Detection and Exploitation of Structural Relationships by A Group of Four Twelfth Graders Working on The Taxicab Problem. Retrieved from https://doi.org/doi:10.7282/t3-jrpa-3r49
DescriptionPurpose/Focus:
The focus of this analytic is to examine how students develop, evolve, and adopt representations and strategies while working in a group on the taxicab problem. We will keep a close eye on how students identify and manipulate the underlying structure of the taxicab problem. In this analytic we see students use their ideas about previous tasks such as the pizza and the towers problem to make sense of the taxicab problem. The taxicab problem has an underlying structure and to see the structure the students use heuristic skills to make and hone their representations. It is noteworthy to observe the environment that the research was being conducted as it had significant implications on students’ mathematical learning. This environment is consistent with what the students are accustomed to since the early years of the longitudinal study. The conditions that support student learning include task strands, opportunities to collaborate with others and revisit problems over time in an environment that respects students’ ideas and ways of reasoning (Francisco & Maher, 2005).
In this analytic, we have four twelfth graders, Brian, Romina, Jeff, and Michael, trying to figure out the solution for the Taxicab problem. All four students, over the years, worked on isomorphic tasks such as Towers and Pizzas problem. “It began with the students’ first investigation of the towers problem in third grade (one episode), continued through their additional investigations of towers problems in fourth grade (six episodes), and then progressed through their work on various versions of the pizza problem in fifth grade (five episodes). The analysis concluded with the students’ work in their sophomore year (four episodes) and junior year of high school (one episode), when they revisited both problems and made the associations that enabled them to generate Pascal’s Identity. (Uptegrove, 2015) The taxicab problem asks the students to find the shortest distance from the taxi stand (the starting point) to the three given pick-up points, A, B, and C, and then they are asked to find the number of shortest paths to each of the three points. Additionally, the problem’s underlying structure was the key to this problem as the task gave students an opportunity to make connections and identify the isomorphisms. The students over the years have become used to building meaning first, creating their own representations, and improving them in the subsequent sessions. It is quite interesting how these environmental factors help the students identify the isomorphisms.
The group initially has no strategy and is keeping track of the number of shortest paths by counting out loud or in their mind. While using this strategy, they realize that it is hard to keep track of the count which prompts them to further analyze the problem and see if they can identify and manipulate the underlying structure. The group initially identified the dyadic nature of the problem and to test their conjecture, they picked points that were closer to the taxi stand which would be easy to work with. When examining the pattern, they came up with a representation that was identical to Pascal’s triangle.
The numbers in the representation are the number of shortest paths to the taxi stand. The first few rows are identical to the initial rows of Pascal’s triangle and uncovering this underlying structure was crucial for the group to identify the isomorphism and make their understanding of this task complete.
This skill to discover and manipulate the underlying structure of the problem is at the heart of learning mathematics because of its importance. Studies by Silver (1979) and others have shown that students strong in mathematics tend to categorize problems on the basis of their underlying structure, whereas students weak in mathematics do so on the basis of surface characteristics. Furthermore, it is a characteristic of mathematicians to search out structural relationships underlying situations with very different surface characteristics. (Greer & Harel, 1998).
For this analytic, we are really interested in how the students justify Pascal’s triangle pattern by comparing and manipulating the underlying structures of previously identified isomorphisms such as the isomorphism between the towers problem and the pizza problem. The students shift and change their representation as the session goes by from representations that are idiosyncratic compared to the ones that use formal register. For example, we see that Romina uses x and y for down and right and Mike uses a binary system of representation to show the underlying structure of the taxicab problem. Uptegrove further elaborates on the students’ representations, noting that,
“The students developed a common notation that allowed them to represent both the towers and pizza problems, helping them recognize their isomorphic relationship (i.e. the fact that the underlying mathematical structures for the solutions were the same), and they also discovered how those problems were related to the mathematics of binary notation, the binomial expansion, Pascal’s Triangle, and Pascal’s Identity.” (Uptegrove, 2015)
Problem Statement: Taxicab Problem
The problem was presented to the students with an accompanying representation on a single (fourth) quadrant of a coordinate grid of squares with the “taxi stand” located at (0,0) and the three “pick-up” points A (blue), B(red) and C(green) at (1, -4), (4, -3) and (5, -5) respectively, implying that movement could only occur horizontally or vertically toward a point. The problem states that: A taxi driver is given a specific territory of a town, as represented by the grid. All trips originate at the taxi stand. One very slow night, the driver is dispatched only three times; each time, she picks up passengers at one of the intersections indicated on the map. To pass the time, she considers all the possible routes she could have taken to each pick-up point and wonders if she could have chosen a shorter route. What is the shortest route from the taxi stand to each point? How do you know it is the shortest? Is there more than one shortest route to each point? If not, why not? If so, how many? Justify your answers.
Maher, C. A., & Weber, K. (2010). Representation systems and constructing conceptual understanding. Mediterranean Journal for Research in Mathematics Education, 9(1), 91–106.
Francisco, J. M. & Maher, C. A. (2005). Conditions for promoting reasoning in problem solving: Insights from a longitudinal study. Special Issue: Mathematical problems solving: What we know and where we are going (Guest Editors: Cai, J, Mamona-Downs, J. & Weber, K.) The Journal of Mathematical Behavior, 24(3-4), 361-372.
Silver, Edward A. (1979). Student perceptions of relatedness among mathematical verbal problems. Journal for Research in Mathematics Education, 10, 195-210.
Greer, Brian, & Harel, Guershon. (1998). The role of isomorphisms in mathematical cognition. The Journal of Mathematical Behavior, 17(1), 5-24
Uptegrove, E. B. (2015). Shared communication in building mathematical ideas: A longitudinal study. Journal of Mathematical Behavior, 40, 106–130.