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Mathematics pedagogical content knowledge (MPCK) of teachers cannot be easily defined but is a complex concept integrating generic pedagogical knowledge, mathematics teaching methodology as well as knowledge of the discipline of mathematics. It is the objective of mathematics methodology courses in teacher preparation programmes to begin the development of MPCK in their pre-service teachers. As part of a research study on the development of MPCK in primary school beginning teachers, a 16-item instrument was developed to measure some aspects of the MPCK for teaching mathematics at primary level. The instrument was administered to the 2003 Intake of the Diploma in Education student teachers at the National Institute of Education, Singapore just at the beginning of their programme. As they completed their methodology course in February 2005, the instrument was administered again. This paper will discuss the findings concerning their performance in these two tests, with reference to the overall performance as well as topic-specific and MPCK construct-specific performance. The findings indicate that student teachers at the beginning of their programmes are generally quite weak in their mathematics pedagogical content knowledge, as might be expected. There was significant improvement in some aspects of their MPCK on completion of their mathematics pedagogy course.

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Most educators would think of heuristics when it comes to solving closed mathematical problems, while many researchers believe that mathematical investigation must be open and is different from problem solving. In this chapter, we discuss the relationship between problem solving and investigation by differentiating investigation as a task, as a process and as an activity, and we show how the process of investigation can occur in problem solving if we view mathematical investigation as a process consisting of specialising, conjecturing, justifying and generalising. By looking at two examples of closed mathematical tasks, we examine how investigation can help teachers and students to solve these problems when they are stuck and how it can aid them to develop a more rigorous proof for their conjectures. We also deliberate whether induction is proof and how heuristics are related to investigation. Finally, we consider the implications of the idea of solving mathematical problems by investigation on teaching.

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