PurposesEffective teaching; Lesson activity; Student elaboration; Student engagement; Student model building; Reasoning; Representation

DescriptionAuthor: Victoria Krupnik

As educators and researchers, we frequently notice students’ struggles when they first encounter higher-order mathematical thinking, such as providing precise and convincing arguments for non-routine and challenging problems and developing formal, mathematical proofs. This occurs for several reasons, including, but not limited to, the following:

1. Math reasoning and argumentation requires more precision in its definitions and assumptions than is usually present in everyday reasoning (Yackel and Hanna, 2003);

2. Proof making, which consists of precise argumentation, is usually introduced in high school geometry or in college level mathematics courses (Maher, 2009);

3. Too often, students experience a rule-oriented, procedural approach to mathematics learning, especially in the elementary school (Davis, 1992);

4. Instruction in elementary and secondary grades often lacks flexible questioning, varied approaches, and mathematical depth (Ma, 1999); and

5. Tasks that stimulate natural mathematical argumentation are not a regular part of mathematical instruction (Maher, 2009).

The good news is that there have been studies showing how, under certain learning conditions, mathematics can be viewed as a sense-making activity, a prerequisite for reasoning about mathematical ideas. Building on previous ideas that make sense is encouraged for growth in mathematical reasoning (Maher, 2009; CCSS 2010). Also, there has been a call for students to transition to learning formal proof through early experiences with reasoning, explaining, and justifying (Yackel and Hanna, 2003). These conditions and recent trends raise the question of how it may be possible to support foundations for higher-order mathematical thinking, such as formal proving, as a part of younger children’s mathematical exploration. Maher and Martino (1999) reported on how types of teacher questioning and instructional moves posed in a timely manner supported students’ arguments into forming justification and generalization for their reasoning to solutions on mathematical tasks.

This analytic focuses on the instances of how one student, Stephanie, provided justifications for her reasoning as she made sense of problem situations and expressed her ideas through the representations she exhibited, such as model building, notations, and verbal explanations. The video clips show how her responses to making sense of problem requirements were triggered by meaningful and strategic teacher-student interaction. Some of the teacher-student interactions that can be viewed in these videos are defined by Herbel-Eisenmann and colleagues (2013), expressed as teacher discourse moves. These include waiting for responses, inviting student participation, revoicing and asking students to revoice, probing student thinking, and creating opportunities for students to engage with each other’s reasoning. On a more global scale, one can see that the teacher-student interactions vary from no moves or from minimal teacher interactions to explicit moves to challenge and encourage students to provide explanations, justifications, and, as appropriate, offer generalizations, as they build their arguments (Maher and Martino, 1999).

It is assumed in this video narrative that instances of students’ early mathematical argumentation and justification for their reasoning can emerge by a combination of task design and the working environment that promote and facilitate collaboration. Attention should be given to researcher questioning about what was convincing, what made sense, and how students developed their solutions to the tasks (Maher, Powell, and Uptegrove, 2010; Uptegrove, 2005). Evidence from the data suggests that an emphasis on justification and explanation can naturally lead to the kind of informal reasoning that takes the form of proof-like arguments, an important outcome for continued mathematics learning (Maher, 2009).

The researchers provide a learning environment that makes possible opportunities for the students to explore ideas, communicate explanations, and justify their findings. Although several of the events show task-based interviews in the form of a teaching experiment between researcher(s) and students, each event might be viewed from a lens that offers examples of instructional strategies that might also extend to a classroom setting (see Maher, 2009).

The video narrative is a compilation of events that document Stephanie’s problem solving on several combinatorics tasks (e.g., as building towers of different heights when selecting from two different colored cubes; exploring relationships between/among Pascal’s Triangle, Pascal’s Identity, and the Binomial Theorem). The tasks, also an important component of the conditions that gave rise to mathematical thinking, were simultaneously mathematically rich and accessible to children. As students began each task, they were “fresh, without pre-taught algorithms,” using intuition and prior knowledge from having investigated precursive tasks. This provided the students with the capacity and potential for “abstraction, systematization, and pattern recognition”, specifically in combinatorial and algebraic content (Maher, Powell & Uptegrove, 2010, p. 10). For detailed analyses of the tasks used in this analytic, see Maher, Powell & Uptegrove, 2010; and Maher and Speiser, 1997.

