PurposesStudent model building; Reasoning; Representation

DescriptionAuthor: Victoria Krupnik, Rutgers University

This analytic is the first of two analytics that showcase the development of an argument by induction to solve a counting task by a student, Milin, over two school years. Both analytics focus on the reasoning, argumentation, and mathematical representations constructed by Milin in a variety of settings, over time, in the fourth (Parts 1 and 2) and fifth (Part 2) grades. The first analytic includes events where Milin is working in a one-on-one interview with a researcher. The second analytic follows with events where Milin participates in a small group formative assessment interview and works on a summative assessment with a partner.

In the current analytic (the first of the two analytics), Milin’s learning progression in building a justification by inductive reasoning to Tower Tasks shows him, in the fourth grade, noticing an even number solution, estimating and seeking numerical patterns for various heights, generating towers inductively (selecting from two and three colors from 1- to 2-tall), and noticing a discrepancy in number solutions between his inductive pattern and a methodology based on cases. His problem solving is presented in a series of events that serve to focus, in detail, on the explanations, reasoning, and argumentation offered by Milin during his problem solving.

For background, prior to the events of this analytic, Milin’s journey, in the fourth grade, included building solutions based on cases in a whole class session occurring on February 6, 1992 and in two one-on-one interviews facilitated by Researcher Alston (R1) occurring on February 7, 1992 and February 21, 1992. In the interviews he had opportunities to estimate and, after building the solutions based on cases and opposites, to consider the numerical solutions for 1-, 2-, 3-, 4-, 5-, and 6-tall Tower Tasks when selecting from two colors. He also built towers 2- and 3-tall, selecting from three colors, although this is not included in the analytic, and sought numerical connections between his solutions for 1- and 2-tall Tower Tasks, selecting from two and from three colors. Refer to Parts 1 and 2 of the following VMCAnalytics: Milin’s development of reasoning by cases to solve Tower Tasks.

This analytic includes six events, which are retrieved from a third one-on-one interview facilitated by R1 that occurred on March 6, 1992, in the fourth grade. Milin reported finding 50 towers, 6-tall, using cases, elevator, and color opposite strategies when he built the towers at home. This third interview included Milin’s discovery of a doubling pattern among Tower Tasks of consecutive heights and the development of his inductive reasoning. Milin developed an inductive argument for an observed doubling pattern for Tower Tasks up to 5-tall and he predicted that the pattern stopped after 5-tall towers. He reflected on his constructions for the 6-tall towers, selecting from two colors, and the 3-tall towers, selecting from three colors, by partial cases (that he did for homework) and by the doubling pattern (that he partly constructed in this interview). Specifically, he noted doubt that his strategy by cases may not have resulted in all possible towers and predicted that, if the doubling pattern worked, then the 6-tall Towers number solution would be 64, rather than 50 (his original findings using a method by cases).

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 (Alston & Maher, 1993; Sran, 2010; Maher, Sran, & Yankelewitz, 2011)

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 two (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 two colors):

You have plastic cubes of two colors available to build towers. Your task is to make as many different looking towers as possible, each exactly three cubes high. Find a way to convince yourself and others that you have found all possible towers three cubes high, and that you have no duplicates. 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 Milin’s journey):

B60, Milin and Michael classwork of the 5-tall towers problem (work view), Grade 4, Feb 6, 1992, raw footage. Retrieved from: https://doi.org/doi:10.7282/T34M985H

B66, Milin third interview, Towers problem, Grade 4, 5-tall, March 6, 1992, Work view-Raw. Retrieved from: https://doi.org/doi:10.7282/t3-84hw-y657

References:

Alston, A., & Maher, C. A. (1993). Tracing Milin’s building of proof by mathematical induction: A case study. In Proceedings of the fifteenth annual meeting for the North American chapter for the psychology of mathematics education (Vol. 2, pp. 1-7).

Maher, C. A., Sran, M. K., & Yankelewitz, D. (2011). Building an inductive argument. In C. A. Maher, A. B. Powell, & E. B. Uptegrove, Combinatorics and Reasoning (pp. 27-43). New York: Springer.

Sran, M. K. (2010). Tracing Milin’s development of inductive reasoning: A case study (Doctoral dissertation, Rutgers University-Graduate School of Education).

