Choy Ban Heng

In this paper, we present a case study of a secondary mathematics teacher, Isaac (pseudonym), and his considerations for teaching with multiple methods for solving missing-value problems. While his students preferred methods that drew more closely on their intuitive understanding of proportionality, Isaac emphasised the algorithmic cross-multiplication method. Analysis of Isaac’s introduction and use of the cross-multiplication method suggest his key considerations were linked demonstrating the efficiency of the cross-multiplication method, while also helping students to make making meaning from the cross-multiplication method.

Successful diffusion of innovation at scale is hard to find, much less one that originates from ‘the ground’. In recent years, the practice of Secondary Mathematics teachers in Singapore designing and using teacher-designed instructional materials (known as “worksheets” locally) has become pervasive across many schools. It is an “innovation” that was not driven by policy mandates; rather, the initiation and spread started from the schools. Taking an oral history approach to elicit recollections from main actors who lived through the spread of worksheet-use in the schools they worked in, this paper is a report of the diffusion processes in these schools. Non-trivial insights can be gleaned from these experiences that may potentially inform efforts to spread domain-specific educational innovations at scale.

Teacher-designed notes and worksheets are common instructional materials used in Singapore mathematics classrooms that are critical to guiding the flow of a lesson. However, making sense of how teachers design these materials is complex and research that reports on their creation is only just emerging. In this paper, we propose a tri-lens approach for capturing teachers’ design processes by using notions of pedagogical reasoning and action, curricular noticing, and resources, orientations and goals. We demonstrate how these frameworks combine to form a tri-lens for unpacking important aspects of teachers’ design work using the example of Mrs Fung, an experienced secondary mathematics teacher. We further illustrate how a tri-lens approach can provide a more comprehensive portrait of the teacher and argue that this approach can potentially address the complexity of teachers’ design processes when crafting instructional materials.

Despite recent calls to adopt practice-embedded approaches to teacher professional learning, how teachers learn from their practice is not clear. What really matters is not the type of professional learning activities, but how teachers engage with them. In this paper, we position learning from teaching as a dialogic process involving teachers’ pedagogical reasoning and actions. In particular, we present a case of an experienced teacher, Mr. Robert, who was part of a primary school’s mathematics professional learning team (PLT) to describe how he learned to teach differently, and how he taught differently to learn for a series of lessons on division. The findings reiterate the complexity of teacher learning and suggest possible implications for mathematics teacher professional development.

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.

This commentary is an extension to the integrated S-T-E-M Quartet Instructional Framework that has been used to guide the design, implementation and evaluation of integrated STEM curriculum. In our discussion of the S-T-E-M Quartet, we have argued for the centrality of complex, persistent and extended problems to reflect the authenticity of real-world issues and hence, the need for integrated, as opposed to monodisciplinary, STEM education. Building upon this earlier work, we propose two additional variationsjsolution-centric and user-centric approaches to the provision of integrated STEM curricular experiences to afford more opportunities that address the meta-knowledge and humanistic knowledge developments in 21st century learning. These variations to the S-T-E-M Quartet aims to expand the scope and utility of the framework in creating curriculum experiences for diverse profiles of learners, varied contextual conditions, and broad STEM education goals. Collectively, these three approaches problem-centric, solution-centric, and user-centricjcan afford more holistic outcomes of STEM education.

We study some essential ideas of Banach spaces which provide n framework for mathematical analysis, and investigate the smoothness of norms and their impact on the topological properties of these spaces. We also take a first look at rough norms and give some equivalent conditions for a norm on a Banach space to be rough.