Combinatorics and Reasoning: Representing, Justifying and Building Isomorphisms

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Member Log-In. E-mail address. Remember me. Account recovery. You are in page, help us by providing your feedback on different features. Select feature Accessibility Type Metadata. Select feature Registration Log-in Account recovery Profile customization. They willingly shared their interpretations and strategies and talked to each other about their findings. In third grade, when the children were again presented with this problem, they did not remember how they had solved the problem earlier, nor did they remember their earlier answers. Of particular interest is that evidence of fur- ther elaboration of earlier strategies emerged.

Students used and built on strategies of their second-grade partners. For example, Stephanie indicated different outfits by drawing lines between drawings of shirts and jeans, as Dana had done in second grade. By third grade, techniques for checking and for keeping track, such as controlling for variables, were complete. Earlier ideas and strategies were refined to produce complete, elegant solutions rather quickly. Students built on their heuristics to solve more complex extensions of the problem to include belts and hats as parts of outfits.

Uptegrove continued their problem solving, driven by earlier heuristics and sense making to produce correct solutions that they could justify. Work together and make as many different towers four cubes tall as is possible when selecting from two colors. See if you and your partner can plan a good way to find all the towers four cubes tall. The definition of a tower is an ordered sequence of Unifix cubes, snapped together.

Each cube can also be called a block. Each tower has a bottom and a top. The height of a tower is the number of its cubes. We say two towers are the same if their colors match, block by block, from top to bottom. Unifix cubes are interlocking cubes that come in various colors typically blue, red, yellow, white, and green. In fourth grade , students worked on the five-tall tower problem.

Then in fifth grade , they revisited the four-tall version. In 10th and 11th grades, they were asked to provide a justification for the n-tall tower problem. Their work on the towers problems also illustrates how their representations changed over the years.

At first, they used Unifix cubes to build towers.

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Eventually, they turned to drawings and codes, for example, using letters R and Y to mean red and yellow cubes. In some cases, a more general code emerged; some students would use X and O or 0 and 1 to indicate any two colors.

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More details on these emerging strategies are given in Sections 2. When the students were first given the problem in fifth grade, they inter- preted the task as allowing different toppings on each half of the pizza, an alternative that they knew that was available in some pizza restaurants. In response to their inter- est in counting the varieties allowing toppings on half a pizza, the researcher asked them to solve it with only two toppings available. This pizza with halves problem is as follows: Kenilworth Pizza has asked up to help them design a form to keep track of certain pizza sales.

Their standard plain pizza contains cheese. On this cheese pizza, one or two toppings could be added to either half of the plain pie or the whole pie. List all possibili- ties. Show your plan for determining the choices. Convince us that you have accounted for all possibilities and there could be no more. The strategy that the students developed for the solution established the heuristic that was applied later when there were five toppings available, again, allowing some or no toppings on half the pizza.

The final problem, the five-topping pizza problem, proved a trivial special case for the pizza with halves problem that they first solved successfully: The local pizza shop offers a plain cheese pizza. On this cheese pizza, you can place up to five different toppings. How many pizzas is it possible to make?

Pizza with Halves was the first of several variations of the pizza problem that the students worked on over the years. It illustrates a basic philosophy of the study — we did not start students off with easier problems and then progress to the more diffi- cult ones. Instead, students began with the more difficult versions of the problems, which required them to tackle several challenges at once — organization making sure no pizzas were repeated and none were omitted , notation how to distinguish between pepperoni and peppers, for example , and forming a valid argument — how to convince the researchers and themselves that they had the right answer.

At first, students drew fairly accurate rendi- tions of pizzas; they drew circles to indicate pizzas, and inside those circles were wavy lines to indicate sausages and smaller circles to indicate pepperoni, for exam- ple. Eventually, they turned to codes, starting with single letters or combinations of let- ters to distinguish between peppers and pepperoni, for example and then moving to more abstract symbols such as 0s and 1s.

These representations and organizational strategies are discussed more fully in Chapter 6. Using the metaphor of the pizza problem, they explained how the triangle grows by explaining how the number of possible pizzas grows as new toppings become available. The problem is to find the shortest route to three specific points on the grid and to determine the number of shortest routes to each point.

Their work on this problem is discussed in greater detail in Chapter It is inter- esting to note that by this time, the students, without prompting, solved the general problem, in addition to answering the specific questions. For any point on the grid, they showed why the general answer was correct, and they demonstrated the con- nection to isomorphic problems towers and pizzas and to the binomial expansion.

What is also interesting from this session is that the students took on the roles of eliciting justifications from each other. Their pursuit of explanations that made sense and that connected to earlier tasks was quite remarkable. Instead, the aim was to establish a culture where the correctness of an answer came from the sensemaking of the students, rather than from the authority of the researcher.

