, certainly, it is. But here, instead, we explore how several people, in collaboration, address a challenge that impels them to extend and reshape what they know, can do, and understand (Speiser, Walter, & Maher, 2003). For this purpose we selected a proverbial tough nut to crack: fraction division, where mute submission to official rules might seem especially entrenched.
∗ Corresponding author. Tel.: +1 801 278 0326. E-mail addresses: [email protected] (B. Speiser), [email protected] (C. Walter). http://dx.doi.org/10.1016/j.jmathb.2014.12.001 0732-3123/© 2014 Elsevier Inc. All rights reserved.
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Fig. 1. A model for the operator product (2/3) × (4/5).
The shift from mimicking official practices to more thoughtful personal engagement (Speiser, Walter, & Lewis, 2004; Speiser et al., 2007) entails a change in how we understand ourselves as thinkers and as social actors. Building arguments from scratch can be a key experience, not just for oneself but perhaps especially in group collaboration. In prior work (Speiser, Walter, & Glaze, 2005) we documented how one learner, working with peers, first learned that mathematical ideas can offer helpful ways of thinking, rather than just templates to reproduce, fill in, and manipulate. As insights emerge, the models we discuss will undergo important shifts. Such shifts might highlight insights, drawn from earlier experience, for further exploration and development. Hence, as we have emphasized (Speiser & Walter, 2011), a model in the kind of practice we discuss should be viewed, right from the outset, as a temporary sketch, available to reconsider, redesign, reject, or reinterpret. Starting from a single, concrete problem, the models we discuss reflect more general concerns. The initial problem begins, in effect, simply as a test case (Speiser et al., 2007) to investigate. But soon, with further understanding, it emerges as a special case (loc. cit.), to support or represent an insight that will hold in general. The kind of knowledge to be built, therefore, has shifted. Hence, for us especially, the mathematics we discuss is mathematics in the making (Latour, 1987; Speiser & Walter, 2000), anchored to evolving models. The work presented here took shape in an extended conversation, beginning in a college mathematics class for future elementary teachers. That conversation would continue, once the class had ended, through the writing that we now complete.
1.1. Setting, concepts, guiding questions We begin with student thinking from a mathematics class1 for future elementary teachers. The students, in groups of five or six, addressed key concepts through extended task investigations, where they built and tested explanations for solutions they had found. Further, they studied videos of young learners, from the Rutgers–Kenilworth longitudinal study (Davis, Maher, & Martino, 1992; Maher and Martino, 1992, 1996a, 1996b; Maher, Powell, & Uptegrove, 2010) who built, presented, and debated mathematically acceptable arguments. Our students, like the learners in these videos, sought explanations and supporting tools, in their own style and language, that would convince them (and us) on the spot, but also could potentially inform their future classroom work. With our students, we agreed on a specific, formal view of multiplication (Speiser & Walter, 2011), that of operator products.2 The first factor of an operator product, the operator, tells how many copies of the second factor, the operand, should be combined. For example, in the product 3 × 6, the operator, 3, counts groups of six. Students in a prior class (around 1999) extended this idea to fractions. The work that we consider here will build from that extension. To be precise, consider 2/3 as an operator. To multiply by 2/3, our students would seek 2/3 of the given operand. They would find, for instance, (2/3) × (4/5), based on a model like that shown in Fig. 1. Here one might, as shown, outline the area for 4/5 first, next draw further lines for 2/3, and finally shade 2/3 of the outlined area for 4/5. One explanation, for example, might proceed as follows. We see 3 × 5 = 15 small rectangles in the large rectangle, each worth 1/15.3 Of these, 2 × 4 = 8 have been shaded, so the sought-for result must be 8/15. The reasoning presented here,
1 Mathematics Education 306, Brigham Young University, January 2009, taught by the first author (Speiser & Walter, 2000, 2011; Speiser et al., 2004, 2007). 2 This idea came from Robert B. Davis (1980), from his classic Postman Stories activity for signed integer arithmetic. We discussed this activity further, for the present classroom context, in Speiser and Walter (2011). 3 Indeed, the array in Fig. 1 has three columns and five rows.
