PurposesLesson activity; Student elaboration; 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 cases to solve a counting task over two school years. The three analytics focus on the reasoning, argumentation, and mathematical representations constructed by Stephanie in a variety of settings, over time, in third (Part 1) and fourth (Parts 1, 2, and 3) grades. The first analytic begins with Stephanie working with a partner, in a whole class setting, building towers with plastic cubes available in two colors that are 4-tall, in third grade, and 5-tall, in fourth grade. This is followed by events from a first one-on-one interview with a researcher. The second analytic follows with events from a second and third one-on-one interview with a researcher. The third analytic follows with events when Stephanie participated in a small group formative assessment interview and then with a partner on a summative assessment.

In this analytic (the first of three), Stephanie’s learning progression is shown as she builds a justification by cases for her solution to Tower Tasks. In third grade, Stephanie solves the 4-tall Tower Task by a guess and check strategy supported by trial and error outcomes, and, in fourth grade, she develops strategies of locally exhaustive enumeration by composite operations and recursion. Her problem solving is presented in a series of 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 third-grade class session, facilitated by Researcher Alston (R1), on October 11, 1990. Partners, Stephanie and Dana, worked on solving the 4-tall Tower Task. These events are shown for the purpose of showing Stephanie’s early third-grade solutions of the Tower task by guess and check and trial and error strategies and identifying opposite and inverse relationships (see definitions below) between pairs of towers.

In Events 4-5, Stephanie, in the fourth grade, with her partner Dana, solves the 5-tall Tower Task on February 6, 1992. They develop a heuristic to generate towers by color opposites and inverses (i.e., a composite operation). In Event 4 they are seen to utilize both pairing strategies to generate and organize the 5-tall towers. In Event 5 Stephanie explains her reasoning for why certainty of the Tower Task solution is not possible. Events 6-7 are retrieved from the same session when the class shared their solutions. The purpose of these events is to display Stephanie’s participation and reasoning as the class explored the cases of exactly one red and exactly two reds adjacent by recursive elevator patterns and the case of exactly two red separated by at least one yellow by the odometer strategy (see definitions below).

Events 8-10 are retrieved from a one-on-one interview facilitated by Researcher Maher (R2) that occurred on the following day on February 7, 1992. Stephanie uses similar patterns that were explored in the class discussion of the previous day, such as a staircase and an elevator (see definition below). She deals with duplication that arise due to these two types of pattern. She also returns to her strategies of inverse and color opposites.

The following definitions and background information about the Tower task are offered.

Duplicate

This occurs when there is a repetition of color cubes in the possible positions. For Towers, this would be two identical towers or images of towers.

Guess and Check

The strategy of guess and check involves first guessing an outcome then checking that the outcome is applicable to the solution. Students can be observed using the Guess and Check method when building a tower pattern in a random order (or with no observable method) and then double-checking for duplicate towers (Maher & Martino, 1996). This occurred during the construction and generation of possibilities to obtain a solution.

Trial and Error

This strategy involves testing of a solution for the Tower Task. It can involve verifying that no outcomes are missing from a solution set or verifying that all existing outcomes are different within a solution set. The trial and error strategy can produce two results: “error” or “no error.” In the situation of verifying no outcomes are missing, “no error” occurs when the tower generated is a duplicate of a tower in the solution set. In the same situation, “error” occurs when a counterexample to the tested solution is found - the tower generated is a new tower pattern that did not formerly exist in the solution set. In the situation when checking for differences or for duplication, “no error” occurs when each tower that is checked against the solution set is different from each other. In the same situation, “error” occurs when two towers are duplicates and one is eliminated. In the latter case, this is a counterexample to the proposed solution.

Strategies of locally exhaustive, systematic enumeration (Maher & Martino, 1996, 2000, 2013):

Color “Opposites” (children’s language)

Each element in a combination containing exactly two types of a particular characteristic, such as color, is replaced with the element of the opposite characteristic. In the combinatorics strand this is known as the strategy of symmetry (Janackova & Janacek, 2006). The opposite of a tower in two colors is a tower of the same height where the cube in each position is the opposite color of the cube in the corresponding position of the first tower. For example, a 4-tall tower with YBBB and a tower with BYYY are considered to be opposites.

