Understanding Middle School Students' Fraction Composition Schemes: A Case Study, Papers of History of Education

Download Understanding Middle School Students' Fraction Composition Schemes: A Case Study and more Papers History of Education in PDF only on Docsity! Hackenberg & Tillema, 1 Constructive Resources for Algebraic Reasoning: Middle School Students’ Construction of Fraction Composition Schemes Amy J. Hackenberg, ahackenb@uga.edu Erik S. Tillema, eriktill@uga.edu Purpose The purpose of our year-long constructivist teaching experiment was to understand how middle school students can construct algebraic reasoning out of their evolving quantitative reasoning. We taught four pairs of sixth graders at a rural middle school in Georgia from October 2003 to May 2004. One area of investigation was the students’ multiplicative reasoning with fractions, including their construction and use of fraction composition schemes (Steffe, 2003). A fraction composition scheme is the way of operating used by a student to produce the part of a fractional whole that is constituted by, say, one-third of two-fifths of the whole. The purpose of this paper is to analyze how two pairs of students constructed and used fraction composition schemes during the experiment. We focus on these schemes because they are crucial in beginning to reason algebraically: We argue that such schemes are a key resource in the construction and solution of basic linear equations of the form ax = b, and also that these schemes are themselves in the province of algebraic reasoning. Theoretical Framework Mathematical Learning and Quantitative Reasoning Following Piaget (1970) and von Glasersfeld (1995), as well as scholars who rely on them (e.g., Confrey, 1995; Steffe, 2002; Thompson, 1994), we view learning as the process by which a person makes modifications and reorganizations in her or his ways of operating in response to perturbations brought about by these current ways of operating. Perturbations are awakenings or disturbances that a person experiences when, for example, her or his ways of operating bring about an unexpected result. Eliminating perturbations in one’s experiential Hackenberg & Tillema, 2 environment opens possibilities for learning, because eliminating perturbations often requires adapting one’s current ways of operating or constructing new ways of operating. Although perturbations are in some sense “controlled” or determined by a person’s current conceptual structures and expectations, they are occasioned (Kieren, 2000) by interaction of various kinds: among people, between people and objects, and of ideas within a person. This view of learning is consistent with an ontogenetic approach to doing research on learning, in which researchers attempt to understand how students’ mathematical ways of operating grow from modifications and reorganizations students make in their previous ways of operating during mathematical interaction with others (cf. Steffe & Thompson, 2000). Researchers who take ontogenetic approaches investigate broad, basic tendencies in students’ spontaneous activity that observers might call mathematical—such as counting and measuring activity (e.g., Steffe, 1994; Steffe & Tzur, 1994). These researchers seek to explain how such activity can be brought forth in interaction with teachers and other students as a basis for engendering further mathematical activity, such as multiplicative reasoning with whole numbers (e.g., Steffe, 1988, 1992), fractional reasoning (e.g., Olive & Steffe, 1999, Steffe, 2002, 2003), and conceptions of ratios and rates (e.g., Thompson, 1995; Thompson, A. G., & Thompson, P. W., 1996; Thompson, P. W., & Thompson, A. G., 1994). In our study, we conceived of students’ algebraic reasoning to be rooted in their ways of reasoning with known and unknown quantities. Our view of quantitative operations (Piaget, 1970) includes actions performed mentally in building and analyzing relationships between two known quantities or between a known and an unknown quantity. Operations are components of schemes, goal-directed ways of operating that include a situation, activity, and result. In our view, quantitative reasoning is the purposeful functioning of schemes in the context of quantities Hackenberg & Tillema, 5 Fraction Composition as Algebraic A fraction composition scheme is involved in solving RMR problems like the Box Problem because students need to form a goal to find one-half of 3/4 of a decameter. That is, since 2/3 of the unknown height is a known quantity (3/4 of a decameter), determining half of the known height will be 1/3 of the unknown height. So finding half of the known height is crucial. To form this goal seems to require an understanding of equivalence (that 2/3 of the unknown quantity is identical to 3/4 of a decameter) and being able to reverse