PurposesStudent model building; Reasoning; Representation

DescriptionAuthor Victoria Krupnik (Rutgers University - graduate)

Overall Description

This analytic is the third of three analytics that showcase Stephanie’s development of an argument by induction to solve a counting task over three school years. The three analytics focus on the reasoning, argumentation, and mathematical representations constructed by Stephanie in a variety of settings over time, in the third (Part 1), fourth (Parts 1 and 2), and fifth grades (Part 3). The first analytic begins with events where third grader Stephanie is working with a partner, in a whole class setting, building towers that are 3- and 4-tall, with plastic cubes available in two colors. This is followed by events in which Stephanie is working in a one-on-one interview with a researcher in the third and then fourth grade. The second analytic follows with events where fourth-grader Stephanie participates in a small group formative assessment and works on a summative assessment with a partner. The third analytic follows with events where fifth grader Stephanie is working with a partner, in a whole class setting, and then presenting her ideas to a small group and then to the whole class.

In this analytic (the third of three), Stephanie’s learning progression in building a justification by inductive reasoning to Tower Tasks shows Stephanie in the fifth grade recalling and testing a doubling pattern between Tower Tasks, observing Michelle and Matt present an inductive method to generate taller towers from shorter towers to justify the doubling pattern, and, finally, presenting an inductive argument for the growth of towers for any height. Her problem solving is presented in events that serve to focus, in detail, on the explanations, reasoning, and argumentation Stephanie offers during her problem solving.

Events 1-7 are retrieved from a fifth-grade class session with partners Stephanie and Matt as they worked on a Guess My Tower (GMT) task on February 26, 1992. Solving the problem promoted revisiting the 3- and 4-tall tower outcome possibilities. Events 1-2 show Stephanie attempting to test a doubling rule that she recalls from prior Towers problem-solving experiences. Milin’s inductive approach to finding taller towers from shorter towers is disseminated to other students during this session, first to Michelle, then to Stephanie and Matt (Event 3), and then to the rest of the class (Events 4-7). The purpose of the GMT events is to explore Stephanie’s learning of his idea. The events presented are evidence of Stephanie’s individual reasoning about the doubling pattern and her reconciliation between the doubling rule and why it worked.

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

An induction argument for the justification of the general 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 2 (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 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”).

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 Stephanie’s journey):

B78,Guess My Tower task,Matt and Stephanie partner work (work view), Grade 5,February 26,1993,Raw footage. Retrieved from: https://doi.org/doi:10.7282/t3-hw4j-2f57

Building Towers, Selecting from two colors for Guess My Tower, Clip 2 of 5: Does the Number Double? Retrieved from: https://doi.org/doi:10.7282/T32V2FBZ

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

Building Towers, Selecting from two colors for Guess My Tower, Clip 4 of 5: Stephanie and Matt Rebuild the Argument. Retrieved from: https://doi.org/doi:10.7282/T3W958DV

Building Towers, Selecting from two colors for Guess My Tower, Clip 5 of 5: Sharing with the Group. Retrieved from: https://doi.org/doi:10.7282/T36M3617

Overall Description

This analytic is the third of three analytics that showcase Stephanie’s development of an argument by induction to solve a counting task over three school years. The three analytics focus on the reasoning, argumentation, and mathematical representations constructed by Stephanie in a variety of settings over time, in the third (Part 1), fourth (Parts 1 and 2), and fifth grades (Part 3). The first analytic begins with events where third grader Stephanie is working with a partner, in a whole class setting, building towers that are 3- and 4-tall, with plastic cubes available in two colors. This is followed by events in which Stephanie is working in a one-on-one interview with a researcher in the third and then fourth grade. The second analytic follows with events where fourth-grader Stephanie participates in a small group formative assessment and works on a summative assessment with a partner. The third analytic follows with events where fifth grader Stephanie is working with a partner, in a whole class setting, and then presenting her ideas to a small group and then to the whole class.

In this analytic (the third of three), Stephanie’s learning progression in building a justification by inductive reasoning to Tower Tasks shows Stephanie in the fifth grade recalling and testing a doubling pattern between Tower Tasks, observing Michelle and Matt present an inductive method to generate taller towers from shorter towers to justify the doubling pattern, and, finally, presenting an inductive argument for the growth of towers for any height. Her problem solving is presented in events that serve to focus, in detail, on the explanations, reasoning, and argumentation Stephanie offers during her problem solving.

Events 1-7 are retrieved from a fifth-grade class session with partners Stephanie and Matt as they worked on a Guess My Tower (GMT) task on February 26, 1992. Solving the problem promoted revisiting the 3- and 4-tall tower outcome possibilities. Events 1-2 show Stephanie attempting to test a doubling rule that she recalls from prior Towers problem-solving experiences. Milin’s inductive approach to finding taller towers from shorter towers is disseminated to other students during this session, first to Michelle, then to Stephanie and Matt (Event 3), and then to the rest of the class (Events 4-7). The purpose of the GMT events is to explore Stephanie’s learning of his idea. The events presented are evidence of Stephanie’s individual reasoning about the doubling pattern and her reconciliation between the doubling rule and why it worked.

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

An induction argument for the justification of the general 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 2 (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 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”).

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 Stephanie’s journey):

B78,Guess My Tower task,Matt and Stephanie partner work (work view), Grade 5,February 26,1993,Raw footage. Retrieved from: https://doi.org/doi:10.7282/t3-hw4j-2f57

Building Towers, Selecting from two colors for Guess My Tower, Clip 2 of 5: Does the Number Double? Retrieved from: https://doi.org/doi:10.7282/T32V2FBZ

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

Building Towers, Selecting from two colors for Guess My Tower, Clip 4 of 5: Stephanie and Matt Rebuild the Argument. Retrieved from: https://doi.org/doi:10.7282/T3W958DV

Building Towers, Selecting from two colors for Guess My Tower, Clip 5 of 5: Sharing with the Group. Retrieved from: https://doi.org/doi:10.7282/T36M3617

Created on2019-09-16T12:13:39-0500

Published on2020-03-03T16:03:44-0500

Persistent URLhttps://doi.org/doi:10.7282/t3-jv0q-p284