Taxicab via Pascal’s Triangle to Towers: Building Isomorphisms to Justify.

PurposesStudent collaboration; Reasoning; Representation
DescriptionFour high-school seniors – Brian, Jeff, Mike, and Romina are engaged in a challenging problem-solving experience with a combinatorics task referred to as The Taxicab Problem. To understand and solve the Taxicab Problem, the group developed a mediated isomorphism, connecting with rows of Pascal’s Triangle and the structure of the previously solved Towers Problems. The students are in their last year of high school and are demonstrating applications of knowledge and ways of reasoning that were developed during their previous years as participants in a longitudinal study (Powell 2010). Their recognition of an isomorphism is what Greer and Harel (1998 p. 5) advocates, “We recommend that awareness of structure, including specifically the recognition of isomorphisms, should be nurtured in children as part of the general development of expertise in constructing representational acts. A balanced view of the goals of mathematics education encompasses both the need to teach mathematics so that its applicability to many contexts is recognized, and a recognition of the importance and power of mathematics as desituated cognition”
The session begins with Researcher Maher introducing the problem, asking whether they understand what is asked. Jeff questions whether one has to stay on the lines (of the grid presented) and whether the lines represent the streets. Researcher Maher responded, “Exactly”. After working on the problem for about thirty minutes, each student realized that from the taxi stand five and seven are respectively the number of blocks it takes to reach point A (1, -4) and B (4, -3) marked on the grid, and that different routes to each point have the same length as long as one does not go beyond the particular pick-up point.
Mathematicians solve problem through a process that involves exploration and justification (Fendel & Resek 1990). The exploration process, which might involve pattern finding, making guesses, or looking at examples, is about the discovery of new ideas. Once a conjecture is made, the mathematician seeks to justify the solution. As the students work on the problem, they generated data that they considered reliable (event 1). They reflected on numerical patterns in their array of data, and observed that the pattern outcomes resembled rows of Pascal’s Triangle, and conjectured that Pascal’s arithmetic array underlies the mathematical structure of the problem. In searching how they might justify this conjecture they embarked on building an isomorphism between the Towers Problem of certain heights (selecting from two colors) and the Taxicab Problem. From earlier problem solving, they recalled that structure of Pascal’s Triangle and the structure of the Towers Problem of certain heights were equivalent (Powell 2010).
In Event 1, students work and figure out how to solve the problem at hand of finding the shortest route from the taxi stand to each of the three different marked destination points. In Event 2, the students experience a watershed moment after working on finding the routes on the points that are closer to the taxi stand instead of the specified three points. From these closer points, students recognize a pattern which resembles a Pascal’s triangle and through generalization they are able to figure out on how to get the shortest routes on the specified destination points (Event 3 and 4). In Event 5 and 6, the students use the Towers Problem of certain heights to justify the Pascal’s Triangle - like pattern recognized in the Taxicab Problem.

The Taxicab problem statement
The problem was presented to the students with an accompanying representation of a grid (Powell, 2003, p. 4) 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 at A(1,-4), B(4,-3) and C(5,-5), 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. 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.

Fendel, D., & Resek, D. (1990). Foundations of higher mathematics: Exploration and proof. Reading, MA: Addison-Wesley Publishing Company.

Greer, B., & Harel, G. (1998). The role of isomorphisms in mathematical cognition. Journal of Mathematics Behavior.

Maher, C, Powell, A & Uptegrove, E. (Eds). (2010). Combinatorics and Reasoning; Representing, Justifying and Building Isomorphisms. New York.

Powell, A. (2003). “So let’s prove it!”: Emergent and elaborated mathematical ideas and reasoning in the discourse and inscriptions of learners engaged in a combinatorial task. Unpublished doctoral dissertation, Rutgers University, NJ.
Created on2019-04-12T23:21:46-0400
Published on2019-07-31T09:30:59-0400
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