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

DescriptionBy Usufu Nyakoojo and Victoria Krupnik

This analytic focuses on the reasoning, argumentation, and mathematical representations showcased by four, fourth-grade students (Jeff, Michelle, Milin, and Stephanie). Explanations of solutions for finding all possible towers, 3-cubes tall when selecting from two colors, are offered by Milin and Stephanie, with input and questioning from other group members. The events of this analytic are retrieved from a small-group assessment interview (Gang of Four), facilitated by Researcher Maher on March 10, 1992. The session was motivated by providing the participating students insight into the different forms of reasoning that surfaced in working with earlier problems of finding the number of towers that could be built selecting from cubes of two colors of a variety of heights. The small-group assessment provided students with an opportunity to share their earlier ideas and reasoning about the Towers Tasks of various heights (Maher & Martino, 1996b).

The students previously worked on tasks to build Towers (4 and 5 cubes tall, selecting from 2 colors), both in grade 3 (4-tall towers) and grade 4 (5-tall towers), during class and in individual interviews (5-tall and higher towers). The task discussed in this assessment (building all possible towers, 3-tall, selecting from cubes available in two colors) was to make visible the different approaches and forms of reasoning offered by the students in providing a justification for their solution. The focus was on eliciting their arguments for accounting for all possible 3-tall towers that could be built. In this assessment, the students chose not to use the Unifix cubes that were available; they used spoken words, charts, diagrams, and symbols.

In grade 3, the students worked in dyads in class on the 4-tall Towers Task and in the month prior, the students worked in dyads in class on the 5-tall Towers Task. During these earlier sessions, students built tower models from the Unifix cubes available in two colors to demonstrate their solutions. Each student had at least one follow-up individual interview with researchers, explaining their ideas, building models, and creating charts and pictures to justify their reasoning for finding towers of various heights, selecting from two colors.

The students were selected for this small group interview because they expressed curiosity about the arguments of others. In this session, the students are asked to convince the researcher and each other of their reasoning. Milin presents two iterations of an inductive argument and Michelle presents one iteration. Stephanie presents two iterations of case-based arguments and an argument by contradiction to offer a convincing solution. The students continue (repeat and refine) their explanations to Jeff and Michelle. They recognize a doubling pattern and generalize a rule to find the solutions for the number of towers of any height, selecting from two colors.

In the first three events, students present their solutions and supporting arguments to the 3-tall Towers Task. In the first event Michelle makes a claim for the numerical solutions to the Towers Task of heights one through five. In the second event Milin presents an argument by induction for building towers of increasing heights. In the third event Stephanie presents an argument by cases for finding all towers and an argument by contradiction to justify the towers of exactly one color. In the fourth event, Stephanie repeats her argument by cases to convince Jeff whose case organization includes all towers with exactly two of a color. She chooses to distinguish towers with exactly two of a color into separate categories: two adjacent colors and two colors separated.

Jeff, who was absent in recent days, raises the question of the need for making a pattern when he asks, “Do you have to make a pattern?” His question stimulates a dialogue among the small group members (Stephanie, Milin, and Michelle) who provide rationales for the value of looking for patterns. The conversation is presented into events that serve to focus, in detail, on the explanations, reasoning, and argumentation of the various ideas offered by the four students. We see Milin and Stephanie repeating earlier justifications in an effort to convince Jeff of their ideas. We also see the use of the mathematics, academic, and everyday register used in their discourse as they present their ideas in an attempt to justify their solution. The following definitions are offered, some presented by the students and others by researchers.

Elevator (student-created)

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. To create a second tower, the cube is then moved to the second position. The cube is continuously lowered one position to create new towers until it is placed in the final position (Maher, Sran & Yankelewitz, 2011).

Argument by Induction

An induction argument for the justification of the general solution 2^n includes the basis step (n=1) in which a participant describes that the total number of one tall towers created when selecting from two colors is 2 (i.e. one of only blue and one of only yellow). The second 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 all of the towers of the previous height, therefore doubling the total number of towers created in the previous height.

