Students’ Cognition of The Induction Step in Proving Inequality Propositions

Abstract

The induction step in proof by induction requires some clever tricks in order to get the expected formula for , especially in statements with inequalities. The purpose of this study was to determine undergraduate students’ cognition regarding the induction step in proving inequality propositions in order to create opportunities to teach the principle of mathematical induction better. A class of 67 students participated in the study on learning proof by induction using the problem-based approach. The Action-Process-Object-Schema theory was used to structure the study and the cyclic activities-class-discussion-exercises instructional approach was used to teach proof by induction. Data for the study was comprised of individual students’ written responses to a task of two questions and the transcriptions of the semi-structured interviews. The findings revealed that students at most showed indication of partial understanding of proof by induction. Executing the induction step sits at the heart of proof by induction and necessitates logical reasoning at the object-level conception. Inadvertently, the implication was the most challenging aspect in proof by induction. The majority of students made inroads in setting up the proof properly but could not succeed in proving that . Some students had challenges of where to begin a proof, so much that they chose to start with direct substitution. In line with that, most students also concluded without deriving the expected formula required to draw a conclusion. 
 Keywords: induction step; proof by induction; APOS-ACE teaching cycles; inequalities.