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in a test case4 where it might have been discovered, will also work in general. Because it underpins the recipe for fraction products that many of us learned by rote, it illustrates the strength of operator products as a working concept. Two specific questions set the research here in motion. The first was mathematical: What might division look like, especially for learners, in an operator product context? With our students, we understood division as the search for a missing factor, when the product and one factor have been given. In an operator product, the known factor might be either the operator or the operand. Suppose first that we know the operand (written A) and the product (written B). We then seek a missing operator (written X) so that X × A = B. In other words, we ask how many A’s, combined, will equal B. Classically, this kind of problem was called quotitive5 division. Or, instead, suppose we know the operator (now written A) and the product (again written B). This time, we seek a missing operand (denoted X) so that A × X = B. Classically, this was known as partitive6 division. We began in class with a quotitive problem. In a concrete situation, what might this entail? To construct an X where X times A gives B, we might try to picture A and B in such a way that we could combine copies of A experimentally, if possible, to produce B. Thus we need not just a model (Speiser & Walter, 2011), but also the beginnings of an action plan, some sense of how the work might go. To make sense of such action plans, it helps to introduce a further central concept, that of number name. In models like the ones we will consider, symbols like A, B and X will stand for quantities that measure objects. (In practice, for example, they might measure lengths, areas, volumes, weights, velocities, or intervals of time.) Following Davis (1980), we will call such measures number names. Here, to find a missing operator X, we need to count how many copies of an object labeled A will produce an object labeled B. But note: to count copies of the object we have labeled A, we necessarily must rename A so that it counts as one. Hence, as a result, our given problem reads: If we rename the object we have labeled A so that its number name is one, what number name should the object we had labeled B now have? In class, our students built from very concrete cases, where the need for number names emerged directly from the models they considered. This motivates our second guiding question: To seek a missing factor in a concrete setting, what models did our students find productive, and, given their models, how did we respond? Our students sketched rough models right away and then refined them to arrive at presentations where the classical division rule emerged from how they reasoned. Based on their presentations, we authors sought to refine and then extend the reasoning in play. We did so by proposing further models, based directly on the ones our students built. Along the way, new questions came into our conversation. We will treat those questions later, once their motivations have emerged.
1.2. Background Our analysis will build on prior work that emphasizes the development, even by quite young learners, of convincing explanations. We focus, here especially, on logical necessity. As mathematicians with research experience, we see reasoning and proof as central to what makes mathematics mathematical. Hence we treat problem-centered exploration, communication, and debate as necessary practices through which learners, with support as needed (Freudenthal, 1991), re-invent the mathematics they will share. Our work reflects some key ideas contributed by the late Robert B. Davis. His approach to fraction arithmetic, division included, appears first, but briefly, in his introduction to Discovery in Mathematics (second ed., 1980). Here Davis used Cuisenaire rods strategically as raw materials for models, and emphasized the fundamental role of number names. “The main point. . . is to give students abundant experience with the relation between reality and mathematical descriptions of reality. . . About 20 years ago, I was introduced to Cuisenaire rods, for which I am eternally grateful. The first use of rods that I found myself making involved fractions. There is abundant evidence that most students do not ever learn to deal confidently, easily, and correctly with fractions (1980, Introduction, section X).” In his teaching, Davis encouraged learners to justify their findings, based on known properties of rods. His emphasis on grounded student explanation, and his personal example, inform the work presented here. Later work by Davis, and by Maher and her collaborators, discuss in detail how students, in collaboration, developed arguments that qualify as mathematical, albeit quite informal, proofs (Davis et al., 1992; Maher and Martino, 1992, 1996a, 1996b; Maher et al., 2010; Maher & Speiser, 1997). These papers emphasize how helpful student-generated (rather than instructed,) models are for building and refining arguments. That elementary school children, as early as grade 4, could grasp, not simply proofs, but also the idea of proof, directly motivates the work presented here. Further research on representational competence, by diSessa and his group (diSessa, 2000; diSessa & Sherin, 2000; Sherin, 2000) at the middle school level, also with learner-generated models, strongly suggests that one can hardly reason scientifically, even in a very concrete setting, without building iconic or symbolic presentations to develop and support
4 Davis suggested a model like this to the first author, then teaching elementary school, in 1970, but (later conversation, around 1996, at Rutgers) he told both authors that he had not thought of it in terms of operator products. 5 If, for example, A = 3, we want to find how many 3’s add up to B, so we are looking at a quotient in the usual sense. 6 To motivate the name, suppose A is 3. Then we would want to split B into three equal parts.