This analytic is the first of two analytics that showcase the development of an argument by induction to solve a counting task by a student, Milin, over two school years. Both analytics focus on the reasoning, argumentation, and mathematical representations constructed by Milin in a variety of settings, over time, in the fourth (Parts 1 and 2) and fifth (Part 2) grades. The first analytic includes events where Milin is working in a one-on-one interview with a researcher. The second analytic follows with events where Milin participates in a small group formative assessment interview and works on a summative assessment with a partner.

In the current analytic (the first of the two analytics), Milin’s learning progression in building a justification by inductive reasoning to Tower Tasks shows him, in the fourth grade, noticing an even number solution, estimating and seeking numerical patterns for various heights, generating towers inductively (selecting from two and three colors from 1- to 2-tall), and noticing a discrepancy in number solutions between his inductive pattern and a methodology based on cases. His problem solving is presented in a series of events that serve to focus, in detail, on the explanations, reasoning, and argumentation offered by Milin during his problem solving.

For background, prior to the events of this analytic, Milin’s journey, in the fourth grade, included building solutions based on cases in a whole class session occurring on February 6, 1992 and in two one-on-one interviews facilitated by Researcher Alston (R1) occurring on February 7, 1992 and February 21, 1992. In the interviews he had opportunities to estimate and, after building the solutions based on cases and opposites, to consider the numerical solutions for 1-, 2-, 3-, 4-, 5-, and 6-tall Tower Tasks when selecting from two colors. He also built towers 2- and 3-tall, selecting from three colors, although this is not included in the analytic, and sought numerical connections between his solutions for 1- and 2-tall Tower Tasks, selecting from two and from three colors. Refer to Parts 1 and 2 of the following VMCAnalytics: Milin’s development of reasoning by cases to solve Tower Tasks.

This analytic includes six events, which are retrieved from a third one-on-one interview facilitated by R1 that occurred on March 6, 1992, in the fourth grade. Milin reported finding 50 towers, 6-tall, using cases, elevator, and color opposite strategies when he built the towers at home. This third interview included Milin’s discovery of a doubling pattern among Tower Tasks of consecutive heights and the development of his inductive reasoning. Milin developed an inductive argument for an observed doubling pattern for Tower Tasks up to 5-tall and he predicted that the pattern stopped after 5-tall towers. He reflected on his constructions for the 6-tall towers, selecting from two colors, and the 3-tall towers, selecting from three colors, by partial cases (that he did for homework) and by the doubling pattern (that he partly constructed in this interview). Specifically, he noted doubt that his strategy by cases may not have resulted in all possible towers and predicted that, if the doubling pattern worked, then the 6-tall Towers number solution would be 64, rather than 50 (his original findings using a method by cases).

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 (Alston & Maher, 1993; Sran, 2010; Maher, Sran, & Yankelewitz, 2011)

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 two (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 two colors):

You have plastic cubes of two colors available to build towers. Your task is to make as many different looking towers as possible, each exactly three cubes high. Find a way to convince yourself and others that you have found all possible towers three cubes high, and that you have no duplicates. 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 Milin’s journey):

B60, Milin and Michael classwork of the 5-tall towers problem (work view), Grade 4, Feb 6, 1992, raw footage. Retrieved from: https://doi.org/doi:10.7282/T34M985H

B66, Milin third interview, Towers problem, Grade 4, 5-tall, March 6, 1992, Work view-Raw. Retrieved from: https://doi.org/doi:10.7282/t3-84hw-y657

References:

Alston, A., & Maher, C. A. (1993). Tracing Milin’s building of proof by mathematical induction: A case study. In Proceedings of the fifteenth annual meeting for the North American chapter for the psychology of mathematics education (Vol. 2, pp. 1-7).

Maher, C. A., Sran, M. K., & Yankelewitz, D. (2011). Building an inductive argument. In C. A. Maher, A. B. Powell, & E. B. Uptegrove, Combinatorics and Reasoning (pp. 27-43). New York: Springer.

Sran, M. K. (2010). Tracing Milin’s development of inductive reasoning: A case study (Doctoral dissertation, Rutgers University-Graduate School of Education).

Created on2019-08-13T00:26:01-0500

Published on2019-12-23T10:10:00-0500

Persistent URLhttps://doi.org/doi:10.7282/t3-hrjs-jq34