The problem-solving sessions that are provided were a component of the Rutgers-Kenilworth longitudinal study between February 6, 1992 and July 15, 2009 of the development of children’s mathematical reasoning (Maher & Martino, 1996). The analytic traces Stephanie’s problem solving during the years when she was in third, fourth, fifth, and eighth grades. It is important to note that Stephanie and her classmates were introduced to counting tasks that investigated variations of tower problems beginning in their early, elementary grades. The full videotape recordings are available in the Robert B. Davis Institute of Learning archive.

Problem Tasks

Building 4-tall towers, selecting from 2 colors:

You have two colors of Unifix cubes available to build towers. Your task is to make as many different looking towers as possible, each exactly four cubes high. Find a way to convince yourself and others that you have found all possible towers four cubes high, and that you have no duplicates. (Remember that a tower always points up, with the little knob at the top.) Record your towers below and provide a convincing argument why you think you have them all. After completing the task for Towers 4-tall, predict the number of towers, 5-cubes tall that could be built when selecting from two colors. Give a reason for your prediction. After recording your prediction, build the towers. Finally, make a prediction for the number of towers, 10-cubes tall, that could be built when selecting from two colors. Give a reason for this prediction.

Guess My Tower:

You have been invited to participate in a TV Quiz Show and have the opportunity to win a vacation to Disneyworld. The game is played by choosing one of the four possibilities for winning and then picking a tower out of a covered box. If the tower matches your choice, you win. You are told that the box contains all possible towers three tall that can be built when you select from cubes of two colors, red and yellow and that there is only one of each tower. There are no duplicates in the box.

You are given the following possibilities for a winning tower: a. All cubes are exactly the same color; b. there is only one red cube; c. exactly two cubes are red; d. at least two cubes are yellow.

Question 1. Which choice would you make and why would this choice be any better than any of the others?

Question 2. Assuming you won, you can play again for the Grand Prize which means you can take a friend to Disneyworld. But now your box has all possible towers that are four tall with no duplicates (built by selecting from the two colors, yellow and red). You are to select from the same four possibilities for a winning tower. Which choice would you make this time and why would this choice be better than any of the others?"

Binomial Questions (verbally stated to Stephanie by the researchers):

1. Consider the meaning of the binomial expansion of (a+b)^6 using the towers by considering a simpler problem, by considering the case of the square of the binomial and then the cube of the binomial, that is, (a+b)^2 and then (a+b)^3.

2. Describe how Pascal’s addition rule relates to the towers by considering specifically the third and fourth row of Pascal’s Triangle.

3. Explain how a particular tower with exactly three green evolves from a two green tower and how a particular tower with exactly two green evolves from a one green tower.

4. Calculate the total number of selected four-tall towers with exactly two-green cubes.

Video Clip References

PUP Math Towers [video]. Retrieved from http://dx.doi.org/doi:10.7282/T3R49Q0D

PUP Math: Gang of four [video]. Retrieved from http://dx.doi.org/doi:10.7282/T3CC0ZND

Building Towers, Selecting from two colors for Guess My Tower, Clip 2 of 5: Does the Number Double? [video]. Retrieved from http://dx.doi.org/doi:10.7282/T32V2FBZ

Building Towers, Selecting from two colors for Guess My Tower, Clip 3 of 5: Milin introduces an inductive argument [video]. Retrieved from http://dx.doi.org/doi:10.7282/T3RN371Z

Building Towers, Selecting from two colors for Guess My Tower, Clip 4 of 5: Stephanie and Matt Rebuild the Argument [video]. Retrieved from http://dx.doi.org/doi:10.7282/T3W958DV

