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Matematiska resonemang i en lärandemiljö med dynamiska matematikprogram
Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science. (SMEER)
2015 (Swedish)Doctoral thesis, comprehensive summary (Other academic)Alternative title
Mathematical Reasoning in a Dynamic Software Environment (English)
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]

Baksidestext

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.
Series
Karlstad University Studies, ISSN 1403-8099 ; 2015:12
Keyword [en]
Mathematical reasoning, Task design, Dynamic mathematics software
Keyword [sv]
matematiska resonemang, uppgiftsdesign, dynamiska matematikprogram
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kau:diva-35037ISBN: 978-91-7063-623-3 (print)OAI: oai:DiVA.org:kau-35037DiVA: diva2:784065
Public defence
2015-03-18, 21A 342 (Eva Erikssonsalen), Karlstad, 10:15 (Swedish)
Opponent
Supervisors
Available from: 2015-03-04 Created: 2015-01-28 Last updated: 2015-03-04Bibliographically approved
List of papers
1. A Model for Task Design with Focus on Exploration,Explanation, and Generalization in a Dynamic GeometryEnvironment
Open this publication in new window or tab >>A Model for Task Design with Focus on Exploration,Explanation, and Generalization in a Dynamic GeometryEnvironment
2014 (English)In: Technology, Knowledge and Learning, ISSN 2211-1662, E-ISSN 2211-1670, Vol. 19, no 3, 287-315 p.Article in journal (Refereed) Published
Abstract [en]

The increasing availability of new technologies in schools provides new possibilities

for the integration of technology in mathematics education. However, research

has shown that there is a need for new kinds of task that utilize the affordances provided by

new technology. Numerous studies have demonstrated that dynamic geometry environments

provide opportunities for students to engage in mathematical activities such as

exploration, conjecturing, explanation, and generalization. This paper presents a model for

design of tasks that promote these kinds of mathematical activity, especially tasks that

foster students to make generalizations. This model has been primarily developed to suit

the use of dynamic environments in tackling geometrical locus problems. The model was

initially constructed in the light of previous literature. This initial model was used to design

a concrete example of such a task situation which was tested in action through a case study

with two doctoral students. Findings from this case study were used to guide revision of the

initial model.

Place, publisher, year, edition, pages
Springer, 2014
Keyword
Mathematics education, Task design, Dynamic geometry
National Category
Educational Sciences
Identifiers
urn:nbn:se:kau:diva-33815 (URN)10.1007/s10758-014-9213-9 (DOI)
Available from: 2014-09-24 Created: 2014-09-24 Last updated: 2017-12-05Bibliographically approved
2. An Expanded Version of Toulmin’s Model to Analyze Students’ Mathematical Reasoning in a Dynamic Software Environment
Open this publication in new window or tab >>An Expanded Version of Toulmin’s Model to Analyze Students’ Mathematical Reasoning in a Dynamic Software Environment
(English)Manuscript (preprint) (Other academic)
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
Toulmin’s model, Mathematical reasoning, Dynamic mathematics software
National Category
Mathematics
Identifiers
urn:nbn:se:kau:diva-35036 (URN)
Available from: 2015-01-28 Created: 2015-01-28 Last updated: 2017-03-30Bibliographically approved
3. Students’ Mathematical Reasoning in a Dynamic Software Environment
Open this publication in new window or tab >>Students’ Mathematical Reasoning in a Dynamic Software Environment
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Mathematical reasoning has been identified as one of the key competencies needed to master mathematics. It is also well documented that dynamic mathematics software can be utilized to provide students with opportunities to practice different aspects of mathematical reasoning. In this study an expanded version of Toulmin’s model of argumentation was used to analyze the reasoning that emerged when three student pairs used GeoGebra to explore how the different parameters influence the graph of the function y = A sin (Bx + C) + D. Several characteristic features, both promising and cautionary, were discerned in the students’ reasoning.

Keyword
Mathematical reasoning, Dynamic mathematics software, Toulmin's model
National Category
Mathematics
Identifiers
urn:nbn:se:kau:diva-35035 (URN)
Available from: 2015-01-28 Created: 2015-01-28 Last updated: 2017-03-30Bibliographically approved
4. Designing Prediction Tasks in a Mathematics Software Environment
Open this publication in new window or tab >>Designing Prediction Tasks in a Mathematics Software Environment
2015 (English)In: International Journal of Technology in Mathematics Education, ISSN 1744-2710, Vol. 22, no 1, 3-18 p.Article in journal (Refereed) Published
Abstract [en]

There is a recognized need in mathematics teaching for new kinds of task which exploit the affordances provided by new technology. This paper focuses on the design of prediction tasks to foster student reasoning about exponential functions in a mathematics software environment. It draws on the first iteration of a design based research study conducted by the authors in collaboration with four upper secondary school teachers. A task sequence was trailed with four 10th grade classes, involving a total of 85 students. The research process pinpointed key didactical variables that proved crucial in designing these tasks. As well as being useful in the task design process, the didactical variables were found to be valuable in the processes of analysis and revision. While the didactical variables identified a priori were informed by the research literature concerning reasoning and functions, those identified a posteriori deal mainly with scaffolding issues that emerged.  

Keyword
Task design, Prediction task, Mathematics software environment
National Category
Mathematics
Identifiers
urn:nbn:se:kau:diva-35034 (URN)
Available from: 2015-01-28 Created: 2015-01-28 Last updated: 2017-08-08Bibliographically approved

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