PurposesLesson activity; Student collaboration; Student elaboration; Student engagement; Student model building; Reasoning; Representation

DescriptionAuthor Victoria Krupnik (Rutgers University - graduate)

Overall Description

This analytic is the first 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 third and then in 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 first of three), Stephanie’s learning progression is shown as she builds a justification for her solution to Tower Tasks by inductive reasoning. In third grade, as she compares solutions to Tower Tasks of two heights, selecting from two colors, and, then in fourth grade, when she discovers a doubling pattern between Tower Tasks, Stephanie applies a doubling rule to solve Tower Tasks up to 11-tall. She is introduced to an inductive method to generate taller towers from shorter towers. 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 and 2 of this analytic are retrieved from a third-grade class session, facilitated by Researcher Alston (R1), on October 12, 1990. Partners Stephanie and Dana explore the 3-tall Tower Task after completing the 4-tall Tower Task. Event 3 is retrieved from a post-interview, facilitated by Researcher Martino (R3), on the same day, when Stephanie uses generic reasoning to justify why the number of 3-tall towers is fewer than the number of 4-tall towers. These events are shown for the purpose of presenting her early third-grade explorations building towers of two different heights.

Events 4-7 are retrieved from a one-on-one interview facilitated by Researcher Maher (R2) that occurred on March 6, 1992. In this interview, Stephanie recognizes a doubling relationship between consecutive tower height solutions and applied this doubling pattern to predict the solutions of Tower Tasks up to 10-tall. She is also introduced by the researcher to an inductive method to account for the doubling pattern. She uses the physical tower models to solve the Outfit Task selecting from two colors of up to five clothing options by generating outfits inductively.

The following definitions and background information about the Tower and Outfit 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”).

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 Stephanie’s journey):

Towers Group Sharing, Clip 3 of 6: Guessing how many towers can be built three cubes high, continued. Retrieved from: https://doi.org/doi:10.7282/T30K26ZR

Towers Group Sharing, Clip 6 of 6: Discussing their findings of how many towers can be built three cubes high. Retrieved from: https://doi.org/doi:10.7282/T3736P9W

Stephanie Grade 3 Towers interview excerpts. Retrieved from: https://doi.org/doi:10.7282/T3FJ2F7X

B64, Stephanie third of three interview sessions when she used a case-based method for all heights below and including four-tall Towers problems (work view), Grade 4, March 6, 1992, raw footage. Retrieved from: https://doi.org/doi:10.7282/T3SF30S5

Overall Description

This analytic is the first 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 third and then in 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 first of three), Stephanie’s learning progression is shown as she builds a justification for her solution to Tower Tasks by inductive reasoning. In third grade, as she compares solutions to Tower Tasks of two heights, selecting from two colors, and, then in fourth grade, when she discovers a doubling pattern between Tower Tasks, Stephanie applies a doubling rule to solve Tower Tasks up to 11-tall. She is introduced to an inductive method to generate taller towers from shorter towers. 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 and 2 of this analytic are retrieved from a third-grade class session, facilitated by Researcher Alston (R1), on October 12, 1990. Partners Stephanie and Dana explore the 3-tall Tower Task after completing the 4-tall Tower Task. Event 3 is retrieved from a post-interview, facilitated by Researcher Martino (R3), on the same day, when Stephanie uses generic reasoning to justify why the number of 3-tall towers is fewer than the number of 4-tall towers. These events are shown for the purpose of presenting her early third-grade explorations building towers of two different heights.

Events 4-7 are retrieved from a one-on-one interview facilitated by Researcher Maher (R2) that occurred on March 6, 1992. In this interview, Stephanie recognizes a doubling relationship between consecutive tower height solutions and applied this doubling pattern to predict the solutions of Tower Tasks up to 10-tall. She is also introduced by the researcher to an inductive method to account for the doubling pattern. She uses the physical tower models to solve the Outfit Task selecting from two colors of up to five clothing options by generating outfits inductively.

The following definitions and background information about the Tower and Outfit 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”).

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 Stephanie’s journey):

Towers Group Sharing, Clip 3 of 6: Guessing how many towers can be built three cubes high, continued. Retrieved from: https://doi.org/doi:10.7282/T30K26ZR

Towers Group Sharing, Clip 6 of 6: Discussing their findings of how many towers can be built three cubes high. Retrieved from: https://doi.org/doi:10.7282/T3736P9W

Stephanie Grade 3 Towers interview excerpts. Retrieved from: https://doi.org/doi:10.7282/T3FJ2F7X

B64, Stephanie third of three interview sessions when she used a case-based method for all heights below and including four-tall Towers problems (work view), Grade 4, March 6, 1992, raw footage. Retrieved from: https://doi.org/doi:10.7282/T3SF30S5

Created on2019-02-15T19:08:09-0500

Published on2020-03-03T15:59:02-0500

Persistent URLhttps://doi.org/doi:10.7282/t3-hfzp-8376