Generalizing and Proving in an Elementary Mathematics Teacher Education Program: Moving Beyond Logic

Abstract

Generalization and proof are a foundation of mathematical practice and as such should be integral to K-12 mathematics instruction. However, if generalizing and proving are to become popular within K-12 mathematics classrooms, we must consider how to effectively enlist pre-service teachers (PSTs) into supporting these activities in their prospective students. Some researchers have suggested that the major challenges to generalization and proving for students in mathematics lie in developing and explicating logical statements. These researchers indicate that children do not have access to these forms of reasoning until adolescence. Those holding such views typically do not advocate for the training of PSTs in elementary education to support mathematical generalizing and proving. This study characterizes the views of and engagement with mathematical generalizing and proving of those principally involved in elementary mathematics education: mathematics faculty, mathematics education faculty, and PSTs. These views and this engagement are analyzed from survey responses and participation in a problem-solving session. Few PSTs provided descriptions of proving as a generalized explanation and demonstrated explicit generalization and proving infrequently. The results suggest that a mathematical focus on logic may be an impediment to proving and generalizing.