Knowledge and beliefs for nurturing reflective learners of rational number concepts

Examples of teaching and learning rational number concepts from Years 2 to 8 (students aged 7 to 14 years) are used as a basis for considering the nature of reflective mathematics learning. It is argued that teachers' knowledge and beliefs are crucial to the development of reflective learners. Work samples and transcripts of learning episodes are used to illustrate behaviours of reflective learners and teacher actions that seem to support reflective learning, and the teacher knowledge and beliefs that appear to underpin these. Teachers need to have a good understanding of the mathematics that they teach and also typical ways that students think about that content, including likely areas of difficulty. They need to be able to analyse students' thinking in the midst of teaching and learning interactions, and respond with appropriate questions, explanations, and representations of the important mathematical ideas. The importance for teachers of reflecting upon their own beliefs about such things as the nature of mathematics and hence what it means to know, learn and teach mathematics, and about the capacities of their students and the importance of them developing a reflective approach to their learning are also considered.