making fractions (that to make 3/3 of a quantity involves finding 1/3 of it and iterating that amount 3 times). Enacting the goal of finding half of the known height transforms the initial relationship between the two quantities into a state that produces the solution. Thus we believe doing so is a fundamental “tool” in the solution of RMR problems, and in the construction of linear equations. Our experience with sixth grade students (as well as with pre-service and in-service teachers) indicates that enacting this goal is a significant achievement. Enacting a fraction composition scheme entails partitioning a fractional quantity recursively (cf. Steffe, 2003). That is, to find 1/2 of 3/4 of a decameter, students may form a goal to find 1/2 of each 1/4 decameter and partition all the fourths into two equal parts—even the “missing” fourth (they may perform these operations solely mentally or also materially). In so doing, they may see that 8 little such pieces make up the entire decameter. Being able to determine that 1/2 of 1/4 of a decameter is 1/8 of a decameter in this way is an example of a unit fraction composition scheme. In this scheme, students effectively use “halving” recursively on the “fourthing” they performed on the 1 decameter (to make the original fourths of a decameter). So the activity of the scheme can be seen as a composition of fractions-as-functions: One-half becomes a “halving function” that operates on 1/4, which in turn was created by a “fourthing Hackenberg & Tillema, 6 function” operating on 1 decameter (cf. Steffe, 2001). Note that this recursive activity is only the first “tier” of activity necessary to accomplish the overarching goal, finding 1/2 of 3/4 of a decameter. That is, to take one-half of three-fourths requires taking one-half of each of the three one-fourths—and recognizing that this result is one half of all three fourths. This activity is clearly distributive. For these reasons, we view the construction of a fraction composition scheme itself as in the realm of algebraic reasoning—as a kernel of function composition that involves distributivity. Methodology and Methods Learning from Students and Conceptual Analysis Ontogenetic approaches to research on mathematical learning are frequently linked to constructive teaching experiment methodology in which teacher-researchers use teaching as a method to engender and explain students’ learning (Steffe & Thompson, 2000). In a constructivist teaching experiment, teacher-researchers are interested in justifying the mathematical constructions that students make, not in justifying their own first-order mathematical knowledge or some a priori notions of mathematics. Teacher-researchers “remain aware that we may not, and probably cannot, account for students’ mathematics using our own mathematical concepts and operations” (p. 268). Instead, each teacher-researcher needs to learn new mathematics to understand students’ mathematics, attempting “to put aside his or her own concepts and operations and not insist that the students learn what he or she knows” (p. 274). To this end, a teacher-researcher acts responsively and intuitively in learning to think like her students—in merging with the students’ experiences to the extent that is possible (Leslie P. Steffe, personal communication, April 25, 2002) and in giving students’ mathematical ways of operating an independent “life” (Cobb & Steffe, 1983). Hackenberg & Tillema, 7 Simultaneously, teacher-researchers engage in conceptual analysis of how students might operate in the context of mathematical interactions. This conceptual analysis is based on teacher- researchers’ mathematical knowledge, previous interactions with these students, and interactions with previous students. Out of the interplay between learning mathematics from students and engaging in conceptual analysis, teacher-researchers design and modify problem situations, formulate and test conjectures, and develop models of students’ mathematical ways of operating. As teacher-researchers refine their models of the students, they can increasingly tailor problem situations to these students’ ways of operating. Teaching practices include presenting students with problem situations, assessing students’ responses as indications of students’ current schemes and operations, and determining new problem situations that might allow students to construct potentially more powerful schemes and operations. On-going and Retrospective Analysis At least one witness-researcher, a critical component of teaching experiment methodology, is present during each teaching episode in a constructivist teaching experiment. The witness-researcher assists in videotaping the episodes with two cameras: One camera captures the interaction between the teacher-researcher and the pair of students, while the other camera