Balancing an Intuitive-Experimental Approach with Mathematical Rigour: A Case Study of an Experienced and Competent Mathematics Teacher in a Singapore Secondary School

Abstract

This chapter reports a case study of an experienced and competent mathematics teacher teaching Angle Properties of Circles to a class of Secondary Three students in the Express course of study. Geometry in the school curriculum serves as a good platform for inducting students into the rigour of mathematical thinking through deductive reasoning, and the world of deductive mathematical arguments in the form of mathematical proof, which forms the common language of mathematicians worldwide. It is this rigour and discipline that students usually encounter much difficulty with. Quite contrary to our stereotyped image of a traditional geometry lesson, the teacher used a variety of approaches to enrich the lesson. She used a series of scaffoldings to lead the students from inductive exploration through discovery activities to deductive reasoning and the formalism of writing of reasoning in geometry, juggling between her belief on the importance of discovery learning and the curriculum requirement of deductive reasoning in geometry. It was interesting to us that the teacher, in transiting from students’ exploration to identifying the geometric properties, made use of rich visual imagery related to circle properties to develop in her students the concept images associated with the geometry property. Through the use of visuals to facilitate her students’ learning, effort was made to ensure her students truly understood the geometrical properties and used the properties in working with problems. Deductive reasoning was introduced in the lesson closure portion of the lesson to stress the interconnectedness across the various geometrical properties. The stages that the teacher went through in guiding the students from the intuitive-experimental stage to the deductive reasoning resonates with the van Hiele levels of students’ learning of geometry. The teacher highlighted during the interview about her conscious attempt to achieve a balance between an intuitive-experimental approach to facilitate her students’ learning and maintaining mathematical rigour that is required of the geometry strand in the Singapore school mathematics curriculum.