Teaching mathematical reasoning: From textbooks to software

Abstract

<p>Background: This project addresses critical challenges associated with learning and teaching mathematical reasoning. Mathematics is not only an enabling science that provides foundations and enhances performance in many disciplines, but also "a critical skill" for "every Australian citizen", and especially for computer science and engineering students. We explore how the effectiveness of learning and teaching mathematics can be enhanced by supplementing traditional teaching material, such as textbooks and exercise collections, with a software package. We use "GrInvIn", open source software developed at Ghent University by Adriaan Peeters, Gunnar Brinkmann, Kris Coolsaet and others, which offers a graphical, interactive, and multi-level approach to teaching and learning. GrInvIn or similar software has been used for teaching at universities in USA, Belgium, Germany, Slovenia, Croatia and Serbia, and in high schools in Belgium (Flanders). <br /><br /> Design/Method: In this project we explore how GrInvIn can be used to enhance mathematical reasoning in students. In a typical teaching scenario, a student chooses (or is given) a graph invariant, and GrInvIn generates a conjecture about that invariant that the student has to prove or disprove. While this works well at postgraduate and senior undergraduate level, at junior undergraduate and high school level a randomly generated conjecture may be too difficult for the student, which may result in a loss of motivation. Consequently, the authors of GrInvIn suggested that in introductory courses it may be necessary to use pre-generated teaching scenarios, to control the level of challenge, so as to not discourage the students. We have developed a basic GrInvIn Instructor's Manual that includes teaching scenarios, graph lists, conjectures and minimum counterexamples. <br /><br /> Results: We ran a GrInvIn instruction session for student volunteers studying towards BCompSc and BEng and the University of Newcastle, and collected their feedback which was then used to evaluate the effectiveness of teaching scenarios and effectiveness of GrInvIn for teaching junior levels. After instruction sessions, students were asked to fill in a questionnaire. The results of the survey and the focus group discussion indicate that GrInvIn is an excellent tool for senior students and that the teaching scenarios are an effective addition to GrInvIn to accommodate junior students. The discussions emphasised the importance of the introductory lecture and identified a need for support throughout the GrInvIn session. <br /><br /> Conclusions: The main strength of GrInvIn is engaging and motivating students to learn mathematics, providing that sufficient support is given to novice learners. GrInvIn, together with teaching scenarios for introductory levels and possibly some additional written material that students can refer to during the GrInvIn sessions, has a potential to foster positive behaviours pertinent to competitive learning style and intrinsic motivation, at least in senior and some junior students.</p>