Student thinking with examples: The criteria-affordances-purposes-strategies framework

A persistent challenge in supporting students' proof activity is fostering the transition from less formal, empirical arguments to formal deductive arguments. A number of researchers have begun to investigate students' thinking with examples, addressing how example use can support conjecture understanding, exploration, and proof. We extend this line of work in presenting the CAPS (Criteria, Affordances, Purposes, and Strategies) Framework, a framework addressing students' choices, uses, strategies, and affordances gained with examples. Based on individual interviews with 38 participants from middle school through the undergraduate level, the CAPS Framework simultaneously addresses multiple aspects of students' example-based exploration and provides a way to analyze how these different aspects mutually inform one another as students conjecture and prove. We present the framework, exemplify its use through vignettes of student thinking, and share findings on how students' thinking with examples shifts across the populations of middle school, high school, and undergraduate students