For the Learning of Mathematics, 32(1), 8-15.

Motivated by the question, “What exactly about a mathematical concept should students discover, when they study it via discovery learning?”, I present and demonstrate an interpretation of discovery pedagogy that attempts to sustain its ideology yet address its criticism. My approach hinges on decoupling the solution-procedure *process* (applying analytic algorithm to a situation under inquiry) from its resultant *product* (material displays or multimodal utterance, e.g., diagrammatic, tabular, or symbolic inscription, that experts interpret as bearing meanings pertaining to properties, relations, patterns, or structures in the situation). Whereas theories of learning often focus on process as the site of discovery, I propose to focus instead on product. Specifically, I view student discovery of mathematical concepts as guided heuristic–semiotic alignment of the product of mathematical analysis process with informal inference from naively seeing situations. I support my thesis with analyses of two vignettes, in which perception-driven design for intensive quantities was implemented as follows: (1) elicit students’ perceptual judgment for a property of a situation created specifically so that the judgment agree with accepted theory; (2) guide students through enacting the analytic process for determining this property; and (3) help students see that the product of this process agrees with, and perhaps amplifies, the original judgment.