Chapter 4 Towers: Schemes, Strategies, and Arguments

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1 Chapter 4 Towers: Schemes, Strategies, and Arguments Carolyn A. Maher, Manjit K. Sran, and Dina Yankelewitz Date and Grade: ; Grades 3 and 4 Tasks: Towers Participants: Dana, Jeff, Michelle I., Milin, and Stephanie Researchers: Carolyn A. Maher and Amy M. Martino 4.1 Introduction In the previous chapter, we examined the representations, strategies, and problem-solving schemes used by four second- and third-grade students to build their solution to the shirts and jean problem (which was to determine how many outfits could be formed from three different shirts and two different pairs of jeans and to provide a convincing argument of the solution). In their effort to make sense of the components of the problem and to monitor their work, the students developed various notations to represent the data and illustrated the use of certain strategies. In this chapter, we examine how those students and others in the longitudinal study build on those representations and strategies in their work on some towers problems. (A towers problem involves determining how many towers can be built of a given height from a specified number of colors of Unifix cubes, small plastic cubes that can be stacked together. Because Unifix cubes have a vertical orientation they have a top and a bottom so do towers. An n-tall tower is one that was built from n Unifix cubes. Appendix A provides an analysis of solutions to the towers problems.) In this chapter, we examine the representations and strategies such as looking for patterns, guess and check, and controlling for variables that were used by students as they worked on the towers task. We trace students use of heuristics and ways of reasoning that were exhibited in their earlier problem solving with shirts and jeans. Finally, we trace the growth in students mathematical reasoning as their arguments and solutions took on proof-like forms. C.A. Maher (B) Graduate School of Education, Rutgers University, New Brunswick, NJ, USA carolyn.maher@gse.rutgers.edu C.A. Maher et al. (eds.), Combinatorics and Reasoning, Mathematics Education Library 47, DOI / _4, Springer Science+Business Media B.V

2 28 C.A. Maher et al. As students were introduced to new problems and worked to make sense of the problem tasks, we observed growth in their knowledge as evidenced by the models they built, the identification of new and more elaborate patterns, and the structure of the arguments they provided in support of their solutions (Maher, 2002). Older ideas were elaborated and expanded upon. Students active engagement in the problem solving gave opportunity to build new ideas and methods of argumentation. As they attempted to resolve issues that could not be solved with their existing schemes, new schemes were built to accommodate the conditions of the problems. The structure of the towers problem served as an assimilation paradigm (Davis, 1984) for students later work with problems of similar structure, providing the students with a conceptualization that we see used in later years to tackle more complex combinatorial problems. 4.2 Stephanie We discuss here Stephanie s emerging strategies as she worked on the towers problem in the third and fourth grades Stephanie Grade 3, Class Session The third-grade students in the study were asked to find all possible combinations of four-tall towers that can be made when selecting from two colors (in this case, red and blue). The strategies of a number of these students are documented and discussed in Martino (1992). We present here a discussion of Stephanie s strategies as she worked on the task in the third grade (see Fig. 4.1). Fig. 4.1 Dana (left)and Stephanie (right), grade 3 tower exploration

3 4 Towers: Schemes, Strategies, and Arguments 29 Fig. 4.2 Stephanie s four-tall opposites BLUE Stephanie and Dana began by working independently. Stephanie built ten towers. She began by making a four-tall tower and then its opposite, that is, a new tower of the same height with the second color in the corresponding position. Her first five towers included two sets of opposites, such as the tower with four blue cubes and its opposite, the tower with four red cubes (see Fig. 4.2). Dana also initially built ten towers, including two pairs of opposites. Then the two girls decided to combine efforts, and Stephanie took each of Dana s towers in turn and checked it against her own to see if it was a duplicate. DANA: Everything we make, we have to check. Everything we make... Let s make a deal. Everything we make, we have to check. All right. I ll always make it and you ll always check it. Okay, you make it and I ll check it. When a duplicate was found, it was dismantled and returned to the pile of cubes. After this process, Dana and Stephanie now had 14 tower combinations. Stephanie suggested that Dana build new towers while she checked each new tower against the existing ones to ensure that it was not a duplicate. They finally eliminated all duplicates, and after attempting to find more combinations but not succeeding, they concluded that there were only 16 combinations, since they had checked many times and could not find new towers. This activity was marked by a number of emerging strategies. First, Stephanie and Dana used trial and error to find as many towers as they could. In addition, both thought of finding a tower and its opposite in an attempt to generate as many towers as possible, but neither used this strategy extensively or consistently. Further, the two decided to compare results and eliminate duplicates, and ultimately used this strategy of elimination to find the remaining tower combinations. Stephanie and Dana s attempt to prevent duplication of combinations as they worked on the towers task is reminiscent of their strategy for solving the shirts and jeans task in the second and third grades (see Chapter 3 for an in-depth discussion). As they worked on the solution to that task, they used lines to ensure that they counted each combination of clothing once and only once. They explained to the researcher:

