To symbols from meaning: students' investigations in counting

This research provides an analysis of how a cohort group of five students learned standard notation for combinatorics over an I8-month period. The students were among the participants in a long-term study of the development of mathematical ideas and reasoning. Over the years, the students worked on open-ended and challenging mathematical problems from a combinatorics strand, such as building-towers of different heights from different colored cubes, counting pizzas with different toppings, and counting taxicab routes. The group was videotaped doing mathematics during their sophomore and junior years of high school, when they revisited combinatorics tasks that they had worked on during middle and elementary school. During these reinvestigations, they were introduced to the standard notation for combinatorics and to Pascal's Triangle and they explored the addition rule for Pascal's Triangle (Pascal's Identity). Three years later, three of the students were videotaped during task-based interviews in which they again revisited the combinatorics problems and the notation. Analysis of their work shows that the students used the combinatorial tasks with which they were already familiar to give meaning to the standard notation and to entries of Pascal's Triangle. They used the understanding of the combinatorics problems that they developed and refined over the years, including their recognition of the isomorphic relationships among the pizza, towers, and taxicab problems (structurally similar problem with different surface features), to build Pascal's Identity. A major contributing factor for the representation of Pascal's Identity in standard notation was their retrieval of earlier images of pizzas and towers that had meaning for them. Other important factors for their success included working together in collaborative investigations and having sufficient time for revisiting and rethinking ideas related to the problems. Follow-up interviews, in which individual students were asked again about Pascal's Triangle, provided evidence that students, independently, were able to rebuild or reconstruct what they had built together earlier as a group. This research provides evidence of the power of giving meaning to symbols. It suggests that students who build meaning first and then develop the symbolic vocabulary can acquire lasting understanding.Ed. D.Includes abstractIncludes bibliographical referencesby Elizabeth B. Uptegrov