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Encountering Proportional Reasoning During a Single Algebra Lesson: A Microgenetic Analysis
Department of Pedagogical, Curricular and Professional Studies, Gothenburg University, Gothenburg, Sweden.ORCID iD: 0000-0001-9110-2851
2022 (English)In: International Electronic Journal of Mathematics Education, E-ISSN 1306-3030, Vol. 17, no 1, article id em0673Article in journal (Refereed) Published
Abstract [en]

This case study explores how 12-13-year-old students encounter proportional reasoning while working with geometric patterning tasks using concrete materials. The focus is on the students’ use of spontaneous concepts when first dealing with such patterns in the context of collaborative work. Based on video recordings of a single lesson, a microgenetic analysis was performed to identify students’ learning trajectories, starting with students familiarizing themselves with pattern structure, followed by engagement in proportional reasoning, and ending with students perceiving a new technique to handle a situation where proportional reasoning did not suffice. While some student groups were able to move along the whole trajectory, most groups, when facing challenges, regressed to simpler techniques. The results provide new insights into students’ learning trajectories, which can be used to support students’ progress in the context of student-teacher interaction.

Place, publisher, year, edition, pages
Eastbourne, United Kingdom: Modestum Publishing Ltd. , 2022. Vol. 17, no 1, article id em0673
Keywords [en]
algebra, geometric pattern, learning trajectory, microgenesis, proportionality, proportional reasoning
National Category
Didactics Other Mathematics Educational Sciences Algebra and Logic
Identifiers
URN: urn:nbn:se:liu:diva-182413DOI: 10.29333/iejme/11571OAI: oai:DiVA.org:liu-182413DiVA, id: diva2:1630052
Projects
VIDEOMAT
Note

Funding: The study is funded by the Joint Committee for Nordic Research Councils for the Humanities and the Social Sciences (NOS-HS)through a grant to the Linnaeus Centre for Research on Learning, Interaction and Mediated Communication in Contemporary Society (LinCS),the Swedish Research Council (dnr 349-2006-146), and the Research School at the Centre for Educational Science and Teacher Research (CUL),Gothenburg University and The Royal Society of Arts and Sciences in Gothenburg.

Available from: 2022-01-19 Created: 2022-01-19 Last updated: 2025-02-18Bibliographically approved
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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