Building Towers, Selecting from two colors for Guess My Tower, Clip 5 of 5: Sharing with the Group [video]. Retrieved from http://dx.doi.org/doi:10.7282/T36M3617

Early algebra ideas about binomial expansion, Stephanie’s interview six of seven, Clip 1 of 11: Stephanie revisits combinatorics notation for building towers [video]. Retrieved from http://dx.doi.org/doi:10.7282/T3X065WG

Early algebra ideas about binomial expansion, Stephanie’s interview six of seven, Clip 2 of 11: Stephanie rebuilds Unifix towers 1-cube, 2-cubes, 3-cubes and 4-cubes tall, selecting from two colors [video]. Retrieved from http://dx.doi.org/doi:10.7282/T31R6PB1

Early algebra ideas about binomial expansion, Stephanie’s interview six of seven, Clip 3 of 11: Comparing towers, selecting from two colors, built inductively and corresponding to the addition rule of Pascal’s Triangle [video]. Retrieved from http://dx.doi.org/doi:10.7282/T35D8QN1

Early algebra ideas about binomial expansion, Stephanie’s interview six of seven, Clip 4 of 11: Developing the correspondence among towers, selecting from two colors, Pascal’s Triangle, and the symbolic algebraic expansions of (a+b) squared and (a+b) cubed [video]. Retrieved from http://dx.doi.org/doi:10.7282/T3959GB9

Early Algebra Ideas About Binomial Expansion, Stephanie’s Interview Six of Seven: Clip 7 of 11: Generating towers 4-cubes tall, selecting from blue and green cubes, from towers with exactly one green cube to towers with exactly two green cubes. [video]. Retrieved from http://dx.doi.org/doi:10.7282/T3PG1QJC

Early algebra ideas about binomial expansion, Stephanie’s interview six of seven, Clip 10 of 11: Developing mathematical expressions for generating the number of towers 4-cubes tall selecting from green and blue cubes for exactly 2 green cubes, exactly 3 green cubes, and for 4 green cubes [video]. Retrieved from http://dx.doi.org/doi:10.7282/T32N513X

References

Common Core State Standards Initiative (CCSSI). (2010). Common Core State Standards for Mathematics. Retrieved from: http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

Davis, R. B. (1992). Understanding "Understanding". Journal for the Research in Mathematics Education, 11, 225-241.

Herbel-Eisenmann, B. A., Steele, M. D., & Cirillo, M. (2013). (Developing) teacher discourse moves: A framework for professional development. Mathematics Teacher Educator, 1(2), 181-196.

Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates.

Maher, C. Children’s Reasoning: Discovering the Idea of Mathematical Proof. In Stylianou, D. A., Blanton, M. L., & Knuth, E. J. (Eds.). (2009). Teaching and learning proof across the grades: A K-16 perspective. Routledge.

Maher, C. A., & Martino, A. M. (1996). 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., Powell, A. B. & Uptegrove, E. (Eds.), (2010). Combinatorics and reasoning: Representing, justifying and building isomorphisms. New York: Springer Publishers.

Maher, C. A. & Speiser, R. (1997). How far can you go with block towers? The Journal of Mathematical Behavior, 16(2), 125-132.

Maher, C. A., Sran, M. & Yankelewitz, D. (2010). Towers: Schemes, Strategies, and Arguments. In C. A. Maher, A. B. Powell, & E. B. Uptegrove (Eds.), Combinatorics and Reasoning: Representing, Justifying, and Building Isomorphisms (pp. 27-44). Springer: New York, NY.

Uptegrove, E. B. (2005). To symbols from meaning: students’ investigations in counting. Unpublished doctoral dissertation, Rutgers, The State University of New Jersey.

Yackel, E., & Hanna, G. (2003). Reasoning and proof. A research companion to Principles and Standards for School Mathematics, 227-236.