We asked students ques- tions about what was convincing, what made sense, and how they developed their answers. In justifying their answers, students usually exceeded our expectations. We were impressed by the seriousness with which students approached the prob- lems and the collegiality of their work, as well as by the forms of reasoning they developed. In the early years of the study, children began to use inductive reason- ing, to organize work by cases, and to think about justification through contradiction.

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  6. By middle school, these forms of reasoning were more sharply defined, and other forms of reasoning emerged, such as controlling for variables. In the following chapters, we provide details on the specific problems, the spe- cific strategies and representations used by the students, and the specific results they generated.

    Maher and Amy M. Martino 3. Convince us that you have them all. When introducing mathematics to children, it is important to invite them to use their personal representations to express their ideas and ways of reasoning NCTM, These representations are the basic elements that chil- dren draw upon to express their ideas as they begin to engage in more abstract and logical reasoning. Maher and D. Yankelewitz 3. According to Davis, representations are mental models that allow for the association between the properties of a mathematical idea and the idea itself.

    These ideas are not stored in the mind in words or pictures, and so when we explore what these intangible representations are, we are only approximating their true nature. Davis stated that doing mathematics involves a series of steps, similar to those of a computer executing a program, through which the student must cycle one or more times.

    First, in an attempt to make sense of the problem, the student builds a representation for the input data. Second, the student searches his memory for knowledge that will assist in solving the problem. Finally, the student maps the data representation with the knowledge representation. When the mapping seems accu- rate enough to tackle the problem at hand, the student uses techniques associated with the knowledge representation to solve the problem. Students use representations that they build to make sense of and attribute mean- ing to the mathematics that they are doing.

    They use mathematical tools, which, according to Davis and Maher include mathematical notation, spoken and written language, physical models, drawings, and diagrams. According to Maher and Martino a , students who are encouraged to build and use multiple representations as they work on problems become sense-makers and active members of the mathematical community. The use of different tools to build and express ideas allows students to make connections between differ- ent representations and understandings and to better understand the mathematics that they are learning.

    Using representations to make sense of problems and using representations to communicate ideas are therefore the building blocks of effective argumentation. Combinatorics problems were well suited to these goals. In the sections that follow, we will consider the specific mathematical ideas, fundamental both to combinatorics in particular and to mathematics in general, that are elicited by the tasks that were used in the longitudinal study.

    The shirts and jeans task above introduces the fundamental counting principle, a key idea in combinatorics. This problem also introduces the need for notation or symbols to represent the real-world items described in the problem. So this problem creates a need for a bridge between the real-world situation presented and the mathematical ideas that will provide a solution.

    The problem also provides students with a chance to realize that an organi- zation of the facts by means of a diagram or an organized list can help them to find a solution; this need for structure is fundamental to mathematics. The problem also requires students to think about how to justify their solution to others and convince others that they have found all the combinations.

    It has the potential to give rise to the need for direct or indirect arguments Fig. There are six combinations. The shirt colors are listed on the left and the jean colors are on the right. The blue shirt can be combined with either the white jeans or the blue jeans to form an outfit; so two and only two outfits can be made using the blue shirt. The same is true for the white shirt as well as for the yellow shirt. All possible combinations are accounted for, since any other attempted combinations will be duplicates of the ones listed above.

    For example, there cannot be a third outfit formed using the blue shirt, because only white and blue jeans are available. The same is true for the other color shirts. The strategies of three students, Dana, Stephanie, and Michael see Fig. Yankelewitz Fig. In addition, selections of the video were included in the Private Universe Project in Mathematics Harvard-Smithsonian Center for Astrophysics, , and the mathematical thinking and representations of these students are discussed there. In the second grade, all three students drew pictures of shirts and jeans to rep- resent the items in the problem, and they used the pictures in their attempts to find different combinations of the shirts and jeans.

    She then numbered the combinations that she found. She then told the researcher that she had found five combinations and she was convinced that she had found them all. It can be concluded from this statement and the subsequent problem-solving steps that she took that she had built a scheme that closely matched the problem solution. Dana used a strategy of connecting her representations of shirts and jeans that she had drawn with lines as is shown in Fig.

    We can conclude that she was aware of all possible outfits but her sense of fashion resulted in her rejecting the yellow shirt and white jeans. He drew diagrams of the different color shirts and jeans, but said that he had arrived at three combinations: a white shirt with white jeans, a blue shirt with blue jeans, and a yellow shirt with yellow jeans see Fig. Although Stephanie and Dana pointed out that the shirts and jeans did not have to be the same color, Michael did not make any changes to his own solution.

    Stephanie and Dana again worked together, and they immediately began to draw diagrams to represent each item in the problem. When ques- tioned about why they drew lines to show the combinations, Stephanie explained that that was to ensure that they do not make any duplicate combinations.