4 30 C.A. Maher et al. What are these lines that you drew? You drew lines between the shirts and the pants. So that we could make sure; so instead of we didn t do that again and say, Oh, that would be seven, eight, nine, 10. We just drew lines so that we can count our lines and say, Oh we can t do that again, we can t do that again. As they worked on the towers task, Stephanie and Dana again were careful to check each combination against the others to ensure that there were no duplicates. How could you be sure that you haven t made any of them twice or that one of you got them all? Is there a way that you could be sure? Well, there is a way. You could take one, like say we could take this one, this red with the blue on the bottom and we could go, we could compare it to every one. And the ones that match that don t match, put back; and the ones that do match, eliminate. Stephanie and Dana were then asked to predict how many three-tall towers they could build. Stephanie first predicted that there would be the same number 16; and other groups predicted that there would be more three-tall towers than fourtall towers. Upon experimentation, they found that removing one cube from each of their four-tall towers resulted in duplicates, or pairs, and they concluded that there were only eight combinations of three-tall towers (see Fig. 4.3). During an interview the next day, Stephanie explained why there were fewer three-tall towers than four-tall towers. What do you think you learned from what you did? Well, we learned that... you might think there d be more because there are less blocks so there s more combinations you can make. There s less because once you take one block off, say you have red, red, red, red, and you have red, red, red, blue. Once you take red, one red away and one blue away, they re the same. Oh...So then you don t have more. You have- Have less. Fig. 4.3 From four-tall to three-tall BLUE

5 4 Towers: Schemes, Strategies, and Arguments 31 Stephanie and Dana began to make conjectures and inferences based on their previous knowledge; both went on to suggest that there would be more five-tall towers than four-tall towers Stephanie: Grade 4, Class Session In the fourth grade, on February 6, 1992, the students were asked to find all possible five-tall towers, selecting from two colors (in this case, yellow and red). Stephanie and Dana began to make towers along with their opposites (see Fig. 4.4); and they checked their work as they progressed to prevent duplication. DANA: DANA: And then I got another idea. Well, tell me it so I can do the opposite. I m going to do the red this, that- Show me. Oh, okay, and I ll do the red and I ll do it with the red at the top. At one point, they realized that an individual tower could be turned upside down to create a new tower. Dana called this new tower cousin (see Fig. 4.5). They used this strategy to find more possible arrangements. After forming as many towers as they could using this strategy of trial and error, they arrived at 32 different towers, arranged in pairs with a tower and its opposite and a tower and its cousin. Dana also considered different ways of arranging specified sets of towers that she referred to as families. An example of Dana s family is the elevator pattern consisting of exactly one red cube (see Fig. 4.6). In her discussion with the researcher, Dana justified that there could only be five towers in this family because it only goes up to five blocks. Her reasoning indicates an argument by contradiction of the Fig. 4.4 Stephanie and Dana s group work

6 32 C.A. Maher et al. Fig. 4.5 Dana s tower and cousin YELLOW Fig. 4.6 Dana s family of one red cube and four yellow cubes YELLOW given condition that the towers should be five-tall. If another red were added, the result would be a six-tall tower. DANA: DANA: Are there any other members of this family? No. Why not? Because it only goes up to five blocks. Stephanie and Dana located other families of towers with exactly two red cubes (see Fig. 4.7). Stephanie explained to the researcher: With two [red cubes] together, you can make four. With one [yellow cube] in between, you can make three. With two [yellow cubes] in between, you can make two. With three [yellow cubes] in between, you can make one. But you can t make four in between or five in between or...anything else because you can only use five blocks.

