From test cases to special cases: Four undergraduates unpack a formula for combinations

Through a case study of four elementary education undergraduates, we seek new analytic constructs that could help make clearer how arguments discovered or tested in quite special cases might come to support assertions that are understood to hold in general. We began analysis from a particular standpoint: to focus fundamentally on learners’ representations and on how the learners reason from them. We have found it helpful to distinguish two perspectives to guide the subsequent analysis. On the one hand, we direct detailed attention to how learners reason, most especially on how they organize the logic of their arguments. On the other hand, we seek to understand the learners’ representations based on the way they structure them, and through the ways such structures might be reshaped or reframed over time. Based on this analysis the student subjects offer further evidence of the depth and power that students often viewed as less mathematically inclined can demonstrate in learning situations that engage them deeply.