DescriptionProblem solving is a cornerstone to the mathematical learning experience that makes possible students' application of creative strategies and logical reasoning while working to complete particular tasks. This analytic is a detailed overview of various problem-solving strategies employed by four twelfth-grade students - Brian, Jeff, Michael, and Romina - when solving the Taxicab Problem, a challenging mathematical task involving combinatorial reasoning and ideas from coordinate geometry. Each clip presents a different focus and its respective contribution to the high school students' Taxicab Problem solution. Among these highlighted problem-solving tactics are peer discussion, solving a simpler problem, pattern discovery, drawing a diagram as a representation, and building isomorphisms. The analytic was developed as an interactive instructional tool not only for K-12 teachers who may wish to integrate a problem solving unit, but also for mathematics teacher educators as a starting point to begin discussing methods to develop insightful problem-solving experiences in the classroom. Classroom teachers and teacher education students can use the Questions to Consider at the end of each video clip to reflect on the effectiveness and significance of the showcased strategies used by the students in solving the Taxicab Problem.

Taxicab Problem Statement
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.

References

Dienes, Z.D. (2002). Zoltan Dienes' six-stage theory of learning mathematics. In Some Thoughts on Mathematics. Retrieved from http://www.zoltandienes.com/?page_id=226

Greer, B. & Harel, G. (1998). The role of isomorphisms in mathematical cognition. Journal of Mathematics Behavior. 17(1), 5-24.

Powell, A. B. (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. (Doctoral dissertation). Retrieved from Rutgers University Community Repository. (UMI Number: 3092981).

Uptegrove, E.B. & Maher, C. A. (2004). Students building isomorphisms. In Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education. (Vol. 4, p. 353-360). Bergen, Norway.