7 4 Towers: Schemes, Strategies, and Arguments 33 Fig. 4.7 Families of five-tall towers with exactly two red cubes YELLOW By the close of this whole class activity, Stephanie and Dana had begun to explore an exhaustive method of finding the combinations of towers that were five cubes tall. This marked Stephanie s first use of a partial argument by cases as she worked on the towers task Stephanie: Grade 4, Interviews Stephanie s work on problems involving towers continued throughout the fourth and fifth grades (see Maher & Martino, 1996a, 1996b, 1997) and again in grade 8 (see Chapter 7). Stephanie s growth in understanding of the idea of a mathematical proof is further documented by Martino and Maher (1999) and Maher and Speiser (1997b). Data from these episodes are presented here with attention to the emergent strategies that Stephanie used while working on the tower tasks. In an interview following the class session described above, Stephanie, using red and blue cubes, extended her family organizations of opposites, cousins, and elevators, to include a new organization, the staircase pattern (see Fig. 4.8). She discovered that introducing additional patterns sometimes resulted in duplicate towers that needed to be eliminated by checking. She said, Yeah, we kept we kept finding different patterns, but we didn t check it with the other patterns. Fig. 4.8 Stephanie s use of different patterns resulting in duplicates BLUE

8 34 C.A. Maher et al. The interviewer asked Stephanie if there was a way she could be sure of how many towers of a specific type could be made. I guess...a very lucky guess. Is there anything else possible for towers with exactly one blue? No. Why are you convinced? Because if there are towers of five, you can only build that many [with one blue cube]. You can t really be convinced for everything because there s no absolute way...you can t go and say I m right. [referring to the set of towers with one blue cube that Stephanie had shown] Well, this is an absolute way. Yeah, this is one of the absolute ways. This absolute way is when you looked at only one blue and I wonder if you could find absolute ways for looking at maybe two blues, three blues, or four blues. You could. Yeah, it is possible to have a certain number and get it right. With this exchange, Stephanie demonstrated that the elevator pattern provided a convincing argument for justifying the number of towers with exactly one (or four) of a color. She also seemed to consider that other organizations, such as exactly two of a color, could be convincing. She began to consider families of towers as belonging to cases that could be justified individually to create the mutually exclusive and exhaustive set of cases for building an argument for finding all five-tall towers. In the latter part of the session, Stephanie used letters O and B to represent two colors. She made a grid with rows and columns to represent different six-tall towers. Notice, in Figs. 4.9 and 4.10, that Stephanie kept the entries in two rows constant, the top two rows in Fig. 4.9 and the bottom two rows in Fig Notice, also, in both figures Stephanie applied her elevator pattern while holding both rows constant. Fig. 4.9 Towers with top two rows constant

9 4 Towers: Schemes, Strategies, and Arguments 35 Fig Towers with bottom two rows constant BLACK WHITE Fig Stephanie s global organization In a subsequent interview, Stephanie shared with her classmates the strategy of controlling for variables, that is, keeping the color of a cube in a particular position constant. This method of controlling for variables was useful to her in keeping track of larger number of towers (Maher & Martino, 1996a). During an individual interview on March 6, 1992, Stephanie presented a complete argument by cases. She was able to produce a global organization for four-tall towers. In her justification, she focused on number of white cubes yielding five categories of towers: towers with no white cubes, towers with exactly one white cube, towers with exactly two white cubes, towers with exactly three white cubes, and towers with exactly four white cubes (see Fig. 4.11). 4.3 Milin Milin: Grade 4, Class Session During the February 6 class session, Milin worked with Michael on the five-tall towers task. Milin s work has been referred to in earlier publications (Alston & Maher, 2003; Maher & Martino, 1996a) and was analyzed in greater detail by Sran (2010). Together with Michael, Milin began by using the strategy of building a tower using trial and error and then making an opposite for each tower. Michael and Milin