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one’s reasoning. In all this prior work, the learners start by building models they can structure. Then they reason, based on structures they have built (Speiser & Walter, 2011). 1.3. Overview Our exposition goes as follows. First we pose the motivating problem that we gave our students: a concrete situation that demands quotitive division by 3/4. Our students, without teacher input, agreed on a first model, and with it fully solved the problem. Then, based on that model, we proposed two further models. We designed the second model to highlight our students’ central observation, in effect a shift of number names. Heuristically, our model localizes the initial task. Then, in a final model, in effect, we reassembled the initial situation. There, perhaps surprisingly, we show how this very model can directly underpin a full solution of the corresponding partitive problem. We owe special thanks to our late friend and mentor, Robert B. Davis, and to Carolyn A. Maher, with whom we have collaborated and exchanged ideas for more than 20 years. Ricardo Nemirovsky, Chris Rasmussen, and Andrea diSessa also shared important insights. Our students, finally deserve a different kind of recognition. They began as authors, too, first of the mathematics they produced with so much interest and enthusiasm, but also of themselves, as thinkers and learners, often in ways that have surprised us all. The analysis presented here, in a sense, began in 1971, when Davis challenged the first author to propose a concrete situation in which fraction quotients might become accessible to student exploration. There the challenge rested until both authors undertook the work presented here. 2. Task7 You’re working in the kitchen, pressed for time. You need to transfer 5 cups of flour from a storage jar into a mixing bowl, right now! You look around in desperation for your one-cup measuring cup, and then you find it, smeared with butter, floating in the sink. Beneath the counter, you find a clean 3/4-cup measuring cup. How can you use it to transfer the 5 cups of flour? This task invites students to invent division by 3/4 in a concrete, familiar setting. In that setting there are strong conventions about how to work with realistic quantities of flour. Measuring cups customarily come marked vertically in fractions of a cup. And ingredients, once measured, are set out in containers horizontally, ready for sequential use. Such conventions, in effect, suggest a model (via cups), and a concrete action plan for moving flour (one given measure at a time). Our everyday language, when we speak of quantities, often reflects this vertical/horizontal interplay. Speaking metaphorically, we refer to larger quantities as “higher”. Further, we tend to think of objects to be counted as arranged in horizontal rows, where we “go on” from one specific item to the next. Such metaphors, deeply embedded psychologically, anchor how ˜ we work mentally with many quantities in daily life (Lakoff & Nunez, 2000). Thus, to model operator products, the operators, like the number 5 above, might most naturally be pictured horizontally, while operands, like each cup above, might, in contrast, be structured vertically. Each model we discuss below will make consistent use of these conventions. 3. First example This sequence of two models was proposed by several of our students, preservice elementary teachers, in 2008. In Fig. 2 we see two models. The first begins with five “official” cups, the quantity we need to move. These are shown in the top line, numbered 1 through 5. So far, we are picturing a solution to the problem, but not (so far) the tool we have, a 3/4 cup measure, the “working cup” we need to use to move8 the flour. To operate with working cups, this initial model will be elaborated first and then replaced. In the second line, we see the same five official cups, but now each cup has been divided into quarter-cups. In the third line, those quarters have been grouped, as far as possible, in threes, with symbolic labels “x” and “o”. These labels prepare for a complete reshaping, a second model, shown at the bottom of Fig. 2. There, each labeled group now constitutes a working cup. These cups, with their contents, form the second model. Each working cup is full except the last, which holds two quarter-cups, each marked “x”. Counting, we therefore find 6 and 2/3 working cups. We can move the five official cups of flour with these. Let us now reflect on the arithmetic these models anchor. From the initial five official cups, we find 5 × 4 = 20 quarter cups. Dividing this 20 by 3, we obtain the quotient 6 and the remainder 2. In effect, we have found 5 × (4/3). But note, we did not use 4/3 directly in the process. Instead, entirely in the context of whole number arithmetic, we multiplied by 4 and then divided the result by 3. Nonetheless, to make sense of the remainder as a count of thirds (not fourths) we need to fold back to the concrete problem. There, as in the second model, we need to count working (rather than official) cups.