focuses on the students’ computer or written work. However, the more important role of a witness-researcher is to provide alternative perspectives during the actual episodes, in on-going analysis that occurs between episodes, and in retrospective analysis that occurs after teaching episodes have ceased. This triangulation of interpretations during analysis is critical in the establishment of viable explanatory models of students’ activity. Central activities of on-going analysis involve making local conjectures about students’ current ways of operating, designing new problems and problem sequences for the next teaching Hackenberg & Tillema, 10 Teaching episodes were thirty minutes long and occurred biweekly during school hours for two to three weeks, followed by a week off. Thus the full data set for the experiment consisted of videotapes from 60 to 70 teaching episodes for each pair of students from October 30, 2003 to May 12, 2004. The two videotapes from each teaching episode were mixed electronically into a single video file where the video of the computer or written work was inset into the video of the interaction between teacher-researcher and students. Most teaching episodes involved the use of computer software—TIMA: Sticks and Javabars. These programs allow students to draw line segments (sticks) or rectangles (bars), respectively, and operate upon these “wholes” in various ways: partition the whole into an equal number of parts, partition partitions, break a whole into its parts, pull a part out of a whole without destroying the whole, copy and join parts, color parts or wholes, and repeat parts so as to join them together in succession. Results and Discussion Sara and Amber At the beginning of the teaching experiment, Sara and Amber had not constructed a unit fraction composition scheme. So in this section we examine Sara and Amber’s activity in situations intended to engender this construction. Through contrasting the two students’ ways of operating in these situations, we highlight how distributive activity is involved in students’ construction of a unit fraction composition scheme, and how in the activity of such a scheme we can see a kernel of function composition. In addition, we examine the close relationship between students’ multiplicative structures and the distributive nature of their mathematical activity. The Cake Problem Amber’s activity. Sara was absent during the teaching episode on February 11th, which was the first time Amber encountered a problem intended to provoke the construction of a unit Hackenberg & Tillema, 11 fraction composition scheme. Amber had already made a “cake” (a stick) and had partitioned it into 15 equal parts using the buttons in the TIMA: Sticks microworld. Then the teacher- researcher posed this problem: Cake Problem, Task 5: At a party, you and fourteen people share a cake fairly. Then you share your piece of cake fairly with a latecomer. How much of the cake does the latecomer get? A witness-researcher asked Amber to pull out one part—one piece of the cake—and to share that piece with him. Amber pulled out one part from the cake (1/15-stick) and partitioned it into two equal parts (see Figure 1). Figure 1, One-fifteenth (1/15-stick) disembedded from the cake and shared with two people. Protocol I: Amber’s solution of the Cake Problem on 2/11/04. 2 W: Can you pull out my share? [A pulls out one-half of the 1/15-stick.] I wonder how much of that cake I have? A: Let’s see that’s… [looks off into space for approximately 25 seconds]. How much of it is yours from like this? [A uses the mouse to point to the 1/15-stick.] W [attempts to redirect A’s statement of the question]: That’s my piece [referring to half of the 1/15-stick], right? I wonder how much of the whole cake I have? A: Okay mine was one fifteenth, so you would have half of one fifteenth. So it would be…[continues to think for 15 seconds, indicating that she believes there may be another fractional name for the part]. I am not sure. T: Could you use his piece to figure it out? A [drags the half of the 1/15-stick next to the whole cake]: Well you could fit two of his pieces in one, so…[subvocally utters fifteen times two is thirty]… T: I think I heard her say it. Do you remember what you said? A [isn’t sure what T is referring to]: Oh, I said fifteen times two. 2 In this protocol, A stands for Amber, S for Sara, T for the teacher-researcher (one of the authors), and W for a witness-researcher. Comments enclosed in brackets describe students’ non-verbal action or interaction from the teacher-researcher’s perspective. Ellipses (…) are used to indicate a sentence or idea that seems to trail off. Four periods (….) are used to indicate omitted dialogue. Hackenberg & Tillema, 12 T: Oh and why would it be fifteen times two? A: Because two of his pieces can fit into one