Enacting genericity: Coming to see the general in the particular while proving with manipulatives, diagrams, and symbols

A pervasive form of mathematical reasoning involves using a specific instance to draw conclusions about a general class. In mathematics education, researchers have conceptualized this practice as reasoning with “generic examples” and proposed it as a scaffold supporting students’ transition from informal exploration to the construction of proofs. Whereas existing research has largely investigated generic examples in terms of their epistemic status or didactic use within proving, it has paid less attention to the cognitive processes by which learners come to perceive a particular instance as standing for the general. Adopting an enactivist perspective, we conceptualize genericity not as an intrinsic property of an inscription but as a subjective cognitive achievement. We report on findings from task-based clinical interviews with five undergraduates in a STEM education course, interleaved with brief micro-phenomenological probes of lived experience. Participants worked through a sequence of proof tasks across registers (manipulatives, diagrams, symbols) designed to elicit generic reasoning. Through micro-genetic, multimodal analysis, we derived a three-phase model of enacting genericity: reaching, in which one case is transformed into neighboring cases; foregrounding, in which case-invariant structures become salient; and contracting, in which these structures are condensed into portable semiotic forms. Illustrative case analyses show how these micro-actions enable learners to treat an example-based argument as general. On this basis, we propose design principles for selecting and sequencing cognitively ergonomic examples that afford reaching, foregrounding, and contracting, and we argue that proofs retain explanatory power for learners when they emerge as contractions of prior example-based exploration.