7 The task dates from January 2003. (In class we called it Kitchen Confidential. No connection to the chef’s memoir, Bourdain, 2000, of that title.) We used it every year thereafter. 8 We do not build models like this physically in kitchens. There, we would need to operate with just one working cup. In the picture, though, our students first imagined five official cups, then seven working ones. In life, the operator works sequentially in time. In the model, though, it plays out horizontally in space.
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Fig. 2. Two models by our students.
Hidden, therefore, in these pictures is a fundamental shift of reference for the number one: This number one began, in Model 1, by counting an official cup. But then, in Model 2, it needs to count a working cup. In the process, 1/4 of an official cup, therefore, needs to be understood as 1/3 of a working one. At the outset, therefore, the official cup had one as number name, while the working cup had number name 3/4. In contrast, to solve the given problem, we need to find how many working cups we would need to measure out. Hence the working cup, not the official one, must count as a one.9 But note: when we rename that working cup, the official cup, by implication, must also be renamed. Its name (implicit here) becomes 4/3. This further shift of names will motivate the next example. 4. Second example We now compare official and working cups directly, as specific objects that can carry different number names. At the top of Fig. 3, is an official cup with one as number name. In it, three-fourths have been shaded. These three fourths form a working cup. Beneath it, the same working cup, the same experiential object, now has one as number name, while the official cup, as a result, has number name 4/3. More concretely, in relation to the last example, this model highlights what the labels “x” and “o” accomplished there: to shift attention from the five official cups we had to break apart, to the newly constituted groups of three component pieces that we needed for the working cups. In our view there’s just one model here, drawn twice: a rectangle, split vertically into four equal pieces, with three shaded. The shift in number names from top to bottom could just as naturally have gone from bottom to top. In this we way we can understand, in concrete terms, the reciprocal relation between 3/4 and 4/3. Or just as naturally, that 3/4 of 4/3, or 4/3 of 3/4, must equal 1. Which piece of a given model, therefore, ought to count as one? That depends, our students emphasized, on what you need to count. First they counted five official cups. Then they found 6 and 2/3 groups of 3, where each such group counted as one. Each time, in other words, there is something new we need to count, we should reassign the use of “one”, and track the resulting further name changes explicitly, supported by models that we have designed to help us. To summarize, the model shown in Fig. 3 has localized the students’ first solution. Once we know how to move just one official cup, then, iterating, we can easily move five. The full solution, in working cups, must therefore be the operator product 5 × (4/3), which readily unpacks to 20/3. Based on Fig. 3, we propose a model for a full solution next.
9 As Davis emphasized repeatedly (personal communication) it is irrelevant, and frequently confusing, to refer to “units” when discussing fractions. Instead, we simply note, as many of our students did, that when we need to count a working cup, it has to count as one.
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Fig. 3. An official cup and a working cup, compared.
