The critical properties of the case of Tamsin and the area of a triangle were that a figure of a triangle inscribed in a rectangle was shared publicly, Tamsin invited a verbal explanation of the area of a triangle being half the area of a rectangle as represented visually by the figure, and a sequence of student—teacher turns followed. In the episode we offered,Tamsin either accepted what the student had said, problematised the vocabulary the student used or problematised the student’s understanding. Our analysis of this case, however, revealed that acceptance or problematisation was always in relation to the particular figure that had been shared publicly, and significantly much of the vocabulary that was accepted or problematised would not have described a generic figure.
Although the case ofTalia and the moment of a force featured several publicly shared figures as well as extended sequences of student-teacher turns with three different students over the course of the episode, it shared the same critical properties as the case of Tamsin in that the figures are of a force and a pivot, Talia invited a verbal explanation of a method for calculating the moment of the force about the pivot as represented visually by the figures, and a sequence of student-teacher turns followed. Our analysis revealed that Talia was more likely to problematise understanding and not problematise vocabulary than in the case ofTamsin. Even when incorrect vocabulary was being used in order to explain, or there was a lack of clarity, Talia often chose to problematise understanding through references back to the figure rather than address the vocabulary. This is not to say that Talia never problematised vocabulary, for there were a number of instances in Extract 4. An important distinction between the two cases, however, was that through problema- tising understanding, including offering different figures as examples, explanations that only related to a particular case were not accepted by Talia.
By combining what is known about promoting vocabulary use with what is known about the role of particular and generic examples, what is understood about the ways in which teachers might pursue vocabulary in situations where explanations are invited that relate to publicly shared figures has been refined through this study. Our pre-analysis categorisation of student-teacher turn couplets was along one dimension only: that of acceptance or problematisation. Where there was prob- lematisation we were aware of drawing on prior research in the field that this could be of the student’s vocabulary use or their understanding, or possibly both (Figure 7.2). We note that in our data we found no examples where both vocabulary and understanding were concurrently problematised, for when this was initially unclear from the teacher turn it became clarified by the way the student treated the teacher turn in their turn that followed. We still do not rule out the possibility of concurrently problematising both vocabulary and understanding and so wish to preserve this as a potential category for now. More importantly through our analysis a second dimension became evident: whether the acceptance or problematisation distinction related only to the particular case of the publicly shared figure or whether what was being accepted or problematised explained the mathematical phenomenon in its generality. However, we found no examples in these two episodes of student turns that were accepted that fully explained the general case of a phenomenon. Nevertheless, we offer in conclusion a representation in Table 7.1 of our categorisation model along two dimensions of student—teacher turn couplets that are part of an explanation of a mathematical phenomenon that has been represented visually through a publicly shared figure.
The categories are not mutually exclusive, and in addition to the possibility of both vocabulary and understanding being concurrently problematised, there are
TABLE 7.1 Categorisation model along two dimensions of student-teacher turn couplets
Categorisation of the student-teacher turn couplet
Student turn relates to
Particular case General case
examples in our data of an explanation of a particular case being accepted but immediately the understanding in the general case being problematised.
For students, learning that a word can label both a particular instantiation of a mathematical phenomenon and the phenomenon itself, such as the way in which ‘moment’ can be used to label the turning effect of the force about point A in the particular case of Figure 7.5 and to label turning effects of forces about pivots generally, is a critical step towards fluent use of technical vocabulary. Drawing on the work of Leung (2005), not only could working in this way promote the use of technical vocabulary it also affords deeper understanding of the mathematical phenomenon itself. We conjecture that both pursuing mathematical vocabulary and also working simultaneously with generic figures may have an important role to play here. Then, as well as seeing the general through the particular (Mason & Pimm, 1984), through saying what they see, students might also articulate the general through the particular, as they will have meaningful labels (or vocabulary) available to them in order to do so.
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MATHEMATICS THROUGH PLAY