Abstract
Realistic Mathematics Education supports students’ formalization of their mathematical activity through guided reinvention. To operationalize “formalization” in a proof-oriented instructional context, I adapt Sjogren's (2010) claim that formal proof explicates (Carnap, 1950) informal proof. Explication means replacing unscientific or informal concepts with scientific ones. I use Carnap's criteria for successful explication – similarity, exactness, and fruitfulness – to demonstrate how the elements of mathematical theory – definitions, axioms, theorems, proofs – can each explicate their less formal correlates. This lens supports an express goal of the instructional project, which is to help students coordinate semantic (informal) and syntactic (formal) mathematical activity. I demonstrate the analytical value of the explication lens by applying it to examples of students’ mathematical activity drawn from a design experiment in undergraduate, neutral axiomatic geometry. I analyze the chains of meanings (Thompson, 2013) that emerged when formal elements were presented readymade alongside those emerging from guided reinvention.
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