Li, an 11 year old boy, participated in the implementation of a mixed-media design for the binomial that combines activities pertaining to theoretical probability (combinatorial analysis) and empirical probability (simulated experiments). This design was engineered to accommodate, corroborate, yet elaborate on students ’ heuristic inferences, and student reasoning was elicited through semi-structured clinical interviews. Applying a cultural–semiotic approach to the analysis of Li’s case study, I discuss a universal pedagogical tradeoff articulated as tension between constructivist and sociocultural perspectives on mathematics education. Li fluctuates between two interpretations of a sample space: event-based attention grounded in intuitive perceptual judgment of a random generator yet oblivious to permutations; and outcome-based attention supporting normative mathematization yet initially unsynthesized with intuition. These apparently vying perspectives are reconciled, if problematically, when Li notices that the entire sample space indexes an expected distribution qualitatively aligned with his perceptual intuition. At a theoretical level, I argue, constructivist and sociocultural perspectives, too, can be reconciled, if problematically, by accepting that mathematical phenomena are phenomenologically akin to scientific phenomena and thus mathematical learning is an inductive process of synthesizing (Schön, 1981) heuristic-based perceptual judgments and artifact-based mediated analytic procedures
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