Michelle’s Longitudinal Problem Solving and Development of Reasoning About Tower Tasks: Part 2 of 3 (Grade 4)

PurposesStudent collaboration; Student elaboration; Student engagement; Reasoning; Representation
DescriptionAuthor: Victoria Krupnik, Rutgers University
Overall Description
This analytic is the second 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 cubes available in 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 second of three analytics), Michelle’s learning progression in building “proof-like” justifications to Towers Tasks shows Michelle, in the fourth grade, listening to the arguments of others by induction and cases and reasoning by cases with her partner. Her problem solving is presented in a series of events that serve to focus, in detail, on the explanations, reasoning, and argumentation she offers during her problem solving.

Events 1-4 are retrieved from a small-group assessment interview (Gang of Four), facilitated by Researcher Maher (R2), on March 10, 1992. In this session, the students are asked to convince the researcher and each other of their solution to the 3-tall Tower Task, selecting from two colors. In Event 1, Michelle makes a claim for the numerical solutions to the Towers Tasks of heights one through five. After Milin and Stephanie offer explanations of solutions for finding all possible 3-tall tower outcomes, when selecting from two colors, Michelle displays her understanding of their arguments, engaging in justifications, as well as the review and evaluation of Milin’s inductive argument (Event 2) and Stephanie’s argument by cases (Event 3). The small group members provide rationales for the value of “looking for patterns.” Michelle contributes to the arguments offered, providing warrants to affirm the validity of the claims offered by her group members. In Event 4 Michelle applies Milin’s inductive reasoning to explain how the 4-tall Towers Task solution emerges from the 3-tall Towers solution.

Events 5-6 are retrieved from a June 15, 1992 session in which Michelle and her partner, Jeff, work to solve the 3-tall Towers Task at the end of the fourth grade during a written assessment. Michelle and Jeff collaborate to jointly produce a written solution that includes an explanation, written by Michelle, along with drawings of towers organized by cases, drawn by Jeff. Event 5 illustrates their methodology by arranging 3-tall towers by elevator and color opposite patterns. Event 6 serves to illustrate their collaboration for their final solution and argument.

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

Strategies of locally exhaustive, systematic enumeration:
Color “Opposites” (children’s language)
Each element in a combination is replaced with the opposite element. The opposite of a tower in two colors is a tower of the same height where each position holds the opposite color of the cube in the corresponding position of the first tower. For example, a four-tall tower with yellow, blue, blue, blue and one with blue, yellow, yellow, yellow are opposites (Maher, Sran, & Yankelewitz, 2011).

“Elevator” strategy (Jeff’s language)
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. To create a second tower, the cube is then moved to the second position. The cube is continuously lowered one position to create new towers until it is placed in the final position (Maher, Sran & Yankelewitz, 2011).

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

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 (towers with a single color); 2) three towers containing one red (towers with exactly one of color); 3) three towers containing 2 red cubes (within cases (2) and (3) can be cubes of the same color adjacent to or separated from each other); 4) one tower containing 3 reds or all red cubes. An argument by cases includes an exhaustive enumeration of the total number of towers in each case.