Representations and Connections

Abstract

In the previous chapter, we viewed a cohort of high school students from the longitudinal study as they explored the towers problems. In this chapter, we observe a different cohort of students also exploring the towers and pizza problems. In the five sessions discussed here, spanning December 1997 through March 1998, the five students in this cohort were reintroduced to the towers and pizza problems, which they last explored in elementary school as described earlier in Chapters 5 and 6. They found general solutions to those problems and a way to organize their solution lists to prove that all solutions were present. They recognized that those problems were related to each other, to the binomial coefficients, and to Pascal’s Triangle. They made use of their understanding of the structure of those problems to form preliminary ideas about the meaning of Pascal’s Identity. We show how their development and use of a sophisticated general representation scheme helped them make these connections and generalize their knowledge.