Inverse towers or “Switched around” or “Duplicate” (Milin’s language)

Two towers are said to be inverses of each other if one tower can be rotated vertically (180 degrees) to form the pattern for the second tower. For example, a 4-tall tower with YBBB and a tower with BBBY are inverses.

“Staircase” (Ankur’s Language)

The staircase pattern is named as such due to its resemblance to a staircase. In towers of two colors, the first tower begins with the first three positions as the same color followed by the 2nd color in the last position. In each new tower, the number of cubes of the 2nd color increases from the bottom by one cube until the final tower is a solid tower of that color (Maher, Sran & Yankelewitz, 2011).

“Elevator” strategy (Jeff’s language; Milin called this “staircases”)

The elevator pattern is used when finding all possible towers containing one cube of one color and the remaining cubes of the other color. The single colored cube is placed in the first position of the first tower (either at the top or the bottom level). To create a second tower, the cube is then moved to the second position (descending, if starting at the top level, or ascending, if starting at the bottom level). The cube is recursively raised or lowered to the next available level to create new towers until it is placed in the final position (Maher, Sran & Yankelewitz, 2011).

Strategy of a globally exhaustive systematic enumeration (Maher & Martino, 1996, 2000, 2013; Batanero et al., 1997):

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 cubes (towers with a single color blue); 2) three towers containing one red (towers with exactly one of a particular color); 3) three towers containing 2 red (within cases (2) and (3) can be cubes of same color adjacent to or separated from each other); 4) one tower containing three red, i.e. with all red. An argument by cases would include an exhaustive enumeration of the total number of towers in each case.

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):

Towers with Stephanie and Dana, Clip 2 of 5: Finding seventeen towers and checking for duplicates. Retrieved from: https://doi.org/doi:10.7282/T3KK995M

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

B61, Stephanie revisits the five-tall Tower task (work view), Grade 4, February 6, 1992, raw footage. Retrieved from: https://doi.org/doi:10.7282/T3MG7SQ3

B61, Stephanie revisits the five-tall Towers problem (work view), Grade 4, February 6, 1992, raw footage. Retrieved from: https://doi.org/doi:10.7282/T3MG7SQ3

References

Batanero, C., Godino, J. D., & Navarro-Pelayo, V. (1997). Combinatorial reasoning and its assessment. In I. Gal, & J. B. Garfield, The assessment challenge in statistics education (pp. 239–252). Amsterdam: IOS Press.

Janácková, M., & Janácek, J. (2006). A classification of strategies employed by high school students in isomorphic combinatorial problems. The Mathematics Enthusiast, 3(2), 128-145. Retrieved from: https://scholarworks.umt.edu/tme/vol3/iss2/2

Maher, C. A. (2010). The Longitudinal Study. In C. A. Maher, A. B. Powell, & E. B. Uptegrove, Combinatorics and Reasoning (pp. 3–14). New York: Springer.

Maher, C. A., Powell, A. B., & Uptegrove, E. B. (2010). Combinatorics and reasoning. In C. A. Maher, A. B. Powell, & E. B. Uptegrove, Representing, justifying and building isomorphisms (Vol. 47). New York: Springer.

Maher, C. A., & Martino, A. M. (1996). The Development of the Idea of Mathematical Proof: A 5-Year Case Study. Journal for Research in Mathematics Education, 27(2), 194-214.

Maher, C. A., & Martino, A. M. (2000). From patterns to theories: Conditions for conceptual change. The Journal of Mathematical Behavior, 19(2), 247-271.

Maher, C. A., & Martino, A. M. (2013). Young children invent methods of proof: The gang of four. In L. Steffe, P. Nesher, P. Cobb, G. Goldin, & B. Greer (Eds.), Theories of mathematical learning (pp. 443-460). Hillsdale, NJ: Erlbaum.

Maher, C. A., Sran, M. K., & Yankelewitz, D. (2011). Towers: Schemes, strategies, and arguments. In C. A. Maher, A. B. Powell, & E. B. Uptegrove, Combinatorics and Reasoning (pp. 27-43). New York: Springer.

Overall Description

This analytic is the first of three analytics that showcase Stephanie’s development of an argument by cases to solve a counting task over two school years. The three analytics focus on the reasoning, argumentation, and mathematical representations constructed by Stephanie in a variety of settings, over time, in third (Part 1) and fourth (Parts 1, 2, and 3) grades. The first analytic begins with Stephanie working with a partner, in a whole class setting, building towers with plastic cubes available in two colors that are 4-tall, in third grade, and 5-tall, in fourth grade. This is followed by events from a first one-on-one interview with a researcher. The second analytic follows with events from a second and third one-on-one interview with a researcher. The third analytic follows with events when Stephanie participated in a small group formative assessment interview and then with a partner on a summative assessment.

In this analytic (the first of three), Stephanie’s learning progression is shown as she builds a justification by cases for her solution to Tower Tasks. In third grade, Stephanie solves the 4-tall Tower Task by a guess and check strategy supported by trial and error outcomes, and, in fourth grade, she develops strategies of locally exhaustive enumeration by composite operations and recursion. Her problem solving is presented in a series of 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 third-grade class session, facilitated by Researcher Alston (R1), on October 11, 1990. Partners, Stephanie and Dana, worked on solving the 4-tall Tower Task. These events are shown for the purpose of showing Stephanie’s early third-grade solutions of the Tower task by guess and check and trial and error strategies and identifying opposite and inverse relationships (see definitions below) between pairs of towers.

In Events 4-5, Stephanie, in the fourth grade, with her partner Dana, solves the 5-tall Tower Task on February 6, 1992. They develop a heuristic to generate towers by color opposites and inverses (i.e., a composite operation). In Event 4 they are seen to utilize both pairing strategies to generate and organize the 5-tall towers. In Event 5 Stephanie explains her reasoning for why certainty of the Tower Task solution is not possible. Events 6-7 are retrieved from the same session when the class shared their solutions. The purpose of these events is to display Stephanie’s participation and reasoning as the class explored the cases of exactly one red and exactly two reds adjacent by recursive elevator patterns and the case of exactly two red separated by at least one yellow by the odometer strategy (see definitions below).

Events 8-10 are retrieved from a one-on-one interview facilitated by Researcher Maher (R2) that occurred on the following day on February 7, 1992. Stephanie uses similar patterns that were explored in the class discussion of the previous day, such as a staircase and an elevator (see definition below). She deals with duplication that arise due to these two types of pattern. She also returns to her strategies of inverse and color opposites.

The following definitions and background information about the Tower task are offered.

Duplicate

This occurs when there is a repetition of color cubes in the possible positions. For Towers, this would be two identical towers or images of towers.

Guess and Check

The strategy of guess and check involves first guessing an outcome then checking that the outcome is applicable to the solution. Students can be observed using the Guess and Check method when building a tower pattern in a random order (or with no observable method) and then double-checking for duplicate towers (Maher & Martino, 1996). This occurred during the construction and generation of possibilities to obtain a solution.

Trial and Error

This strategy involves testing of a solution for the Tower Task. It can involve verifying that no outcomes are missing from a solution set or verifying that all existing outcomes are different within a solution set. The trial and error strategy can produce two results: “error” or “no error.” In the situation of verifying no outcomes are missing, “no error” occurs when the tower generated is a duplicate of a tower in the solution set. In the same situation, “error” occurs when a counterexample to the tested solution is found - the tower generated is a new tower pattern that did not formerly exist in the solution set. In the situation when checking for differences or for duplication, “no error” occurs when each tower that is checked against the solution set is different from each other. In the same situation, “error” occurs when two towers are duplicates and one is eliminated. In the latter case, this is a counterexample to the proposed solution.

Strategies of locally exhaustive, systematic enumeration (Maher & Martino, 1996, 2000, 2013):

Color “Opposites” (children’s language)

Each element in a combination containing exactly two types of a particular characteristic, such as color, is replaced with the element of the opposite characteristic. In the combinatorics strand this is known as the strategy of symmetry (Janackova & Janacek, 2006). The opposite of a tower in two colors is a tower of the same height where the cube in each position is the opposite color of the cube in the corresponding position of the first tower. For example, a 4-tall tower with YBBB and a tower with BYYY are considered to be opposites.

Inverse towers or “Switched around” or “Duplicate” (Milin’s language)

Two towers are said to be inverses of each other if one tower can be rotated vertically (180 degrees) to form the pattern for the second tower. For example, a 4-tall tower with YBBB and a tower with BBBY are inverses.

“Staircase” (Ankur’s Language)

The staircase pattern is named as such due to its resemblance to a staircase. In towers of two colors, the first tower begins with the first three positions as the same color followed by the 2nd color in the last position. In each new tower, the number of cubes of the 2nd color increases from the bottom by one cube until the final tower is a solid tower of that color (Maher, Sran & Yankelewitz, 2011).

“Elevator” strategy (Jeff’s language; Milin called this “staircases”)

The elevator pattern is used when finding all possible towers containing one cube of one color and the remaining cubes of the other color. The single colored cube is placed in the first position of the first tower (either at the top or the bottom level). To create a second tower, the cube is then moved to the second position (descending, if starting at the top level, or ascending, if starting at the bottom level). The cube is recursively raised or lowered to the next available level to create new towers until it is placed in the final position (Maher, Sran & Yankelewitz, 2011).

Strategy of a globally exhaustive systematic enumeration (Maher & Martino, 1996, 2000, 2013; Batanero et al., 1997):

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 cubes (towers with a single color blue); 2) three towers containing one red (towers with exactly one of a particular color); 3) three towers containing 2 red (within cases (2) and (3) can be cubes of same color adjacent to or separated from each other); 4) one tower containing three red, i.e. with all red. An argument by cases would include an exhaustive enumeration of the total number of towers in each case.

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):

Towers with Stephanie and Dana, Clip 2 of 5: Finding seventeen towers and checking for duplicates. Retrieved from: https://doi.org/doi:10.7282/T3KK995M

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

B61, Stephanie revisits the five-tall Tower task (work view), Grade 4, February 6, 1992, raw footage. Retrieved from: https://doi.org/doi:10.7282/T3MG7SQ3

B61, Stephanie revisits the five-tall Towers problem (work view), Grade 4, February 6, 1992, raw footage. Retrieved from: https://doi.org/doi:10.7282/T3MG7SQ3

References

Batanero, C., Godino, J. D., & Navarro-Pelayo, V. (1997). Combinatorial reasoning and its assessment. In I. Gal, & J. B. Garfield, The assessment challenge in statistics education (pp. 239–252). Amsterdam: IOS Press.

Janácková, M., & Janácek, J. (2006). A classification of strategies employed by high school students in isomorphic combinatorial problems. The Mathematics Enthusiast, 3(2), 128-145. Retrieved from: https://scholarworks.umt.edu/tme/vol3/iss2/2

Maher, C. A. (2010). The Longitudinal Study. In C. A. Maher, A. B. Powell, & E. B. Uptegrove, Combinatorics and Reasoning (pp. 3–14). New York: Springer.

Maher, C. A., Powell, A. B., & Uptegrove, E. B. (2010). Combinatorics and reasoning. In C. A. Maher, A. B. Powell, & E. B. Uptegrove, Representing, justifying and building isomorphisms (Vol. 47). New York: Springer.

Maher, C. A., & Martino, A. M. (1996). The Development of the Idea of Mathematical Proof: A 5-Year Case Study. Journal for Research in Mathematics Education, 27(2), 194-214.

Maher, C. A., & Martino, A. M. (2000). From patterns to theories: Conditions for conceptual change. The Journal of Mathematical Behavior, 19(2), 247-271.

Maher, C. A., & Martino, A. M. (2013). Young children invent methods of proof: The gang of four. In L. Steffe, P. Nesher, P. Cobb, G. Goldin, & B. Greer (Eds.), Theories of mathematical learning (pp. 443-460). Hillsdale, NJ: Erlbaum.

Maher, C. A., Sran, M. K., & Yankelewitz, D. (2011). Towers: Schemes, strategies, and arguments. In C. A. Maher, A. B. Powell, & E. B. Uptegrove, Combinatorics and Reasoning (pp. 27-43). New York: Springer.

Created on2019-02-07T14:25:11-0500

Published on2020-03-03T15:53:11-0500

Persistent URLhttps://doi.org/doi:10.7282/t3-9zz5-za71