PROCEEDINGS OF THE 18TH ANNUAL CONFERENCE ON RESEARCH IN UNDERGRADUATE MATHEMATICS EDUCATION
2015
Pittsburgh, Pennsylvania
Modeling outcomes in combinatorial problem solving: The case of combinations
Page: 690
In an effort to understand ways to help students solve counting problems successfully, we conducted a paired teaching experiment in which two students reinvented four counting formulas by generalizing their work from an initial set of basic problems. Subsequent to reinventing these four formulas, they solved all but one counting problem correctly, regularly drawing upon outcomes and displaying a set-oriented perspective. In this paper, we report on the problem that they missed, which involved combinations (the Bits problem: How many 256-bit binary strings contain exactly 75 0’s?). We describe a key aspect of their activity that we refer to as combinatorial encoding of outcomes, and we use this language to analyze the student work. We discuss the importance of encoding as an informal way to articulate bijections, and we suggest avenues for future work and pedagogical implications.