Transposition of Knowledge: Encountering Proportionality in an Algebra Task

This article reports on an analysis of the process in which "knowledge to be taught" was transposed into "knowledge actually taught," concerning a task including proportional relationships in an algebra setting in a grade 6 classroom. We identified affordances and constraints of the task by describing the mathematical praxeology of the two different types of knowledge exposed, in the task as such and in the activity of the classroom. Through the teacher's explicit process of reasoning, modeling, revising, solving, and repeatedly explaining the task, we found that the transposition of knowledge was seriously affected by the contextualization of the task. Modeling word problems about everyday situations has its limitations and can, as in this case, make the problem unsolvable unless it is accepted as a "textbook task" disguised as real but adjusted to the norms of school mathematics. Such constraints may obscure mathematical ideas afforded by the task. We conclude that learning opportunities embedded in a task do not necessarily surface when a task is treated in a classroom setting.