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doi:10.7282/t3-dwqy-mg03

Author: Victoria Krupnik, Rutgers University

This analytic is the second of two analytics that showcase the development of an argument by induction to solve a counting task by a student, Milin, over two school years. Both analytics focus on the reasoning, argumentation, and mathematical representations constructed by Milin in a variety of settings over time, in the fourth (Parts 1 and 2) and fifth (Part 2) grades. The first analytic includes events where Milin is working in a one-on-one interview with a researcher. The second analytic follows with events where Milin participates in a small group formative assessment interview and works on a summative assessment with a partner.

In the current analytic (the second of the two analytics), Milin’s learning progression in building a justification by inductive reasoning to Tower Tasks shows him, in the fourth and fifth grades, presenting a complete argument for the doubling pattern of number solutions to Tower Tasks by inductive reasoning and applying the doubling rule to solve Tower Tasks. His problem solving is presented in a series of events that serve to focus, in detail, on the explanations, reasoning, and argumentation offered by Milin during his problem solving.

Events 1–3 are retrieved from a small-group formative assessment interview (“Gang of Four”), facilitated by Researcher Maher (R2) on March 10, 1992. Milin presented his generalization of a doubling rule to number solutions of Tower Tasks up to 6-tall by an inductive argument. Specifically, he demonstrated how from each tower of a given height one could generate two distinct towers of a higher consecutive height when choosing from two colors. On multiple occasions throughout this session, he demonstrated his understanding of the doubling pattern, or as he previously called it, the “family strategy,” with an inductive argument. He used Michelle’s and his own tower drawings. In each instance Milin explained how a tower of a given height could be used to generate two different towers of a consecutively higher height by first adding a cube of the first color on top of the base and then adding a cube of the second color on top of a second tower with the same base. In this group interview he portrayed no doubt, as he had done in the previous individual interview, that this pattern would work for other heights. Specifically, he conjectured the doubling pattern should work for towers taller than 5-tall.

Events 4–5 are retrieved from a fourth-grade class session, facilitated by a classroom teacher, in which Milin and his partner, Stephanie, solve the 3-tall Tower Task as a summative assessment occurring on June 15, 1992. His primary strategies continue to be an elevator pattern for the towers with exactly one green cube, and color opposite pairings for the single-colored towers, even when Stephanie utilizes a different organization for the towers with exactly one black. The purpose of this event is to show his continuation, three months later, to find patterns among the towers, using the elevator and color opposite strategies. Milin and Stephanie also apply the doubling rule to justify their solution.

Events 6–7 are retrieved from a fifth-grade class session, facilitated by Researcher Alston (R1), with partners Milin and Michelle as they worked on the Guess My Tower (GMT) Task on February 26, 1993. Solving the problem depended on the students revisiting and rebuilding the 3- and 4-tall tower outcome possibilities. In these events Milin explained his inductive reasoning for the doubling pattern to Michelle, facilitated by Researcher Maher (R2).

The following definitions and background information about the Tower and Guess My Tower Tasks are offered:

Doubling rule:
The total number of different tower combinations of height k would be double the total number of tower combinations of height k–1.

Argument by Induction (Alston & Maher, 1993; Sran, 2010; Maher, Sran, & Yankelewitz, 2011)
An induction argument for the justification of the general Tower Task solution 2^n includes the basic step (n=1) in which a participant states that the total number of 1-tall towers created when selecting from two colors is two (i.e. one of only blue and one of only yellow). The second step describes that the total number of towers of a given height can be found by placing either a yellow or blue cube on the top of each of the towers of the previous height, therefore doubling the total number of towers created in the previous height.

Three-tall Tower Task (selecting from two colors):
You have plastic cubes of two colors available to build towers. Your task is to make as many different looking towers as possible, each exactly three cubes high. Find a way to convince yourself and others that you have found all possible towers three cubes high, and that you have no duplicates [repetition of same color and order]. Record your towers below and provide a convincing argument why you think you have them all. After completing the Task for Towers 3-tall, describe and justify the approach you have chosen. (The Tower Task can be generalized to towers of any height “n-tall”).

Guess My Tower Task:
You have been invited to participate in a TV Quiz Show and have the opportunity to win a vacation to Disney World. The game is played by choosing one of the four possibilities for winning and then picking a tower out of a covered box. If the tower matches your choice, you win. You are told that the box contains all possible towers three tall that can be built when you select from cubes of two colors, red and yellow and that there is only one of each tower. There are no duplicates in the box.
You are given the following possibilities for a winning tower:
a. All cubes are exactly the same color;
b. there is only one red cube;
c. exactly two cubes are red;
d. at least two cubes are yellow.
Question 1. Which choice would you make and why would this choice be any better than any of the others?
Question 2. Assuming you won, you can play again for the Grand Prize which means you can take a friend to Disney World. But now your box has all possible towers that are four tall with no duplicates (built by selecting from the two colors, yellow and red). You are to select from the same four possibilities for a winning tower. Which choice would you make this time and why would this choice be better than any of the others?

Video and Transcript References (in chronological order of Milin’s journey):

B42, The Gang of Four (Michelle and Milin view), Grade 4, March 10, 1992, raw footage [video]. Retrieved from https://doi.org/doi:10.7282/T3833Q5P

B75, Towers Assessment, WV, Grade 3, Jun 15, 1992, raw. Retrieved from: https://doi.org/doi:10.7282/t3-tpqc-b719

Building Towers, Selecting from two colors for Guess My Tower, Clip 3 of 5: Milin introduces an inductive argument. Retrieved from: https://doi.org/doi:10.7282/T3RN371Z

References:
Alston, A., & Maher, C. A. (1993). Tracing Milin’s building of proof by mathematical induction: A case study. In Proceedings of the fifteenth annual meeting for the North American chapter for the psychology of mathematics education (Vol. 2, pp. 1-7).

Maher, C. A., Sran, M. K., & Yankelewitz, D. (2011). Building an inductive argument. In C. A. Maher, A. B. Powell, & E. B. Uptegrove, Combinatorics and Reasoning (pp. 27-43). New York: Springer.

Sran, M. K. (2010). Tracing Milin’s development of inductive reasoning: A case study (Doctoral dissertation, Rutgers University-Graduate School of Education).

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