PurposesStudent model building; Reasoning; Representation

DescriptionAuthor: Victoria Krupnik, Rutgers University

This analytic is the first of two analytics that showcase the development of reasoning by cases to solve a counting task by a student, Milin, over one school year. Both analytics focus on the reasoning, argumentation, and mathematical representations constructed by Milin in a variety of settings, over time, in the fourth grade. The first analytic begins with events where Milin is working with a partner, in a whole class setting, building towers with plastic cubes that are 5-tall. This is followed by events from a first one-on-one interview with researchers. The second analytic continues with events from the first interview, as well as with events from a second interview.

In the current analytic (the first of the two analytics), Milin’s learning progression in building a justification by cases to Tower Tasks shows him using a Guess and Check strategy supported by trial and error outcomes, justifying why the solution is always an even number, developing heuristics to enumerate and locally exhaust towers by partial and complete case-based reasoning, and recognizing equivalent cases. His problem solving is presented in a series of events that serve to focus, in detail, on the explanations, reasoning, and argumentation he offers during his problem solving.

In Events 1–3, Milin, in the fourth grade, with his partner Michael, solves the 5-tall Tower Task on February 6, 1992, in a whole class setting facilitated by Researcher Maher (R2). Milin and Michael develop a heuristic to generate towers by color opposites and inverses (i.e., a composite operation), which is evidenced by some of the towers they group together. In Event 1 they explain the color opposite strategy to Researcher Martino (R3). In Event 2, Milin explains to Researcher Alston (R1) his reasoning about how the results of trial and error made him sure that their solution was complete. In Event 3 of this analytic Milin explains during a whole class discussion why the solution to the 5-tall Tower Task must be an even number.

Events 4–9 show how Milin develops a methodology that identifies cases (e.g., same attributes) of towers, their opposite cases, and exhaustively enumerates each case by patterns (i.e. by “staircases”) to solve the 5-tall Tower Task in a first interview facilitated by R1 and visiting teacher O’Brien (R4), on February 7, 1992, the day following the classroom work with his partner. Event 4 shows Milin beginning with a color opposite pairing strategy and then, recalling the previous day’s class discussion that utilized elevator patterns to generate particular cases of towers. He is asked to explain how he solved the 5-tall Tower Task. In Events 4 and 5, Milin shows how he built towers with none, one, and two adjacent cubes of a particular color (and of the opposite color). When he initiates exploration of towers with exactly three of a color in Events 6–7, the researcher encourages him to compare these towers with the towers he has already created. He recognizes that some 5-tall towers with exactly three adjacent yellow cubes can be identified as a tower with exactly two adjacent red (e.g., YYYRR* or YYYRR but not RYYYR). What follows is a dialogue about his consideration of other towers within the case of three of a color that were not accounted for in his earlier cases. In Event 8 he is encouraged to generate other unique towers. He then identifies towers with exactly three of a color with some separation of the cubes of that color. In this event he also acknowledged that some towers with some separation of a particular color replicates his earlier cases (e.g., in tower YYRRY the three yellows have some separation, but the two reds do not, and therefore replicate a tower in the two adjacent red case). In Event 9 Milin attempts to organize the towers with three yellows and two reds with some separation in an elevator pattern.

*A tower is denoted by the first letter of each color that takes on positions from top to bottom.

The following are definitions and background information about the Tower Task:

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.

“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.

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, 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 5-tall towers with the attribute of exactly two of a color 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).

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 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

B74, Combinatorics, T5T, Steph & Dana (People view), Grade 4, Feb 6,1992, Raw Footage. Retrieved from https://doi.org/doi:10.7282/t3-hvhr-mw75

B76, Milin’s first of three interviews with Researcher Alston on the 5-tall Tower Task (Work view), Grade 4, February 7, 1992, Raw footage. Retrieved from: https://doi.org/doi:10.7282/t3-e248-d631

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.

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., & Martino, A. (1996). The development of the idea of mathematical proof: A 5-year case study. Journal for Research in Mathematics Education, 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.

