Leong, Yew Hoong

This project involves the development and implementation of a problem solving package (M-ProSE) in the secondary school mathematics curriculum. It aims to induct secondary school mathematics students into the discipline of mathematics via a programme that turns well established theories of mathematical problem solving into praxis. In contrast with conventional training for mathematics competitions which tend to be restricted to a small number, M-ProSE is designed for all mathematics students Development of the project: In a pilot study conducted over two years in an Integrated Programme of a junior college, the research team observed that students were generally resistant to following the stages of Polya's model. In an attempt to 'make' the students follow the Polya model, especially when they were clearly struggling with the problem, we decided to construct a worksheet like that used in science practical lessons and told the students to treat the problem solving class as a mathematics 'practical' lesson. In this way, we hoped to achieve a paradigm shift in the way students looked at these 'difficult, unrelated' problems which had to be done in this 'special' class. Practical work to achieve the learning of the scientific processes has a long history of at least a hundred years. It is certainly conceivable that similar specialised lessons and materials for mathematics may be necessary to teach the mathematical processes, including and via problem solving. Implementation of the project: M-ProSE is an attempt to teach problem solving in 'practical' setup. Students will be taught Polya's model and problem solving in general in two or three dedicated lectures. The main mode of learning is then through a series of 'mathematics practical' lessons. Students work on usually one or at most two problems which have to be worked out on a special worksheet which requires the student to systematically and metacognitively go through the Polya model. M-ProSe is to be implemented as part of the mathematics curriculum and will be assessed. In order to implement M-ProSE, we need to build the teachers' capacity first to solve non-routine mathematics problems and thereafter to teach problem solving to their students. This involves the researchers conducting a series of workshops for the school teachers to widen their repertoire of problem solving resources. Next, we will develop with the teachers the instructional strategies to teach problem solving to their students, by means of a lesson study approach. Some of the researchers will initially teach some student classes as a model for the teachers before they take over entirely. To contribute to the understanding of teaching mathematical problem solving in general, the researchers will collect data over some cohorts which will enable them to further improve the package and make the package useful to other schools. The evidence collected will provide the basis for pedagogical practices in the mathematics classrooms.

In the field of education, research projects that involve both the researcher and teacher being the same person are common today, as attested by the significant number of teacher-researcher studies. One issue confronting the dual role of teacher-researcher is the nature of interaction between the underlying goals that come with each of these roles. There are some researchers who express concern that the combination of these goals within the teacher-researcher may compromise either or both of the work of teaching and research in an unproductive way. This paper is an account of my' adventure in attempting to fulfil both teaching and research goals in my work as teacher-researcher in a year 7 (Secondary One)geometry class in Singapore. My experience is then re-interpreted in the context of the ongoing conflicting-versus-complementary talk on the interaction between teacher/researcher'selves'. A model is proposed to account for the seemingly opposite sides of the camp as reported in the literature on this issue.

While reports of teachers’ use of curriculum materials are common, that of teachers as designers of their own materials are far less so. We argue that these (rare) instructional materials, defined as materials that are classroom-ready and that carry the teachers’ actual instructional goals, are ‘objects’ that are suitable as records of teachers’ planning and learning when developed alongside professional development. We provide supporting evidence of this claim and unpack the complexities of interacting instructional goals through a case study of a teacher who (re-)designed her own instructional materials as she participated in professional development. From the findings of the case, we reflect on the educational and methodological implications of pursing this research approach.

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.

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.

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.

This paper presents a re-design of an undergraduate mathematics content course on Introductory Differential Equations for pre-service secondary school mathematics teachers. Based on the science practical paradigm, mathematics practical lessons emphasizing problem-solving processes via the undergraduate content knowledge were embedded within the curriculum delivered through the traditional lecture-tutorial system. The pre-service teachers' performance in six mathematics practical lessons and the mathematics practical test was examined. They were able to respond to the requirements of the mathematics practical to go through the entire process of problem solving and to carry out "Look Back" at their solution: checking the correctness of their solution, offering alternative solutions, and expanding on the given problem. The use of Mathematics Practical has altered the pre-service teachers’ approach in tackling mathematics problems in a positive direction.