As educators and researchers, we frequently notice students’ struggles when they first encounter higher-order mathematical thinking, such as providing precise and convincing arguments for non-routine and challenging problems and developing formal, mathematical proofs. This occurs for several reasons, including, but not limited to, the following:

1. Math reasoning and argumentation requires more precision in its definitions and assumptions than is usually present in everyday reasoning (Yackel and Hanna, 2003);

2. Proof making, which consists of precise argumentation, is usually introduced in high school geometry or in college level mathematics courses (Maher, 2009);

3. Too often, students experience a rule-oriented, procedural approach to mathematics learning, especially in the elementary school (Davis, 1992);

4. Instruction in elementary and secondary grades often lacks flexible questioning, varied approaches, and mathematical depth (Ma, 1999); and

5. Tasks that stimulate natural mathematical argumentation are not a regular part of mathematical instruction (Maher, 2009).

The good news is that there have been studies showing how, under certain learning conditions, mathematics can be viewed as a sense-making activity, a prerequisite for reasoning about mathematical ideas. Building on previous ideas that make sense is encouraged for growth in mathematical reasoning (Maher, 2009; CCSS 2010). Also, there has been a call for students to transition to learning formal proof through early experiences with reasoning, explaining, and justifying (Yackel and Hanna, 2003). These conditions and recent trends raise the question of how it may be possible to support foundations for higher-order mathematical thinking, such as formal proving, as a part of younger children’s mathematical exploration. Maher and Martino (1999) reported on how types of teacher questioning and instructional moves posed in a timely manner supported students’ arguments into forming justification and generalization for their reasoning to solutions on mathematical tasks.

This analytic focuses on the instances of how one student, Stephanie, provided justifications for her reasoning as she made sense of problem situations and expressed her ideas through the representations she exhibited, such as model building, notations, and verbal explanations. The video clips show how her responses to making sense of problem requirements were triggered by meaningful and strategic teacher-student interaction. Some of the teacher-student interactions that can be viewed in these videos are defined by Herbel-Eisenmann and colleagues (2013), expressed as teacher discourse moves. These include waiting for responses, inviting student participation, revoicing and asking students to revoice, probing student thinking, and creating opportunities for students to engage with each other’s reasoning. On a more global scale, one can see that the teacher-student interactions vary from no moves or from minimal teacher interactions to explicit moves to challenge and encourage students to provide explanations, justifications, and, as appropriate, offer generalizations, as they build their arguments (Maher and Martino, 1999).

It is assumed in this video narrative that instances of students’ early mathematical argumentation and justification for their reasoning can emerge by a combination of task design and the working environment that promote and facilitate collaboration. Attention should be given to researcher questioning about what was convincing, what made sense, and how students developed their solutions to the tasks (Maher, Powell, and Uptegrove, 2010; Uptegrove, 2005). Evidence from the data suggests that an emphasis on justification and explanation can naturally lead to the kind of informal reasoning that takes the form of proof-like arguments, an important outcome for continued mathematics learning (Maher, 2009).

The researchers provide a learning environment that makes possible opportunities for the students to explore ideas, communicate explanations, and justify their findings. Although several of the events show task-based interviews in the form of a teaching experiment between researcher(s) and students, each event might be viewed from a lens that offers examples of instructional strategies that might also extend to a classroom setting (see Maher, 2009).

The video narrative is a compilation of events that document Stephanie’s problem solving on several combinatorics tasks (e.g., as building towers of different heights when selecting from two different colored cubes; exploring relationships between/among Pascal’s Triangle, Pascal’s Identity, and the Binomial Theorem). The tasks, also an important component of the conditions that gave rise to mathematical thinking, were simultaneously mathematically rich and accessible to children. As students began each task, they were “fresh, without pre-taught algorithms,” using intuition and prior knowledge from having investigated precursive tasks. This provided the students with the capacity and potential for “abstraction, systematization, and pattern recognition”, specifically in combinatorial and algebraic content (Maher, Powell & Uptegrove, 2010, p. 10). For detailed analyses of the tasks used in this analytic, see Maher, Powell & Uptegrove, 2010; and Maher and Speiser, 1997.