Fig. 4. Five copies.
5. Third example Fig. 4 shows five copies of the local model arranged side by side, with number names as in the lower half of Fig. 3. On the one hand, by construction, the full rectangle presents five times 4/3. On the other hand, because the shaded area, viewed as a single object, has number name 5, the large rectangle shows 4/3 of 5. Now let us go back to the kitchen. Writing 5 × (4/3) as 6 and 2/3, we can solve the concrete problem by moving 6 and 2/3 working cups of flour. In other words, we just found how many times 3/4 goes into 5. Our model therefore anchors a proof that, to obtain a correct numerical solution, we can invert 3/4 and then multiply by 5. More generally, recall our earlier discussion of division as a search for missing factors. The problem we just solved was quotitive: we looked for a missing operator, given 3/4 as operand and 5 as product. In a partitive problem, however, we would look for a missing operand instead. In the present case, we would try to find a number (call it A) such that 3/4 of A gives 5. The diagram above, perhaps surprisingly, displays exactly such a number. Indeed, we see the shaded area both as 5 and as 3/4 of the full rectangle, whose number name is 5 × (4/3), by construction. To sum up, not only does the same rule (invert and multiply) solve both the quotitive and the partitive division problems, but the same model (in particular with number names unchanged) can underpin a common explanation. 6. Discussion The work described above exemplifies what we have called a method (Speiser et al., 2004) rather than a rote procedure (like the recipe “invert and multiply”) while it justifies, in several ways, the latter as a working rule. One hallmark of a method is a quite specific kind of mutability, not just of models, but also of the language and procedure details that the models anchor. Here, for each model we considered, its mutability reflects emerging understandings that can hold in general. Remember that the students’ models in the first example helped them solve the given problem just as well as the reshaped, reconsidered models of the following examples. Those first models, as we understand them now, illustrate a theme or insight shared by every model we considered: the need to change the name of an official cup, based on a new name for the working cup. What we have called a method demonstrates its presence through just such a range of valid, socially accepted variations on a common theme or insight. Our examples also clarify important changes in the kind of knowledge that emerging models can support. Specifically, we see a test case (the concrete situation of the first example) reshaped in stages to become a special case of a more general,
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Fig. 5. Same cups, new number names.
emerging understanding (Speiser et al., 2007). This reshaping began locally, in the second example, where one model isolates the core of a more global understanding, to be developed further in the third example. When a test case evolves into a special case, as we saw here, its model can return to its initial status as a rough sketch, potentially for later use, if helpful, to address new situations. In particular, its mutability returns, but now to underpin a newly constituted understanding. We see such mutability as central to what has been called meta-representational competence (diSessa, 2000; diSessa & Sherin, 2000; Sherin, 2000), the ability to design and purposefully modify specific models in the course of an investigation. To illustrate, let us look again at partitive division, where this time we know the operator and the product, and look for a missing operand. Suppose again the operator is 3/4, but this time that the product is 5/7. We need to find an unknown number such that 3/4 of that number gives 5/7. Our rough sketch of the official cup, with working cup shaded inside, now comes directly into play, but with new number names, as shown in Fig. 5. We can now find directly, with the local model shown in Fig. 3, that the missing operand must be 4/3 of 5/7. But reflect: Is this example just a test case for us now, or has it emerged, based on further work that we have done, as a special case of something we sensed tacitly, through work already done, and now can make explicit? Suppose the latter. If so, the key transition, from test case to special case, must be more than just a pattern in a narrative of personal development. Something has shifted, not simply in our