Leong, Yew Hoong

This article presents a case study on a secondary mathematics teacher, Mary (pseudonym), and her design of a set of instructional tasks in the context of proportional reasoning. In keeping with the way Singapore teachers generally conceive of instructional planning, we investigated the connections between four comparison tasks she designed through her instructional goals. We adopt the use of an item-level lens to analyse the instructional goals of individual tasks, followed by a set-level lens to determine the movement in her instructional goals across tasks. The metaphors of overlaps and shifts describe how the movement of her instructional goals helped to advance students’ understanding of proportionality and to develop their proportional reasoning. Findings illustrate the usefulness of adopting the metaphors of overlaps and shifts and using two lenses to gain insights on teachers’ complex design processes. Additionally, they suggest that teachers’ understanding of student’s knowledge at the primary level is valuable for designing tasks that will ease students’ transition to secondary mathematics.

Self-efficacy is a subject of ongoing intense research in cognitive psychology. Studies within this tradition are focused on how this agentic aspect of human functioning fits within and contributes to a network of other sociocognitive functionalities. From a mathematics education perspective, we seek a theoretical re-framing of self-efficacy that accounts for and advances thinking in students’ (lack of) learning of mathematics in the classroom. This study takes this perspective by starting with a student’s actual experiences in switches of self-efficacy states. Through a case study of a student who has a profile that matches one with low mathematics self-efficacy, I examine the contours of self-efficacy across nested mathematical domains. This nuanced view provides an alternative to static presumptions of self-efficacy models common in research reports in this area, and is in keeping with the dynamic ebb and flow of actual classroom experiences.

This introductory article of the Special Issue themed Teachers as Designers of Instructional Tasks provides an overview of the research reported in the six articles of the issue. The articles show teachers playing a more active role in choosing and designing materials for their own instructional work thus illuminating new affordances in a number of fronts. We summarise these new utilities along the lines research, design, and professional development.

This article focuses on a challenging geometry problem that was originally posed to primary school students. Four solution approaches, ranging from elementary to advanced, are discussed. Reflections on these approaches and the problem solving processes are also shared.

This paper establishes the context for the debate to oppose the motion that "Research should not inform practice". The paper first defines what is meant by the terms research and practice in the context they are used in this debate. Four key points are then offered which illustrate the importance of research informing practice.

When mathematics teachers plan lessons, they interact with curriculum materials in various ways. In this paper, we draw on Brown’s (2009) Design Capacity for Enactment framework to explore the practice of adapting curriculum materials in the case of a Singapore secondary mathematics teacher. Problems from the textbook used and the worksheets she crafted were compared to determine how she adapted the content. Video-recordings of the lessons and post-lesson interviews were used to clarify how her personal teacher resources contributed to her design decisions. The findings suggest that her seemingly casual use of problems from the textbook are in fact unique variations of adapting curriculum materials.

In this paper, we discuss the development of a very specific problem solving curriculum in an independent school in Singapore as part of the first phase of our research project. We are using a design research methodology to fine-tune the problem solving curriculum in which we are introducing the mathematics practical, an idea borrowed from science education.

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.

Mathematical problem solving has been at the core of the Singapore mathematics curriculum framework since the 1990s. We report here the features of the Mathematical Problem Solving for Everyone (M-ProSE) project which was carried out in a Singapore school to realise the learning of mathematical problem solving and as described by Pólya and Schoenfeld. A mathematics problem solving package comprising “mathematics practical” lessons and assessment rubric was trialled in the school for Grade 8 in 2009. Responses from three students show mixed perceptions to the module, but an end-of-module assessment shows that the students were able to present their solutions along Pólya’s four stages. We also describe teacher preparation for teaching the module. After the trial period, the school adopted the module as part of the curriculum and it is now a compulsory course for all Grade 8 students in that school.