DescriptionThe focus of this analytic is twofold. First, it is to examine Researcher Davis’s deliberate teaching moves that promote students to build on their ideas to solve the Tower of Hanoi(TOH) problem. Second, it is to trace the collaborative problem solving of a class of sixth graders. Maher and Weber (2010) suggest that opportunities be given to learners to express their own ideas and represent them in multiple ways in the context of a collaborative, supportive environment. Children, naturally, have wonderful ideas and they are empowered when they can express them. This is the type of environment Researcher Davis promotes where students are working in small groups and we see them use different heuristic skills and natural powers of reasoning to come up with different mathematical ideas.

According to Davis (1992) discussing students’ problem solving, “We would then leave it up to the students to invent a way to carry it out, or to solve the problem. We do not show how to do things – inventing ways to solve problems is the job of the student!”. This approach gives the responsibility to the students to develop more ways to solve the tasks. It does it by giving appropriate experiences where children talk to peers about the tasks at hand and listen to their peers’ questions and respond appropriately to further make meaning of the task at hand. The researcher’s style is to emphasize individual students’ mathematical thinking and based on how the students respond, ask questions that shift the responsibility to the students to find the solution without seeking an external source for verification. This approach enables the researcher to see what would otherwise be the students’ invisible “mental representations” (Davis, 1992, p. 14).

Since these teaching moves are dependent on students’ thinking, a second focus is on how the students collaborate, grow, and justify their solutions to build mathematical ideas without seeking the researcher’s verification. As time goes by, the students’ understanding evolves from one that is instrumental to one that is relational. According to Skemp (2006), relational understanding is “knowing not only what method worked but why it worked” (p. 89) and instrumental understanding is viewed as providing “rules without reasons” (p. 92). Both types of understanding have their advantages. Instrumental understanding is easier to understand, and one can often get the right answer much faster. Relational understanding is more adaptable to new tasks, and it is more organic and durable (Skemp, 2006). Relational understanding enables the students to make connections in a way that students have a more complete idea of the concept associated with the task.

Researcher Davis’s teaching philosophy and teaching style are manifested in this analytic where he focuses on students’ learning by closely listening to their ideas. We see “deliberate creation and application of assimilation paradigms” (p. ii) by Researcher Davis which is a consequence of his philosophy put into practice. According to Piaget, “assimilation occurs whenever an organism utilizes something from its environment and incorporates it. Assimilation occurs when new learning fits into a cognitive operational structure and requires no change in that structure. Accommodation occurs when new learning requires some modification of an operational structure, the balance of the processes of assimilation and accommodation may be called a psychological equilibrium” (Piaget, 1980, p. 8). When Researcher Davis asks Michelle in event 2 to come at the front of the class and explain her ideas, it creates a great opportunity for Mike and Milin to accommodate and assimilate ideas shared by Michelle. We see how durable the relational understanding is in Event 4 where Mike is asked about the problem after 7 years when he was a freshman in college. It is important to note the classroom setting which encourages students to work together, share ideas, and test conjectures as they collaborate and communicate with each other (Mayansky, 2007, p. 91).

Researcher Davis introduces a legend associated with Towers of Hanoi (TOH). In the legend it is predicted that when the Tower of Hanoi Task with 100 disks is solved, the world will come to an end. When Researcher Davis mentions this legend, a lot of emotions come out of students. This introduction to a real-world problem is important as it makes the TOH task meaningful to the students.

Problem Statement: Tower of Hanoi

The task was based on the classic game, the Tower of Hanoi. It consists of three rods and several disks of different sizes, which can slide onto any rod. For this task, the number of disks were 100. The puzzle starts with the disks in a neat stack in ascending order of size on one rod, the smallest at the top, thus making a conical shape. The objective of the puzzle is to move the entire stack to another rod, obeying the following simple rules:

1. Only one disk can be moved at a time.

2. Each move consists of taking the upper disk from one of the stacks and placing it on top of another stack or on an empty rod.

3. No larger disk may be placed on top of a smaller disk.

References (Text)

Davis, Robert B. (1992). Understanding “Understanding”. Journal of Mathematical Behavior, 11, 225-241

GeeksforGeeks. (2021, March 26). Program for Tower of Hanoi. https://www.geeksforgeeks.org/c-program-for-tower-of-hanoi/

Maher, Carolyn A. & Weber, Keith (2010). Representation Systems and Constructing Conceptual Understanding. Mediterranean Journal for Research in Mathematics Education, 9-1, pp. 91-106

Mayansky, Elmira. An analysis of the pedagogy of Robert B. Davis: Young children working on the tower of Hanoi problem. Diss. Rutgers University, 2007.