10 36 C.A. Maher et al. Fig Milin s cousin and opposite ways of making pairs YELLOW Fig Milin s cases of red cubes separated by one, two, and three yellow cubes YELLOW Fig Class discussion and sharing of solutions always paired their towers. Milin sometimes used the opposite strategy to make a pair and other times he utilized the cousin pairing, by inverting a tower. At the conclusion of the group work, the boys found all 32 towers. The strategies used by the two students on February 6, 1992, included: trial and error, building an opposite tower to complete a pair by switching the color of each cube, building an opposite tower to complete a pair by inverting the original tower, and monitoring work by checking for duplicates by comparing to previous towers (see Fig. 4.12). Milin noted that there were three possible combinations of towers in which the red cubes were separated by one yellow cube, two in which the red cubes were separated by two yellow cubes, and one in which the red cubes were separated by three yellow cubes (see Fig. 4.13). During the sharing session (see Fig. 4.14), the children were attentive as they listened to the findings and strategies used by their classmates.

11 4 Towers: Schemes, Strategies, and Arguments Milin: Grade 4, Interviews In an interview on February 7, the day after the class session, Milin made sets of towers using the elevator method for moving a cube of one color to each floor of the tower and by moving two cubes of one color the same way. He then found the remaining combinations by trial and error, and by grouping towers together with their opposites. Although Milin believed that he had found all combinations, he was only able to provide a convincing argument for his elevator patterns and his solid towers (see Fig. 4.15). Later during this interview, Milin began to consider simpler cases, and he said that there were four towers that could be built that were two cubes tall, and two that could be built that were one cube tall. Milin continued exploring simpler cases after this interview and brought the cubes home to further explore his idea. During the second interview 2 weeks later (on February 21), Milin reported that there were 16 four-tall towers. Later on in the interview, Milin showed towers that were one-, two-, and three-cubes tall, and he recorded the number of combinations that were possible for each (see Fig. 4.16). Fig Milin s partial organization by cases and opposites YELLOW Fig Milin s one-, two-, and three-cube tall towers

12 38 C.A. Maher et al. Fig Milin s inductive reasoning with families BLACK LIGHT BLUE Then, in a third interview (on March 6), he showed that larger towers could be built from smaller ones. For example, one can build four two-tall towers from the two one-tall towers (a blue cube or a black cube) by placing either a blue or a black cube on the blue cube and then placing either a blue or a black cube on the black cube. Milin showed that groups of larger towers could be included in the family of the smaller tower from which it was built (see Fig. 4.17). Later during the March 6 interview, Milin suggested that his rule for generating taller towers from shorter towers breaks down after five-tall towers. Toward the end of this interview, he retracted this claim, and he suggested that there were 64 possible combinations of six-tall towers. When asked if his pattern would hold for towers taller than five, he said it should, indicating, We followed the pattern till five. Why can t it follow the pattern to six? Small Group Interview: March 10, 1992 Grade 4 Three weeks later, in a small group interview, arguments were presented by a group of four children for accounting for all possible towers, three-tall, selecting from two colors. This group sharing is referred to as The Gang of Four (see Fig. 4.18). It was conducted so that the children could share their strategies and arguments for Fig Milin, Michelle, Jeff, and Stephanie (left to right)

13 4 Towers: Schemes, Strategies, and Arguments 39 building towers of a variety of heights in earlier investigations. A simpler version of the problem was chosen deliberately for this session, as the evaluation was intended to identify the forms of reasoning and methods of justification that the children used to convince themselves and one another of the validity of their solutions (Maher & Martino, 1996b). The session began with the researcher asking the students how many sixtall towers could be built. Milin answered, probably 64. He was asked to explain why, and he described his inductive rule: multiply the previous answer by two. MILIN: Well, because there was a pattern. What s that? MILIN: You just times them by two. MILIN: Times what by two? The towers by two, because one is two, and then we figured out two is two, and then, I mean four, and then- Milin used inductive reasoning to justify his solution, extending the problem beyond the three-tall case given to the group. He said that there were two one-tall towers, four two-tall towers, and eight three-tall towers. He was asked to re-explain how he progressed from four to eight towers. In this clip, he noted that a cube of each color could be added to the top of the shorter tower to build the taller tower. Why eight? That s what Jeffrey asked about. MILIN: I know. Go ahead. Let Milin persuade Jeff. MILIN: If you do that, you just have to add for each one of those you have to add- Each one of what? These four? MILIN: Yeah. You have to add one more color for each one. Which way are you adding it? Where are you putting that one more color, Milin? MILIN: No, two more colors for each one. See- So this one with red on the bottom and blue on the top. MILIN: You could put another blue or another red. Later, Milin explained the logic behind the leap from two-tall towers to three-tall towers using inductive reasoning. He was able to demonstrate his doubling rule with each individual tower. In his own words, he noted that there were two possible cubes to be added to each three-tall tower to make a four-tall tower: This was for three, so you could add two for each one of the three. Milin explained this doubling rule by drawing two three-tall towers from a two-tall tower by adding a different color cube on top. He took his first two-tall tower with a blue cube on the bottom floor and a red cube on the top floor and generated two three-tall towers by first adding a red cube on the third floor and then adding a blue cube on the third floor (see Fig. 4.19). It is interesting to notice how Milin chose to draw the towers rather than use actual cubes as he had during his individual interview.

