The Discovery Approach Strategy in Solving Hard Mathematics Problems: A Case Study

Abstract

Four goals of this report: (a) To examine whether Mathematics Pre-Service Teachers (MPSTs) have the ability to solve correctly hard mathematical problems taken from the International Mathematics Olympiad (IMO), (b) To examine the main strategies used by MPSTs to solve these problems, (c) To examine whether MPSTs’ attitudes and beliefs about mathematics teaching and learning, could be influenced by a course in “Math Problem Solving Seminar”, and (d) what strategies might promote these beliefs. This has direct implications for the teaching and learning of mathematics. To answer these questions, sample that included a class of 36 students participate in this study which included two groups 14 excellent and 22 ordinary MPSTs. All participants took the same above course and worked in pairs (17) or individually (2). The participants choose a problem from a questionnaire which included 19 IMO problems. All the participants presented their solutions in front of the other students. Our case study- Samia’s (female) solution to “The napkins problem (IMO 2011, C7)”. The results showed that all students in our example have the ability to solve hard mathematical problems taken from the IMO. The results showed as well, that the Discovery Approach Strategy together with the Visual Approach Strategy was found to contribute positively to the solution of her problem (need to get a conjecture without the proof). In the process of solving these problems, the MPSTs reorganized their information and reconstructed their arguments. Samia supported this by using The Discovery Approach Strategy. This result is in partial accordance with the study of Tripathi (2009). The results of the semi–constructed interview showed, that the attitudes and beliefs were positively influenced by the lecturer “ An amazing and professional lecturer can contribute positively to MPSTs’ solving of difficult IMO problems via his/her direction, enthusiasm, general encouragement, setting up of challenges and encouragement of creativity, motivation, self-confidence and mathematical thinking”. We believe that training by professional mathematical teachers, using The Discovery Approach Strategy, to solve difficult mathematical problems contributes positively to MPSTs’ cognitive development, and as a direct implication, will contribute positively to the cognitive development of their students.