Yeo, Joseph B. W.

Equivalence is one of the eight clusters of big ideas proposed for the 2020 secondary school mathematics syllabus in Singapore. In this chapter, I will unpack the meaning of equivalence: the idea of an equivalence relation, the differences between equivalence and equality, and the usefulness of equivalent equations and equivalent statements in solving mathematical problems. In particular, I will examine the solutions of some types of equations which do not seem to produce equivalent equations in subsequent steps, e.g. the introduction of an extraneous solution when solving some equations involving surds or logarithms, and the elimination of a variable when solving a pair of simultaneous equations in two variables, and explain how the solutions of these equations can still produce equivalent equations in subsequent steps. In other words, the transformation or conversion from one equation to another equivalent equation is still the basis of the method for solving any kinds of equations. In addition, this chapter will discuss how to teach secondary school students the big idea of equivalence without teaching the abstract idea of an equivalence relation.

While reports of teachers’ use of curriculum materials are common, that of teachers as designers of their own materials are far less so. We argue that these (rare) instructional materials, defined as materials that are classroom-ready and that carry the teachers’ actual instructional goals, are ‘objects’ that are suitable as records of teachers’ planning and learning when developed alongside professional development. We provide supporting evidence of this claim and unpack the complexities of interacting instructional goals through a case study of a teacher who (re-)designed her own instructional materials as she participated in professional development. From the findings of the case, we reflect on the educational and methodological implications of pursing this research approach.