Pascal’s Identity

PurposesStudent representation; Student reasoning; Effective teaching
DescriptionThis is an after-school session in which Jeff, Michael, Romina, and Brian volunteer to participate. This particular session is called the Night Session because it took place on an evening late in their junior year. These students have been working on many combinatorics problems since second grade, some of which are isomorphic to each other. In this session, they were discussing the meaning of combinatorial notation and discussing the addition rule of Pascal’s Identity in terms of that notation. They were asked to write the general form of Pascal’s Identity starting from discussing the coefficients of the expansion of (a+b)^n. Their work indicates recognition of an isomorphism between the two problems. In this Analytic, I will show how the four students worked together to connect and explain the isomorphism, how they derived Pascal’s Identity, and how they represented the relationship using formal mathematical notation.
References:
Greer, B., And Harel, G. (1998). The role of isomorphisms in mathematical cognition. Journal of Mathematical Behavior, 17, 5-24.
Lo, W. (2010). Task analysis: The inherent mathematical structures in students' problem-solving processes (Doctoral dissertation). Rutgers University.
Maher, C. A., Powell, A. B., And Uptegrove, E. B. (2010). Combinatorics and reasoning: Representing, justifying and building isomorphisms Springer Verlag.
Uptegrove, E. B. (2005). To symbols from meaning: Students' investigations in counting (Doctoral dissertation). Rutgers University.
Created on2011-10-26
Published on2011-10-26T09:46:05-05:00
Persistent URLhttp://dx.doi.org/doi:10.7282/T3VX0FMM