The Journal of the Learning Sciences, 21(4), 626-649. doi: 10.1080/10508406.2011.633838

ABSTRACT: Some intensive quantities, such as slope, velocity, or likelihood, are perceptually privileged, in the sense that we experience these quantities as holistic, irreducible sensations. However, the formal expression of these quantities uses *a/b*analytic metrics, for example, the slope of a line is the quotient of its rise and run. Thus, whereas students’ sensation of an intensive quantity could serve as a powerful resource for grounding its formal expression, accepting the mathematical form requires of students to align the sensation with a new way of reasoning about the phenomenon. We offer a case analysis of a middle-school student who successfully came to understand the intensive quantity of likelihood. Our analysis highlights a form of reasoning called abduction and suggests that sociocognitive processes can guide and mediate students’ abductive reasoning. Interpreting the child and tutor’s multimodal action through the lens of abductive inference, we demonstrate emergence of a proportional concept as guided mediated objectification of tacit perception. This “gestalt first” process is contrasted with traditional “elements first” approaches to building proportional concepts, and we speculate on epistemic and cognitive implications of this contrast for the design and instruction of these important concepts. In particular, our approach highlights an important source of epistemic difficulty for students as they learn intensive quantities; the difficulty is in shifting from intuitive perceptual conviction to mediated disciplinary analysis. Our proposed conceptualization of learning can serve as an effective synthesis of traditional and reform-based mathematics instruction.