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In this paper we describe a problem solving course for first year undergraduate mathematics students who would be future school teachers. Subject Classification: 97B50, 97B70, 97D50, 97D60, 97F60, 97U30

In this work, we report the emergence of the Hands, Head and Heart framework that arose within the curriculum review for subject knowledge courses for primary school pre-service teachers in the National Institute of Education, Singapore. Through an initial grounded analysis of a survey of pre-service teachers and faculty focus group meeting data, the responses were broadly categorised into hands, head and heart domains and these formed an initial framework for discussions in the review committee meetings. By revisiting the data from the survey, an analysis through a complexity lens revealed the emergence of a characteristic nested self-similarity of the framework. Over the course of several committee meetings, further self-similarity was discovered. We conjecture that the Hands, Head and Heart framework and its self-similarity property provide a potential basis for a holistic approach to curriculum review. We used this framework to revise the learning objectives of the subject knowledge curriculum by resolving perspectives which previously seemed contradictory.

The skewness of a graph G, denoted by sk(G), is the minimum number of edges in G whose removal results in a planar graph. It is an important parameter that measures how close a graph is to planarity, and it is complementary, and computationally equivalent, to the Maximum Planar Subgraph Problem. For any connected graph G on p vertices and q edges with girth g, one can easily verify that sk(G) ≥ π(G), where $$\pi(G)=\lceil q-\frac{g}{g-2}(p-2)\rceil$$ , and the graph G is said to be π-skew if equality holds. The concept of π-skew was first proposed by G. L. Chia and C. L. Lee. The π-skew graphs with girth 3 are precisely the graphs that contain a triangulation as a spanning subgraph. The purpose of this paper is to explore the properties of π-skew graphs. Some families of π-skew graphs are obtained by applying these properties, including join of two graphs, complete multipartite graphs and Cartesian product of two graphs. We also discuss the threshold for the existence of a spanning triangulation. Among other results some sufficient conditions regarding the regularity and size of a graph, which ensure a spanning triangulation, are given.

The more ambitious an educational innovation, the greater the challenge in scaling up. In this paper, we focus on the scaling up of an ambitious pedagogical practice—mathematics problem solving as a regular feature in the classroom. We adopt a long-term approach to continual professional development (CPD) that began with intensive work with one school before we broadened the programme to four other schools which span the spectrum of schools in Singapore. To evaluate this overall design, we examine the current state of each school’s capacity in sustaining mathematics problem-solving instruction. In particular, we study and report findings on these areas: the readiness of teachers, the instructional materials and supporting structures. Based on the findings, we reflect on our CPD strategies and our theory of action which guided the CPD programme.

A study of the enacted secondary school mathematics curriculum in Singapore schools, was a programmatic research project at the National Institute of Education (NIE) funded by the Ministry of Education (MOE) in Singapore through the Office of Education Research (OER) at NIE. The project had two aims. The first was to document how experienced and competent teachers enacted the school mathematics curriculum in secondary schools. It did this by examining: (i) pedagogies adopted by experienced and competent mathematics teachers when enacting the curriculum, and (ii) experienced and competent teachers’ use of instructional materials for the enactment of the curriculum. The second was to establish how uniform these adopted pedagogies and use of instructional materials by experienced and competent teachers were practised in the mathematics classrooms of Singapore schools. The project had two phases. The first was the video-segment and the second was the survey-segment. The survey-segment was dependent on the findings of the video-segment. The video-segment documented the pedagogy of experienced and competent secondary mathematics teachers while the survey-segment helped to establish how uniform the pedagogy of experienced and competent teachers was in the mathematics classrooms of Singapore schools. Thirty experienced and competent mathematics teachers and their students participated in the first phase, while another 691 mathematics teachers from across the schools in Singapore participated in the second phase. Data collected have been subjected to purposeful analysis, using appropriate methods. The following chapters in the book, present evidence-based portraits of mathematics teaching and learning in Singapore secondary schools.

