Choy, Ban Heng

Mathematics education research has focused on developing teachers’ knowledge or other visible aspects of the teaching practice. This paper contributes to conversations around making a teacher’s thinking visible and enhancing a teacher’s pedagogical reasoning by exploring the use of pedagogical documentation. In this paper, we describe how a teacher’s pedagogical reasoning was made visible and highlight aspects of his thinking in relation to his instructional decisions during a series of lessons on division. Implications for professional learning are discussed.

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.

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.

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.

The recent revision of the Singapore secondary mathematics syllabuses emphasises seeing mathematics as a tool and as a discipline. Doing this requires teachers to design and implement inquiry-based learning activities with their students. However, it is not always clear what inquiry-based learning entails and what it means for students to learn mathematics as a discipline. In this paper, we discuss what it means to think like a mathematician and illustrate three forms of inquiry-based teaching approaches with some examples for teachers to consider.

The call for teachers to engage students to learn mathematics through learning experiences (Tyler, 1949) is not through a change in content coverage; but instead, on the way Mathematics is taught and learnt (Ministry of Education-Singapore, 2012). Learning experiences were thus included as part of the current mathematics syllabus to provide opportunities for students to engage in mathematical processes, and to “influence the ways teachers teach and students learn”, so that the key objectives of mathematics education in Singapore could be achieved (Ministry of Education-Singapore, 2012, p. 20). They are stated in the form “students should have opportunities…” to signal the type of activities expected for each topic. However, as Tyler (1949) had highlighted, different students may experience these tasks differently even though the tasks are set up in the same way. This, according to Tyler (1949), presents the challenge of setting up the learning experiences to orchestrate learning.
Although the intentions and even the descriptions of learning experiences are given, teachers have the autonomy to design, select, and adapt tasks to provide these experiences for students. To realise the learning experiences as intended by the tasks, teachers would also need to orchestrate the implementation of the tasks in their mathematics classrooms (Tyler, 1949). This is deliberate work, and while most of the current support given to teachers is to help them make sense of the learning experiences, how teachers can orchestrate learning experiences to teach mathematics remains largely unexplored. Furthermore, given that what teachers notice—attend to, and how they interpret and respond to students’ reasoning (Sherin, Jacobs, & Philipp, 2011)—during the implementation of tasks is critical if we were to develop students’ competencies in the mathematical processes, it is therefore crucial for mathematics educators to investigate what teachers notice when orchestrating these learning experiences.

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.