This analytic is the first of two analytics that showcase the development of reasoning by cases to solve a counting task by a student, Milin, over one school year. Both analytics focus on the reasoning, argumentation, and mathematical representations constructed by Milin in a variety of settings, over time, in the fourth grade. The first analytic begins with events where Milin is working with a partner, in a whole class setting, building towers with plastic cubes that are 5-tall. This is followed by events from a first one-on-one interview with researchers. The second analytic continues with events from the first interview, as well as with events from a second interview.

In the current analytic (the first of the two analytics), Milin’s learning progression in building a justification by cases to Tower Tasks shows him using a Guess and Check strategy supported by trial and error outcomes, justifying why the solution is always an even number, developing heuristics to enumerate and locally exhaust towers by partial and complete case-based reasoning, and recognizing equivalent cases. His problem solving is presented in a series of events that serve to focus, in detail, on the explanations, reasoning, and argumentation he offers during his problem solving.

In Events 1–3, Milin, in the fourth grade, with his partner Michael, solves the 5-tall Tower Task on February 6, 1992, in a whole class setting facilitated by Researcher Maher (R2). Milin and Michael develop a heuristic to generate towers by color opposites and inverses (i.e., a composite operation), which is evidenced by some of the towers they group together. In Event 1 they explain the color opposite strategy to Researcher Martino (R3). In Event 2, Milin explains to Researcher Alston (R1) his reasoning about how the results of trial and error made him sure that their solution was complete. In Event 3 of this analytic Milin explains during a whole class discussion why the solution to the 5-tall Tower Task must be an even number.

Events 4–9 show how Milin develops a methodology that identifies cases (e.g., same attributes) of towers, their opposite cases, and exhaustively enumerates each case by patterns (i.e. by “staircases”) to solve the 5-tall Tower Task in a first interview facilitated by R1 and visiting teacher O’Brien (R4), on February 7, 1992, the day following the classroom work with his partner. Event 4 shows Milin beginning with a color opposite pairing strategy and then, recalling the previous day’s class discussion that utilized elevator patterns to generate particular cases of towers. He is asked to explain how he solved the 5-tall Tower Task. In Events 4 and 5, Milin shows how he built towers with none, one, and two adjacent cubes of a particular color (and of the opposite color). When he initiates exploration of towers with exactly three of a color in Events 6–7, the researcher encourages him to compare these towers with the towers he has already created. He recognizes that some 5-tall towers with exactly three adjacent yellow cubes can be identified as a tower with exactly two adjacent red (e.g., YYYRR* or YYYRR but not RYYYR). What follows is a dialogue about his consideration of other towers within the case of three of a color that were not accounted for in his earlier cases. In Event 8 he is encouraged to generate other unique towers. He then identifies towers with exactly three of a color with some separation of the cubes of that color. In this event he also acknowledged that some towers with some separation of a particular color replicates his earlier cases (e.g., in tower YYRRY the three yellows have some separation, but the two reds do not, and therefore replicate a tower in the two adjacent red case). In Event 9 Milin attempts to organize the towers with three yellows and two reds with some separation in an elevator pattern.

*A tower is denoted by the first letter of each color that takes on positions from top to bottom.

The following are definitions and background information about the Tower Task:

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.

“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.

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, 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 5-tall towers with the attribute of exactly two of a color 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).

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 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

B74, Combinatorics, T5T, Steph & Dana (People view), Grade 4, Feb 6,1992, Raw Footage. Retrieved from https://doi.org/doi:10.7282/t3-hvhr-mw75

B76, Milin’s first of three interviews with Researcher Alston on the 5-tall Tower Task (Work view), Grade 4, February 7, 1992, Raw footage. Retrieved from: https://doi.org/doi:10.7282/t3-e248-d631

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.

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., & Martino, A. (1996). The development of the idea of mathematical proof: A 5-year case study. Journal for Research in Mathematics Education, 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 on2019-08-13T14:15:03-0500

Published on2019-12-23T09:42:24-0500

Persistent URLhttps://doi.org/doi:10.7282/t3-vvfk-d817