- In preparing learners to participate in and contribute to a democratic society, we will need to give them opportunities to experience such processes. It is increasingly important, particularly in South Africa, for people to realise that democracy involves responsibility. In order to avoid a situation where ‘citizens and public officials can use democratic principles to destroy democracy’ (Guttmann, 1987, p. 4), citizens need to see that democracy grants all people the right to voice their views. This aspect, where contestation is not only allowed but actively sought, is a valuable component of democracy. As Guttman argues:

- Mathematical knowledge for teaching has received much attention in recent times (Adler & Davis, 2006; Adler, Pournara, Taylor, Thorne & Moletsane, 2009; Ball & Bass, 2000; Ball, Thames & Phelps, 2008; Ball et al., 2005; Perressini, Borko, Romagnano, Knuth & Willis, 2004). Ball et al. (2005) define four domains of mathematical knowledge for teaching: common content knowledge (mathematical knowledge of the school curriculum), specialised content knowledge (mathematical knowledge teachers make use of in teaching), knowledge of students and content (knowledge that combines knowledge of content and learners – in this domain, teachers need to be able to anticipate learner errors and common misconceptions, interpret learners’ incomplete thinking and predict what learners are likely to do with specific tasks and what they will find interesting or challenging), and knowledge of teaching and content (knowledge about instructional sequencing of particular content, and about salient examples for highlighting mathematical issues). In these definitions the role of the learners in contributing to the development of the knowledge is relegated to the sidelines, just in terms of predicting what they would do in a particular situation. Let us look at the following statement more closely:

- How do we create authentic opportunities for learners’ voices, so that we learn about the learners, what they value and what they want? Vithal (1999) has shown that project work can be used as a site for learners to play a meaningful role in their own learning. She comments that:

- In Vithal’s (1999) study about project work in a primary mathematics classroom, it was found that the concerns driving the learners were different from those of the teacher:

- Learning to listen can be seen as a catalyst for teacher learning, but what kinds of listening must a teacher engage in? Davis (1997) distinguishes between three types of listening:

- Thomson and Thomson (1996) remark that ‘how one teaches a subject is influenced greatly by the many ways one understands it’ (p. 16). We expand on that: how one teaches should be influenced greatly by what one knows of the many ways the learners understand it. Henderson (1996) suggests that a teacher needs to be in touch with the different ways learners understand particular concepts:

- Davis (1997) also argues that: