Teachers’ mathematical discourses in instruction (MDIs), essentially the mathematical aspects of what teachers say, do and write as they interact with learners in mathematics classrooms, are a key feature of classroom practice. Typically, these MDIs include a problem, a selected representation that is subsequently transformed, and explanations and justifications for the representations selected and transformations performed. Our interest in this article is in developing a language that can be used to describe a range of MDIs. This interest is driven by our need to understand MDIs that seem to us to disrupt coherence and connection in mathematical text in a range of ways, and thus impact on what mathematics is made available to learn.
Transformation of representations, through manipulations within and across different representation forms, is a central feature of mathematical activity (Duval, 2006) and, therefore, of MDIs. Solving a problem in school mathematics often involves a set of steps through which one representation is transformed into another. For example, completing the square is comprised of a series of transformation steps that can act upon a quadratic function as input representation, if the stated problem is to find the turning point of the function. Consider the problem:
Find the turning point of f (x) = x2 − 8x + 9
The first step to solving this stated problem could be to recognise that rewriting a quadratic expression as a perfect square, plus or minus some constant, allows us to ‘see’ vertical and horizontal shifts with respect to the parent function, and so the turning point, more easily. We would thus rewrite the function in the form f (x) = a(x − p)2 + q by completing the square:
f (x) = x2 − 8x + 9
What is important for this transformation activity1 is that within the MDI, the input representation introduced, the representations produced through transformation activity, and the accompanying explanations connect with each other and cohere with the stated problem. Our observations, across primary and secondary classrooms within our respective projects, suggest that such coherence or connection is frequently, but varyingly, disrupted within MDIs.
In this article, we share our development of an empirically derived analytical language (elements of which are italicised above) that allows us to make visible a range of disruptions to connection and coherence that come into play across four contrasting teaching episodes. We focus here on input objects, transformational activity and accompanying explanations in order to ‘see into’ the micro-level production of mathematics in classrooms through describing differences in the nature and degree of coherence and connection and to consider the consequences for what is made available to learn. This micro-level focus on specific episodes within lessons follows our observation of the occurrence of disruptions at this level, rather than at the broader level of lessons or lesson sequences that have been taken up in prior research (e.g. Sekiguchi, 2006).
In order to present our thinking on making aspects of connection and coherence visible within transformation activity, we begin with a brief overview of the literature. We draw on writing focused on transformation activity and representations as these actions and objects are at the centre of all our episodes and, as noted already, at the heart of mathematical activity more generally. We also summarise evidence that points to the shortcomings that characterise practices in which transformation steps are emphasised at the expense of gaining understanding of the representations they act upon. From this review, we outline the key concepts that we found helpful in beginning to pull apart some of the range of procedural practices that we were working with. Centrally, we home in on the stated problem, the selected input representation, subsequent sequences of transformation steps, and the interim and final representations produced in this sequence. These concepts are all covered in the literature we review.
Somewhat absent in this literature is a focus on the MDIs that accompany transformation activity. Teaching involves the giving of accompanying explanations alongside transformation and so, unlike the (often predictably) piecemeal learner discourses that are in focus in much of the literature on transformation-oriented activity, one expects MDIs to be both coherent and connected, and to provide some of the rationales for the representations selected and transformation activity that is enacted. As noted above however, we see this expectation disrupted relatively frequently, and in a range of different ways. In order to consider the nature of these disruptions to coherence, we use a tentative set of framing questions, drawn from our grounded analysis of the episodes presented in this article, to analyse and differentiate the transformation activity in four selected teaching episodes. This could be criticised as somewhat circular: developing grounded framing questions from a dataset, and then using them to analyse the same dataset. Our aim in doing this is to share this set of literature-drawn concepts and grounded framing questions in order to start conversations across the mathematics teaching and teacher education communities that can help to build a more robust language for thinking about what constitutes coherence and connection within mathematics teaching. We have already been through several iterations of concepts and framing questions, and have seen that our current formulation can be applied to a significantly broader group of episodes that we have encountered.