understanding, but perhaps also in the way that we embody mathematics through our action. More explicitly, consider how Robert Bringhurst (2011), thinking for the moment as an anthropologist, views the Haida artists who transmit their culture’s constituting narratives through the lively, personal, surprising ways that they convey them. “I would be happy to have Claude Lévi-Strauss’ gift for seeing narrative patterns too—since I suppose, as he did, that stories often tell us more than their tellers ever know. . . I want to know, as he did, how myths think themselves in people, but I also want to know how people think themselves in myths. These are two quite different modes of thinking. Only one of them is, strictly speaking, human, though both are in the broad sense humanistic, and both take place in human hearts and heads.” (p. 15) In a similar way, we might suggest, with Levi-Strauss, that mathematics thinks itself in people, while our work here goes further, as with Bringhurst, to recognize, and indeed emphasize, how people can think themselves in mathematics. The work reported here began when students in our classroom thought themselves into a kitchen, and then followed how that kitchen, in effect, began to think itself through them, and, later, through several of us. Because the thought was mathematical, so was the sense that each of us could make. Because we made that sense as humans, with important human limitations, we began with models that we first sketched roughly and, in an extended conversation, modified. That we ended with rough sketches therefore seems appropriate as well as necessary: through their mutability and roughness we have thought ourselves in mathematics. References Bourdain, A. (2000). Kitchen confidential. New York: Harper Perennial. Bringhurst, R. (2011). A story as sharp as a knife. The classical Haida mythtellers and their world. Vancouver/Toronto/Berkeley: Douglas & McIntyre. Davis, R. B. (1980). Discovery in mathematics: A text for teachers. White Plains, NY: Cuisenaire. Davis, R. B., Maher, C. A., & Martino, A. M. (1992). Using videotapes to study the construction of mathematical knowledge by individual children working in groups. Journal of Science Education and Technology, 1(3), 177–189. diSessa, A. A. (2000). Changing minds: Computers, learning, and literacy. Cambridge, MA: MIT Press. diSessa, A. A., & Sherin, B. L. (2000). Meta-representation: An introduction. Journal of Mathematical Behavior, 19, 385–398. Freudenthal, H. (1991). Revisiting mathematics education: China lectures. New York: Springer. ˜ R. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books. Lakoff, G., & Nunez, Latour, B. (1987). Science in action. Cambridge, MA: Harvard University Press. Maher, C. A., & Martino, A. M. (1992). Teachers building on students’ thinking. Arithmetic Teacher, 39(7), 32–37.
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Maher, C. A., & Martino, A. M. (1996a). Young children invent methods of proof: The gang of four. In P. Nesher, P. Nesher, et al. (Eds.), Theories of mathematical learning (pp. 431–447). Hillsdale, NJ: Erlbaum. Maher, C. A., & Martino, A. M. (1996b). The development of the idea of mathematical proof: A 5-year case study. Journal for Research in Mathematics Education, 27(2), 194–214. Maher, C. A., & Speiser, B. (1997). How far can you go with block towers: Stephanie’s intellectual development. Journal of Mathematical Behavior, 16, 125–132. Maher, C. A., Powell, A. B., & Uptegrove, E. B. (2010). Combinatorics and reasoning: Representing, justifying and building isomorphisms. Dordrecht/Heidelberg/ London/New York: Springer. Sherin, B. L. (2000). How students invent representations of motion: A genetic account. Journal of Mathematical Behavior, 19, 399–441. Speiser, B., & Walter, C. (2000). Five women build a number system. Stamford, CT: Ablex. Speiser, B., & Walter, C. (2011). Models for products. Journal of Mathematical Behavior, 30, 271–290. Speiser, B., Walter, C., & Glaze, T. (2005). Getting at the mathematics: Sara’s journal. Educational Studies in Mathematics, 58(2), 189–207. Speiser, B., Walter, C., & Lewis, T. (2004). Talking through a method. For the Learning of Mathematics, 24(3), 40–45. Speiser, B., Walter, C., & Maher, C. A. (2003). Representing motion: An experiment in learning. Journal of Mathematical Behavior, 22, 1–35. Speiser, B., Walter, C., & Sullivan, C. (2007). From test cases to special cases: Four undergraduates unpack a formula for combinations. Journal of Mathematical Behavior, 26, 11–26.