Tin Lam Toh

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.

Reasoning, Communication and Connections in Mathematics: An Introduction

This introductory chapter provides an overview of the chapters in the book. The chapters are organised according to three broad themes that are central to reasoning, communication and connections. The themes are mathematical tasks, classroom discourse and connections within and beyond mathematics. It ends with some concluding thoughts that readers may want to be cognizant of while reading the book and also using it for reference and further work.

Convergence of a linear recursive sequence

A necessary and sufficient condition is found for a linear recursive sequence to be convergent, no matter what initial values are given. Its limit is also obtained when the sequence is convergent. Methods from various areas of mathematics are used to obtain the results.

Reasoning, Communication and Connections in Mathematics : Yearbook 2012, Association of Mathematics Educators

This introductory chapter provides an overview of the chapters in the book. The chapters are organised according to three broad themes that are central to reasoning, communication and connections. The themes are mathematical tasks, classroom discourse and connections within and beyond mathematics. It ends with some concluding thoughts that readers may want to be cognizant of while reading the book and also using it for reference and further work.

Mathematical Problem Solving: Linking Theory and Practice

Mathematical problem solving is the primary goal of school mathematics curriculum in Singapore. Prospective secondary school mathematics teachers, as part of their teacher education at the National Institute of Education, undertake a 96 hour course called Teaching and Learning of Mathematics. Throughout the course, as part of the study of content and pedagogy of various topics of secondary mathematics, they are engaged in solving mathematical problems. A formal introduction to mathematical problem solving and review of the relevant literature is done at the beginning of the course. As an introduction to mathematical problem solving, we engage our teachers in two tasks, The Circular Flower Bed and Solve 4 Problems, to jump start discussion on mathematical problem solving and bridge theory into practice. The goals of the tasks are as follows. The Circular Flower Bed task engages prospective teachers in problem solving and initiates discussion on the process of finding a solution, specifically the feelings, emotions and regulation of thinking during the process. The Solve 4 Problems task engages prospective teachers in clarifying the definition of a problem, distinguishing heuristics from strategies and making connections with Polya’s (1973) four phases of problem solving.

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.

Preliminary study of integrated physics and mathematics bridging course

Creating an environment to allow students to appreciate the linkage between subjects has been gaining increasing importance because interdisciplinary approaches are necessary to address socio-technological challenges. Subjects like Physics and Mathematics have often been taught as separate subjects. This results in students viewing various subjects as individual subjects, which is not ideal because there is no clear distinction between the subjects when dealing with real-life problems. For example, a number of students have a tendency to view Mathematics as only formula without applications, which result in them losing interest as they are unable to appreciate the vast number of applications that Mathematics can be applied in. In addition, it has been observed that a number of students are able to solve the Mathematics portion during Mathematics lesson, but are unable to evaluate similar Mathematics questions during Physics lessons. Hence, this paper proposes an integrated Physics and Mathematics learning and aims to help students establishing linkage between the two subjects. In order to achieve the learning objective of the students being able to appreciate the linkage between the two mentioned subjects, the syllabus is planned such that Physics is used as an application of Mathematics. The team has performed a preliminary study and implemented the proposed idea with a group of students in a bridging course. The bridging course is conducted for a duration of five days, whereby both Mathematics and Physics topics are covered. The students involved are incoming undergraduate students, and the data collected is analyzed. The preliminary results indicate a clear shift in enabling the students to appreciate the linkage between Physics and Mathematics with the integrated teaching of Physics and Mathematics.

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.