- The most recent reform in curriculum and assessment policy in South Africa aims at producing learners who are able to communicate effectively using visual, symbolic and/or language skills in various modes (Department of Basic Education, 2011, p. 2). This is expected to occur throughout their learning where opportunity for representation arises. Teachers need to engage in meaningful discourse with their learners so as to better recognise and appreciate the learners’ use and understanding of specific representations. Such shared exchanges result in a process in which two groups come to understand the other’s viewpoints as well as the discursive resources and mathematical representations employed to communicate those viewpoints. In this way, deliberative interaction is exemplified. However, to build on and from the representations of learners, teachers must have both a deep understanding of the different representations (including the affordances and drawbacks of each) and the flexibility to use the representation that is most appropriate for the mathematical situation and the learners. The issue under investigation in this study is whether teachers have deep, flexible understandings of mathematical representation that enable them to create democratic environments in their mathematics classrooms. This concern will inform mathematics teacher educators and the relevant educational department authorities whether these teachers are prepared for changing educational policies (Department of Basic Education, 2011). Accordingly, this study was designed to explore teachers’ understanding of mathematical representations. For this study, we formulated the following research questions:

- The semi-structured questionnaire, with twelve items, was administered to the 76 participants in the second semester of their study. For this article we consider only the teachers’ responses to the first two items of the questionnaire, namely:

- The theme Examples was most commonly noted: almost 80% of the teachers gave examples of representations in their responses. In considering the Examples theme, it is noteworthy that 84 responses (32 to Item 1 and 52 to Item 2) provided examples of representations although only 59 teachers used examples on one item only. This means that a significant proportion (84 − 59 = 25) of the teachers used examples in their responses to both items. Some teachers mentioned only examples of a single representation, for example, T6 wrote ‘writing mathematics in graphical form’ for Item 1 and ‘graphs’ for Item 2. On the other hand, many other teachers listed a variety of examples of representations, such as T31:

- The Representation theme was second most common, addressed by 54% of the teachers. In this theme, teachers described what representations are and how they are used, especially in response to Item 1. A typical example is T14:

- The teachers in this sample displayed knowledge of numerous types of representations (as seen in the responses coded under the Examples theme) as well as a belief that mathematical ideas can be represented in different ways. This Variety theme was the third most common, with 40% of the teachers showing evidence of it in their responses. The Variety theme was applied to teachers’ responses that explained that there are many different ways to represent mathematical concepts, ideas or relationships. Such responses show that the teachers strongly believe that there are different ways to represent mathematical concepts. This is evidence that these teachers do not subscribe to an absolutist view of classroom communication. One example of the Variety theme was displayed by teacher T17:

- The Communication theme was noted by 38% of the teachers. Whilst mathematical concepts are important, representations are the vehicles through which these concepts are shared with others. As evidenced by responses that fall under the Communication theme, 21 teachers saw representations as things that convey or express mathematical information. An example of a response in this theme is T31’s response to Item 1:

- Here the perception of T31 is that mathematical representation is learner driven. This is in keeping with the principles of the South African school curriculum (Department of Education, 2003). T26 put it another way:

- Twenty-six per cent of the teachers had responses that were coded as evidence of the Aid for Understanding theme. This reveals that the teachers see representations as facilitating the understanding of mathematical concepts or relationships. T3 expressed the view that mathematical representation means:

- With the recent emphasis worldwide on the need for links between mathematics education and real-life situations, it is no surprise that 24% of teachers identified the role of representations in portraying Real Life situations. The linking the learning of mathematics to real life is of paramount importance. This is highlighted in the curriculum and assessment policy statements (Department of Basic Education, 2011), which state in the first specific aim that real-life situations should be incorporated into all sections whenever appropriate. Such linking will prevent the classroom from being a micro-society in which only mathematical abstractions prevail. T51 places emphasis on real life in his response to Item 1:

- Teacher T61 was one who associated mathematical representations with problem solving:

- Across the two items, nine teachers associated representations with the Tools (equipment or resources) used to create them. For example, T21 wrote:

- whilst T46 wrote: