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On Mathematical Reasoning: being told or finding out
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.ORCID iD: 0000-0002-7594-5602
2016 (English)Doctoral thesis, comprehensive summary (Other academic)Alternative title
Om Matematiska resonemang : att få veta eller att få upptäcka (Swedish)
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 [en]
mathematics education, creative reasoning, reasoning, cognitive proficiency, fMRI
National Category
Educational Sciences Mathematics Didactics Learning
Research subject
didactics of mathematics
Identifiers
URN: urn:nbn:se:umu:diva-124677ISBN: 978-91-7601-525-4OAI: oai:DiVA.org:umu-124677DiVA: diva2:954413
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
List of papers
1. Learning mathematics through algorithmic and creative reasoning
Open this publication in new window or tab >>Learning mathematics through algorithmic and creative reasoning
2014 (English)In: Journal of Mathematical Behavior, ISSN 0732-3123, E-ISSN 1873-8028, no 36, 20-32 p.Article in journal (Refereed) Published
Abstract [en]

There are extensive concerns pertaining to the idea that students do not develop sufficient mathematical competence. This problem is at least partially related to the teaching of procedure-based learning. Although better teaching methods are proposed, there are very limited research insights as to why some methods work better than others, and the conditions under which these methods are applied. The present paper evaluates a model based on students’ own creation of knowledge, denoted creative mathematically founded reasoning (CMR), and compare this to a procedure-based model of teaching that is similar to what is commonly found in schools, denoted algorithmic reasoning (AR). In the present study, CMR was found to outperform AR. It was also found cognitive proficiency was significantly associated to test task performance. However the analysis also showed that the effect was more pronounced for the AR group.

Place, publisher, year, edition, pages
Elsevier, 2014
Keyword
mathematical reasoning, reasoning, cognitive proficiency, memory retrieval
National Category
Psychology Didactics
Identifiers
urn:nbn:se:umu:diva-95773 (URN)10.1016/j.jmathb.2014.08.003 (DOI)2-s2.0-84907851938 (ScopusID)
Available from: 2014-11-05 Created: 2014-11-05 Last updated: 2016-08-25Bibliographically approved
2. Do explanations increase the efficiency of procedural tasks?
Open this publication in new window or tab >>Do explanations increase the efficiency of procedural tasks?
(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:nbn:se:umu:diva-124672 (URN)
Available from: 2016-08-22 Created: 2016-08-22 Last updated: 2016-08-25
3. Learning mathematics without a suggested solution method: durable effects on performance and brain activity
Open this publication in new window or tab >>Learning mathematics without a suggested solution method: durable effects on performance and brain activity
Show others...
2015 (English)In: Trends in Neuroscience and Education, ISSN 2211-9493, Vol. 4, no 1-2, 6-14 p.Article in journal (Refereed) Published
Abstract [en]

A dominant mathematics teaching method is to present a solution method and let pupils repeatedly practice it. An alternative method is to let pupils create a solution method themselves. The current study compared these two approaches in terms of lasting effects on performance and brain activity. Seventythree participants practiced mathematics according to one of the two approaches. One week later, participants underwent fMRI while being tested on the practice tasks. Participants who had created the solution method themselves performed better at the test questions. In both conditions, participants engaged a fronto-parietal network more when solving test questions compared to a baseline task. Importantly, participants who had created the solution method themselves showed relatively lower brain activity in angular gyrus, possibly reflecting reduced demands on verbal memory. These results indicate that there might be advantages to creating the solution method oneself, and thus have implications for the design of teaching methods.

Place, publisher, year, edition, pages
Elsevier, 2015
Keyword
Mathematics, Learning, fMRI, Parietal cortex, Angular gyrus, Education
National Category
Psychology (excluding Applied Psychology)
Identifiers
urn:nbn:se:umu:diva-109088 (URN)10.1016/j.tine.2015.03.002 (DOI)000363545300002 ()
Available from: 2015-09-17 Created: 2015-09-17 Last updated: 2016-08-25Bibliographically approved
4. Unraveling students’ reasoning: analyzing small-group discussions during task solving
Open this publication in new window or tab >>Unraveling students’ reasoning: analyzing small-group discussions during task solving
(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:nbn:se:umu:diva-124674 (URN)
Available from: 2016-08-22 Created: 2016-08-22 Last updated: 2016-08-25

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