Teacher Questioning and Invitations to Participate in Advanced Mathematics Lectures

In this study, we were interested in exploring the extent to which advanced mathematics lecturers provide students opportunities to play a role in considering or generating course content. To do this, we examined the questioning practices of 11 lecturers who taught advanced mathematics courses at the university level. Because we are unaware of other studies examining advanced mathematics lecturers’ questioning, we first analyzed the data using an open coding scheme to categorize the types of content lecturers solicited and the opportunities they provided students to participate in generating course content. In a second round of analysis, we examined the extent to which lecturers provide students opportunities to generate mathematical contributions and to engage in reasoning researchers have identified as important in advanced mathematics. Our findings highlight that although lecturers asked many questions, lecturers did not provide substantial opportunities for students to participate in generating mathematical content and reasoning. Additionally, we provide several examples of lecturers providing students some opportunities to generate important contributions. We conclude by providing implications and areas for future research

Stephanie’s Development of Reasoning by an Inductive Argument to Solve Tower Tasks: Part 2 of 3 (Grade 4)

Author Victoria Krupnik (Rutgers University - graduate) Overall Description This analytic is the second of three analytics that showcase Stephanie’s development of an argument by induction to solve a counting task over three school years. The three analytics focus on the reasoning, argumentation, and mathematical representations constructed by Stephanie in a variety of settings over time, in the third (Part 1), fourth (Parts 1 and 2), and fifth grades (Part 3). The first analytic begins with events where third grader Stephanie is working with a partner, in a whole class setting, building towers that are 3- and 4-tall, with plastic cubes available in two colors. This is followed by events in which Stephanie is working in a one-on-one interview with a researcher in the third and then fourth grade. The second analytic follows with events where fourth-grader Stephanie participates in a small group formative assessment and works on a summative assessment with a partner. The third analytic follows with events where fifth grader Stephanie is working with a partner, in a whole class setting, and then presenting her ideas to a small group and then to the whole class. In this analytic (the second of three), Stephanie’s learning progression is shown as she builds a justification for her solution to Tower Tasks by inductive reasoning. In fourth grade Stephanie presents an observed doubling pattern between Tower Tasks to a small group of students, presents a shortcut method to finding taller towers, and applies the doubling pattern as a rule to solve Tower Tasks up to 11-tall, selecting from two colors, in the partner summative assessment. Her problem solving is presented in 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 small-group assessment interview (“Gang of Four”), facilitated by Researcher Maher (R2) on March 10, 1992. Stephanie presented her generalization of the doubling rule to taller towers up to 10-tall and a shortcut method for obtaining the number of towers of any height. Event 4 is retrieved from a June 15, 1992 session in which Stephanie and her partner, Milin, solved the 3-tall Tower Task written summative assessment at the end of the fourth grade. They applied the doubling rule to verify the numerical solution of the 3-tall Tower Task. The following definitions and background information about the Tower Task are offered: Doubling rule: The total number of different tower combinations of height k would be double the total number of tower combinations of height k–1. Argument by Induction An induction argument for the justification of the general solution 2^n includes the basic step (n=1) in which a participant states that the total number of 1-tall towers created when selecting from two colors is 2 (i.e. one of only blue and one of only yellow). The second step 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 each of the towers of the previous height, therefore doubling the total number of towers created in the previous height. 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): B41, The Gang of Four (Jeff and Stephanie view), Grade 4, March 10, 1992, raw footage. Retrieved from: https://doi.org/doi:10.7282/T3CV4FWP B75, Towers Assessment, WV, Grade 4, Jun 15, 1992, raw. Retrieved from: https://doi.org/doi:10.7282/t3-tpqc-b71

A Longitudinal Case Study of Stephanie’s Growth in Mathematical Reasoning through the Lens of Teacher Discourse Moves

Author: 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

Stephanie’s Development of Reasoning by an Inductive Argument to Solve Tower Tasks: Part 3 of 3 (Grade 5)

