The Mathematical Proof Steps of Mathematics Study Program Students in the Subject of Real Analysis

Abstract

the present paper describes the second year-research entitled the learning trajectory on mathematical proof of the students majoring in mathematics education. It applies the product of the first-year research namely the framework of the learning trajectory of mathematical proof and the design of its means in the topic of the real number system and the real number sequence. The present research aims at obtaining the mathematical proof steps in the learning trajectory which has been already set in the first-year research. Design research was determined as the research method aimed at, according to Gravemeijer (2013), developing a Local Instruction Theory (LIT). The LIT is expected to improve the quality of learning by developing the sequence of activities and understanding the empirical phenomena about how something works. In this research, the steps of mathematical proof were hypothetically designed i.e., (1) understanding the statement which will be proved, (2) deciding to choose either direct proof or indirect proof, (3) writing the proof specifically, and (4) verifying the validity of the proof. After the activities had been applied, retrospective analysis was undertaken to identify whether the stages were in line to the learning trajectory set in the first-year research. The learning trajectory itself includes: (a) proof with simple procedures of which the proof of a statement is done with single thinking procedure, (b) existential proof covering existential proof and non-existential fact or concept, (c) proof with complex thinking procedure, and (d) proof through construction, The findings suggest that the steps of the previously hypothesized proof occur with a support which is called scaffolding by visualizing a diagram of proof as the guide for writing the proof and there are still difficulties of writing the proof symbolically and describing mathematical argument which relates each step of the proof process.