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An Expanded Version of Toulmin’s Model to Analyze Students’ Mathematical Reasoning in a Dynamic Software Environment
Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science. (SMEER)
(English)Article in journal (Refereed) Submitted
Abstract [en]

Toulmin’s model of argumentation has been used to analyse mathematical reasoning in a wide range of contexts. While conducting a recent case study of students’ mathematical reasoning in a dynamic software environment, it proved advantageous to develop an expanded version of this model to deepen the data analysis. Since this expanded model served well to discern and illustrate characteristic features in students’ reasoning, the question arose of whether it could also be useful in other cases. The present paper addresses this question, by considering further examples of students working in pairs in a dynamic software environment. The model was examined using data from two different studies, varying in the types of task and level of mathematics concerned. Transcribed data from these studies were interpreted in terms of the model to examine its applicability. Assessment of its use in all three studies shows that the expanded version of Toulmin’s model can enhance its capacity to analyse reasoning in this type of setting. At the same time, the operationalization of the model in concrete situations can sometimes be debatable, emphasising the importance of elucidating the interpretation principles used.

Keyword [en]
Toulmin’s model, Mathematical reasoning, Dynamic mathematics software
National Category
URN: urn:nbn:se:kau:diva-35036OAI: diva2:784039
Available from: 2015-01-28 Created: 2015-01-28 Last updated: 2016-08-23
In thesis
1. Matematiska resonemang i en lärandemiljö med dynamiska matematikprogram
Open this publication in new window or tab >>Matematiska resonemang i en lärandemiljö med dynamiska matematikprogram
2015 (Swedish)Doctoral thesis, comprehensive summary (Other academic)
Alternative title[en]
Mathematical Reasoning in a Dynamic Software Environment
Abstract [en]

The overall problem that formed the basis for this thesis is that students get limited opportunity to develop their mathematical reasoning ability while, at the same time, there are dynamic mathematics software available which can be used to foster this ability. The aim of this thesis is to contribute to knowledge in this area by focusing on task design in a dynamic software environment and by studying the reasoning that emerges when students work on tasks in such an environment. To analyze students’ mathematical reasoning, a new analytical tool was developed in the form of an expanded version of Toulmin’s model.

Results from one of the studies in this thesis show that exploratory tasks in a dynamic software environment can promote mathematical reasoning in which claims are formulated, examined and refined in a cyclic process. However, this reasoning often displayed a lack of the more conceptual, analytic and explanatory reasoning normally associated with mathematics. This result was partly confirmed by another of the studies. Hence, one key question in the thesis has been how to design tasks that promote conceptual and explanatory reasoning. Two articles in the thesis deal with task design. One of them suggests a model for task design with a focus on exploration, explanation, and generalization. This model aims, first, to promote semantic proof production and then, after the proof has been constructed, to encourage further generalizations. The other article dealing with task design concerns the design of prediction tasks to foster student reasoning about exponential functions. The research process pinpointed key didactical variables that proved crucial in designing these tasks.

Abstract [sv]


Det övergripande problem som legat till grund för denna avhandling är att elever får begränsad möjlighet att utveckla sin resonemangsförmåga samtidigt som det finns dynamiska matematikprogram som kan utnyttjas för att stimulera denna förmåga. Syftet med avhandlingen är att bidra till den samlade kunskapen inom detta problemområde, dels genom att fokusera på design av uppgifter i en lärandemiljö med dynamiska matematikprogram och dels genom att studera och karakterisera de resonemang som utvecklas när elever jobbar med olika uppgifter i denna miljö. För att analysera elevernas resonemang utvecklades ett nytt analysverktyg i form av en utökad version av Toulmins modell.

Resultat från en av studierna i avhandlingen visar att dynamiska matematikprogram i kombination med utforskande uppgifter kan stimulera till matematiska resonemang där hypoteser formuleras, undersöks och förfinas i en cyklisk process. Samtidigt visar samma studie att de resonemang som utvecklas i stor utsträckning saknar matematiskt grundade förklaringar. Detta resultat bekräftas till viss del av ytterligare en studie.  Frågan hur uppgifter bör designas för att främja matematiskt grundade resonemang har därför varit central i avhandlingen. Två av artiklarna behandlar uppgiftsdesign, men utifrån olika utgångspunkter.

Place, publisher, year, edition, pages
Karlstad: Karlstads universitet, 2015. 77 p.
Karlstad University Studies, ISSN 1403-8099 ; 2015:12
Mathematical reasoning, Task design, Dynamic mathematics software, matematiska resonemang, uppgiftsdesign, dynamiska matematikprogram
National Category
Research subject
urn:nbn:se:kau:diva-35037 (URN)978-91-7063-623-3 (ISBN)
Public defence
2015-03-18, 21A 342 (Eva Erikssonsalen), Karlstad, 10:15 (Swedish)
Available from: 2015-03-04 Created: 2015-01-28 Last updated: 2015-03-04Bibliographically approved

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