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Transposition of Knowledge: Encountering Proportionality in an Algebra Task
Department of Pedagogical, Curricular and Professional Studies, University of Gothenburg,Gothenburg,Sweden. (VIDEOMAT)
Department of Pedagogical, Curricular and Professional Studies, University of Gothenburg, Gothenburg, Sweden.
2018 (English)In: International Journal of Science and Mathematics Education, ISSN 1571-0068, E-ISSN 1573-1774, Vol. 16, no 3, p. 559-579, article id 1571-0068Article in journal (Refereed) Published
Abstract [en]

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 knowl- edge 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 mathe- matical 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

Place, publisher, year, edition, pages
2018. Vol. 16, no 3, p. 559-579, article id 1571-0068
National Category
Other Mathematics Didactics
Identifiers
URN: urn:nbn:se:liu:diva-154909DOI: 10.1007/s10763-016-9781-3OAI: oai:DiVA.org:liu-154909DiVA, id: diva2:1293427
Projects
VIDEOMAT
Funder
Nordic Council of Ministers, 349-2006-146Available from: 2019-03-04 Created: 2019-03-04 Last updated: 2024-12-12
In thesis
1. Att lära om statisk och dynamisk proportionalitet: En studie av den didaktiska transpositionen av svenska matematikuppgifter med proportionalitet
Open this publication in new window or tab >>Att lära om statisk och dynamisk proportionalitet: En studie av den didaktiska transpositionen av svenska matematikuppgifter med proportionalitet
2024 (Swedish)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The purpose of this study is to shed light on how static and dynamic proportionality is treated by authors of teaching materials, on national tests and by teachers and students in the classroom, as well as how students encounter mathematics tasks where proportional reasoning is an option. The research is based on two sets of empirical data. In concrete terms, the thesis includes three studies examining three themes that relate to proportionality in classroom interaction and in texts. The first study analyses how proportionality is presented in some Swedish textbooks, in curricular texts and national course tests in mathematics for students in upper secondary school. The second study is a case study of how a teacher instructs and explains a task in a class in Grade 6, where proportional reasoning is a possible solution technique. Finally, the third study concerns how students in Grade 6 handle proportional reasoning when they encounter a patterning task involving proportional relationships. The analyses of textbooks and national course tests show that proportionality is handled differently in these two settings in the context of “Mathematics A” at the upper secondary school. About a quarter of the tasks in the textbooks and the national course tests involved proportionality tasks of one specific kind (missing value). Other types of proportionality tasks were infrequent. The results of the classroom studies show that students are able to engage in early forms of proportional reasoning before being taught about proportionality as a mathematical concept. The concept of learning trajectory is used to identify situations in the learning process during instruction where students meet obstacles and need scaffolding and teacher support. It is shown how a teacher dealing with a mathematical task involving mixtures of liquids encounters a task that has the possibility of making proportional reasoning visible for the students, and how she struggles to make the modelling required intelligible to herself and to the students. The instructional strategy of using everyday problems as a basis for learning implies that the initial modelling phase becomes crucial, and the students have to be aware of the conditions and limitations under which proportional reasoning is applicable. In conclusion, students engage in early forms of proportional reasoning well ahead of formal instruction. The difficulties they experience as they are to develop their proficiency, and where they require support from the teacher, concern how to model the familiar, everyday situations they encounter in exercises in mathematically precise and productive ways. In addition, in textbooks and national course tests proportionality is presented in a standardized, and rather simplified, form, and it is not sufficiently connected to the various areas of mathematics teaching and learning where it is applicable

Place, publisher, year, edition, pages
Göteborg: Göteborgs universitet, 2024. p. 221
Series
Göteborg studies in educational sciences, ISSN 0436-1121 ; 495
Keywords
proportionality, proportional reasoning, learning trajectory, textbooks, interaction, video analysis, microgenesis
National Category
Didactics
Identifiers
urn:nbn:se:liu:diva-210405 (URN)2077/83652 (DOI)9789179630164 (ISBN)9789179630171 (ISBN)
Public defence
2024-12-03, Sal: BE 015, Hus B, Pedagogen, Läroverksgatan 15, Göteborg, 09:00 (Swedish)
Opponent
Available from: 2024-12-12 Created: 2024-12-12 Last updated: 2026-02-17Bibliographically approved

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Lundberg, Anna, L., V.

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