Argument by Contradiction

When a situation arises that is inconsistent or contrary to known or inherent facts/assumptions, a contradiction has been reached. For the 3-tall Tower Task, when selecting from two colors (i.e. red and blue), an argument by contradiction can be used to justify the total number of towers that can be built in the case of exactly one blue cube. The blue cube can be placed in either first, second or third position. If other towers can be built with one blue cube, then the cube would have to be in a different position, say, the fourth position or below the first (zeroth) position. Placing a cube in the fourth or zeroth position would require the tower to be a height of at four or building a tower that doesn’t exist. This is a contradiction of the requirement that the tower has a height four (Maher & Martino, 1996).

Argument by cases

In a proof by cases, a statement is proved by proving all the smaller subsets of statements that make up the whole. For example, the solution to the 3-tall Towers Task when selecting from two colors (i.e. blue or 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: (a) one tower containing no red (towers with a single color); (b) three towers containing one red (towers with exactly one of color); (c) three towers containing 2 red (within cases (2) and (3) can be cubes of same color adjacent to or separated from each other); (d) one tower containing 3 red or all red. A complete proof by cases would include an exhaustive proof of the total number of towers in each case.

Problem Task:

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 (adapted from Krupnik, 2017).

Video and Transcript References:

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

B42, The Gang of Four (Michelle and Milin view), Grade 4, March 10, 1992, raw footage doi link: https://doi.org/doi:10.7282/T3833Q5P

References (articles and books):

Krupnik, V. (2017). A longitudinal case study of Stephanie’s grown in mathematical reasoning through the lens of teacher discourse moves. http://dx.doi.org/doi:10.7282/T34X5B6C

Maher, C. A., & Martino, A. M. (1993). Four case studies of the stability and durability of children’s methods of proof. In Proceedings of the Fifteenth Annual Meeting for the North American Chapter for the Psychology of Mathematics Education (Vol. 2, pp. 33-39).

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, 194-214.

Maher, C. A., & Martino, A. M. (1996b). Young children invent methods of proof: the gang of four. In P. Nesher, L. P. Steffe, P. Cobb, B. Greer, & J. Golden (Eds.). Theories of mathematical learning (pp. 431–447). Mahwah, NJ: Lawrence Erlbaum Associates.

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

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

This analytic focuses on the reasoning, argumentation, and mathematical representations showcased by four, fourth-grade students (Jeff, Michelle, Milin, and Stephanie). Explanations of solutions for finding all possible towers, 3-cubes tall when selecting from two colors, are offered by Milin and Stephanie, with input and questioning from other group members. The events of this analytic are retrieved from a small-group assessment interview (Gang of Four), facilitated by Researcher Maher on March 10, 1992. The session was motivated by providing the participating students insight into the different forms of reasoning that surfaced in working with earlier problems of finding the number of towers that could be built selecting from cubes of two colors of a variety of heights. The small-group assessment provided students with an opportunity to share their earlier ideas and reasoning about the Towers Tasks of various heights (Maher & Martino, 1996b).

The students previously worked on tasks to build Towers (4 and 5 cubes tall, selecting from 2 colors), both in grade 3 (4-tall towers) and grade 4 (5-tall towers), during class and in individual interviews (5-tall and higher towers). The task discussed in this assessment (building all possible towers, 3-tall, selecting from cubes available in two colors) was to make visible the different approaches and forms of reasoning offered by the students in providing a justification for their solution. The focus was on eliciting their arguments for accounting for all possible 3-tall towers that could be built. In this assessment, the students chose not to use the Unifix cubes that were available; they used spoken words, charts, diagrams, and symbols.

In grade 3, the students worked in dyads in class on the 4-tall Towers Task and in the month prior, the students worked in dyads in class on the 5-tall Towers Task. During these earlier sessions, students built tower models from the Unifix cubes available in two colors to demonstrate their solutions. Each student had at least one follow-up individual interview with researchers, explaining their ideas, building models, and creating charts and pictures to justify their reasoning for finding towers of various heights, selecting from two colors.

The students were selected for this small group interview because they expressed curiosity about the arguments of others. In this session, the students are asked to convince the researcher and each other of their reasoning. Milin presents two iterations of an inductive argument and Michelle presents one iteration. Stephanie presents two iterations of case-based arguments and an argument by contradiction to offer a convincing solution. The students continue (repeat and refine) their explanations to Jeff and Michelle. They recognize a doubling pattern and generalize a rule to find the solutions for the number of towers of any height, selecting from two colors.

