Milin’s Learning Progression in Reasoning by Cases to Solve Tower Tasks: Part 2 of 2 (Grade 4)

PurposesStudent model building; Reasoning; Representation
DescriptionAuthor: Victoria Krupnik, Rutgers University

This analytic is the second of two analytics that showcase the development of a variation of reasoning by cases to solve a counting task by a student, Milin, over one school year in fourth grade. Both analytics focus on the reasoning, argumentation, and mathematical representations constructed by Milin in a variety of settings, over time, in the fourth grade. The first analytic begins with events where Milin is working with a partner, in a whole class setting, building towers with plastic cubes that are 5-tall. This is followed by events from a first one-on-one interview with researchers. The second analytic continues with events from the first interview, as well as with events from a second interview.

In the current analytic (the second of the two analytics), Milin’s learning progression in building a justification to Tower Tasks shows Milin extending his method by cases and opposites for towers 1-, 2-, 3-, and 4-tall and comparing the number solutions. 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.

Event 1 shows how Milin uses a methodology that combines cases (i.e. by “staircases”) and color opposite strategies to solve the 4-tall Tower Task (after solving the 5-tall Tower Task) in a first interview facilitated by Researcher Alston (R1) and visiting Teacher O’Brian (R4) on February 7, 1992, the day following the classroom work with his partner.

Events 2–6 illustrate Milin’s refinement, modification, and application of the case and/or color opposite strategies to solve the 2-, 3-, and 4-tall Tower Tasks in a second interview facilitated by R1 on February 21, 1992. In Events 2–4 he builds the cases of 4-tall towers with exactly one cube of a particular color, towers with all cubes the same color, towers with exactly two cubes of a particular color adjacent to each other, and towers with alternating color cubes. In Event 5 he builds the solution by cases for 2-tall and 1-tall towers and compares them to the 4-tall tower cases. In Event 6, Milin returns to the color opposite and inverse pairs of towers strategies to build towers that are 3-tall. Not shown in the analytic, Milin later predicts the number of 6-tall towers to be “forty something” and is given a homework assignment to solve the 6-tall Tower Task.

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

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.

*A tower is denoted by the first letter of each color that takes on positions from top to bottom.

“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 final position (Maher, Sran & Yankelewitz, 2011).

Strategy of a globally exhaustive systematic enumeration (Maher & Martino, 1996, 2000, 2013; Batanero et al., 1997):
Case organization and/or argument:
In an organization and/or argument by cases, a statement is demonstrated by showing all the smaller subsets of statements that make up the whole. For example, the solution to the 3-tall Tower Task when selecting from two colors (i.e. blue and red) can be justified by separating the towers into cases using a characteristic of the tower. One such characteristic is the number of cubes of a specific color that the towers contain. In this situation, the towers can be broken down into four cases: 1) towers containing no red cubes (towers with a single color blue); 2) three towers containing one red (towers with exactly one of a particular color); 3) three towers containing two reds (within cases (2) and (3) can be cubes of same color adjacent to or separated from each other); 4) one tower containing three red, i.e. with all red. An argument by cases would include an exhaustive enumeration of the total number of towers in each case.

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”)

Video and Transcript References (in chronological order of Milin’s journey):
B60, Milin and Michael classwork of the 5-tall towers problem (work view), Grade 4, Feb 6, 1992, raw footage. Retrieved from:

B76, Milin’s first of three interviews with researcher Alston on the five-tall Tower Task (Work view), Grade 4, February 7, 1992, Raw footage. Retrieved from:

B62, Stephanie’s and Milin’s second of three interview sessions and Michelle’s second of two interview sessions revisiting five-tall Towers and other heights (work view), Grade 4, Feb 21, 1992, raw footage. Retrieved from:

Batanero, C., Godino, J. D., & Navarro-Pelayo, V. (1997). Combinatorial reasoning and its assessment. In I. Gal, & J. B. Garfield, The assessment challenge in statistics education (pp. 239–252). Amsterdam: IOS Press.

Maher, C. A. (2010). The Longitudinal Study. In C. A. Maher, A. B. Powell, & E. B. Uptegrove, Combinatorics and Reasoning (pp. 3–14). New York: Springer.

Maher, C. A., Powell, A. B., & Uptegrove, E. B. (2010). Combinatorics and reasoning. In C. A. Maher, A. B. Powell, & E. B. Uptegrove, Representing, justifying and building isomorphisms (Vol. 47). New York: Springer.

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.
Created on2019-09-14T19:32:13-0500
Published on2019-12-23T09:44:34-0500
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