Maher C. A., Kiczek R. D. (2000)
Long Term Building of Mathematical Ideas Related to Proof Making.

Contribution to: Paolo Boero, G. Harel, C. Maher, M. Miyazaki (organisers) Proof and Proving in Mathematics Education. ICME9 TSG 12. Tokyo/Makuhari, Japan.

© Carolyn A. Maher & Regina D. Kiczek

We have engaged in a research program of teaching experiments designed to create classroom environments within which sense-making is a cultural norm for over a decade. In the course of these experiments, a particularly striking outcome of the culture has been the emergence of argumentation, justification, and proof-making in students' discourse. Such processes have developed in the context of coherent strands of mathematics -- combinatorics, probability, and mathematical modeling. We have been particularly interested in observing the development of representations and explanations provided by the students, as well as the reasoning offered to support their ideas. The arguments, justifications, and proof making contributions of a focus group of students who have been participants in a study, now in its twelfth year, will be traced in this paper. This paper describes the building of mathematical ideas over a seven-year period, fifth through eleventh grade. Four episodes that come from videotapes of problem-solving sessions in grades 5, 10, 11, and 12 are reported. The "Pizza Problem" , a task originally explored by the children in the fifth grade (10-11 year olds), became an important metaphor that guided students' later justification of the addition rule of Pascal's triangle and the generalization of that rule.

Episode 1: Grade 5, 4/2/93. Romina, Jeff, Brian and Ankur worked as a group on the Pizza Problem as fifth graders and used a variety of strategies and representations to produce the sixteen combinations. These included a partial tree diagram, lists, and an organization that systematically controlled for variables. Michael worked in parallel with the others and represented the various pizzas by drawing circles and labeled each "pizza" with its toppings. All of the students created codes using letters or abbreviations to represent the four toppings (for example, pe = pepperoni; m = mushrooms) and all decided to code for a pizza with no toppings (pl = plain or c = cheese). The pizzas were categorized as "whole" (plain and one-topping pizzas) and "mixed" (two or more toppings). The students found all sixteen pizzas, justifying their solution by the way they organized their results.

Episode 2: Grade 10, 12/12/97. Four years later, the same five students were again given the "Pizza Problem." They talked aloud about combinations of toppings and of the patterns they were observing, using a code of letters to represent the toppings. Some of the students shared their lists of combinations; they switched their notation, using the numerals one through five, and concluded that if five toppings were available, thirty different pizzas could be made with at least one topping, plus one plain cheese pizza, for a total of thirty-one. Meanwhile Michael worked alone for at least fifteen minutes quietly developing his own solution. He found there were thirty-two pizzas when choosing from five toppings and challenged the solution of the other students. He used a representation based on a binary-coding scheme to account for all possibilities, and explained what the zeros and ones meant in his representation and how they were used to write base ten numbers in base two notation. He then related his coding scheme to the pizza problem, summarizing his conclusion as follows:

Okay, here's what I think. You know like a binary system we learned a while ago? Like with the ones and zeros - binary, right? The ones would mean a topping; zero means no topping. So if you had a four-topping pizza, you have four different places - in the binary system, you'd have - the first one would be just one. The second one would be that [He wrote 10]; that's the second number up. You remember what that was? This was like two, and this was three [He wrote 11]….Well, you get, I think&emdash;I have a thing in my head. It works out in my head… You've got four toppings. This is like four places of the binary system. It all equals up to fifteen. That's the answer for four toppings…So you go from this number [He pointed to 0001], which is in the binary system, it's one, to this number [He pointed to 1111], which is fifteen, and that's how many toppings you have. There's fifteen different combinations of ones and zeros if you have four different places. … I don't know how to explain it, but it works out. That's in my head&emdash;these weird things going on in my head. And if you have an extra topping, you just add an extra place and that would be sixteen, that would be thirty-one.

Michael's representation using binary numbers did not include the representation for a plain cheese pizza [0 0 0 0]. He later corrected for this by adding one to the fifteen combinations (for making pizzas when selecting from four toppings) and to the thirty-one combinations (for making pizzas with five toppings available), thus accounting for all possibilities. Jeff interjected: "And then you add the cheese?", to which Michael acknowledged: "Plus the cheese would be thirty-two." With his classmates, Michael presented his binary coding scheme to the teacher/researcher, saying, "This is the way I interpret it into the pizza problem." When the teacher/researcher probed further about Michael's solution, the other students joined Michael in his justification, pointing out, for example that the difference between one, zero, zero, zero [1 0 0 0] and zero, one, zero, zero [0 1 0 0] would represent "the difference between an onion pizza and a pepperoni pizza". Then, the five students considered the case where ten toppings were available and recognized that a string containing all zeros represented a plain cheese pizza. When asked to generalize to the case of n toppings, after a few minutes, determined that there would be 2n different pizzas when there are n topping choices.

