Leong, Yew Hoong

This paper examines some of the complexities involved in the actual work of classroom instruction by examining interactions among the goals of teaching. The research is part of a case study of teaching a Year 7 Singapore class comprising students of average mathematical ability. Among the complexities of teaching analysed here are the problems associated with trying to fulfil the multiple goals of teaching and the conflict experienced by the teacher as he attempts to carry out these goals. This provides insight into how a teacher performs the act of balancing different goals while carrying out instruction in class. The implications of these insights into teaching practice for the wider education community are also discussed.

In designing a set of instructional materials to use in his classroom, a teacher heavily offloaded items (e.g., worked examples, practice questions, exercises) from school-based materials and textbooks. At a cursory level, one may easily dismiss this as a thoughtless lifting of curricular materials. But upon careful analysis – as is detailed in this paper – a different picture emerges. In this paper, we describe and analyse how this teacher adapted one of many worked examples, beyond its typical use, during instruction to develop students’ conceptual understanding of proportionality. We argue that he noticed and harnessed multiple affordances in a single item that most teachers may overlook, without the need to modify the example, and propose a notion of “affordance space” as a lens to view teachers’ design of instructional materials.

In this symposium, we discuss some preliminary data collected from our problem solving project which uses a design experiment approach. Our approach to problem solving in the school Curriculum is in tandem with what Schoenfeld (2007) claimed: “Crafting instruction that would make a wide range of problem-solving strategies accessible to students would be a very valuable contribution … This is an engineering task rather than a conceptual one” (p. 541). In the first paper, we look at how two teachers on this project taught problem solving. As good problems are key to the successful implementation of our project, in the second paper, we focus on some of the problems that were used in the project and discuss the views of the participating students on these problems. The third paper shows how an initially selected problem led to a substitute problem to meet our design criteria.

We recognize that though teachers may participate in various forms of professional development (PD) programmes, learning that they may have gained in the PD may not always lead to corresponding perceivable changes in their classroom teaching. We offer a theoretical re-orientation towards this issue by introducing a construct we term “concretisation”. Concretisations are resources developed in PD settings which can be converted into tangible tools for classroom use. In theorising such resources, we contribute in informing the design process of teacher professional development for better impact into actual classroom practice. We purport principles of design which render concretisations effective. Subsequently, we test these principles by presenting a specific case of teaching mathematical problem solving.

The longstanding criticism against education research is: Has it made a difference to actual classroom practice? In this chapter, I present a case for the affirmative in the context of mathematics education research in Singapore – not merely by describing cases but also extracting common underlying features that contribute to impact. These examples include the now well-known ‘model method’, mathematics problem-solving and the concrete-pictorial-abstract instructional heuristic.

The Concrete-Pictorial-Abstract (CPA) approach, based on Bruner’s conception of the enactive, iconic and symbolic modes of representation, is a well-known instructional heuristic advocated by the Singapore Ministry of Education since early 1980’s. Despite its ubiquity as a teaching strategy throughout the entire mathematics education community in Singapore, it is somewhat surprising to see a lack of an account of its theoretical roots. This paper is an attempt to contribute to this discussion on the CPA strategy and its potential role in continuing advancement of quality mathematics education.