14 40 C.A. Maher et al. Fig Milin s representation showing his doubling rule Fig Stephanie s representation of an argument by cases In this session, Stephanie presented an argument by cases to account for building all possible towers, three-tall. She represented the towers in a grid using letters B and R, for blue and red cubes as indicated in Fig Details of the session are described in Maher and Martino (1996b). She showed that there was only one way to form a tower without any blues. Then she showed that there were three combinations of two red cubes and one blue cube using the staircase pattern. She then used an argument by contradiction to show that this pattern could be used to show that there were only three possible combinations. Stephanie said, Well, there s no, there s no more of these because if you had to go down another one you d have to have another block on the bottom. But then you have with three blues well, not with three blues. I ll go like this. Stephanie used the staircase pattern to argue by contradiction that there could not be a fourth arrangement of two red cubes and one blue cube. What is of interest here is that Stephanie felt the need to prove that her argument by cases was complete and convincing, even though no one had challenged her answer. Stephanie continued her argument by cases by describing all the possible combinations of two blue cubes and one red cube. Stephanie s organization was interesting in that she separated the case of two blue cubes into sub cases: two adjacent blue cubes and two non-adjacent blue cubes. When her classmates pointed out that these two cases could be grouped into one broader group, Stephanie insisted on continuing her explanation as she had originally presented it. The entire conversation follows, starting with Stephanie s description of the all red tower. All right, first you have without any blues, which is red, red, red. Okay, no blues. Then you have with one blue Okay.

15 4 Towers: Schemes, Strategies, and Arguments 41 Blue, red, red; or red, blue, red; or red, red, blue. All right. You could put blue, blue, red; you could put red, blue, and blue. MILIN: You could put blue, red, and blue. You could put... Yeah, but that s not what I am doing. I m doing it so that they re stuck together. JEFF: MILIN: JEFF: MILIN: MICHELLE: Okay. There should be one there could be one with one red and then you could break it up and there s one with two reds and there s one with three reds and then... Ah, but see you did the same thing, but there s the blue. See, there s all reds and there s three reds, two reds. There should be one with one red. And then you change it to blue. Well, that s not how I do it. Let s hear how Steph we ll hear that other way; that s interesting. Okay, now, so what you ve done so far is One blue, two blue. Okay, no blues One blue, two blue. One blue, and two blues, but Milin just said you don t have all two blues, and you said that why is that? All right, so show me another two blues. With them stuck together, because that s what I am doing. In that case, no. Okay, so now what are you doing, Stephanie? What if you just had two blues and they weren t stuck together, you could But that s what I m doing. I m doing the blues stuck together. Okay. Then we have three blues, which you can only make one of. Then you want two blues stuck apart not stuck apart; took apart. Separated? Yeah, separated. And you can go blue, red, blue right here. Although Stephanie insisted on explaining her method of using two categories of towers with two blue cubes during this session, she later indicated (in a written assessment) that she understood the arguments of Milin and Michelle. At that time, Stephanie organized her cases as her classmates had suggested, producing a more elegant proof by cases (Maher & Martino, 1996a). Toward the end of the session, the students used Milin s argument by induction as a stepping-stone to generalize the solution to the towers problem. Their progression of this understanding is documented by Maher and Martino (1997, 2000) and Maher (1998).

