- 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:

- 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:

- Paying attention to the representations selected and produced through transformation activity is described by Haapasalo and Kadijevich (2000), cited in Haapasalo (2003), as important within transformation activity underlain by strong procedural knowledge. Strong procedural knowledge, for them, involves:

- An important second thread in the mathematics education literature is highly critical of the ways in which transformation activity has come to be configured within classrooms. Artigue (2011), in her article for UNESCO on the challenges of extending basic mathematics and science education for all students, refers to international surveys to describe how schooling is very often unstimulating because the teaching of mathematics is framed by:

- The framing questions are presented below:

- . This serves as motivation for the greater simplicity of the dual intercept method for drawing straight-line graphs. He returns to a function they had worked on,

- , and begins a discussion of ‘dual’ meaning ‘two’, eliciting from learners that the two intercepts are where the graph cuts the x and y axes. He demonstrates how to find the coordinates of the y-intercept by calculating the value of y when x = 0 and, similarly, the coordinates of the x-intercept. He writes (0; -3) and (2; 0) on the chalkboard and proceeds to sketch the axes, explaining how you can ‘estimate’ where the points are on each of the axes. He plots the two points and continues:

- Then a learner asks a question:

- After a brief discussion on the labelling of points on a graph, Learner 2 and Learner 3 ask Nash:

- In this episode, drawn from Askew, Venkat and Mathews (2012), a Grade 2 teacher is working on missing addend problems using a wheel representation with three concentric circles: 7 written on the inner circle, and the numbers 0–7 placed in random order around the outermost circle in separate sectors. Askew et al. state:

- The stated problem of the lesson, indexed by the title on the board Hlanganisa (‘Addition’ in Zulu), is for the class to fill in the numbers that need to be added to the numbers on the edge to make 7. Initial answers from some of the children indicate that they are interpreting the task in terms of addition of the numbers shown. In one episode, the teacher is focused on the problem: ‘What number is added to 3 to make 7?’ She shows the class three open fingers on her hand as she asks this, pointing to the 3 on the circle rim, and then shows seven fingers as she indicates the need to make 7, pointing to the 7 in the centre. Some children are seen counting out seven on their fingers. When no correct answers are forthcoming from the class, the following exchange takes place:

- Prior to and following this episode, we see instances of some learners able to give correct answers. However, we also see several learners who appear unaware of how many fingers to open, and what to do once they have one of the given numbers showing. Here, a stated problem that is given in terms of missing addends comes to be ‘funnelled’ into a subtraction problem through a transformation step and associated explanation, and then verified by adding the two numbers as an addition problem. The teacher appears aware of the equivalence between missing addend problems and subtraction, but this equivalence is not established for learners; rather, the equivalence is simply assumed, and subsequently verified empirically. Thus, a problem stated in terms of missing addends is worked out in terms of subtraction-based transformation activity, and checked through addition. Essentially, the sum below is presented as the stated problem to be solved (though not in this form):

- whilst the transformation activity instead involves solving the following subtraction problem:

- In this episode, drawn from Davis (2010), a Grade 10 teacher is working on integer addition sums, such as: -7 + 5. Davis describes the teacher’s instructions to the class as follows (p. 384):

- As was the case in Episode 1, an ordered set of instructions is relayed to the class – ‘first you take ... and then you take ...’. Some conditions for the application of transformation sequences are established at the outset: essentially ways to distinguish the input representation in order to recognise which transformation must be selected. Davis (2010), discussing this episode in terms of operations (addition) and objects (integers), notes that:

- In terms of our analytical concepts, -7 + 5 is the input representation that is transformed through a series of TA steps that provide an algorithm for solving the problem. In the interim stages, following the instructions would produce these representations:

- In the process of asking generally about ways in which data can be presented, a student mentions the notion of a ‘tally table’. Venkat and Mhlolo note the subsequent return by the teacher, after several interim episodes focused on a range of other stated problems, to the notion of tallying. The teacher shifts attention from a focus on the meaning of the frequency values in Table 1 with the following question:

- Having asked the question, she then adds a further column to her frequency table and gives it the title ‘tallies’. She then shows the class how to tally the number ‘8’, this being the first frequency value in her table. Then, pausing to ask the class if they have seen this (pointing to her tally) before, she explains further and demonstrates: