PurposesLesson activity; Student elaboration; Student model building; 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 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 second of three), Stephanie’s learning progression is shown as she builds a justification by cases for her solution to Tower Tasks. In fourth grade, Stephanie solves the 6-tall Tower Task by partial cases, controlling for variables, recognizing duplication within equivalent cases, and then she solves the 4-tall Tower Task by complete cases. Her problem solving is presented in a series of events that serve to focus, in detail, on the explanations, reasoning, and argumentation she offers during her problem solving.

Events 1-5 illustrate Stephanie’s refinement, modification, and application of the case and/or color opposite strategies to solve the 6-tall Tower task in a second interview facilitated by Researcher Maher (R2) on February 21, 1992. She also uses the odometer strategy to account for all towers with exactly two of a color separated by at least one of the other color. She finds 50 towers by partial cases and is asked to complete another case for homework.

Events 6-11 show a third interview facilitated by R2 on March 6, 1992 where Stephanie reports finding 20 towers, 4-tall, using cases, elevator, and color opposite strategies. She reproduces the solution (without having her homework in front of her) by cases of towers with exactly no white cubes, one white cube, two white cubes together, two white cubes separated, three white cubes together, three white cubes separated, and four white cubes. In Events 19-20 she reconciles the mismatch between her home solution by both cases and opposite strategies and her solution of 16 by cases only.

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.

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.

“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).

Controlling for variables - Strategy of the odometer (English, 1991; 1993)

Controlling for variables is a method in which one variable is held constant in position x while the other variable progressively occupies all remaining positions from the highest to the lowest or the lowest to the highest (but not both) without repeating previously discovered combinations (English, 1993; Janackova & Janacek, 2006). After exhausting all possibilities, next position x+1 is chosen for the constant variable and the process repeats (Janackova & Janacek, 2006). This strategy ends when all possibilities for the choice of the constant variable are exhausted. An example of this when building towers is when one color of the tower is held constant in one position while the color arrangements in all other positions are varied recursively (Maher & Martino, 1996).

Case organization and/or argument (an example of globally exhaustive systematic enumeration; Maher & Martino, 1996, 2000, 2013; Batanero et al., 1997).

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.

Equivalent cases

The case when selecting m times of a color to place into n positions is equivalent to the case when selecting n–m of the opposite color into n positions. In the example for Towers combinations, selecting two blues to place into five available positions is equivalent to the case of selecting three reds to place into five available positions. The towers with the attribute of exactly two of a color in towers 5-tall and the towers with the attribute of exactly three of the opposite color result in duplicates (e.g., RRBBB have both two reds and three blues). In the figure the first and last tower of each set are duplicates and occur when the strategy of (color opposite) symmetry is applied for each case of adjacent blues and then exhausting all adjacent blue cases, thereby repeating combinations. To avoid repetition, the strategies in combination must be taken with caution.

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):

B62, Stephanie’s and Milin’s second of three interview sessions and Michelle’s second of two interview sessions revisiting five-tall Towers and other heights (work view), Grade 4, Feb 21, 1992, raw footage. Retrieved from:

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

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.

English, L. D. (1993). Children’s Strategies for Solving Two-and Three-Dimensional Combinatorial Problems. Journal for Research in Mathematics Education, 24(3), 255-73.

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 second 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 second of three), Stephanie’s learning progression is shown as she builds a justification by cases for her solution to Tower Tasks. In fourth grade, Stephanie solves the 6-tall Tower Task by partial cases, controlling for variables, recognizing duplication within equivalent cases, and then she solves the 4-tall Tower Task by complete cases. Her problem solving is presented in a series of events that serve to focus, in detail, on the explanations, reasoning, and argumentation she offers during her problem solving.

Events 1-5 illustrate Stephanie’s refinement, modification, and application of the case and/or color opposite strategies to solve the 6-tall Tower task in a second interview facilitated by Researcher Maher (R2) on February 21, 1992. She also uses the odometer strategy to account for all towers with exactly two of a color separated by at least one of the other color. She finds 50 towers by partial cases and is asked to complete another case for homework.

Events 6-11 show a third interview facilitated by R2 on March 6, 1992 where Stephanie reports finding 20 towers, 4-tall, using cases, elevator, and color opposite strategies. She reproduces the solution (without having her homework in front of her) by cases of towers with exactly no white cubes, one white cube, two white cubes together, two white cubes separated, three white cubes together, three white cubes separated, and four white cubes. In Events 19-20 she reconciles the mismatch between her home solution by both cases and opposite strategies and her solution of 16 by cases only.

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.

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.

“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).

Controlling for variables - Strategy of the odometer (English, 1991; 1993)

Controlling for variables is a method in which one variable is held constant in position x while the other variable progressively occupies all remaining positions from the highest to the lowest or the lowest to the highest (but not both) without repeating previously discovered combinations (English, 1993; Janackova & Janacek, 2006). After exhausting all possibilities, next position x+1 is chosen for the constant variable and the process repeats (Janackova & Janacek, 2006). This strategy ends when all possibilities for the choice of the constant variable are exhausted. An example of this when building towers is when one color of the tower is held constant in one position while the color arrangements in all other positions are varied recursively (Maher & Martino, 1996).

Case organization and/or argument (an example of globally exhaustive systematic enumeration; Maher & Martino, 1996, 2000, 2013; Batanero et al., 1997).

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.

Equivalent cases

The case when selecting m times of a color to place into n positions is equivalent to the case when selecting n–m of the opposite color into n positions. In the example for Towers combinations, selecting two blues to place into five available positions is equivalent to the case of selecting three reds to place into five available positions. The towers with the attribute of exactly two of a color in towers 5-tall and the towers with the attribute of exactly three of the opposite color result in duplicates (e.g., RRBBB have both two reds and three blues). In the figure the first and last tower of each set are duplicates and occur when the strategy of (color opposite) symmetry is applied for each case of adjacent blues and then exhausting all adjacent blue cases, thereby repeating combinations. To avoid repetition, the strategies in combination must be taken with caution.

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):

B62, Stephanie’s and Milin’s second of three interview sessions and Michelle’s second of two interview sessions revisiting five-tall Towers and other heights (work view), Grade 4, Feb 21, 1992, raw footage. Retrieved from:

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

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.

English, L. D. (1993). Children’s Strategies for Solving Two-and Three-Dimensional Combinatorial Problems. Journal for Research in Mathematics Education, 24(3), 255-73.

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 on2019-09-14T20:08:54-0500

Published on2020-03-03T15:55:16-0500

Persistent URLhttps://doi.org/doi:10.7282/t3-gj4x-wr97