    Figure 3. Dana, in her grade 3 drawing, again used a tree representation to form all shirts and jeans outfits as indicated in Fig. This time, Michael used the strategy of connecting lines between the shirts and jeans to represent the possible outfits, but, unlike Stephanie and Dana, he drew lines between the words in the problem, rather than between drawings of the shirts and jeans as indicated in Fig.

    Using this strategy, Michael also arrived at a solution of six combinations. He used a strategy similar to that used by Stephanie in the second grade: he listed the combinations by writing the letter representing the color shirt above the letter representing the color jeans. Yankelewitz building schemes that could account for some or all of the outfits.

    All three students drew pictures in order to model the shirts and jeans, and all three used notation first letter abbreviations to indicate the colors of the shirts and jeans. Their represen- tations of the problem included letters and line diagrams. It is important to note that Dana showed evidence of building the scheme for controlling for variables in the second grade.

    If not for her sense of style, Dana would have arrived at the cor- rect answer of six outfits in the second grade.

    Representing, Justifying and Building Isomorphisms

    In the third grade, all three students arrived at a correct solution and none recalled that their correct solution was different from the solution they found in grade 2. This time, Stephanie offered a justification, explaining that the lines that they drew between the shirts and the jeans ensured that they accounted for all possibilities and also that they had not counted any combina- tion more than once.

    Stephanie also used a system of counting that enabled her to keep track of her outfits. It invites stu- dents to bring forth personal representations and it offers opportunities for sense making, so that students can begin to discuss how they arrived at their solutions and how they know their solutions are correct. Also, different ways of reasoning can be explored while students can learn how to formulate organizational schemes that can help them solve other problems in mathematics.

    In addition, the real-world setting of the problem shows the direct connection between the mathematical ideas and the world that students know. Thinking about real-world considerations which are impor- tant and which can be ignored is a necessary step on the journey to mathematical understanding and abstraction. It is interesting to note the variety of strategies and approaches that were used in this problem. For example, in second grade, the three children used three different strategies to solve the problem. Although none of the strategies produced a correct answer, it should be emphasized that arriving at the correct answer was not the goal.

    In fact, when asked as third graders what answer they gave in grade 2, these students all responded that they found six outfits. For the researchers, the goal of the sessions was not for the children to give the correct answer; we were confident that they would eventually succeed. Our primary goal was to engage the children in thoughtful problem solving that could trigger growth in schema. We wanted solutions to be meaningfully constructed by the stu- dents. As the following chapters will reveal, a pattern of working with the students in the longitudinal study was to revisit problems and solutions in cycles so that ear- lier ideas could be built upon and new representations could be revealed.

    We observed the heuristics they used and the schemes that were later retrieved and modified. Over the months, there is evidence of durable learning. More importantly, perhaps, is that each student incorporated strategies in unique ways. These episodes also demonstrate some benefits of group work. The contribution of each student to the cumulative body of knowledge enables students to arrive at their own solutions while simultaneously benefiting from the knowledge of others. As the data indicate, the children built durable schemes to solve the mathemat- ical tasks that they were given.

    In addition, the data show that the representations and arguments that were originally built by individual students were used to effectively communicate to others the schemes upon which they had been built. In Chapter 4 , we follow two of these students — Dana and Stephanie — along with a third student, Milin, as they work on more problems designed to help them explore fundamental ideas in algebra and combinatorics.

    Martino 4. 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 prob- lems.

    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.

    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, Older ideas were elaborated and expanded upon. 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 strategies of a number of these students are documented and discussed in Martino Stephanie built ten towers. Dana also initially built ten towers, including two pairs of opposites. Everything we make. Everything we make, we have to check. DANA: All right. 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.

    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. You drew lines between the shirts and the pants. Is there a way that you could be sure? 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. 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 four- tall towers. During an interview the next day, Stephanie explained why there were fewer three-tall towers than four-tall towers. Stephanie and Dana began to make towers along with their opposites see Fig. At one point, they realized that an individual tower could be turned upside down to create a new tower.

    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. If another red were added, the result would be a six-tall tower. DANA: No. DANA: Because it only goes up to five blocks. 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. Data from these episodes are presented here with attention to the emergent strategies that Stephanie used while working on the tower tasks. She discovered that introducing additional patterns sometimes resulted in duplicate tow- ers that needed to be eliminated by checking. The interviewer asked Stephanie if there was a way she could be sure of how many towers of a specific type could be made.

    Yeah, it is possible to have a certain number and get it right.

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    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 belong- ing 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. Notice, also, in both figures Stephanie applied her elevator pattern while holding both rows constant. During an individual interview on March 6, , Stephanie presented a com- plete 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 cat- egories 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.

    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. At the conclusion of the group work, the boys found all 32 towers. The strategies used by the two students on February 6, , 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. 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.

