Tay, Eng Guan

In this paper, we propose and describe in some detail a framework for helping low achievers in mathematics that attends to the following areas: Mathematical content resources, Problem solving disposition, Feelings towards the learning of mathematics, and Study habits.

Lee Shulman (1986) brought to the forefront the need to distinguish the different dimensions of teacher knowledge that will help guide teacher preparation. Instead of the then prevalent view of ensuring that the student teacher enters teacher education with adequate subject content knowledge and then equipping her with generic pedagogical knowledge, Shulman introduced the dimension at the nexus of subject content knowledge and generic pedagogical knowledge as pedagogical content knowledge. We, as Mathematics educators, are happy to claim Shulman as one of our own, as he based much of his research and writing on the discipline of Mathematics.

The more ambitious an educational innovation, the greater the challenge in scaling up. In this paper, we focus on the scaling up of an ambitious pedagogical practice—mathematics problem solving as a regular feature in the classroom. We adopt a long-term approach to continual professional development (CPD) that began with intensive work with one school before we broadened the programme to four other schools which span the spectrum of schools in Singapore. To evaluate this overall design, we examine the current state of each school’s capacity in sustaining mathematics problem-solving instruction. In particular, we study and report findings on these areas: the readiness of teachers, the instructional materials and supporting structures. Based on the findings, we reflect on our CPD strategies and our theory of action which guided the CPD programme.

For low cost and more flexibility and choice, instructional materials can be prepared by using a spreadsheet. This paper shows how the software Near Miss written on a spreadsheet is used to simulate the popular gambling game 4-D so that students may investigate probability and a little psychology.

The skewness of a graph 𝐺 is the minimum number of edges in 𝐺 whose removal results in a planar graph. It is a parameter that measures how nonplanar a graph is, and it also has important applications to VLSI design, but there are few results for skewness of graphs. In this paper, we first prove that the skewness is additive for the Zip product under certain conditions. We then present results on the lower bounds for the skewness of Cartesian products of graphs with trees and paths, respectively. Some exact values of the skewness for Cartesian products of complete graphs with trees, as well as of stars and wheels with paths are obtained by applying these lower bounds.

Koh and Tay proved a fundamental classification of G vertex-multiplications into three classes C0, C1 and C2. In this paper, we prove that vertex-multiplications of Cartesian products of graphs G × H lie in C0 (respectively, C0∪C1) if G(2) ∈ C0 (respectively, C1), d(G) ≥ 2 and d(G×H) ≥ 4, providing further support for a conjecture by Koh and Tay. We also focus on Cartesian products involving trees, paths and cycles and show that most of them lie in C0.

In a design experiment, the feedback from the teacher-implementer is crucial to the success of the innovation simply because the teacher is finally the one that brings the innovation to life in front of the students. We describe in this paper the feedback made by the teacher-implementer after teaching one cycle of the problem solving module in a mainstream school, and the modifications the researchers and the teacher-implementer have made in our design of the module to fit into the requirement of the school.

This research project is an attempt to realise the ideals of mathematical problem solving, which is at the heart of the Singapore mathematics curriculum in the daily practices of mainstream mathematics classrooms. This work builds on the foundation of M-ProSE (OER 32/08 TTL) to diffuse the findings to the mainstream school curriculum. Our work involves three steps: (1) initialisation of problem solving as an essential part of the mathematics curriculum in a school at the foundational year; (2) infusion of problem solving as an embedded regular curricular and pedagogical practice across all year levels in the school, and (3) diffusion of this innovation from this school to the full range of schools in Singapore. In each of the above steps, we take a complex systems approach and include curriculum, instructional practices, assessment and teacher professional development in our overall design research process. Our current project builds upon the initial foundation of MProSE to scale out (infuse) and scale up (diffuse) the innovation to mainstream schools in Singapore, hence the project is named MInD. With the experience and data collected from MProSE research school, the design needs to be re-adjusted in order for problem solving to be diffused throughout the mainstream schools. The importance and relevance of this research project to schools is readily observed by the schools' responses: To the researchers' pleasant surprise, four mainstream schools readily expressed their commitment to participate in this research as the school leaders see the relevance of this project to their school curriculum. Further, the Principal of MProSE research school expressed his interest to get his school involved for the infusion phase(step (2)) of the research. The research team of MInD consists of the original researchers from MProSE and two more new team members. The entire team consists of expertise from different fields: mathematicians, mathematics educator, educational psychologist, curriculum specialist, senior teacher, a school principal (who is also a mathematician), an expert of change management and leadership studies, a senior MOE curriculum specialist.

Although there is an acknowledged need to attend to the learning needs of low achievers in mathematics, there is relatively scant research in this area locally. This proposed project aims to contribute to this sub-field within mathematics education. In particular, this study will focus on helping Normal Academic (NA) students make improvements in their learning of Mathematics.