Author Victoria Krupnik (Rutgers University - graduate) Overall Description This analytic is the third of three analytics that showcase Stephanie’s development of an argument by induction to solve a counting task over three school years. The three analytics focus on the reasoning, argumentation, and mathematical representations constructed by Stephanie in a variety of settings over time, in the third (Part 1), fourth (Parts 1 and 2), and fifth grades (Part 3). The first analytic begins with events where third grader Stephanie is working with a partner, in a whole class setting, building towers that are 3- and 4-tall, with plastic cubes available in two colors. This is followed by events in which Stephanie is working in a one-on-one interview with a researcher in the third and then fourth grade. The second analytic follows with events where fourth-grader Stephanie participates in a small group formative assessment and works on a summative assessment with a partner. The third analytic follows with events where fifth grader Stephanie is working with a partner, in a whole class setting, and then presenting her ideas to a small group and then to the whole class. In this analytic (the third of three), Stephanie’s learning progression in building a justification by inductive reasoning to Tower Tasks shows Stephanie in the fifth grade recalling and testing a doubling pattern between Tower Tasks, observing Michelle and Matt present an inductive method to generate taller towers from shorter towers to justify the doubling pattern, and, finally, presenting an inductive argument for the growth of towers for any height. Her problem solving is presented in events that serve to focus, in detail, on the explanations, reasoning, and argumentation Stephanie offers during her problem solving. Events 1-7 are retrieved from a fifth-grade class session with partners Stephanie and Matt as they worked on a Guess My Tower (GMT) task on February 26, 1992. Solving the problem promoted revisiting the 3- and 4-tall tower outcome possibilities. Events 1-2 show Stephanie attempting to test a doubling rule that she recalls from prior Towers problem-solving experiences. Milin’s inductive approach to finding taller towers from shorter towers is disseminated to other students during this session, first to Michelle, then to Stephanie and Matt (Event 3), and then to the rest of the class (Events 4-7). The purpose of the GMT events is to explore Stephanie’s learning of his idea. The events presented are evidence of Stephanie’s individual reasoning about the doubling pattern and her reconciliation between the doubling rule and why it worked. The following definitions and background information about the Tower and Guess My Tower tasks are offered: Doubling rule The total number of different tower combinations of height k would be double the total number of tower combinations of height k–1. Argument by Induction An induction argument for the justification of the general solution 2^n includes the basic step (n=1) in which a participant states that the total number of 1-tall towers created when selecting from two colors is 2 (i.e. one of only blue and one of only yellow). The second step 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 each of the towers of the previous height, therefore doubling the total number of towers created in the previous height. 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”). Guess My Tower Task: 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?" Video and Transcript References (in chronological order of Stephanie’s journey): B78,Guess My Tower task,Matt and Stephanie partner work (work view), Grade 5,February 26,1993,Raw footage. Retrieved from: https://doi.org/doi:10.7282/t3-hw4j-2f57 Building Towers, Selecting from two colors for Guess My Tower, Clip 2 of 5: Does the Number Double? Retrieved from: https://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. Retrieved from: https://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. Retrieved from: https://doi.org/doi:10.7282/T3W958DV Building Towers, Selecting from two colors for Guess My Tower, Clip 5 of 5: Sharing with the Group. Retrieved from: https://doi.org/doi:10.7282/T36M361

Stephanie’s Development of Reasoning by Cases to Solve Tower Tasks: Part 2 of 3 (Grade 4)

Author 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

Milin’s Learning Progression in Reasoning by Cases to Solve Tower Tasks: Part 2 of 2 (Grade 4)

