- The envisaged guidance must propel

- Key to this guidance is the shift in power relations so that the teacher listens to pupils in depth and allows them to make and express judgements whilst at the same time valuing their contributions. This article problematises this learner–teacher relationship and the shift (or not) in power relations by analysing how teachers who participated in this study responded to learner contributions. In doing so we worked with classroom transcripts and raised the following questions to guide our analyses:

- Our concern in this article can be couched in the form of a question, borrowed from Stinson (2004):

- According to Ernest (2002):

- Secondly, epistemological empowerment concerns personal power over the creation and validation of knowledge. It has to do with the teacher not only encouraging learners to make contributions to the classroom discourse but also acknowledging and supporting such learners’ ways of understanding. It is this personal creation and validation of knowledge by the learners that we are putting under the spotlight in this article as we question the extent to which mathematics teachers acknowledge, support and build on learners’ personal routes or ways of knowing. According to Boaler (2000):

- What this model suggests is that to achieve the epistemological empowerment of learners through mathematics, it is not enough for them to gain mastery over some mathematical knowledge and skills. There needs to be a personal engagement with mathematics so that it becomes an integral part of the learner’s personal identity. Based on both a theoretical analysis and on personal experience, Ernest (2002) identified seven different factors that are associated with a shift towards engagement, confidence and epistemological empowerment. However, this article specifically focuses on the following two that we found to be complementary and at the same time consistent with our objectives in this article:

- Within the literature, researchers have shown how it is possible to relate the learning of mathematics to empowerment and democracy (Ernest, 2002; Muller, 2005; Stinson, 2004; Volmink, 1994). This relationship has been examined as a means to underpin a more equitable mathematics education system and to promote a more just society. However, mathematics education can also involve both overt and covert disempowerment, which can occur at all levels of the schooling system. Skovsmose (2000) refers to this as an aporia in that on one hand mathematics education could mean inclusion and empowerment, yet on the other it could also mean suppression, exclusion and disempowerment. It is from this paradoxical observation that some researchers have used the metaphor of mathematics being either a pump or filter: a pump for some by propelling learners into educational opportunities and economic access, and a filter for others by limiting their access to careers and professions. In this context, Volmink (1994) says:

- We borrowed from Fernandez, Yoshida and Stigler’s (1992) proposition of relatability of events, which builds on the presumption that not all relations between events must be presented for learning to occur because some events are not relevant to the content of the lesson. Fernandez et al. suggest that following the teachers’ objectives (explicit or implied) for the lessons, and their responses to learner productions, one can judge whether or not learner productions were:

- When teachers do not listen to or do not understand their learners’ thinking, they are likely to be ineffective; hence, Ball and Forzani (2010) posit that:

- The teacher wrote the following trinomial on the board: a2 + 14a + 48. The lesson was on factorisation and the method involved finding factors of the last term (48) which would add to the middle term (14), hence:

- A few more examples were worked out including the following one with a negative middle term:

- This approach can create many misconceptions, as explained in Tall’s (1991) generic extension principle: