# Framework and rationale of RaPiTE

It is a recurrent explanation of school students’ lack of proficiency with R&P that their teachers have limited knowledge of these processes and that their beliefs run counter to current reform efforts. This somewhat acquisitionist perspective primarily locates the educational problems of R&P within the individual teacher. This is so even when it is acknowledged that these individual constructs may be challenged by dominant norms or practices at the schools where prospective teachers do their practicum or where novices take up teaching (e.g. Gabriel J. Stylianides, Stylianides & Shilling-Traina, 2013).

In RaPiTE we adopt a somewhat more participatory stance to human functioning. The framework that we use, PoP, draws on social practice theory (e.g. Holland, Skinner, Lachicotte & Cain, 1998; Lave, 1997; Wenger, 1998), symbolic interactionism (Blunter, 1969; Mead, 1934) and Sfard’s theory of commognition (Sfard, 2008). The first and the last of these approaches focus, respectively, on emerging social processes (e.g. romance at a US university campus, cf. Holland & Eisenhart, 1990) and on well-structured cultural practices (e.g. mathematics, cf.

Sfard, 2008). PoP, however, does not focus on any one such practice per se and on how an individual moves towards more comprehensive participation in it. Rather, PoP re-centres the individual and asks how a teacher’s involvement in unfolding school and classroom events relates to and is transformed by her or his re-engagement in other past and present practices and discourses (Skott, 2018b). We have found the I-me distinction in the symbolic interactionist notion of self helpful for this purpose. The distinction points to how a person in an interaction acts (does, says, thinks, etc.); that is, the person performs as an I but instantaneously adjusts the act as (s)he takes the attitude to her- or himself of others and becomes a me. The attitude (s)he takes to her- or himself may be that of immediate interlocutors or of significant individual or generalized others that come to mind at the instant. The distinction between the I and the me allows us to focus on how the teacher may see her- or himself from the perspective of the teacher’s students or other individual and generalized others as classroom processes unfold.

If, for instance, a teacher seeks to develop a good mathematical argument with a group of students, who appear to be weak and vulnerable in the situation, (s)he may simultaneously take the attitude to her- or himself of the students in question; of colleagues, who focus on creating trusting relationships with the students; of the school leadership or of parents, who emphasize students’ performance on standardized tests; or of her or his teacher education programme that focuses on the use of manipulatives to facilitate student learning with understanding (Skott, 2013, 2015; Skott, Larsen & 0stergaard, 2011). The teacher’s engagement with each of these social constellations - or others - may transform or subsume her or his involvement in the practice of mathematical R&P and, for instance, have the teacher accept justifications that do not qualify as mathematical. PoP provides a perspective on if and how this is the case.

PoP has so far framed studies conducted “in the perspective of teacher education” (Krainer & Goffree, 1998). These studies are not on MTE, but they develop understandings of teaching-learning practices in schools and may raise questions about MTE and inform decisions on how to address them. As indicated in the previous section, the results suggest that even when teachers engage students in elements of the R&P cycle, most notably what NCTM (2000) calls “examining patterns and noting regularities” (p. 262), modes ofjustification may lose their subject specificity. To avoid this, it seems that MTE needs to fulfil two requirements. First, it must be close to teaching-learning processes in schools, as the community in which R&P practices develop are otherwise too distant from classroom interaction for it to function as a generalized other in instruction. This is in line with the suggestion to emphasize *proving why* using generic arguments (Rowland, 2002). Second, and in spite of that, MTE must be close to the disciplinary practice of R&P and include significant elements of *proving that* so as to limit the risk of classroom processes losing their subject specificity. This is important also because it is often more complicated to *prove why* than to *prove that*, and if the teacher is unable to produce a valid mathematical justification in response to an unforeseen student conjecture, (s)he has no alternative but to accept the students’ empirical arguments. The assumption of RaPiTE, then, is that MTE needs to avoid the two extremes of focusing either on academic mathematics or school mathematics, not by reducing the emphasis on either but by transforming both (Skott, 2018a).

To be “sufficiently close” to both school mathematics and academic mathematics, we use tasks and conjectures that may be used in or developed from tasks used in school and take them beyond the level of the school students in question. Examples include:

- 1 Does 8 always divide n
^{2}- 1, if n is odd? (From an interaction in Larry’s Grade 5; cf. the previous example on perfect squares, Skott, 2018a) - 2 30 = 6 + 7 + 8 + 9; 31 = 15 + 16; 32 = . . .; 35 = 17 + 18 = 5 + 6 + 7 + 8 + 9 = . . .; What positive integers are the sum of other consecutive, positive integers? (Arose from a reversal of the question of how to find the sum of the first n positive integers)
- 3 Assume that you have a set of rods similar to Cuisenaire rods representing the positive integers from 1 to n. For what values of n can you make two “trains” of rods of equal length? Three trains? m trains?