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Do explanations increase the efficiency of procedural tasks?
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]

Studies in mathematics education often point to the necessity for students to engage in more cognitively demanding activities than just solving tasks by applying given solution methods. Lithner’s (2008) framework on mathematical reasoning address this by studying which reasoning a task promotes. Previous studies have shown that students that create their own solution methods, denoted creative reasoning (CMR), perform significantly better in follow up tests than students that are given the solution method and engage in algorithmic reasoning (AR). However, teachers and textbooks at least occasionally provide explanations together with a solution method (XAR) and this could possibly be more efficient than creative reasoning. In this study three matched groups practiced with either AR-, XAR- or CMR-tasks. The study showed that students that practiced with AR- and XAR-tasks performed similarly during both practice and test. The CMR-group did, although a worse practice score, outperform the XAR- group in the test. Additionally, there were differences in which variables predicted the test result with similar results between the XAR- and AR-group where cognitive proficiency and mathematics grade where significant. For the CMR-group the test score was predicted by the practice score alone. This would indicate that students that practice with CMR-tasks are not as dependent on their cognitive abilities for test performance as students that practice with AR- tasks. 

National Category
Educational Sciences
Research subject
didactics of mathematics
Identifiers
URN: urn:nbn:se:umu:diva-124672OAI: oai:DiVA.org:umu-124672DiVA: diva2:954301
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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Department of Mathematics and Mathematical StatisticsUmeå Mathematics Education Research Centre (UMERC)
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