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.

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.

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.

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.

Typical mathematics problems, such as examination-type questions, are often used in classrooms to develop students' procedural fluency. In this article, we describe and analyse what a secondary school mathematics teacher noticed about the affordances of such a problem, as well as how she orchestrated a mathematically productive discussion using the adapted problem in class. The findings suggest that a teacher's productive noticing of the affordances offered by typical problems can enhance the learning experiences of mathematics students.

In this paper, we re-examine the commonly-held notion that typical problems, such as textbook exercises and examination questions, are not useful for orchestrating mathematically-rich learning experiences. Drawing from a larger design-based research project, we present a case study of Alice, a secondary school teacher, who orchestrated a productive discussion by using examination questions. We describe how she perceived and harnessed the affordances of such typical problems before and during her lesson. Findings suggest teacher noticing as a key mechanism to enable teachers to unlock the mathematical potential of such problems.

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.