Choy Ban Heng

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.

Teachers have several challenges when designing and implementing mathematically-rich tasks, and hence, these tasks are not prevalent in many mathematics classrooms. Instead, teachers often use typical problems, such as standard textbook tasks and examination questions, to develop students’ procedural fluency. This begs the question of whether, and if so, how teachers can think about, and use these typical problems differently to develop conceptual understanding. In this paper, we report findings drawn from a two-year design-based research project and highlight two teaching vignettes to illustrate how typical problems were used to orchestrate instructional activities. Our findings suggest three important principles for teachers to consider when using typical problems.

Excellence in mathematics education is often linked with high performance in international achievement tests such as TIMSS. In this short paper, I broaden the notion of excellence by considering how the different aspects of mathematics education come together instead of only focusing on what these aspects are. Using confluence as a metaphor to describe excellence, I examine Singapore’s excellence in mathematics education by showing how the “big things” of education such as societal expectations, policy formulation and implementation, and how the “small things” of classroom practices—scheme of work, tasks (especially typical problems), and examinations—flow together towards the same vision of ambitious teaching articulated by the Singapore Mathematics Curriculum Framework.

The urgency of teaching diverse learners is aptly demonstrated in many parts of the world as the ethnic, racial, class, and linguistic diversity grows rapidly. Such diversity not only brings about opportunities for creative teaching, but also challenges for ensuring educational equity and providing high-quality teaching for all students from diverse backgrounds, especially those presently underserved by the educational system (Buehl, & Beck, 2014; Civitillo, Juang, & Schachner, 2018). Researchers have found that teachers prepared for working with students from diverse cultural backgrounds need to embrace beliefs that recognize the strengths of cultural diversity (Anagnostopoulos, 2006; Banks et al., 2005; Fives & Buehl, 2014; Gay, 2010). Thus, exploring and challenging teachers’ beliefs about cultural diversity should constitute a major objective in teacher professional learning. However, only a few studies have examined how in-service teachers’ beliefs are enacted and shaped in professional learning community practices (Little, 2003; Tam, 2015; Turner, 2011), and focused even less on teachers’ beliefs about cultural diversity (Pang, 2005; Sleeter, 1992). There are a few studies examining teachers’ cultural beliefs about diversity in Singapore, and found that Singaporean teachers are influenced by prevailing political ideologies, and have ambiguous perceptions towards students from less advantaged backgrounds (Anderson, 2015; Alviar-Martin & Ho, 2011; Dixon & Liang, 2009; Ho & Alviar-Martin, 2010; Ho et al., 2014; Lim & Tan, 2018). However, these studies discussed teachers’ individual perceptions of disadvantaged learners without further exploring how these perceptions are mediated by influences from professional development practices, where teachers’ cultural beliefs about diversity issues are in (inter)action as ideas emerge, clash, change, and (dis)agree with each other when teachers work together.

It is challenging to design and structure lessons to maximize high-quality opportunities to learn mathematics in the classrooms. This paper presents a case study of Mary, a beginning mathematics teacher in Singapore, to illustrate how she noticed opportunities to learn during the planning and enacting of a lesson on decimal fractions for Primary 4 students. The case highlights the importance of noticing affordances of typical problems and opportunities to orchestrate productive discussions to provide quality opportunities to learn.

In designing a set of instructional materials to use in his classroom, a teacher heavily offloaded items (e.g., worked examples, practice questions, exercises) from school-based materials and textbooks. At a cursory level, one may easily dismiss this as a thoughtless lifting of curricular materials. But upon careful analysis – as is detailed in this paper – a different picture emerges. In this paper, we describe and analyse how this teacher adapted one of many worked examples, beyond its typical use, during instruction to develop students’ conceptual understanding of proportionality. We argue that he noticed and harnessed multiple affordances in a single item that most teachers may overlook, without the need to modify the example, and propose a notion of “affordance space” as a lens to view teachers’ design of instructional materials.

Much of a teacher’s practice and professional learning remains unseen despite recent calls to incorporate practice-based and inquiry-based approaches to improve mathematics instruction. Although the idea of pedagogical reasoning and action can provide a way to unpack these unseen aspects of practice, it remains to be seen how a teacher’s actions and thinking can be made visible. In this paper, we present a case of how a teacher’s pedagogical reasoning is made visible through pedagogical documentation, which suggests the possibility of using documentation to unpack these unseen aspects of a teacher’s practices.

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.