Transposition of Knowledge: Encountering Proportionality in an Algebra Task

Abstract

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 Btextbook 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.

Investigation of Proportional Reasoning Problem Solving Processes of Seventh Grade Students: A Mixed Method Research

“…Ayan and Isiksal-Bostan (2018) also indicated that students make comparisons in the process of solving proportional reasoning problems related to geometric shapes, while giving arbitrary values to the lengths. Lundberg and Kilhamn (2018) emphasized that students are dealing with proportional reasoning problems through trial and error. In addition, Lundberg and Kilhamn (2018) determined that students also use the strategies like direct proportion and doubling or halving during the proportional reasoning process.…”

“…Lundberg and Kilhamn (2018) emphasized that students are dealing with proportional reasoning problems through trial and error. In addition, Lundberg and Kilhamn (2018) determined that students also use the strategies like direct proportion and doubling or halving during the proportional reasoning process. Langrall and Swafford (2000) stated that students use pictures, models or manipulatives to understand proportional reasoning problems and to make a qualitative comparison.…”

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“…In the present study, I have employed praxeology to analyse the didactic transposition of rational numbers. The study focuses only on knowledge to be taught and taught knowledge (Lundberg & Kilhamn, 2018;Pansell & Boistrup, 2018). The knowledge to be taught is investigated through analysing a mathematics textbook for teacher education.…”

Comparing the Lower Secondary Textbooks of Japan and England: a Praxeological Analysis of Symmetry and Transformations in Geometry

“…Moreover, following ATD, our approach regarded textbooks as an empirical source which can reveal knowledge to be taught in the didactic transposition process; this differs from the approaches of previous textbook research studies. Although there are some existing studies using ATD for textbook analysis (González-Martîn, Giraldo, & Souto, 2013;Lundberg & Kilhamn, 2018;Wijayanti, 2015;Wijayanti & Winsløw, 2017), the innovation this paper provides is in the precision achieved at the logos level. The theoretical and methodological aspects of this study can contribute to future research on mathematics textbooks.…”