Pre-Service Teachers’ Benefits from an Inquiry-Based Transition-to-Proof Course with a Focus on Generic Proofs

Abstract

At the University of Paderborn, the course “Introduction into the culture of mathematics” is required for all first-year students who enter the study program for future mathematics teachers at lower secondary level (grade 5–10). In this inquiry-based transition-to-proof course, we use four different kinds of proofs (the generic proof with numbers, the generic proof with figurate numbers, the proof with figurate numbers using geometric variables, and the so-called “formal proof”) to engage students in exploration, reasoning, and proving. In this paper, we report findings from an empirical study in winter term 2014/15 (pre- and posttest) concerning proof validation and acceptance. We used different kinds of ‘reasoning’ taken from Healy and Hoyles (2000) to assess students’ proof validation. At the beginning of the course, about a third of the students judged the purely empirical verifications and wrong algebraic operations as correct proof. These forms of reasoning being judged as correct proofs decreased greatly in the posttest. To investigate proof acceptance, the students had to rate different aspects – such as “conviction”, “explanatory power”, or “validity” - of the four kinds of proofs. “Proof acceptance scales” with very high reliabilities (Cronbach’s α> .864) were constructed using factor analysis. While in the pretest most of the students did not accept the generic proofs and the proof with geometric variables as general valid verifications, their acceptance increased during the course. However, in the posttest, the ratings of the different aspects vary greatly concerning the four kinds of proofs.