Bridging Theory: Activities Designed to Support the Grounding of Outcome-Based Combinatorial Analysis in Event-Based Intuitive Judgment—A Case Study Abrahamson, Dor University of California, Berkeley Berkeley, USA dor@berkeley.edu Summary 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. Introduction Leading education researchers of probabilistic cognition generally agree as to the pedagogical value of enabling students to explore the complementarity of two genres of investigative activities, ‘theoretical’ and ‘empirical,’ targeted at random generators: (a) combinatorial analysis as a procedure for determining expected outcome distributions in actual experiments with the generators; and (b) experimentation, typically computer simulated (Jones, Langrall, & Mooney, 2007). Yet how should this complementarity be facilitated? Namely, which media may best support such inquiry? What are the perceptual experiences, cognitive requisites, and semiotic contexts that optimize a connecting of the knowledge associated with each of these activities? What pedagogical approaches might best foster participant students’ learning in such designs? How might normative mathematical concepts and procedures emerge from these activities? Design solutions vary widely. Some researchers present students with computer-based models of familiar random generators, e.g., dice, and enable the students to alter properties of simulated experiments to explore the impact of these alterations on the distribution of actual outcomes (Drier, 2000; Konold, Harradine, & Kazak, 2007; Pratt, 2000). Other researchers have the learners themselves author computer programs, thus to “debug” their own misconceptions regarding randomness, sampling, and distribution (Abrahamson, Berland, Shapiro, Unterman, & Wilensky, 2006; Abrahamson & Wilensky, 2003; Papert, 1980; Wilensky, 1995). Yet other researchers engage students in competitive games in which chances have been pre-distributed unevenly, so that the students are motivated to discover the sample space as an ICME 11 2008 – Topic Study Group 13: Research and development in the teaching and learning of probability ICME 11, TSG 13, Monterrey, Mexico, 2008: Dor Abrahamson