fifteenth and there’s fifteen pieces in the whole cake so… W: So how much would it be? A: It’d be… [stares at the ceiling] thirty. T: Thirty of his would fit in the whole cake? Is that what…so how much of— W: So how much of the whole thing is my piece? A: Okay so thirty of his pieces would fit in the whole cake, so that would be thirty fifteenths. [She wrinkles her forehead and looks at T.] No I don’t think that’s right. T: How many pieces has he got? A: He has one. W: And how many little pieces in the whole cake? A: Thirty. W: You got it solved. It’s thirty little pieces, then what would one little piece be? A: One thirtieth. In order to solve the Cake Problem, we conjecture that Amber mentally partitioned each of the fifteenths into two equal parts. She contributed this activity independently—that is, she engaged in partitioning a partition to serve a non-partitioning goal (determining the size of the smallest part in relation to the unit bar). In conceiving of the situation in this way, Amber’s multiplying schemes were activated and she knew that she needed to multiply two by fifteen in order to find out how many parts were in the whole cake. Steffe (2002, 2003) has referred to such activity as recursive partitioning and has discussed its importance in the construction of a unit fraction composition scheme. He has also commented that in engaging in this activity, he sees the echoes of the composition of two functional processes (2001). We concur, but we would like to emphasize the distributive, not just the recursive, nature of Amber’s activity. That is, Amber effectively enacted “halving” on the “fifteenthing” she had first performed on the cake in an effort to determine the size of the smallest part. In this sense she distributed halving across the fifteen parts of the cake. We see a kernel of a distributive operation in her activity, although we don’t claim that she was aware of a distributive pattern in her activity. Hackenberg & Tillema, 15 immediately wrote down seventeen times three in a vertical format, computing the result with her standard computational algorithm for whole number multiplication. She said there would be fifty-one pieces in the sub sandwich and that one piece would be “one fifty-oneth.” In contrast to her activity on the Cake Problem, Sara seemed to assimilate the Sub Problem to her multiplying scheme. We infer that by partitioning each of the seventeen parts of the sandwich into three equal parts, Sara knew that she had made seventeen three’s. For her, finding the resulting number of parts was accomplished by multiplying seventeen and three using her computational algorithm for whole number multiplication. Then, after she had figured out the total number of parts, Sara certainly established the sandwich as a unit of fifty-one units, and she used her fraction scheme to determine the size of one of the parts in relation to the whole. However, a clear difference between the Sub Problem and the Cake Problem is that the former indicates that partitioning each part of the initial partition should occur. So we cannot infer that Sara engaged in recursive partitioning as Amber had—in fact, partitioning a partition in service of a non-partitioning goal seemed unavailable to Sara for much of the teaching experiment. In addition, based on Sara’s work in both the Cake and Sub Problems, we cannot infer that Sara “saw” the sandwich as a unit of seventeen units each containing three units. In fact, Sara’s solutions to the Cake and Sub Problems prompted us to conjecture that Sara was coordinating only two levels of units at that time. That is, she could first work with seventeen as a unit of seventeen units. Instead of counting by ones to seventeen, she could think of each “one” as a “three” and keep track of them until she had counted seventeen three’s. Furthermore, this situation was multiplicative for her, and she knew that her computational algorithm for whole number multiplication would allow her to find the result. Once she found the result, fifty-one, she could then conceive of the sub sandwich as a unit that contained fifty-one units, but in doing Hackenberg & Tillema, 16 so we cannot infer that she maintained a conception of the sandwich as a unit of seventeen units, each containing units of three. It was as if she could coordinate two levels of units at two different times, but could not hold the two structures together. So, in operating in situations of fraction composition, determining the number of parts in the whole and identifying the fractional size of the part in relation to the whole appeared to be separate but associated problems for Sara. To solve them Sara seemed to use her multiplying and fraction schemes sequentially, rather than embed her multiplying scheme into her fraction scheme as Amber had done. Because she could not yet make this