In the first three events, students present their solutions and supporting arguments to the 3-tall Towers Task. In the first event Michelle makes a claim for the numerical solutions to the Towers Task of heights one through five. In the second event Milin presents an argument by induction for building towers of increasing heights. In the third event Stephanie presents an argument by cases for finding all towers and an argument by contradiction to justify the towers of exactly one color. In the fourth event, Stephanie repeats her argument by cases to convince Jeff whose case organization includes all towers with exactly two of a color. She chooses to distinguish towers with exactly two of a color into separate categories: two adjacent colors and two colors separated.

Jeff, who was absent in recent days, raises the question of the need for making a pattern when he asks, “Do you have to make a pattern?” His question stimulates a dialogue among the small group members (Stephanie, Milin, and Michelle) who provide rationales for the value of looking for patterns. The conversation is presented into events that serve to focus, in detail, on the explanations, reasoning, and argumentation of the various ideas offered by the four students. We see Milin and Stephanie repeating earlier justifications in an effort to convince Jeff of their ideas. We also see the use of the mathematics, academic, and everyday register used in their discourse as they present their ideas in an attempt to justify their solution. The following definitions are offered, some presented by the students and others by researchers.

Elevator (student-created)

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. To create a second tower, the cube is then moved to the second position. The cube is continuously lowered one position to create new towers until it is placed in the final position (Maher, Sran & Yankelewitz, 2011).

Argument by Induction

An induction argument for the justification of the general solution 2^n includes the basis step (n=1) in which a participant describes that the total number of one tall towers created when selecting from two colors is 2 (i.e. one of only blue and one of only yellow). The second 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 all of the towers of the previous height, therefore doubling the total number of towers created in the previous height.

Argument by Contradiction

When a situation arises that is inconsistent or contrary to known or inherent facts/assumptions, a contradiction has been reached. For the 3-tall Tower Task, when selecting from two colors (i.e. red and blue), an argument by contradiction can be used to justify the total number of towers that can be built in the case of exactly one blue cube. The blue cube can be placed in either first, second or third position. If other towers can be built with one blue cube, then the cube would have to be in a different position, say, the fourth position or below the first (zeroth) position. Placing a cube in the fourth or zeroth position would require the tower to be a height of at four or building a tower that doesn’t exist. This is a contradiction of the requirement that the tower has a height four (Maher & Martino, 1996).

Argument by cases

In a proof by cases, a statement is proved by proving all the smaller subsets of statements that make up the whole. For example, the solution to the 3-tall Towers Task when selecting from two colors (i.e. blue or 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: (a) one tower containing no red (towers with a single color); (b) three towers containing one red (towers with exactly one of color); (c) three towers containing 2 red (within cases (2) and (3) can be cubes of same color adjacent to or separated from each other); (d) one tower containing 3 red or all red. A complete proof by cases would include an exhaustive proof of the total number of towers in each case.

Problem Task:

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 (adapted from Krupnik, 2017).

Video and Transcript References:

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

B42, The Gang of Four (Michelle and Milin view), Grade 4, March 10, 1992, raw footage doi link: https://doi.org/doi:10.7282/T3833Q5P

References (articles and books):

Krupnik, V. (2017). A longitudinal case study of Stephanie’s grown in mathematical reasoning through the lens of teacher discourse moves. http://dx.doi.org/doi:10.7282/T34X5B6C

Maher, C. A., & Martino, A. M. (1993). Four case studies of the stability and durability of children’s methods of proof. In Proceedings of the Fifteenth Annual Meeting for the North American Chapter for the Psychology of Mathematics Education (Vol. 2, pp. 33-39).

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, 194-214.

Maher, C. A., & Martino, A. M. (1996b). Young children invent methods of proof: the gang of four. In P. Nesher, L. P. Steffe, P. Cobb, B. Greer, & J. Golden (Eds.). Theories of mathematical learning (pp. 431–447). Mahwah, NJ: Lawrence Erlbaum Associates.

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

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

Created on2019-06-09T22:35:17-0400

Published on2019-08-13T13:41:58-0400

Persistent URLhttps://doi.org/doi:10.7282/t3-67gc-9s26