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Unraveling students’ reasoning: analyzing small-group discussions during task solving
Mälardalens Högskola.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Umeå University, Faculty of Science and Technology, Umeå Mathematics Education Research Centre (UMERC).ORCID iD: 0000-0002-7594-5602
(English)Manuscript (preprint) (Other academic)
Abstract [en]

The aim of this study is to examine students’ mathematical reasoning, suggested by Lithner (2008), to see how reasoning sequences will unfold in actual classroom situations. We visited two classrooms in an upper secondary school and observed two student groups in each classroom for the time it took them to complete a task, constructed and presented to them by their teacher. Initial analysis showed that there were two interesting dimensions to regard, group characteristics (i.e., motivation and persistence) and task design (i.e., reasoning promoted by the task). We recorded conversations between the students and after transcribing we utilized Lithner’s (2008) framework of mathematical reasoning to analyze students’ reasoning. We classified the moments (vertices) when the students’ reasoning took a new trajectory and characterized the segment (edge) between two such vertices according to the students’ reasoning (i.e., either creative mathematically founded reasoning or algorithmic reasoning). We then visualized the students’ reasoning in graphs (see Figure 10) and analyzed the patterns, the progress and types of reasoning, as well as how the group characteristics and task design would influence reasoning and progress. The result showed that task design is important for which reasoning the students will use. Although an algorithmic-task does not exclude creative reasoning, it only occurs in our data if the students have difficulties and strive to handle them by themselves. We also observed that group characteristics were important for the chosen reasoning type. 

National Category
Educational Sciences
Research subject
didactics of mathematics
Identifiers
URN: urn:nbn:se:umu:diva-124674OAI: oai:DiVA.org:umu-124674DiVA: diva2:954338
Available from: 2016-08-22 Created: 2016-08-22 Last updated: 2016-08-25
In thesis
1. On Mathematical Reasoning: being told or finding out
Open this publication in new window or tab >>On Mathematical Reasoning: being told or finding out
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Alternative title[sv]
Om Matematiska resonemang : att få veta eller att få upptäcka
Abstract [en]

School-mathematics has been shown to mainly comprise rote-learning of procedures where the considerations of intrinsic mathematical properties are scarce. At the same time theories and syllabi emphasize competencies like problem solving and reasoning. This thesis will therefore concern how task design can influence the reasoning that students apply when solving tasks, and how the reasoning during practice is associated to students’ results, cognitive capacity, and brain activity. In studies 1-3, we examine the efficiency of different types of reasoning (i.e., algorithmic reasoning (AR) or creative mathematically founded reasoning (CMR)) in between-groups designs. We use mathematics grade, gender, and cognitive capacity as matching variables to get similar groups. We let the groups practice 14 different solution methods with tasks designed to promote either AR or CMR, and after one week the students are tested on the practiced solution methods. In study 3 the students did the test in and fMRI-scanner to study if the differing practice would yield any lasting differences in brain activation. Study 4 had a different approach and focused details in students’ reasoning when working on teacher constructed tasks in an ordinary classroom environment. Here we utilized audio-recordings of students’ solving tasks, together with interviews with teachers and students to unravel the reasoning sequences that students embark on. The turning points where the students switch subtask and the reasoning between these points were characterized and visualized. The behavioral results suggest that CMR is more efficient than AR, and also less dependent on cognitive capacity during the test. The latter is confirmed by fMRI, which showed that AR had higher activation than CMR in areas connected to memory retrieval and working memory. The behavioral result also suggested that CMR is more beneficial for cognitively less proficient students than for the high achievers. Also, task design is essential for both students’ choice of reasoning and task progression. The findings suggest that: 1) since CMR is more efficient than AR, students need to encounter more CMR, both during task solving and in teacher presentation, 2) cognitive capacity is important but depending on task design, cognitive strain will be more or less high during test situations, 3) although AR-tasks does not prohibit the use of CMR they make it less likely to occur. Since CMR-tasks can emphasize important mathematical properties, are more efficient than AR- tasks, and more beneficial for less cognitively proficient students, promoting CMR can be essential if we want students to become mathematically literate. 

Place, publisher, year, edition, pages
Umeå: Umeå universitet, 2016. 55 p.
Series
Doctoral thesis / Umeå University, Department of Mathematics, ISSN 1102-8300
Keyword
mathematics education, creative reasoning, reasoning, cognitive proficiency, fMRI
National Category
Educational Sciences Mathematics Didactics Learning
Research subject
didactics of mathematics
Identifiers
urn:nbn:se:umu:diva-124677 (URN)978-91-7601-525-4 (ISBN)
External cooperation:
Public defence
2016-09-16, MA121, MIT-huset, Umeå, 13:00 (Swedish)
Opponent
Supervisors
Available from: 2016-08-25 Created: 2016-08-22 Last updated: 2016-09-02Bibliographically approved

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Norqvist, Mathias
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