It is easy to dismiss the work of “teaching students to apply formula” as a low-order priority and thus trivialises the professional knowledge associated with this practice. Our encounter with an experienced teacher—through the examination of her practices and elaborations—challenges this simplistic assumption. There are layers of complexities that are as yet under-discussed in the existing literature. This paper reports a case study of her practices that reflect a complex integration of relevant theories in task design. Through examining her praxis around the theme of “recognise the form”, we discuss theoretical ideas that can potentially advance principles in the sequencing of examples for the purpose of helping students develop proficiency in applying formula.

Teaching students to apply formula using instructional materials: A case of a Singapore teacher's practice

Over the years, concreteness fading has been used to design learning materials and educational tools for children Unfortunately, it remains an underspecified technique without a clear guideline on how to design it, resulting in varying forms of concreteness fading and conflicting results due to the design inconsistencies To our knowledge, no research has analyzed the existing designs of concreteness fading implemented across different settings, formulated a generic framework, or explained the design dimensions of the technique This poses several problems for future research, such as lack of a shared vocabulary for reference and comparison, as well as barriers to researchers interested in learning and using this technique Thus, to inform and support future research, we conducted a systematic literature review and contribute: (1) an overview of the technique, (2) a discussion of various design dimensions and challenges, and (3) a synthesis of key findings about each dimension We open source our dataset to invite other researchers to contribute to the corpus, supporting future research and discussion on concreteness fading

How do we design for concreteness fading?: survey, general framework, and design dimensions

The mathematical reasoning ability is one of the essential hard skills of a student, especially in solving everyday problems. In supporting the achievement of this mathematical reasoning ability, meaningful teaching materials are an essential part of keeping exciting and not dull a learning process. This study aimed to design and develop an e-module in the ePub format with the problem-based learning approach to improve students' mathematical reasoning ability on arithmetic sequences and series concepts. This study used the design research method of development research in two stages—preliminary design and formal evaluation design—with tryouts performed on 32 grade XI students as research subjects. The data collection techniques used were documentation, observation, interviews, and tryouts. Data analysis was carried out quantitatively and qualitatively. The results showed that the e-module developed was valid and practical to improve students' mathematical reasoning ability, especially in solving problems related to sequences and series. The indicators of mathematical reasoning ability better improved using this ePub module were proposing problem-solving assumptions and concluding solutions logically. The development of the e-module in the e-Pub format offers alternative solutions to improve students' mathematical reasoning ability.

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An ePub learning module and students' mathematical reasoning ability: A development study

This chapter examines how experienced and competent Singapore secondary school mathematics teachers select and modify materials for instructional practice. For the empirical section, we begin by analysing survey responses of 677 participants across a wide range of secondary schools to determine the extent of modification among teachers before identifying which instructional materials were used as reference materials in their modification. The findings showed that the teachers relied heavily on their school-based materials as reference materials. We next analyse the instructional materials of 30 experienced and competent teachers which reveal that the teachers’ selection and modification of instructional materials were carried out in such a way as to integrate into their own instructional conceptions. The characteristics of the instructional materials that help teachers enact worthy instructional goals of teaching mathematics, such as making connections, reasoning, and challenge, were distilled from the 30 experienced and competent teachers’ interview transcripts and their instructional materials.

Singapore Secondary School Mathematics Teachers’ Selection and Modification of Instructional Materials for Classroom Use

Do Singapore secondary mathematics teachers include challenging items regularly in their instructional materials, and if so, how do they help students engage productively with them? We attempted to inquire into these questions in the study. Using some of the actual challenging items used by the teachers in the first phase of the project, we design a chronologically grounded survey that aimed at inquiring the extent of use of challenging items among teachers across a wide range of secondary schools. The findings revealed that teachers’ inclusion of challenging items range on average between “Sometimes” and “Frequently”. Also, the picture that emerges from their written comments is one where teachers are sensitive to students’ cognitive ability and affective disposition to the extent that deliberate planning and supportive mechanisms are the norm when they plan to use challenging items in the instructional materials.

Use of Challenging Items in Instructional Materials by Singapore Secondary School Mathematics Teachers

The phrase ‘make it explicit’ is a common advice given to teachers. It is, however, not clear to us what this actually means when translated into classroom practice. Our review found that we are not alone: “explicit” is used in different ways in the education literature. This paper explores, through a case study of a teacher who stated “making things explicit” as an ostensible goal of his instructional practice, how the explicitation is realised in teaching mathematics. In particular, we examine how he used the instructional materials that he crafted to fulfil his goal of explicitation. We were able to uncover three strategies he used: explicit-from, explicit-within, and explicit-to.

Making things explicit using instructional materials: a case study of a Singapore teacher’s practice

It is widely acknowledged that making connections is an important part of learning mathematics—instead of seeing mathematics as comprising merely isolated procedures to follow, it is desirable that students learn the distinctiveness of mathematics as being a tightly connected subject. In fact, the Singapore mathematics curriculum framework listed “connections” as part of mathematical processes—one of the five areas of major foci. In the study reported here, we look specifically at how an experienced and competent secondary mathematics teacher listed “making connections” as one of her ostensible principles in the design of the instructional materials for her lessons on quadratic equations. The method used can be summarised as one of progressive widening of the analytical lens: we started by conducting an in-depth examination of one unit of her instructional material to uncover the connecting strategies she built into it. Based on the strategies we uncovered, we widened the analysis to include its adjoining unit. From here, not only did we test the applicability of these strategies on the next unit, we also explored her design principles on how she connected between units. Finally, we further widened the lens of focus to the whole set of instructional material to study other connecting strategies she used across all the units in the material. The four design principles she used are: connections across multiple modes of representation, conceptual connections, temporal connections, and connections across different solution strategies. This teacher’s design principles with respect to making connections challenge conventional stereotypes of how Singapore mathematics teachers carry out instruction—it is not merely repeated practice of unrelated procedures; rather, it is a careful structuring of instruction such that the underlying mathematical connections are made explicit. Not only so, the deliberate design was not just carried out in-class; it was, as reflected in the careful crafting of the instructional material, an intentional plan prior to the teaching of the unit. The principles used by the teacher hold useful lessons for mathematics teachers, especially within the context of teacher professional development. These are discussed towards the end of the chapter.

Wei Yeng Karen Toh

Papers

Making things explicit using instructional materials: a case study of a Singapore teacher’s practice

Teaching students to apply formula using instructional materials: A case of a Singapore teacher's practice

Singapore Secondary School Mathematics Teachers’ Selection and Modification of Instructional Materials for Classroom Use

Use of Challenging Items in Instructional Materials by Singapore Secondary School Mathematics Teachers

Designing Instructional Materials to Help Students Make Connections: A Case of a Singapore Secondary School Mathematics Teacher’s Practice