Two lessons were selected from a wider corpus of 41 video-recorded and transcribed lessons that have been collated over time by two of the authors. All the lessons in the corpus are naturalistic in the sense that choices about what and how to teach lie with the teacher, subject to their normal teaching expectations. Furthermore, it was also the teacher’s choice as to when a lesson is video-recorded. Three lessons were initially selected for closer analysis because the teachers were working with language and images simultaneously but also because there appeared to be a contrast in the way the teachers were working with these. The whole-class teaching from each lesson was then reviewed independently by each of the authors in order to identify episodes where mathematical vocabulary was a key feature in some way. The authors then came together and discussed these episodes to identify one episode from each lesson for fine-grained analysis.These episodes were selected on the basis that they featured student use of mathematical vocabulary, an extended sequence of student and teacher turns that follow from the teacher inviting an explanation of a mathematical phenomenon, and a publicly shared visual representation of that phenomenon. Two of these episodes are drawn upon in this chapter to illustrate the different ways in which teachers simultaneously worked on vocabulary and meaning from visual representations. One episode is shorter and features only one student’s explanation whereas the other episode is longer and features a series of student explanations of the same phenomenon. It is important to recognise that the features of these two episodes on which we focus are highlighted as choices that may be available more generally to teachers and that no judgement is being made about the classroom practices in doing so. The two teachers have been given pseudonyms beginning with the letter T (Tamsin andTalia) to mark them out as teachers and students have been given pseudonyms beginning with S.

Drawing on IRE structure (Mehan, 1979a), teacher initiation turns were identified in each episode, along with the sequence of‘couplets’ of student response and teacher evaluation turns that followed it. We treat couplets as an adjacency pair (Schegloff, 2007) because they arise within the IRE structure and have a reflexive relationship: the teacher evaluation turn always follows the student response turn and responds to it in some way, and the students’ response turn constrains the teacher evaluation turn that follows. The couplet is categorised as ‘accepted’ when the teacher evaluation turn is an acceptance of the student response turn, such as the turn 6 and 7 couplet when referring back to Extract 1. Otherwise, the couplet is categorised as ‘problematised’. In Extract 1, the turn 2 and 3 couplet is prob- lematised because the teacher turn questions the student’s vocabulary use in their turn.The turn 4 and 5 couplet is problematised because the teacher turn is probing student’s understanding. It is important to note that the categorisation of couplets is not conducted in isolation of the rest of the interaction. For example, following Simon’s turn 2 in Extract 1 Tamsin might have asked “Why is it a square?” in the following turn and yet Simon might still have said “Sorry, rectangle”. Even though in this imagined situation Tamsin posed a ‘why?’ question, it is not treated by Simon as problematising his understanding (as he did when asked a ‘why?’ question in turn 5 of Extract 1) but rather problematising his vocabulary use and so would still be categorised as problematised because of vocabulary. The categorisations and sub-categorisations of couplets are summarised in Figure 7.2, where the possibility of a couplet problematised both because of vocabulary and understanding concurrently is included.

The case of Tamsin and the area of a triangle

Extract 1 was only superficially analysed in the introduction section in order to exemplify the fundamental categories in Figure 7.2. While Simon’s use of‘square’ in turn 2 is problematised by Tamsin in turn 3, he uses this word in a non-technical sense as it is placed alongside the unequal side lengths to which he refers. Simon’s

Categorisation of couplets

FIGURE 7.2 Categorisation of couplets

clear reasoning for why Figure 7.1 is not a square is evident in turn 6 and from this we can infer that he knows how to discern a rectangle from a square. He is not appealing to the properties of a square, other than those it shares with a rectangle, when making the assertion in turn 8. His use of‘square’ in turn 2 might therefore have been treated as a slip rather than an issue of understanding: it was not critical to his line of reasoning.

Extract 1 only featured the first eight turns of the episode between Tamsin and Simon. The rest of the episode, starting again with turn 8, is given in Extract 2.

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