A model for conceptualising algebraic activity is proposed by Kieran (2004), where she introduces three interrelated principal activities of school algebra: generational activity‚ transformational activity‚ and global/meta-level activity. The global/meta-level activities involve activities for which algebra is used as a tool, and include: problem solving; modelling and predicting; studying structure and change; analysing relationships; and, generalising and proving (Kieran, 2004). I have studied three student teachers as they engaged in a global/meta-level activity: It is a problem solving situation which involves incidents of specialising, generalising and justification. The focus is on the transformation between different representations of the involved mathematical object. The relevance of changing between different registers of semiotic representation is asserted by Duval (2006), who claims that this is the threshold of mathematical comprehension for all learners. The task in the analysed classroom episode is about the increase of the area of a square as a consequence of its side length being increased by an arbitrary percentage. The target knowledge is the connection: “Increasing a plane figure’s linear dimensions by a scale factor s, increases the area by the scale factor s2”. The addressed research question is: What factors are constraining the change between three registers of semiotic representation: iconic, natural language, and algebraic representation? Research participants are three student teachers in their first year of a four-year undergraduate teacher education programme for primary and lower secondary education in Norway, and one of their mathematics teachers. The data is the transcript of the video recorded lesson from the students’ problem solving process, with teacher involvement. The transcript has been analysed with respect to participants’ use of different registers of representation, and transformation between them. The students used manipulative material that represented the original square and its additional parts in cases where the side length increased by specific percentages (50 %, 25 % and 10 %). It is shown how the ability to manipulate the material in accordance with the linear scale factor (i.e., to represent the increased area appropriately in practice), does not make it easy for them to express it in natural language or in algebraic notation. Brousseau’s (1997) theory of didactical situations is used to suggest elements that are important for an appropriate adidactical milieu related to the target knowledge.

Brousseau, G. (1997). The theory of didactical situations in mathematics. Dordrecht, The Netherlands: Kluwer. Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 103-131. Kieran, C. (2004). The core of algebra: Reflections on its main activities. In K. Stacey, H. Chick, & M. Kendal (Eds.), The future of the teaching and learning of algebra: The 12th ICMI Study (pp. 21-33). Dordrecht, The Netherlands: Kluwer.