Author: Victoria Krupnik, Rutgers University This analytic is the second of two analytics that showcase the development of a variation of reasoning by cases to solve a counting task by a student, Milin, over one school year in fourth grade. 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 second of the two analytics), Milin’s learning progression in building a justification to Tower Tasks shows Milin extending his method by cases and opposites for towers 1-, 2-, 3-, and 4-tall and comparing the number solutions. His problem solving is presented in a series of events that serve to focus, in detail, on the explanations, reasoning, and argumentation offered by Milin during his problem solving. Event 1 shows how Milin uses a methodology that combines cases (i.e. by “staircases”) and color opposite strategies to solve the 4-tall Tower Task (after solving the 5-tall Tower Task) in a first interview facilitated by Researcher Alston (R1) and visiting Teacher O’Brian (R4) on February 7, 1992, the day following the classroom work with his partner. Events 2–6 illustrate Milin’s refinement, modification, and application of the case and/or color opposite strategies to solve the 2-, 3-, and 4-tall Tower Tasks in a second interview facilitated by R1 on February 21, 1992. In Events 2–4 he builds the cases of 4-tall towers with exactly one cube of a particular color, towers with all cubes the same color, towers with exactly two cubes of a particular color adjacent to each other, and towers with alternating color cubes. In Event 5 he builds the solution by cases for 2-tall and 1-tall towers and compares them to the 4-tall tower cases. In Event 6, Milin returns to the color opposite and inverse pairs of towers strategies to build towers that are 3-tall. Not shown in the analytic, Milin later predicts the number of 6-tall towers to be “forty something” and is given a homework assignment to solve the 6-tall Tower Task. The following definitions and background information about the Tower Tasks are offered. 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. *A tower is denoted by the first letter of each color that takes on positions from top to bottom. “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 two reds (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 two colors): You have plastic cubes of two colors available to build towers. Your task is to make as many different looking towers as possible, each exactly three cubes high. Find a way to convince yourself and others that you have found all possible towers three 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 B76, Milin’s first of three interviews with researcher Alston on the five-tall Tower Task (Work view), Grade 4, February 7, 1992, Raw footage. Retrieved from: https://doi.org/doi:10.7282/t3-e248-d631 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: https://doi.org/doi:10.7282/T3X3523K 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

Stephanie’s Development of Reasoning by Cases to Solve Tower Tasks: Part 1 of 3 (Grades 3 & 4)

Author 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

Stephanie’s Development of Reasoning by Cases to Solve Tower Tasks: Part 3 of 3 (Grade 4)

Author Victoria Krupnik (Rutgers University - graduate) Overall Description This analytic is the third 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 third 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 3-tall Tower Task by cases in an attempt to convince a small group of students of her solution and modifies her method in a partner assessment. 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 1-4 are retrieved from a small group assessment interview that occurred on March 10, 1992 with four students, Milin, Michelle, Jeff, and Stephanie, and Researcher Maher (R2). Other researchers, a classroom teacher, and mathematics supervisor were observers. Milin, during his third individual interview, expressed interest to researchers about how other students were solving the problems. Each of the four students in this small group assessment had been interviewed at least once after they participated in the fourth-grade classroom session to solve the 5-tall Tower Task. Their approaches varied and so the researchers decided that sharing approaches with each other was timely. During this group assessment, Stephanie provides a justification by cases for the 3-tall Tower task. Specifically, in Events 1-3 she organizes the eight tower outcomes into cases of no blue cubes, one blue cube, two blue cubes “stuck together,” three blues cubes, and two blue cubes separated. In Event 4 she then repeats her argument by cases, with the help of Michelle, a second time in an attempt to convince Jeff that her towers are all different. Event 5 is retrieved from a June 15, 1992 session in which Stephanie and her partner, Milin, work to solve the 3-tall Tower task at the end of the fourth grade. They construct the solution by a modified case organization and color opposite grouping that includes two single-colored towers, the case of two green cubes, and the opposite case of two black cubes. In this solution she does not separate the cases of two of a color together and two of a color separated. 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. 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). 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): B41, The Gang of Four (Jeff and Stephanie view), Grade 4, March 10, 1992, raw footage. Retrieved from: https://doi.org/doi:10.7282/T3CV4FWP B75, Towers Assessment, WV, Grade 4, Jun 15, 1992, raw. Retrieved from: https://doi.org/doi:10.7282/t3-tpqc-b719 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

Milin’s Learning Progression in Reasoning by an Inductive Argument to Solve Tower Tasks: Part 2 of 2 (Grades 4 & 5)