The problem-solving sessions that are provided were a component of the Rutgers-Kenilworth longitudinal study between February 6, 1992 and July 15, 2009 of the development of children’s mathematical reasoning (Maher & Martino, 1996). The analytic traces Stephanie’s problem solving during the years when she was in third, fourth, fifth, and eighth grades. It is important to note that Stephanie and her classmates were introduced to counting tasks that investigated variations of tower problems beginning in their early, elementary grades. The full videotape recordings are available in the Robert B. Davis Institute of Learning archive.

Problem Tasks

Building 4-tall towers, selecting from 2 colors:

You have two colors of Unifix cubes available to build towers. Your task is to make as many different looking towers as possible, each exactly four cubes high. Find a way to convince yourself and others that you have found all possible towers four cubes high, and that you have no duplicates. (Remember that a tower always points up, with the little knob at the top.) Record your towers below and provide a convincing argument why you think you have them all. After completing the task for Towers 4-tall, predict the number of towers, 5-cubes tall that could be built when selecting from two colors. Give a reason for your prediction. After recording your prediction, build the towers. Finally, make a prediction for the number of towers, 10-cubes tall, that could be built when selecting from two colors. Give a reason for this prediction.

Guess My Tower:

You have been invited to participate in a TV Quiz Show and have the opportunity to win a vacation to Disneyworld. The game is played by choosing one of the four possibilities for winning and then picking a tower out of a covered box. If the tower matches your choice, you win. You are told that the box contains all possible towers three tall that can be built when you select from cubes of two colors, red and yellow and that there is only one of each tower. There are no duplicates in the box.

You are given the following possibilities for a winning tower: a. All cubes are exactly the same color; b. there is only one red cube; c. exactly two cubes are red; d. at least two cubes are yellow.

Question 1. Which choice would you make and why would this choice be any better than any of the others?

Question 2. Assuming you won, you can play again for the Grand Prize which means you can take a friend to Disneyworld. But now your box has all possible towers that are four tall with no duplicates (built by selecting from the two colors, yellow and red). You are to select from the same four possibilities for a winning tower. Which choice would you make this time and why would this choice be better than any of the others?"

Binomial Questions (verbally stated to Stephanie by the researchers):

1. Consider the meaning of the binomial expansion of (a+b)^6 using the towers by considering a simpler problem, by considering the case of the square of the binomial and then the cube of the binomial, that is, (a+b)^2 and then (a+b)^3.

2. Describe how Pascal’s addition rule relates to the towers by considering specifically the third and fourth row of Pascal’s Triangle.

3. Explain how a particular tower with exactly three green evolves from a two green tower and how a particular tower with exactly two green evolves from a one green tower.

4. Calculate the total number of selected four-tall towers with exactly two-green cubes.

Video Clip References

PUP Math Towers [video]. Retrieved from http://dx.doi.org/doi:10.7282/T3R49Q0D

PUP Math: Gang of four [video]. Retrieved from http://dx.doi.org/doi:10.7282/T3CC0ZND

Building Towers, Selecting from two colors for Guess My Tower, Clip 2 of 5: Does the Number Double? [video]. Retrieved from http://dx.doi.org/doi:10.7282/T32V2FBZ

Building Towers, Selecting from two colors for Guess My Tower, Clip 3 of 5: Milin introduces an inductive argument [video]. Retrieved from http://dx.doi.org/doi:10.7282/T3RN371Z

Building Towers, Selecting from two colors for Guess My Tower, Clip 4 of 5: Stephanie and Matt Rebuild the Argument [video]. Retrieved from http://dx.doi.org/doi:10.7282/T3W958DV