Episode 3: Grade 11, 12/14/98. One year later, in a task-based interview, Michael was invited to explain his binary representation with respect to pizzas with three toppings. In this interview, Michael referred to the list of binary numbers he had written, Michael explained that for pizzas, 000 was the code for no toppings, while 111 was the code for a pizza with "everything on it." When asked if he could describe the distribution of kinds of toppings, Michael referred to the rows of Pascal's triangle. He demonstrated the addition rule by using the pizza problem as a metaphor. Michael was then invited to present his ideas in writing. In a subsequent e-mail message, Michael provided a written justification for why the numbers in Pascal's Triangle mapped to the variety of choices for making pizzas, and illustrated this by moving from pizzas with four toppings to pizzas with five.

Episode 4: Grade 11, 5/12/99. In a subsequent after-school session, when invited to explain his ideas to the other four students in the group, Michael gave an argument for moving from pizzas with n toppings to pizzas with n+1 toppings. When presented his ideas both informally and formally, Michael treated the numbers which appeared both in Pascal's Triangle and in the pizza problem as the result of underlying structures which he demonstrated, in the general case, were isomorphic.

Michael, Jeff, Romina, and Ankur created and verified the meaning of the expression, nCr, proposed by another student, Robert. They explained their ideas using two examples: arranging people in a row and building towers when selecting from plastic cubes available in two colors (Maher & Martino, 1996). The students referred to their notation as "choose." They were then asked to represent the entries in the triangle using the "choose" notation. Michael went to the chalkboard and described how this would be done. Referring to the fourth row of the triangle, Michael used the pizza example once again to indicate that all the one topping pizzas that have a new topping added plus all the two topping pizzas that don't get a new topping added equal all the two topping pizzas when one more topping is available. Pointing to 4C1 and 4C2 [written in vertical notation format], he indicated that the entries above combine to give 5C2, explaining as follows:

Wherever this guy [4C1] goes, he's gonna get another topping 'cause he's moving this way [indicates to the right], so this bottom one's gonna change to 2. This guy's [4C2] not going anywhere &endash; this bottom number stays the same. It's gonna be 5 - the next one's gonna be 5 and it has to be a 2 [5C2]. You understand why you add?

Jeff joined Michael in the conversation and reiterated that they were explaining the addition rule for Pascal's Triangle, "using chooses to fill out the triangle." On the chalkboard, Jeff sketched the nth row [using vertical format for nC0 nC1 ... nCx ... nCn] and explained that the row above this would be (n-1)C0 (n-1)C1 ... (n-1)Cx ... (n-1)C(n-1). He wrote:

n
x

n
x

n+1
x+1

He then wrote the equation:

n   +   n   =  n+1
x       x+1   x+1

Jeff explained that the first expression was changing but the second was not and Michael joined him to emphasize: "The top number changes 'cause you have one more thing ... one more topping." Jeff continued the explanation and indicated:

That's [n+1] the increase in n and then the x+1 - if you added another topping onto your whole - Say we're doing pizzas, you add another topping onto it. ... [He & Michael start speaking at the same time.] ... When you add another topping onto it ... if it gets a topping, it goes up to the x+1. If it doesn't get anything, it would stay the same.

Jeff commented to another student: "So that would be their general addition rule, in this case."

The students responded by writing an equation for the addition rule using factorial notation on the chalkboard. As Michael wrote, Jeff commented: "You know how intimidating this equation must be? You just pick up a book and look at that!"

Conclusions

The pizza task became a powerful metaphor for explaining the addition rule of Pascal's triangle in general terms. Michael's coding scheme enabled him to list all possibilities. On the basis of a systematic organization, Michael built arguments, first informally for special cases and later in greater generality. Michael worked alone before he shared his coding scheme with others. When he did share it, despite apologies ("I don't know how to explain it, but it works."), Michael showed quite clearly how the code was used to solve the given problem. While the other students acknowledged its usefulness in later problem situations, the code continued to be designated Michael's "binary thing." Over time, Michael's application of the code was more general. For example, when Michael first used his code for pizzas, he made a chart with topping choices as the column headings. In later sessions, when listing possibilities, Michael omitted column labels for a given, particular situation. Instead, he determined how many places needed to be filled and then listed the corresponding binary numbers sequentially, beginning with zero. This explicit list of numbers, without column labels, enabled him to track all possibilities easily.
  These students, both as fourth and tenth graders, provided a justification for their solution that took the form of a proof by cases (Muter and Maher, 1998; Kiczek and Maher, 1998; Muter, 1999). However, their representations and notations became increasingly more abstract and general over time. The properties of combinations, for these students, grew from very concrete images, such as towers (Maher & Speiser, 1997; Maher & Martino, 1996a & b) and pizzas (Maher, 1993). Once these properties emerged, as they did here in Pascal's Triangle, they linked the prior images, towers and pizzas, into a larger framework that connected quite readily to other ideas in algebra and combinatorics. Michael's representation, triggered by the need to find and justify a particular solution, served as a tool, for him and the others, to connect mathematical situations which they explored for a number of years. The students proposed thoughtful and strong arguments at a young age and built upon those ideas in later years. Their representations were modified over time, indicating the use of symbols to represent the objects. While the structure of the arguments were durable over time, the particular representational systems became increasingly more elegant and powerful.

References

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