Author: Victoria Krupnik, Rutgers University This analytic is the second of two analytics that showcase the development of an argument by induction to solve a counting task by a student, Milin, over two school years. Both analytics focus on the reasoning, argumentation, and mathematical representations constructed by Milin in a variety of settings over time, in the fourth (Parts 1 and 2) and fifth (Part 2) grades. The first analytic includes events where Milin is working in a one-on-one interview with a researcher. The second analytic follows with events where Milin participates in a small group formative assessment interview and works on a summative assessment with a partner. In the current analytic (the second of the two analytics), Milin’s learning progression in building a justification by inductive reasoning to Tower Tasks shows him, in the fourth and fifth grades, presenting a complete argument for the doubling pattern of number solutions to Tower Tasks by inductive reasoning and applying the doubling rule to solve Tower Tasks. His problem solving is presented in a series of events that serve to focus, in detail, on the explanations, reasoning, and argumentation offered by Milin during his problem solving. Events 1–3 are retrieved from a small-group formative assessment interview (“Gang of Four”), facilitated by Researcher Maher (R2) on March 10, 1992. Milin presented his generalization of a doubling rule to number solutions of Tower Tasks up to 6-tall by an inductive argument. Specifically, he demonstrated how from each tower of a given height one could generate two distinct towers of a higher consecutive height when choosing from two colors. On multiple occasions throughout this session, he demonstrated his understanding of the doubling pattern, or as he previously called it, the “family strategy,” with an inductive argument. He used Michelle’s and his own tower drawings. In each instance Milin explained how a tower of a given height could be used to generate two different towers of a consecutively higher height by first adding a cube of the first color on top of the base and then adding a cube of the second color on top of a second tower with the same base. In this group interview he portrayed no doubt, as he had done in the previous individual interview, that this pattern would work for other heights. Specifically, he conjectured the doubling pattern should work for towers taller than 5-tall. Events 4–5 are retrieved from a fourth-grade class session, facilitated by a classroom teacher, in which Milin and his partner, Stephanie, solve the 3-tall Tower Task as a summative assessment occurring on June 15, 1992. His primary strategies continue to be an elevator pattern for the towers with exactly one green cube, and color opposite pairings for the single-colored towers, even when Stephanie utilizes a different organization for the towers with exactly one black. The purpose of this event is to show his continuation, three months later, to find patterns among the towers, using the elevator and color opposite strategies. Milin and Stephanie also apply the doubling rule to justify their solution. Events 6–7 are retrieved from a fifth-grade class session, facilitated by Researcher Alston (R1), with partners Milin and Michelle as they worked on the Guess My Tower (GMT) Task on February 26, 1993. Solving the problem depended on the students revisiting and rebuilding the 3- and 4-tall tower outcome possibilities. In these events Milin explained his inductive reasoning for the doubling pattern to Michelle, facilitated by Researcher Maher (R2). The following definitions and background information about the Tower and Guess My Tower Tasks are offered: Doubling rule: The total number of different tower combinations of height k would be double the total number of tower combinations of height k–1. Argument by Induction (Alston & Maher, 1993; Sran, 2010; Maher, Sran, & Yankelewitz, 2011) An induction argument for the justification of the general Tower Task solution 2^n includes the basic step (n=1) in which a participant states that the total number of 1-tall towers created when selecting from two colors is two (i.e. one of only blue and one of only yellow). The second step 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 each of the towers of the previous height, therefore doubling the total number of towers created in the previous height. Three-tall Tower Task (selecting from two colors): You have plastic cubes of two colors available to build towers. Your task is to make as many different looking towers as possible, each exactly three cubes high. Find a way to convince yourself and others that you have found all possible towers three 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”). Guess My Tower Task: You have been invited to participate in a TV Quiz Show and have the opportunity to win a vacation to Disney World. 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 Disney World. 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? Video and Transcript References (in chronological order of Milin’s journey): B42, The Gang of Four (Michelle and Milin view), Grade 4, March 10, 1992, raw footage [video]. Retrieved from https://doi.org/doi:10.7282/T3833Q5P B75, Towers Assessment, WV, Grade 3, Jun 15, 1992, raw. Retrieved from: https://doi.org/doi:10.7282/t3-tpqc-b719 Building Towers, Selecting from two colors for Guess My Tower, Clip 3 of 5: Milin introduces an inductive argument. Retrieved from: https://doi.org/doi:10.7282/T3RN371Z References: Alston, A., & Maher, C. A. (1993). Tracing Milin’s building of proof by mathematical induction: A case study. In Proceedings of the fifteenth annual meeting for the North American chapter for the psychology of mathematics education (Vol. 2, pp. 1-7). Maher, C. A., Sran, M. K., & Yankelewitz, D. (2011). Building an inductive argument. In C. A. Maher, A. B. Powell, & E. B. Uptegrove, Combinatorics and Reasoning (pp. 27-43). New York: Springer. Sran, M. K. (2010). Tracing Milin’s development of inductive reasoning: A case study (Doctoral dissertation, Rutgers University-Graduate School of Education)

