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We consider designs for conceptual learning where students first engage in pre-symbolic problem solving and then articulate their solutions formally. An enduring problem in these designs has been to support students in accessing their pre-conceptual situated process, so that they can reflect on it and couch it in mathematical form. Contemplative practices may offer practical solutions to this epistemic bottleneck by orienting students not on their explicit thoughts but on nuanced somatic sensations within pre-conceptual liminal space. We support this proposal for contemplative mathematics reflecting through process approaches on Mason and Roth's work in this area, and a report on two pilot studies.

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 address its criticism. My approach hinges on decoupling the solution process from its resultant product. 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 their guided heuristic-semiotic aligning of the product of analysis process with informal inference from naively seeing situations. I support my thesis with two vignettes.