PurposesStudent collaboration; Student elaboration; Student engagement; Reasoning; Representation

DescriptionAuthor Victoria Krupnik (Rutgers University - graduate)

Overall Description

This analytic is the second 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 second of three), Stephanie’s learning progression is shown as she builds a justification for her solution to Tower Tasks by inductive reasoning. In fourth grade Stephanie presents an observed doubling pattern between Tower Tasks to a small group of students, presents a shortcut method to finding taller towers, and applies the doubling pattern as a rule to solve Tower Tasks up to 11-tall, selecting from two colors, in the partner summative assessment. 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 1-3 are retrieved from a small-group assessment interview (“Gang of Four”), facilitated by Researcher Maher (R2) on March 10, 1992. Stephanie presented her generalization of the doubling rule to taller towers up to 10-tall and a shortcut method for obtaining the number of towers of any height.

Event 4 is retrieved from a June 15, 1992 session in which Stephanie and her partner, Milin, solved the 3-tall Tower Task written summative assessment at the end of the fourth grade. They applied the doubling rule to verify the numerical solution of the 3-tall Tower Task.

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

Video and Transcript References (in chronological order of Stephanie’s journey):

B41, The Gang of Four (Jeff and Stephanie view), Grade 4, March 10, 1992, raw footage. Retrieved from: https://doi.org/doi:10.7282/T3CV4FWP

B75, Towers Assessment, WV, Grade 4, Jun 15, 1992, raw. Retrieved from: https://doi.org/doi:10.7282/t3-tpqc-b719

Overall Description

This analytic is the second 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 second of three), Stephanie’s learning progression is shown as she builds a justification for her solution to Tower Tasks by inductive reasoning. In fourth grade Stephanie presents an observed doubling pattern between Tower Tasks to a small group of students, presents a shortcut method to finding taller towers, and applies the doubling pattern as a rule to solve Tower Tasks up to 11-tall, selecting from two colors, in the partner summative assessment. 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 1-3 are retrieved from a small-group assessment interview (“Gang of Four”), facilitated by Researcher Maher (R2) on March 10, 1992. Stephanie presented her generalization of the doubling rule to taller towers up to 10-tall and a shortcut method for obtaining the number of towers of any height.

Event 4 is retrieved from a June 15, 1992 session in which Stephanie and her partner, Milin, solved the 3-tall Tower Task written summative assessment at the end of the fourth grade. They applied the doubling rule to verify the numerical solution of the 3-tall Tower Task.

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

Video and Transcript References (in chronological order of Stephanie’s journey):

B41, The Gang of Four (Jeff and Stephanie view), Grade 4, March 10, 1992, raw footage. Retrieved from: https://doi.org/doi:10.7282/T3CV4FWP

B75, Towers Assessment, WV, Grade 4, Jun 15, 1992, raw. Retrieved from: https://doi.org/doi:10.7282/t3-tpqc-b719

Created on2019-09-16T12:10:01-0500

Published on2020-03-03T16:00:46-0500

Persistent URLhttps://doi.org/doi:10.7282/t3-g20s-0d46