    During the sharing session see Fig. 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. 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. 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. 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. It was conducted so that the children could share their strategies and arguments for Fig.

    The session began with the researcher asking the students how many six- tall towers could be built. MILIN: 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.

    Combinatorics and reasoning : representing, justifying and building isomorphisms

    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. Let Milin persuade Jeff. These four? You have to add one more color for each one. Where are you putting that one more color, Milin?

    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. 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.

    It is interesting to notice how Milin chose to draw the towers rather than use actual cubes as he had during his individual interview. 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 b. 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. But then you have with three blues — well, not with three blues. 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. 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. All right. You could put blue, blue, red; you could put red, blue, and blue. You could put. There should be one with one red. And then you change it to blue. Then you want two blues stuck apart — not stuck apart; took apart.

    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. Their progression of this understanding is documented by Maher and Martino , and Maher 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 jus- tification. Stephanie looked for patterns and controlled for variables to eventually formulate her justification using cases. Milin considered simpler cases and then rec- ognized 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. 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. In the second grade, Stephanie listed the outfit combinations by using the initials of each color and recording the combina- tions in a vertical format see Fig. 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 orga- nized her tower combinations, a strategy that her partner Dana had used in the shirts and jeans task. 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. In this chapter, we trace how Stephanie and her classmates tried to make sense of the inductive method of generating towers.

    This strategy was originated by Milin, but it was eventually adopted by many other students. We attempt to identify the moments at which individual students gained ownership of the inductive argument and explained their new understanding to others. To support their solutions, students followed two different approaches. Stephanie and others made extensive use of argument by cases.

    Milin, over a series of task-based interviews, built an inductive argument and he was able to use an inductive argument to show how to generate the number of combinations of towers of any height. Stephanie also C. Stephanie conjectured a doubling rule from her successful case-building justifications of towers of different heights.

    The researcher showed Stephanie the first two levels of the tree diagram and then asked Stephanie to extend the tree to include three-tall and four-tall towers. Stephanie responded by producing a partial extension of the tree organization as indicated in Fig. The children worked in pairs on this assessment to provide a convincing justification of the towers task.

    Stephanie and Milin were partners and provided individual written work of their solution. See Fig. In her letter, Stephanie gave an elegant argument by cases to show that she found all the towers, and then used a doubling pattern to predict taller towers, offering a general method. He also demonstrated an understanding of the doubling rule in his written work. Notice, too, Fig. The children worked alone and produced individual written work Fig. Milin, in his October 25 letter, again explained his doubling rule and drew all eight three-tall towers in a grid.

    He then also included the one- and two-tall towers see Fig. We discuss below the traveling of ideas within a small community of students, initiated first, by Milin; then, from Milin to Michelle I. Since solving the Guess My Tower problem required the building of a sample space for all possible events, it required that the students revisit the question of finding the total number of four-tall towers.

    We have seen that earlier, Stephanie had already built and justified by cases all possible four-tall towers and used the doubling method for determining the total number of towers. Stephanie and Matt worked together on this problem first, using paper and pencil and then by building actual towers using Unifix cubes. They found all eight towers see Fig. The researcher asked Stephanie and Matt to predict how many four-tall towers they would find. Stephanie remembered the pattern that she had noticed the previous year.

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    It was Fig. Stephanie insisted that there should be 16 based upon her confidence in the doubling rule. She shared with Matt the doubling pattern. If you multiply the last problem by two, you get the answer for the next problem. But you have to get all the answers. MATT: I thought we did. I mean all the answers, all the answers we can get. Because I am positive it works. I have my papers at home that say it works. I know that you had to multiply it [the total number of towers of a given height] by something.

    Maybe it was adding two. Stephanie and Matt continued to attempt to find more four-tall towers, but they could not find more than Stephanie continued to assert that the total would have to be I know it worked.

    Combinatorics and Reasoning

    MATT: Steph! As Stephanie and Matt worked to find more towers, Milin attempted to convince his partner Michelle I. Recall that both Milin and Michelle I. Refer to Chapter 4 for details. Michelle I. In this session, however, Michelle told Milin that she did not understand his explanation of how he generated new towers using an inductive argument.

    Milin then explained his reasoning for a second time see Fig. Once Milin finished explaining his reasoning for building three-tall towers by adding a red cube or a yellow cube to the tower from the previous stage, Michelle I. Michelle also shared her understanding of the inductive method with Stephanie and Matt see Fig. As Michelle I. The researcher suggested that Matt be given an oppor- tunity to explain to the group how Milin and Michelle I. Matt eagerly complied, showing that one could find the total number of towers of any height see Fig.

    Stephanie and Matt moved on to talk to Bobby and Michelle R. We were only able to find. So you go to the next number. And the next number is two. So you have four of two. As Matt explained to Michelle R.