Michelle’s Longitudinal Problem Solving and Development of Reasoning About Tower Tasks: : Part 3 of 3 (Grade 5)

Author: Victoria Krupnik, Rutgers University Overall Description This analytic is the third of three analytics that showcase the problem solving of a counting task by a student, Michelle, over two school years. The analytics focus on the development of reasoning, argumentation, and mathematical representations constructed by Michelle in a variety of settings, over time, in the fourth (Parts 1 & 2) and fifth (Part 3) grades. The first analytic begins with Michelle working with a partner, in a whole class setting, building towers with plastic cubes that are 5-tall, selecting from cubes available in 2 colors. This is followed by events from a first and second one-on-one interview with researchers. The second analytic continues with events when Michelle participated in a small group formative assessment interview and with a partner on a summative assessment. The third analytic includes events when Michelle worked with a partner, Milin, in a whole class setting, using towers to solve an application task. In this analytic (the third of three analytics), Michelle’s learning progression in building “proof-like” justifications to Towers Tasks shows Michelle, in the fifth grade, listening to other arguments by induction and reasoning by induction to solve 3- and 4-tall Tower Tasks, selecting from two colors. 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 1-4 are retrieved from a fifth-grade class session with partners Michelle and Milin as they work on a Guess My Tower (GMT) task on February 26, 1993. Solving the problem promotes revisiting the 3- and 4-tall tower outcome possibilities. Michelle indicates uncertainty of the 4-tall tower total, whereas Milin claims it to be 16. The events illustrate the dissemination of Milin’s idea about the doubling pattern that is recognized as they build towers of consecutive heights, supporting Milin’s inductive reasoning. Facilitated by R2, Milin explains his idea to Michelle in Event 1, Michelle displays her understanding in Events 2-3, and then demonstrates her reasoning with Stephanie and Matt in Event 4. The following definitions and background information about the Towers and Guess My Tower Tasks are offered. Argument by induction An induction argument for the justification of the general solution 2^n includes the basic step (n=1) in which a participant describes that the total number of 1-tall towers created when selecting from two colors is 2 (i.e. one of only blue and one of only yellow). The second states that the total number of towers of a given height can be found by placing either a yellow cube or a 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. 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 (towers with a single color); 2) three towers containing one red (towers with exactly one of color); 3) three towers containing 2 red cubes (within cases (2) and (3) can be cubes of the same color adjacent to or separated from each other); 4) one tower containing 3 reds or all red cubes. An argument by cases includes an exhaustive enumeration of the total number of towers in each case. Tower 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 (n) cubes high. Find a way to convince yourself and others that you have found all possible towers 3 (n) 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. Guess My Tower Task: 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?" Video and Transcript References (in chronological order of Michelle’s journey): Building Towers, Selecting from two colors for Guess My Tower, Clip 3 of 5: Milin introduces an inductive argument. Retrieved from: https://doi.org/doi:10.7282/T3RN371