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doi:10.7282/t3-c5ja-qn71

Author: Victoria Krupnik, Rutgers University
Overall Description
This analytic is the first of three analytics that showcase the problem solving of a counting task by a student, Michelle, over two school years. The analytics focus on the development of reasoning, argumentation, and mathematical representations constructed by Michelle in a variety of settings, over time, in the fourth (Parts 1 & 2) and fifth (Part 3) grades. The first analytic begins with Michelle working with a partner, in a whole class setting, building towers with plastic cubes that are 5-tall, selecting from two colors. This is followed by events from a first and second one-on-one interview with researchers. The second analytic continues with events when Michelle participated in a small group formative assessment interview and with a partner on a summative assessment. The third analytic includes events when Michelle worked with a partner, Milin, in a whole class setting, using towers to solve an application task.

In this analytic (the first of three analytics), Michelle’s learning progression in building “proof-like” justifications to Towers Tasks shows Michelle, in the fourth grade, solving towers by local strategies of creating partner towers that she called opposites and towers with elevator patterns as she worked with a partner. Michelle, in individual interviews, is shown making connections between the Outfit and Tower Tasks up to 6-tall, selecting from cubes available in 2 colors. Her problem solving is presented in a series of events that focus, in detail, on the explanations, reasoning, and argumentation Michelle offers during her problem solving.

Events 1-2 are retrieved from a fourth-grade class session with partners Michelle and Jeff as they work on the 5-tall Tower Task on February 6, 1992. Michelle and Jeff use color opposite and elevator pattern strategies when building tower models with Unifix cubes selected from two colors to illustrate their solution to the 5-tall Tower Task.

Events 3-4 and 5-8 are retrieved from two individual interviews facilitated by Researcher Maher (R2) with the presence of Researcher Alston (R1) on February 7, 1992 and February 21, 1992, respectively. Michelle is seen in both interviews making a connection between the solutions of the Outfit Task and the corresponding Tower Task and reasoning by analogy to solve varying Tower Tasks.

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

Duplicate
This occurs when there is a repetition of color cubes in the possible positions. For Towers, this would be two identical towers or images of towers.

Guess and Check
The strategy of guess and check involves first guessing an outcome then checking that the outcome is applicable to the solution. Students can be observed using the Guess and Check method when building a tower pattern in a random order (or with no observable method) and then double-checking for duplicate towers (Maher & Martino, 1996). This occurred during the construction and generation of possibilities to obtain a solution.


Strategies of locally exhaustive, systematic enumeration (Maher & Martino, 1996, 2000, 2013):
Color “Opposites” (children’s language)
Each element in a combination containing exactly two types of a particular characteristic, such as color, is replaced with the element of the opposite characteristic. In the combinatorics strand this is known as the strategy of symmetry (Janackova & Janacek, 2006). The opposite of a tower in two colors is a tower of the same height where the cube in each position is the opposite color of the cube in the corresponding position of the first tower. For example, a 4-tall tower with YBBB and a tower with BYYY are considered to be opposites.

“Elevator” strategy (Jeff’s language; Milin called this “staircases”)
The elevator pattern is used when finding all possible towers containing one cube of one color and the remaining cubes of the other color. The single colored cube is placed in the first position of the first tower (either at the top or the bottom level). To create a second tower, the cube is then moved to the second position (descending, if starting at the top level, or ascending, if starting at the bottom level). The cube is recursively raised (or lowered) to the next available level to create new towers until it is placed in the last available position (Maher, Sran & Yankelewitz, 2011).

Three-tall Tower Task (selecting from 2 colors):
You have plastic cubes of 2 colors available to build towers. Your task is to make as many different looking towers as possible, each exactly 3 cubes high. Find a way to convince yourself and others that you have found all possible towers 3 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”).

The original 3rd grade Outfit (aka Shirts and Pants) Task:
Stephen has a white shirt, a blue shirt and a yellow shirt. He has a pair of blue jeans and a pair of white jeans. How many different outfits can he make?

Video and Transcript References (in chronological order of Michelle’s journey):
B65, Jeff & Michelle class work on 5-tall Towers (WV), Grade 4, Feb 6, 1992, raw. Retrieved from: https://doi.org/doi:10.7282/T3D50RK8

B74, Combinatorics: Towers, work view, Grade 4, February 7, 1992, Raw footage. Retrieved from: https://doi.org/doi:10.7282/t3-7m88-3180

B62, Michelle’s 2 of 2 interviews revisiting Towers (WV), Grade 4, Feb 21, 1992, raw. Retrieved from: https://doi.org/doi:10.7282/T3X3523K

References

Janácková, M., & Janácek, J. (2006). A classification of strategies employed by high school students in isomorphic combinatorial problems. The Mathematics Enthusiast, 3(2), 128-145. Retrieved from: https://scholarworks.umt.edu/tme/vol3/iss2/2

Maher, C., & Martino, A. (1996). The development of the idea of mathematical proof: A 5-year case study. Journal for Research in Mathematics Education, 194-214.

Maher, C. A., & Martino, A. M. (2000). From patterns to theories: Conditions for conceptual change. The Journal of Mathematical Behavior, 19(2), 247-271.

Maher, C. A., & Martino, A. M. (2013). Young children invent methods of proof: The gang of four. In L. Steffe, P. Nesher, P. Cobb, G. Goldin, & B. Greer (Eds.), Theories of mathematical learning (pp. 443-460). Hillsdale, NJ: Erlbaum.

Maher, C. A., Sran, M. K., & Yankelewitz, D. (2011). Towers: Schemes, strategies, and arguments. In C. A. Maher, A. B. Powell, & E. B. Uptegrove, Combinatorics and Reasoning (pp. 27-43). New York: Springer.

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