kind of coordination, she seemed unable to construct a unit fraction composition scheme at this time. Amber’s Progress Taking a non-unit fraction of a unit fraction. During the rest of February and early March, the teacher-researcher investigated whether Amber could extend her unit fraction composition scheme into taking any proper fractional amount of a unit fraction. Samples of the problems the teacher-researcher posed to Amber are the following: Task 7: You are at a party and a cake is cut into nine pieces. Two people show up to the party late and you decide to share your piece of the cake with them. What fraction of the whole cake do the latecomers get together? Task 8: Can you make 2/5 of 1/3 of the cake? How much is that of the whole cake? Amber solved these problems by partitioning the cake into fractional parts, partitioning one fractional part into the required number of “mini-parts,” pulling out the number of these mini-parts requested in the problem, and identifying the fractional size of the result in relation to the whole cake. For instance in Task 7, Amber partitioned the first ninth of the cake into three parts, pulled out two of them, determined that each one was one twenty-seventh of the original Hackenberg & Tillema, 17 cake, and therefore concluded that the latecomers would get two twenty-sevenths of the whole cake. Although Task 7 and Task 8 do not require significantly different ways of operating, we posed Task 8 to engender Amber’s awareness of making a fraction composition. We believed this goal could be achieved through the explicit use of fraction language. We conjecture that the explicit use of fraction language could contribute to Amber becoming more aware of using fractions like two-fifths as an operation. That is, using her concept of two-fifths on quantities other than a whole meant two-fifths had an operational meaning as opposed to solely the meaning of a resulting quantity. Taking a unit fraction of a non-unit fraction. The teacher-researcher wanted to investigate whether Amber could construct a more general fraction composition scheme by asking her to take a unit fraction of a non-unit fraction (e.g. take 1/5 of 2/3). Such problems are more challenging than Tasks 7 and 8 because they require students to take a unit fractional part of a quantity consisting of multiple fractional parts. Thus such problems require students to distribute their concept of taking a unit fraction, like one-fifth, across more than one fractional part, like two-thirds. Using their unit fraction composition scheme, students also need to keep track of the relationship of the “mini-parts” (one-fifth of one-third) to the whole in order to find the resulting fractional part of the whole. So on March 10th, a month after Amber solved the Cake Problem (Task 5), the teacher-researcher posed this problem: Task 9: Can you make 1/3 of 3/4 of that cake and find out how much of the whole cake that is? Amber solved this problem by partitioning each of the three fourths into three equal parts and taking one part from each of the three fourths. She then named the resulting fraction three twelfths of the whole cake. In solving this problem, she appeared to distribute her concept of Hackenberg & Tillema, 20 distributive operation in her multiplying schemes with whole numbers so as to regularly and impressively reason strategically.4 From such activity we infer that she likely had a rich set of images when operating with whole numbers, but they were probably entirely implicit in her thinking. Her response to the Box Problem indicates that she had not yet developed that level of sophistication—or private imagery—with her fractional calculation (and did not know the algorithm for fraction division, or if she did, she did not recognize the Box Problem as a potential situation of fraction division). Over the approximately five minutes we spent on the problem she appeared to be in a state of perturbation that was consciously conflictive and quite uncomfortable, and she did not find a way of operating to eliminate the perturbation. Deborah had entered the teaching experiment with sophisticated fraction schemes and operations as shown by her ability to solve Task 4, the Money Problem at the start of the experiment. Furthermore, she modified her schemes and operations swiftly during our interactions. Yet in working on the Box Problem, these ways of operating seemed almost entirely blocked or suppressed, and she was unable to initiate activity that satisfied her, let alone that solved the problem. Seeing Bridget achieve some success with the problem via reasoning with a drawing may have only exacerbated Deborah’s relative paralysis in the situation and persistence in not drawing. Bridget’s response. Unlike Deborah, Bridget seemed to find drawing pictures useful, and she independently drew a picture right after the teacher-researcher posed the Box Problem. To represent the Cobras’ height,5 3/4 of a decameter, Bridget drew a rectangle partitioned 4 For example, to determine 78 divided by 13, Deborah reasoned that 7 would be too big because 7 times 13 was 70 + 21. She determined that the answer was 6, because 6 times 13 was 60 + 18 = 78. 