Yew Hoong Leong

Making mathematics practical : an approach to problem solving

Mathematical Problem Solving Scheme of Work and Assessment of the Mathematics Practical Detailed Lesson Plans Scaffolding Suggestions, Solutions to the Problems and Assessment Notes.

Relooking ‘Look Back’: a student's attempt at problem solving using Polya's model

Against the backdrop of half a century of research in mathematics problem solving, Pólyas last stage is especially conspicuous – by the scarcity of research on it! Much of the research focused on the first three stages (J.M. Francisco and C.A. Maher, Conditions for promoting reasoning in problem solving: Insights from a longitudinal study, J. Math. Behav. 24 (2005), pp. 361–372; J.A. Taylor and C. Mcdonald, Writing in groups as a tool for non-routine problem solving in first year university mathematics, Int. J. Math. Educ. Sci. Technol. 38(5) (2007), pp. 639–655.), with little or no successful attempts at following through with the subjects. In this article, we describe a case study of how the innovation of a ‘Practical Worksheet’ within a new paradigm of a ‘Mathematics Practical’ enabled a high-achieving student to push beyond getting a solution for a problem to extending, adapting and generalizing his solution. The findings from this study indicate promise in achieving the learning of Polyas model with notable success in the fourth stage, Look Back.

The problem-solving approach in the teaching of number theory

Mathematical problem solving is the mainstay of the mathematics curriculum for Singapore schools. In the preparation of prospective mathematics teachers, the authors, who are mathematics teacher educators, deem it important that pre-service mathematics teachers experience non-routine problem solving and acquire an attitude that predisposes them to adopt a Pólya-style approach in learning mathematics. The Practical Worksheet is an instructional scaffold we adopted to help our pre-service mathematics teachers develop problem-solving dispositions alongside the learning of the subject matter. The Worksheet was initially used in a design experiment aimed at teaching problem solving in a secondary school. In this paper, we describe an application and adaptation of the MProSE (Mathematical Problem Solving for Everyone) design experiment to a university level number theory course for pre-service mathematics teachers. The goal of the enterprise was to help the pre-service mathematics teachers develop problem-solving dispositions alongside the learning of the subject matter. Our analysis of the pre-service mathematics teachers’ work shows that the MProSE design holds promise for mathematics courses at the tertiary level.

Encouraging problem-solving disposition in a Singapore classroom

In this article, we share our learning experience as a Lesson Study team. The Research Lesson was on Figural Patterns taught in Year 7. In addition to helping students learn the skills of the topic, we wanted them to develop a problem-solving disposition. The management of these two objectives was a challenge to us. From the lesson observation and the students’ classwork, it turned out better than we expected.

Infusing Mathematical Problem Solving in the Mathematics Curriculum: Replacement Units

There are many reports on how problem solving is successfully carried out in specialised settings; relatively few studies report similar successes in regular mathematics teaching in a sustainable way. The problem is, in part, one of boundary crossings for teachers: the boundary that separates occasional (fun-type) problem solving lessons from lessons that cover substantial mathematics content. This chapter is about an attempt to cross this boundary. We do so by designing “replacement units” that infuse significant problem solving opportunities into the teaching of standard mathematics topics.

Boundary Objects Within a Replacement Unit Strategy for Mathematics Teacher Development

We recognise that, for instructional innovations to take root in mathematics classrooms, curriculum redesign and teachers’ professional development are two necessary and mutually-reinforcing processes: a redesigned curriculum needs to be seen as an improvement in order to facilitate teachers’ buy-in—an ingredient for effective professional development; on the other hand, teachers’ professional development content needs to be directed towards actual useable classroom implements through the enterprise of collaborative curriculum redesign. In this chapter, we examine the interaction between researchers and teachers in this collaborative enterprise through the metaphor of boundary crossing. In particular, we study a basic model of how “boundary objects” located within a “Replacement Unit” strategy interact to advance the goals of professional development.

Mathematics Teacher Professional Development: An Asian Perspective

This introductory chapter sets the context for the book. It also provides an overview of the chapters in the two parts of the book. The first part comprises eight chapters on policies, structures, frameworks, and contexts. The chapters provide us with some ideas about the professional development (PD) of mathematics teachers in eight Asian countries, namely China, Indonesia, Japan, Korea, Malaysia, the Philippines, Singapore, and Taiwan. It is evident from these eight chapters that the countries are at different phases of development of teacher professionalism. In some countries there are mandatory acts and regulations governing the continuous teacher PD while in others the situation is lax, and in such cases PD would greatly depend on a teacher’s own motivation and also the availability of resources. The second part comprises eight chapters that showcase innovative approaches to mathematics teacher PD in Asian countries, namely India, Japan, Korea, Pakistan, Singapore, and Taiwan. It is apparent that all of these PD programs have similar characteristics and exemplify a critical development in teacher PD in Asia. This development reflects a gradual shift in the center of gravity away from the university-based, supply-side, off-line forms of knowledge production conducted by university researchers for teachers toward emergent school-based, demand-side, on-line, in situ forms of knowledge production by teachers with support from university scholars.

Using Video Clubs to Learn for Mathematical Problem-Solving Instruction in the Philippines: The Case of Teaching Extensions

The study reported here is situated within a 7-month professional development programme aimed at supporting Secondary 1 mathematics teachers in the teaching of mathematical problem-solving in the Philippines. A video club was part of the programme. In a video club, the teachers view video clips of their own or their peer’s classroom teaching and discuss certain aspects of teaching. Evaluations at the end of the programme revealed that teachers considered the video club as one of the components of the programme that had the most impact on them. This chapter examined whether and how the video club influenced teachers’ mathematical problem-solving classroom instruction particularly in the area of teaching extensions. The findings offer a provisional theory for the trajectory of teacher learning in video clubs.

Teaching of Problem Solving in School Mathematics Classrooms

The 1980s saw a world-wide push for problem solving to be the central focus of the school mathematics curriculum since the publication of Polya’s book about solving mathematics problems in 1954. However, attempts to teach problem solving typically emphasised the learning of heuristics and not the kind of mathematical thinking used by mathematicians. There appears to be a lack of success of any attempt to teach problem solving within school curriculum. Problem solving strategies learned at lower levels tended to be ignored instead of being applied in their mathematical engagements at the higher levels, possibly because of the routine nature of the high-stake national examinations.

The Impact of Online Video Suite on the Singapore Pre-service Teachers’ Buying-In to Innovative Teaching of Factorisation via AlgeCards

A group of pre-service teachers at one institute of education was assigned to record their reflections as they viewed an online video suite which recorded (1) a secondary mathematics teacher employing a Concrete-Pictorial-Abstract (CPA)-based approach to teaching quadratic expansion and factorisation to a group of low-ability secondary 2 (14+) mathematics students; (2) students using concrete manipulative, AlgeCards, and pictorial representations, Rectangle Diagram, to perform quadratic expansion and factorisation; and (3) teachers, including a Head of Department (HOD), sharing their teaching experience in the use of the teaching package based on the abovementioned CPA approach. This chapter studied the impact of the online video suite on the pre-service teachers’ willingness to buy-in to innovative teaching of factorisation via AlgeCards through inspecting the written responses of these teachers to preset Milestone Tasks placed at different points along the progression of the video watching. The term ‘buy-in’ refers to the evidential shift of teacher’s belief in the applicability of the lesson innovation. We identified the salient features of the video suite which were responsible for bringing about pre-service teachers’ buying-in to the use of AlgeCards in teaching factorisation. Based on these findings, this chapter makes some recommendations on the design of teacher preparation method course using video technology.