16 42 C.A. Maher et al. Stephanie, during interviews preceding The Gang of Four, noticed a relationship between the height of towers and the total number produced and conjectured a doubling rule. During the Gang of Four session, she made reference to her doubling pattern and offered that there would be 1,024 ten-tall towers that could be built selecting from cubes of two colors. However, during that session, she chose to justify her solution to the three-tall tower problem with an argument by cases (Maher & Martino, 2000). In Chapter 5, while working on another problem, Guess My Tower, we see Stephanie learn why the doubling rule works as she investigates Milin s inductive argument. 4.4 Summary of Strategies and Justifications Figure 4.21 outlines the strategies, representations, and forms of justification used by Stephanie and Milin during the five sessions on the towers problems. Both Stephanie and Milin began by using trial and error and justifying their solution empirically. They both progressed to more sophisticated strategies and forms of justification. Stephanie looked for patterns and controlled for variables to eventually formulate her justification using cases. Milin considered simpler cases and then recognized the recursive nature of the problem, arriving at his inductive justification. Both Milin and Stephanie arrived at a complete justification of their solution during The Gang of Four session. In addition, both students chose not to use the Unifix cubes to represent their towers but instead used notations in a grid (Stephanie) and drawings (Milin) to represent the different tower combinations. Class Session Interview 1 Interview 2 Interview 3 Gang of Four STEPHANIE MILIN Strategies Tools Justification Strategies Tools Justification Trial and error; Unifix cubes Opposites; cousins, Staircase Pattern recognition Drawings and symbols Controlled for variables Staircase; Controlled for variables Staircase, Controlled for variables; Pattern recognition Grid with symbols Patterns, partial cases Partial cases Drawings Partial cases and symbols Drawings Emergent and symbols cases Case argument Trial and error; Opposites; Staircase Patterns; partial cases; simpler problem Considered simpler cases Inductive pattern recognition Inductive pattern recognition Unifix cubes Unifix cubes Unifix cubes Unifix cubes Drawing and symbols Partial cases Partial Cases Partial induction Emergent induction Inductive argument Fig Strategies, representations, and justifications used by Stephanie and Milin 4.5 Discussion The Gang of Four session evidenced particular structures and modes of reasoning by Milin and Stephanie in their justification of their solutions to the towers task. The students built and refined their representations over a period of time in which they

17 4 Towers: Schemes, Strategies, and Arguments 43 had the chance to reflect upon the task, recognize emergent patterns, and choose schemes that best matched the representation that they had formed. Stephanie used symbols within a matrix to organize the towers by cases; Milin used drawings of towers to explain how they grew. The call for justification of the three-tall tower task enabled Stephanie and Milin to make public the schemes that they had built earlier. There are some similarities in Stephanie s early use of representations for both the towers task and shirts and jeans tasks. In the second grade, Stephanie listed the outfit combinations by using the initials of each color and recording the combinations in a vertical format (see Fig. 3.2). When working on the towers task, she again used initials for the colors of cubes using a grid organization to show the different towers. Stephanie also used the heuristic of controlling for variables as she organized her tower combinations, a strategy that her partner Dana had used in the shirts and jeans task. Milin s strategies of considering simpler cases and pattern recognition were powerful tools in his building of an inductive argument. As will be seen in Chapter 5, both students schemes proved durable as Stephanie and her classmates folded back to reflect on their earlier work to make sense of more complex combinatorial tasks in later grades. Importantly, these data show the advantage to revisiting tasks, group discussions about ideas, and sharing strategies. All of these components play a key role in the formulation and refinement of justifications. Stephanie and Milin, after having had multiple opportunities to think about and justify their ideas, presented a compelling argument to classmates during the group evaluation setting. As is evidenced in later years, unique aspects of the discussions that continued among this community of learners further triggered the development of more complex cognitive structures, triggered by the students need to produce justifications for combinatorial tasks of ever-increasing complexity. In Chapter 5, we follow Stephanie and other classmates as they continue to work on understanding Milin s inductive argument for building towers as they retrieve earlier frames and cognitive structures revealed during the Gang of Four work.