Building Towers, Selecting from two colors for Guess My Tower, Clip 5 of 5: Sharing with the Group [video]. Retrieved from http://dx.doi.org/doi:10.7282/T36M3617

Early algebra ideas about binomial expansion, Stephanie’s interview six of seven, Clip 1 of 11: Stephanie revisits combinatorics notation for building towers [video]. Retrieved from http://dx.doi.org/doi:10.7282/T3X065WG

Early algebra ideas about binomial expansion, Stephanie’s interview six of seven, Clip 2 of 11: Stephanie rebuilds Unifix towers 1-cube, 2-cubes, 3-cubes and 4-cubes tall, selecting from two colors [video]. Retrieved from http://dx.doi.org/doi:10.7282/T31R6PB1

Early algebra ideas about binomial expansion, Stephanie’s interview six of seven, Clip 3 of 11: Comparing towers, selecting from two colors, built inductively and corresponding to the addition rule of Pascal’s Triangle [video]. Retrieved from http://dx.doi.org/doi:10.7282/T35D8QN1

Early algebra ideas about binomial expansion, Stephanie’s interview six of seven, Clip 4 of 11: Developing the correspondence among towers, selecting from two colors, Pascal’s Triangle, and the symbolic algebraic expansions of (a+b) squared and (a+b) cubed [video]. Retrieved from http://dx.doi.org/doi:10.7282/T3959GB9

Early Algebra Ideas About Binomial Expansion, Stephanie’s Interview Six of Seven: Clip 7 of 11: Generating towers 4-cubes tall, selecting from blue and green cubes, from towers with exactly one green cube to towers with exactly two green cubes. [video]. Retrieved from http://dx.doi.org/doi:10.7282/T3PG1QJC

Early algebra ideas about binomial expansion, Stephanie’s interview six of seven, Clip 10 of 11: Developing mathematical expressions for generating the number of towers 4-cubes tall selecting from green and blue cubes for exactly 2 green cubes, exactly 3 green cubes, and for 4 green cubes [video]. Retrieved from http://dx.doi.org/doi:10.7282/T32N513X

References

Common Core State Standards Initiative (CCSSI). (2010). Common Core State Standards for Mathematics. Retrieved from: http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

Davis, R. B. (1992). Understanding "Understanding". Journal for the Research in Mathematics Education, 11, 225-241.

Herbel-Eisenmann, B. A., Steele, M. D., & Cirillo, M. (2013). (Developing) teacher discourse moves: A framework for professional development. Mathematics Teacher Educator, 1(2), 181-196.

Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates.

Maher, C. Children’s Reasoning: Discovering the Idea of Mathematical Proof. In Stylianou, D. A., Blanton, M. L., & Knuth, E. J. (Eds.). (2009). Teaching and learning proof across the grades: A K-16 perspective. Routledge.

Maher, C. A., & Martino, A. M. (1996). 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., Powell, A. B. & Uptegrove, E. (Eds.), (2010). Combinatorics and reasoning: Representing, justifying and building isomorphisms. New York: Springer Publishers.

Maher, C. A. & Speiser, R. (1997). How far can you go with block towers? The Journal of Mathematical Behavior, 16(2), 125-132.

Maher, C. A., Sran, M. & Yankelewitz, D. (2010). Towers: Schemes, Strategies, and Arguments. In C. A. Maher, A. B. Powell, & E. B. Uptegrove (Eds.), Combinatorics and Reasoning: Representing, Justifying, and Building Isomorphisms (pp. 27-44). Springer: New York, NY.

Uptegrove, E. B. (2005). To symbols from meaning: students’ investigations in counting. Unpublished doctoral dissertation, Rutgers, The State University of New Jersey.

Yackel, E., & Hanna, G. (2003). Reasoning and proof. A research companion to Principles and Standards for School Mathematics, 227-236.

Created on2013-03-16T14:52:54-0400

Published on2017-04-28T11:45:39-0400

Persistent URLhttps://doi.org/doi:10.7282/T34X5B6C