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A Model for Task Design with Focus on Exploration,Explanation, and Generalization in a Dynamic GeometryEnvironment
Karlstad University, Faculty of Health, Science and Technology (starting 2013).ORCID iD: 0000-0003-4553-0630
Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science.ORCID iD: 0000-0003-2897-9188
2014 (English)In: Technology, Knowledge and Learning, ISSN 2211-1662, E-ISSN 2211-1670, Vol. 19, no 3, p. 287-315Article 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. Vol. 19, no 3, p. 287-315
Keywords [en]
Mathematics education, Task design, Dynamic geometry
National Category
Educational Sciences
Identifiers
URN: urn:nbn:se:kau:diva-33815DOI: 10.1007/s10758-014-9213-9OAI: oai:DiVA.org:kau-33815DiVA, id: diva2:749412
Available from: 2014-09-24 Created: 2014-09-24 Last updated: 2017-12-05Bibliographically approved
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]

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. p. 77
Series
Karlstad University Studies, ISSN 1403-8099 ; 2015:12
Keywords
Mathematical reasoning, Task design, Dynamic mathematics software, matematiska resonemang, uppgiftsdesign, dynamiska matematikprogram
National Category
Mathematics
Research subject
Mathematics
Identifiers
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)
Opponent
Supervisors
Available from: 2015-03-04 Created: 2015-01-28 Last updated: 2015-03-04Bibliographically approved
2. Designing for the integration of dynamic software environments in the teaching of mathematics
Open this publication in new window or tab >>Designing for the integration of dynamic software environments in the teaching of mathematics
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis concerns the challenge of integrating dynamic software environments into the teaching of mathematics. It investigates particular aspects of the design of tasks which employ this type of computer-based system, with a focus on improvement, both of the tasks themselves and of the design process through which they are developed and refined.

The thesis reports two research projects: a small initial one preceding a larger main project. The initial case study, involving two graduate students in mathematics, develops a task design model for geometrical locus problems. The main study constitutes the first iteration of a design-based study, conducted in collaboration with four upper-secondary school teachers and their classes. It seeks to identify task design characteristics that foster students’ mathematical reasoning and proficient use of software tools, and examines teachers’ organisation of ‘follow-up’ lessons.

The findings concern three particular aspects: features of tasks and task environment relevant to developing a specific plan of action for a lesson; orchestration of a particular task environment to support the instrumental genesis of specific dynamic software tools; how to follow up students’ work on computer-based tasks in a whole-class discussion.

Place, publisher, year, edition, pages
Karlstad: Karlstads universitet, 2015. p. 107
Series
Karlstad University Studies, ISSN 1403-8099 ; 50
Keywords
design research, dynamic mathematics software, intrumental orchestration, mathematics education, task design
National Category
Educational Sciences
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-38213 (URN)978-91-7063-669-1 (ISBN)
Public defence
2015-12-15, Eva Erikssonsalen, 21A 342, Karlstad University, Karlstad, 10:15 (English)
Opponent
Supervisors
Available from: 2015-11-24 Created: 2015-10-19 Last updated: 2017-11-29Bibliographically approved

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Fahlgren, MariaBrunström, Mats

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