Preservice Mathematics Teachers’ Determination Skills of the Proof Techniques: The Case of Integers

Abstract

The aim of the study is to reveal the preservice mathematics teachers’ ability to determine the techniques of proofs on integers. A qualitative case study approach was adopted in this study. The participants of the study consisted of 172 preservice teachers enrolled in an elementary mathematics teaching program in their second and third years at a state university in Turkey. The data of the study were obtained from the Proof Techniques Determination Form (PTDF) which consists of six proofs on integers proven by different techniques and semi-structured interviews with five preservice teachers who were successful in different achievement levels of PTDF. The preservice teachers were asked to determine the proof techniques presented to them and express their warrants. The proof techniques used in PTDF are; direct proof, proof by induction, proof by contradiction, proof by contraposition, proof by counterexample and proof by confliction. At the end of the study, the preservice teachers were successful in determining proof by induction and direct proof technique. However, they were unsuccessful in determining proof by contraposition and proof by contradiction technique. While there was no difficulty in determining proof by counterexample technique, most of the preservice teachers had struggled in determining proof by confliction used to show that the proposition was false. The preservice teachers mostly used direct proof instead of proof by contraposition, proof by contraposition instead of proof by contradiction, and direct proof instead of proof by confliction. It was determined that the preservice teachers tended to evaluate the technique of any proof as direct proof. Sources of these difficulties were: deficiencies in understanding the differences between proof by contradiction and proof by contraposition, usage of same warrants of both techniques, and accepting general warrants which are valid for every proof as the property of direct proof.