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Lärande i matematik: Om resonemang och matematikuppgifters egenskaper
Karlstad University, Faculty of Arts and Social Sciences (starting 2013), Department of Educational Studies.ORCID iD: 0000-0002-7956-8795
2014 (Swedish)Doctoral thesis, comprehensive summary (Other academic)Alternative title
Learning Mathematics : On the Features of Reasoning and Mathematical Tasks (English)
Abstract [en]

Since mathematical tasks are central to the teaching of mathematics, it is crucial to extend our knowledge of the characteristic features of tasks that are conducive to student development of problem-solving and reasoning abilities as well as conceptual understanding.

 

The aim of the dissertation is to investigate how different types of mathematical tasks affect student learning and choice of learning strategies. This is done through a twofold approach: 1) to test the hypothesis that tasks affording students the opportunity and responsibility for constructing knowledge are more effective learning tools than tasks for which the solution is presented, and 2) to analyse the educational message embedded in the teacher’s formulation of the mathematical tasks on the Internet.

 

The main conclusion is that the type of task students engage with is important for their learning of new things. The participants who were engaged in creating their own solutions were less successful during practice but performed better on the tests in comparison with the participants who were involved in solving the tasks with a given method. The results of the sub-studies indicate that in a learning situation consisting of repeated practice of a solution method, the results are closely related to the students’ cognitive ability. The investigation shows that tasks inviting the opportunity to be solved through creative reasoning, to a certain extent serve a compensatory function in relation to students’ cognitive resources. This means that the participants need not put in so much effort in the test situation if they have practiced creative reasoning.

 

One conclusion to be drawn from the study of the educational message in Internet documents, when it comes to teachers’ formulation of tasks, is that there are many teachers who design tasks that encourage young students’ creative reasoning. However, the educational message in the documents shows that the teachers demand relatively little of the students in the majority of the tasks. The result indicates that there is some uncertainty about how to formulate and use tasks to support the older student’s mathematical development. The way the tasks are formulated indicates a lack of discursive tools to clarify the intended educational situation. Thus, the qualities in the tasks are hidden resources.

 

 

Abstract [sv]

Det övergripande syftet med avhandlingen är att undersöka hur olika typer av matematikuppgifter påverkar elevers möjligheter till lärande och val av lärandestrategi.

Resultaten visar att i en lärandesituation som består av upprepad träning av en lösningsmetod är elevernas resultat starkt beroende av deras kognitiva förmåga. Uppgifter som ger eleverna möjligheter att lösa dem med kreativa resonemang verkar i viss mån kompensatoriskt i förhållande till elevers kognitiva resurser. Testdeltagare som tränat via kreativa resonemang har en lägre kognitiv belastning.

Studien av lärarkonstruerade uppgifter på internet visar exempel på uppgifter som upplevs som användbara i undervisningen. Uppgiftskonstruktörerna tenderar att ställa förhållandevis låga krav på eleverna i merparten av uppgifterna. Resultatet tyder även på en osäkerhet inför hur uppgifter skall kommuniceras för att stötta elevers matematiska utveckling. De kvalitéer som finns i uppgifterna kommer därför i skymundan.

Matematikuppgifter är centrala i matematikundervisningen. Det är därför viktigt att öka kunskapen om matematikuppgifter och hur dessa kan bidra till elevers möjligheter att utveckla sin matematiska kompetens. Väl utformade matematikuppgifter kan bidra till att minska elevers utantillärande.

Place, publisher, year, edition, pages
Karlstad: Karlstads universitet, 2014. , 65 p.
Series
Karlstad University Studies, ISSN 1403-8099 ; 2014:16
Keyword [en]
imitative and creative reasoning, educational design research
Keyword [sv]
Imitativa och kreativa resonemang, designforskning
National Category
Educational Sciences
Research subject
Educational Work
Identifiers
URN: urn:nbn:se:kau:diva-31456ISBN: 978-91-7063-546-5 (print)OAI: oai:DiVA.org:kau-31456DiVA: diva2:696528
Public defence
2014-03-28, Nyqvistsalen 9C 203, Karlstads universitet, Karlstad, 13:00 (Swedish)
Opponent
Supervisors
Available from: 2014-03-12 Created: 2014-02-14 Last updated: 2014-03-12Bibliographically approved
List of papers
1. Reasoning in alignment with the task design? Students actual handling of algorithmic and creative mathematical reasoning
Open this publication in new window or tab >>Reasoning in alignment with the task design? Students actual handling of algorithmic and creative mathematical reasoning
(English)Manuscript (preprint) (Other academic)
National Category
Educational Sciences
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-31460 (URN)
Projects
Learning by imitative and creative reasoning
Available from: 2014-02-14 Created: 2014-02-14 Last updated: 2014-03-12Bibliographically approved
2. 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, Vol. 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.

Keyword
Mathematical reasoning, Reasoning, Cognitive proficiency, Memory retrieval
National Category
Social Sciences Learning
Research subject
Education
Identifiers
urn:nbn:se:kau:diva-31461 (URN)10.1016/j.jmathb.2014.08.003 (DOI)
Projects
Learning by imitative and creative reasoning
Note

The article was not yet published at the time of the defense.

Available from: 2014-02-17 Created: 2014-02-14 Last updated: 2017-12-06Bibliographically approved
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 2452-0837, E-ISSN 2211-9493, Vol. 4, no 1-2, 6-14 p.Article, book review (Other academic) Published
Abstract [en]

A dominant mathematics Leaching 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. Seventy-three participants practiced mathematics according to one of the two approaches. One week later, participants underwent BM 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 gyros, 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
Neurosciences Educational Sciences
Research subject
Education
Identifiers
urn:nbn:se:kau:diva-31464 (URN)10.1010/j.tine.2015.03.002 (DOI)000363545300002 ()
Projects
Learning by imitative and creative reasoning
Note

The article was included as manuscript in Yvonne Liljekvists thesis. It was then titled: Learning mathematics without a given solution method have beneficial effects on subsequent performance and modulates brain activity

Available from: 2014-02-17 Created: 2014-02-14 Last updated: 2017-12-06Bibliographically approved
4. Teacher-shared documents on the Internet: Didactical message and mathematical tasks
Open this publication in new window or tab >>Teacher-shared documents on the Internet: Didactical message and mathematical tasks
(English)Manuscript (preprint) (Other academic)
National Category
Educational Sciences
Research subject
Educational Work
Identifiers
urn:nbn:se:kau:diva-31463 (URN)
Available from: 2014-02-14 Created: 2014-02-14 Last updated: 2014-03-12Bibliographically approved

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