5 I use “Cobras’ height” to refer to the height of the Cobras’ tower, and similarly for the Lizards. Hackenberg & Tillema, 21 horizontally into three equal parts. She stated that the other tower was bigger because the Cobras’ height was two-thirds of the Lizards’ height. Then she partitioned the middle of the three equal parts of the Cobras’ height into two equal parts and crossed out the lines that indicated the fourths of a decameter. To represent the Lizards’ height she drew another rectangle consisting of three rectangles, each equal to half of the Cobras’ height (see Figure 2). Figure 2, Bridget’s drawing of the Cobras’ height (3/4 dm), which was two thirds of the Lizards’ height. Upon questioning, she articulated that one of the two equal parts of the Cobras’ height was one- third of the Lizards’ height. We infer that she used her reversible fraction scheme to draw the height of the Lizards’ tower in relation to the height of the Cobras’ tower. The teacher-researcher then challenged her to find the Lizards’ height. She said that it was one decameter “because you’ve got one extra fourth.” But immediately she thought that was wrong, and then she seemed stumped. The teacher-researcher tried to focus her on the three fourths of a decameter and what she had done to it in making the Lizards’ height. Then the teacher-researcher intervened more strongly by asking Bridget about half of the Cobras’ height. Bridget responded that “you can’t divide three fourths into half—well you can, but you’d get one Hackenberg & Tillema, 22 point five over two.” The teacher-researcher probed to see if Bridget could “figure it out as a fraction.” After a few moments, Bridget drew a rectangle to represent a whole decameter and partitioned it into fourths (see Figure 3). Figure 3, Bridget’s determination of half of three fourths of a decameter. She then partitioned each fourth into two equal parts and said “three eighths.” Although Bridget’s response was impressive, simultaneously Deborah was experiencing considerable discomfort in not finding a way to act in the problem. So the teacher-researcher began to curtail the episode, complimenting both girls on a good start to a hard problem. “Wait a minute,” Bridget said excitedly, pointing to the Lizards’ height, “This is—this is nine eighths!” Bridget’s work here is significant because she partitioned 4/4 of a decameter recursively to determine half of the 3/4 of a decameter, obviously with considerable interventions from the teacher-researcher. These interventions assisted her in forming a goal to find half of 3/4 of a decameter, which she had indicated was relevant through how she made the Lizards’ height, but which she had not explicitly articulated. But she independently contributed the drawing of the decameter and the further partitioning of it to accomplish this goal. At the time of this episode, the research team inferred that although Bridget’s reversible fraction scheme was quite solid, she Hackenberg & Tillema, 25 “fully fractional” in that she did not determine the length of the composition as an outgrowth of making it. However, that Bridget made sense of, and agreed with, Deborah’s response of six twentieths indicates that Bridget could make three levels of units in activity, in reviewing the bar she had made. That is, Bridget could take the 3/4-bar as a unit in relation to the unit bar after she made the composition, and she could use her recursive partitioning operation to relate the small parts to the unit bar. So Bridget’s ways of operating with fraction composition seemed to be a “two step” process: make the composition by coordinating whole numbers of parts, and then determine the measure of it (in relation to the unit bar) as a “separate” problem. In this sense, Bridget’s ways of operating with fraction compositions seemed to be constrained by the levels of units she could coordinate prior to activity, and we cannot claim that at this point she had constructed a general fraction composition scheme. Deborah’s distributive activity. Like Bridget, Deborah sometimes coordinated whole numbers of parts in making fraction compositions. In addition, perhaps partly as a result of her affinity for computation over making pictures, Deborah frequently made the composition after she had calculated the result, often by iterating individual parts. Doing so could hide the structure that she imputed to the situations. For example, on March 10th, Deborah worked on this problem: Task 11: Make 11/9 