PROCEEDINGS OF THE 16TH ANNUAL CONFERENCE ON RESEARCH IN UNDERGRADUATE MATHEMATICS EDUCATION (Vol 1)
2013
Denver, Colorado
Explication as a Lens for the Formalization of Mathematical Theory Through Guided Reinvention
Page: 1-160
Realistic Mathematics Instruction 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 is an explication (Carnap, 1950) of informal proof. Explication is the process of replacing unscientific concepts with scientific ones. I use Carnap’s criteria for successful explication to demonstrate how each element of mathematical theory (definitions, axioms, theorems, proofs) explicates its less formal correlate. I provide examples of students’ proving activity in an axiomatic geometry course to motivate the need for explication, meaning that students should see formal theory as a precise expression of their less formal understandings. I conjecture that students who understand formal theory as an explication will better coordinate semantic and syntactic reasoning toward proof. I provide supporting evidence for this claim from a teaching experiment in which students reinvented axioms and definitions of planar geometry.