We recognise that though teachers may participate in various forms of professional development (PD) programmes, learning that they may have gained in the PD may not always lead to corresponding perceivable changes in their classroom teaching. We offer a theoretical re-orientation towards this issue by introducing a construct we term “concretisation”. Concretisations are resources developed in PD settings which can be converted into tangible tools for classroom use. In theorising such resources, we contribute in informing the design process of teacher professional development for better impact into actual classroom practice. We purport principles of design which render concretisations effective. Subsequently, we test these principles by presenting a specific case of teaching mathematical problem solving.

A study of school mathematics curriculum enacted by competent teachers in Singapore secondary schools is a programmatic research project at the National Institute of Education (NIE) funded by the Ministry of Education (MOE) in Singapore through the Office of Education Research (OER) at NIE. The main goal of the project is to collect a set of data that would be used by two studies to research the enacted secondary school mathematics curriculum. The project aims to examine how competent experienced secondary school teachers implement the designated curriculum prescribed by the MOE in the 2013 revision of curriculum. It does this firstly by examining the video recordings of the classroom instruction and interactions between secondary school mathematics teachers and their students, as it is these interactions that fundamentally determine the nature of the actual mathematics learning and teaching that take place in the classroom. It also examines content through the instructional materials used—their preparation, use in classroom and as homework. The project comprises a video segment and a survey segment. Approximately 630 secondary mathematics teachers and 600 students are participating in the project. The data collection for the video segment of the project is guided by the renowned complementary accounts methodology while the survey segment adopts a self-report questionnaire approach. The findings of the project will serve several purposes. They will provide timely feedback to mathematics specialists in the MOE, inform pre-service and professional development programmes for mathematics teachers at the NIE and contribute towards articulation of “Mathematics pedagogy in Singapore secondary schools” that is evidence based.

This book serves the purpose of discussing the state of mathematics education in Singapore at the time it is written, and an update of a previous edition of the book on Singapore mathematics education published about a decade ago. This chapter identifies the central message that runs across the chapters in this book: Singapore is striving for excellence in mathematics education while addressing her imperfections; Singapore is also learning from good practices of other countries and at the same time bases her practices on sound educational theory. In addition, Singapore places great emphasis on teacher professional development. Further, this chapter discusses the future trends of mathematics education in Singapore: an increasing emphasis on big ideas, big data and computational thinking.

The National Institute of Education (NIE) is the sole teacher education institution in Singapore. This chapter considers the various kinds of knowledge that make up the knowledge base of a mathematics teacher of substance, and illustrates in detail the various programme designs which NIE implements to ensure the development of such a knowledge base in pre-service and in-service mathematics teachers. The special features of NIE, such as the symbiotic relationship of its mathematicians and mathematics educators and its adaptation of good practices from around the world, are described in some detail to cast light on how it has been generally successful in carrying out this purpose. Challenges ahead and possible future directions for improvement are also discussed.

In this chapter, we present a conceptualisation of mathematics teaching and learning which we term realistic ambitious pedagogy. We locate this pedagogy within the domains of teaching goals and teaching enactment, and the interactions between them. We argue that it is a suitable pedagogy for use in teacher development enterprises because it takes into deliberate consideration the realistic constraints within which teachers work while pursuing ambitious goals of mathematics teaching. To illustrate, we provide an example taken from our work of redesigning a curriculum unit on simultaneous linear equations in two variables with some Year 8 mathematics teachers in Singapore.

Problem solving has been the heart of the Singapore school mathematics curriculum since the early 1990s after being adopted as the goal of school mathematics education. Since its adoption, it has captured the interest of many Singapore educators and researchers. It appears that problem solving will continue to be a very active research area since there is great interest in the very high level of performance of Singapore students in international comparative studies such as TIMSS and PISA. This chapter begins with a re-categorization of the research work done to date on problem solving in Singapore using the Singapore mathematics curriculum framework by integrating two classifications done by Foong in 2009 and Chan in 2014, respectively, and including work done since 2011 that was not reported in either survey. The earlier research focused on addressing the readiness of students for mathematical problem solving (MPS) from the perspective of the Singapore mathematics curriculum framework; the later research tended to emphasize the enactment of MPS in the Singapore mathematics classroom and teacher education. This chapter gives more detail to this later research with an emphasis on the enactment of Polya’s stages in solving structured problems.