Working with Concept-Based Curricula for Mathematics
The purpose of this chapter aims to address some of the challenges of delivering a concept-based curriculum in a typical mathematics classroom in a secondary school in Singapore, as well as documenting some of the teaching strategies that educators can us
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bjective The purpose of this chapter aims to address some of the challenges of delivering a concept-based curriculum in a typical mathematics classroom in a secondary school in Singapore, as well as documenting some of the teaching strategies that educators can use to promote conceptual understanding in students.
Background The definition of mathematics as a discipline has changed and evolved over time. One that has stood for centuries was by Greek mathematician and philosopher Aristotle, who defined mathematics as the science of quantity. Indeed, early works in the discipline had been focused in counting (arithmetic) and measurement (geometry) and were widely used in fields such as construction and navigation in ancient times. It was not until the nineteenth century that new abstract areas of mathematics were studied, such as analysis (calculus), non-Euclidean geometry and set theory (logic). Subsequently, the definition of mathematics varied from one scholar to another in varying perspectives. For example, Russell (1903) claimed that mathematics was, essentially, symbolic logic. However, not all share the same view. Sawyer (1955), for example, focused on observations of patterns and structure, and he defined the discipline as the classification and study of all possible patterns.
C.W. Tan (*) Hwa Chong Institution, Singapore, Singapore e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2017 L.S. Tan et al. (eds.), Curriculum for High Ability Learners, Education Innovation Series, DOI 10.1007/978-981-10-2697-3_12
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Understanding the Structure of Mathematical Knowledge Mathematical knowledge refers to the relationships between theorems, concepts and sub-concepts that seek to explain how the world functions in its nonliving form. We see mathematical knowledge as an important tool in several fields including physical science, accounting, economics and statistics. It is also widely applied in engineering, medicine and research in social sciences. Without mathematical knowledge we would not be able to make everyday decisions such as the amount of money we spend or even understand the days and months on the calendar or know how to construct buildings and machines to become the modern society we are now. Literature and research on the structure of mathematical knowledge have been limited or otherwise vague. Michener (1978) described in his report The Structure of Mathematical Knowledge that mathematical knowledge comprises theorems and proofs, and relations between theorems, showcasing examples that highlight the application of the theorems and concepts which contain ‘mathematical definitions and pieces of heuristic advice’ (p. 5) when he was in the Artificial Intelligence Laboratory in Massachusetts Institute of Technology (MIT). He introduced the three representative spaces of mathematical knowledge: Results-space, which comprises results (theorems) together with relationships of logical support—such as how one theorem is used to support another; Examples-space, which comprises illustra
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