Orr-Woods, C. (2018, June 28). The Teaching Moves of Professor Robert B. Davis with Six- graders Working on the Tower of Hanoi Problem. Https://Rucore.Libraries.Rutgers.Edu/. https://rucore.libraries.rutgers.edu/rutgers-lib/61541/

Skemp R. (2006). Relational Understanding and Instrumental Understating. Mathematics Teacher, 12(2), 88-95

According to Davis (1992) discussing students’ problem solving, “We would then leave it up to the students to invent a way to carry it out, or to solve the problem. We do not show how to do things – inventing ways to solve problems is the job of the student!”. This approach gives the responsibility to the students to develop more ways to solve the tasks. It does it by giving appropriate experiences where children talk to peers about the tasks at hand and listen to their peers’ questions and respond appropriately to further make meaning of the task at hand. The researcher’s style is to emphasize individual students’ mathematical thinking and based on how the students respond, ask questions that shift the responsibility to the students to find the solution without seeking an external source for verification. This approach enables the researcher to see what would otherwise be the students’ invisible “mental representations” (Davis, 1992, p. 14).

Since these teaching moves are dependent on students’ thinking, a second focus is on how the students collaborate, grow, and justify their solutions to build mathematical ideas without seeking the researcher’s verification. As time goes by, the students’ understanding evolves from one that is instrumental to one that is relational. According to Skemp (2006), relational understanding is “knowing not only what method worked but why it worked” (p. 89) and instrumental understanding is viewed as providing “rules without reasons” (p. 92). Both types of understanding have their advantages. Instrumental understanding is easier to understand, and one can often get the right answer much faster. Relational understanding is more adaptable to new tasks, and it is more organic and durable (Skemp, 2006). Relational understanding enables the students to make connections in a way that students have a more complete idea of the concept associated with the task.

Researcher Davis’s teaching philosophy and teaching style are manifested in this analytic where he focuses on students’ learning by closely listening to their ideas. We see “deliberate creation and application of assimilation paradigms” (p. ii) by Researcher Davis which is a consequence of his philosophy put into practice. According to Piaget, “assimilation occurs whenever an organism utilizes something from its environment and incorporates it. Assimilation occurs when new learning fits into a cognitive operational structure and requires no change in that structure. Accommodation occurs when new learning requires some modification of an operational structure, the balance of the processes of assimilation and accommodation may be called a psychological equilibrium” (Piaget, 1980, p. 8). When Researcher Davis asks Michelle in event 2 to come at the front of the class and explain her ideas, it creates a great opportunity for Mike and Milin to accommodate and assimilate ideas shared by Michelle. We see how durable the relational understanding is in Event 4 where Mike is asked about the problem after 7 years when he was a freshman in college. It is important to note the classroom setting which encourages students to work together, share ideas, and test conjectures as they collaborate and communicate with each other (Mayansky, 2007, p. 91).

Researcher Davis introduces a legend associated with Towers of Hanoi (TOH). In the legend it is predicted that when the Tower of Hanoi Task with 100 disks is solved, the world will come to an end. When Researcher Davis mentions this legend, a lot of emotions come out of students. This introduction to a real-world problem is important as it makes the TOH task meaningful to the students.

Problem Statement: Tower of Hanoi

The task was based on the classic game, the Tower of Hanoi. It consists of three rods and several disks of different sizes, which can slide onto any rod. For this task, the number of disks were 100. The puzzle starts with the disks in a neat stack in ascending order of size on one rod, the smallest at the top, thus making a conical shape. The objective of the puzzle is to move the entire stack to another rod, obeying the following simple rules:

1. Only one disk can be moved at a time.

2. Each move consists of taking the upper disk from one of the stacks and placing it on top of another stack or on an empty rod.

3. No larger disk may be placed on top of a smaller disk.

References (Text)

Davis, Robert B. (1992). Understanding “Understanding”. Journal of Mathematical Behavior, 11, 225-241

GeeksforGeeks. (2021, March 26). Program for Tower of Hanoi. https://www.geeksforgeeks.org/c-program-for-tower-of-hanoi/

Maher, Carolyn A. & Weber, Keith (2010). Representation Systems and Constructing Conceptual Understanding. Mediterranean Journal for Research in Mathematics Education, 9-1, pp. 91-106

Mayansky, Elmira. An analysis of the pedagogy of Robert B. Davis: Young children working on the tower of Hanoi problem. Diss. Rutgers University, 2007.

Orr-Woods, C. (2018, June 28). The Teaching Moves of Professor Robert B. Davis with Six- graders Working on the Tower of Hanoi Problem. Https://Rucore.Libraries.Rutgers.Edu/. https://rucore.libraries.rutgers.edu/rutgers-lib/61541/

Skemp R. (2006). Relational Understanding and Instrumental Understating. Mathematics Teacher, 12(2), 88-95

Created on2020-12-13T15:59:52-0400

Published on2021-07-09T14:56:15-0400

Persistent URLhttps://doi.org/doi:10.7282/t3-bjex-h668