of a yard of ribbon. Your friend wants 4/3 of that piece. Make it and tell how much your friend needs. Deborah divided each of the eleven ninths into three equal parts. Then she pulled out one part and repeated it 44 times to make 44 parts. When the teacher-researcher asked about the size of one of the parts in relation to the unit yard, Bridget responded “one thirty-third.” Deborah insisted that 1/33 was wrong. After five seconds, she stated that it was 1/27, so the new piece of Hackenberg & Tillema, 26 ribbon was 44/27 of a yard. In explanation of how she had made the composition, she said, “take one out of each of those boxes [the eleven ninths] and that equals eleven and that would be one third, so I multiplied that by four ‘cause she needs four.” This explanation is significant for two reasons. First, Deborah indicated that she was (at least sometimes) using a distributive operation to make the unit fraction of the given fraction. To make 1/3 of the entire 11/9 she imagined taking 1/3 from each of the 11/9. So 11 parts was 1/3 of the whole bar. This way of operating is pivotal in the construction of a general fraction composition scheme because it allows recursive partitioning to be embedded in the scheme for making the composition (i.e., one third of each ninth is 1/27, so the entire 11 parts is 11/27.) We conjecture that Deborah’s use of an explicit distributive operation in making 1/3 of 11/9 is a central reason why she emphatically rejected Bridget’s idea that each part was one thirty-third and concluded that each part was one twenty-seventh. Second, Deborah used her iterative fraction scheme to make four thirds by iterating one- third four times, but she did not use the tools of JavaBars to reflect this construction (i.e., she did not repeat one part to make eleven parts and then repeat the eleven parts four times.) In fact, she did not often independently use the tools of JavaBars to demonstrate her “structural” ways of operating with fraction composition problems, perhaps because she found making the bars somewhat tedious—or perhaps because making the bars remained largely an illustration of the end results of her mental activity. Nevertheless, from Deborah’s explanation of her solution of Task 11 we infer that she had made an initial construction of a general fraction composition scheme. Conclusion Hackenberg & Tillema, 27 A central conclusion of our research is that students’ multiplicative structures—the extent to which they coordinate levels of units prior to activity—significantly influence their distributive activity with fractions. As shown in the contrast between Amber’s and Sara’s ways of operating in fraction composition situations, coordinating three levels of units in activity seems necessary for constructing a unit fraction composition scheme. That is, coordinating three levels of units seems necessary for inserting composite units, such as a unit of two, into a unit of units, such as a whole that has been partitioned to show fifteenths. This description summarizes our interpretation of Amber’s mental activity in the Cake Problem (Task 5), and the result was that she had made a unit of fifteen units, each containing two units, which meant that the bar was both a unit of thirty units (the result of her multiplying scheme) and a unit of fifteen units each containing two units. In contrast, Sara could coordinate at most two levels of units, and so the second “view” of the bar that Amber had was not available to Sara. Thus Sara’s activity did not yet include a distributive operation because she could not insert composite units into a unit of units and retain the structure of the number (or bar) as a unit of units of units. This constraint in her ways of operating is corroborated by her activity in the Sub Problem (Task 6). Our central conclusion gains further support from the activity of Deborah and Bridget. Coordinating three levels of units prior to activity seems necessary for constructing a more general fraction composition scheme. In a fully general scheme, students may enact at least two “tiers” of distribution in making a composition and determining the result as an outgrowth of making it. That is, for Deborah to determine 4/3 of 11/9 (Task 11), she first took 1/3 of 11/9 by taking 1/3 of each of the eleven ninths. Taking 1/3 of 1/9 already involves distributive activity through the use of recursive partitioning, as explained in Amber’s activity with unit fraction composition. But Deborah also explicitly used distribution across all eleven ninths by taking 1/3 Hackenberg & Tillema, 30 References Clement, J. (2000). Analysis of clinical interviews: Foundations and model viability. In R. Lesh & A. E. Kelly (Eds.), Handbook of research design in mathematics and science education (pp. 547-589). 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