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doi:10.7282/t3-rjdv-ar64

Author: Victoria Krupnik, Rutgers University
Overall Description

This analytic is the third 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 2 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 third of three analytics), Michelle’s learning progression in building “proof-like” justifications to Towers Tasks shows Michelle, in the fifth grade, listening to other arguments by induction and reasoning by induction to solve 3- and 4-tall Tower Tasks, selecting from two colors. 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 fifth-grade class session with partners Michelle and Milin as they work on a Guess My Tower (GMT) task on February 26, 1993. Solving the problem promotes revisiting the 3- and 4-tall tower outcome possibilities. Michelle indicates uncertainty of the 4-tall tower total, whereas Milin claims it to be 16. The events illustrate the dissemination of Milin’s idea about the doubling pattern that is recognized as they build towers of consecutive heights, supporting Milin’s inductive reasoning. Facilitated by R2, Milin explains his idea to Michelle in Event 1, Michelle displays her understanding in Events 2-3, and then demonstrates her reasoning with Stephanie and Matt in Event 4.

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

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.

Tower Task:
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 (n) cubes high. Find a way to convince yourself and others that you have found all possible towers 3 (n) 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.

Guess My Tower Task:
You have been invited to participate in a TV Quiz Show and have the opportunity to win a vacation to Disneyworld. 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 Disneyworld. 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 Michelle’s journey):
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

Student collaboration

Student elaboration

Student engagement

Student model building

Reasoning

Representation

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