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Björk, Kjell

Open this publication in new window or tab >>Generating the patterns of variation with GeoGebra: the case of polynomial approximations### Attorps, Iiris

### Björk, Kjell

### Radic, Mirko

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_0_j_idt188_some",{id:"formSmash:j_idt184:0:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_0_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_0_j_idt188_otherAuthors",{id:"formSmash:j_idt184:0:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_0_j_idt188_otherAuthors",multiple:true}); 2016 (English)In: International journal of mathematical education in science and technology, ISSN 0020-739X, E-ISSN 1464-5211, Vol. 47, no 1, p. 45-57Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

mathematics, Taylor polynomial, teaching, university education, Variation theory
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:hig:diva-19944 (URN)10.1080/0020739X.2015.1046961 (DOI)000366674100003 ()2-s2.0-84930156781 (Scopus ID)
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Available from: 2015-06-30 Created: 2015-06-30 Last updated: 2019-09-27Bibliographically approved

University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.

University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.

University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.

In this paper, we report a teaching experiment regarding the theory of polynomial approximations at the university mathematics teaching in Sweden. The experiment was designed by applying Variation theory and by using the free dynamic mathematics software GeoGebra. The aim of this study was to investigate if the technology-assisted teaching of Taylor polynomials compared with traditional way of work at the university level can support the teaching and learning of mathematical concepts and ideas. An engineering student group (n = 19) was taught Taylor polynomials with the assistance of GeoGebra while a control group (n = 18) was taught in a traditional way. The data were gathered by video recording of the lectures, by doing a post-test concerning Taylor polynomials in both groups and by giving one question regarding Taylor polynomials at the final exam for the course in Real Analysis in one variable. In the analysis of the lectures, we found Variation theory combined with GeoGebra to be a potentially powerful tool for revealing some critical aspects of Taylor Polynomials. Furthermore, the research results indicated that applying Variation theory, when planning the technology-assisted teaching, supported and enriched students’ learning opportunities in the study group compared with the control group.

Open this publication in new window or tab >>Teaching Inverse Functions at Tertiary level### Attorps, Iiris

University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.### Björk, Kjell

University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.### Radic, Mirko

University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.### Viirman, Olov

University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_some",{id:"formSmash:j_idt184:1:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_otherAuthors",{id:"formSmash:j_idt184:1:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_otherAuthors",multiple:true}); 2013 (English)In: CERME 8: Proceedings of the Eight Congress of the European Society for Research in Mathematics Education, Antalya: Middle East Technical University , 2013, p. 2524-2533Conference paper, Published paper (Refereed)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Antalya: Middle East Technical University, 2013
##### Keywords

Concept definition, Concept image, Inverse functions, Technology, Variation theory
##### National Category

Educational Sciences Mathematics
##### Identifiers

urn:nbn:se:hig:diva-18023 (URN)
##### Conference

The 8th Congress of the European Society for Research in Mathematics education, 6-10 February, Antalya, Turkiet
#####

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##### Projects

Learning Studies - Praxisnära forskning i matematik
Available from: 2014-11-20 Created: 2014-11-20 Last updated: 2019-09-27Bibliographically approved

This study is a part of an ongoing research that attempts to explain the relationship between the teachers’ instructional practise and students’ learning in the context of functions and function inverses. The question in this paper is how the use of technology as a pedagogical tool may contribute to the understanding of the inverse function concept. An engineering student group (n =17) was taught functions and inverse functions with the assistance of GeoGebra. In our theoretical framework we apply Variation theory together with the theory of Concept image and Concept definition. The data were gathered by doing a pre and post test concerning inverse functions. Our experiment revealed that students’ concept images in the post test were more developed compared with the results in the pre test.

Open this publication in new window or tab >>Varied ways to teach the definite integral concept### Attorps, Iiris

University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.### Björk, Kjell

University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.### Radic, Mirko

University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.### Tossavainen, Timo

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_some",{id:"formSmash:j_idt184:2:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_otherAuthors",{id:"formSmash:j_idt184:2:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_otherAuthors",multiple:true}); 2013 (English)In: International Electronic Journal of Mathematics Education, ISSN 1306-3030, Vol. 8, no 2-3, p. 81-99Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Turkey: Gökkuşağı Ltd. Şti., Turkey, 2013
##### Keywords

definite integral, GeoGebra, learning study, tertiary education, variation
##### National Category

Other Mathematics
##### Identifiers

urn:nbn:se:hig:diva-15942 (URN)2-s2.0-84891647831 (Scopus ID)
#####

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Available from: 2014-01-07 Created: 2014-01-07 Last updated: 2019-09-27Bibliographically approved

University of Eastern Finland, Finland.

In this paper, we report on a collaborative teaching experiment based on the Learning Study model (LS model) which grounds on the Variation Theory. Until today, most of such studies have focused on the teaching and learning of elementary school mathematics; ours was carried out in undergraduate mathematics education. In the following, we discuss how we managed to promote students’ conceptual learning by varying the treatment of the object of learning (the concept of definite integral and the Fundamental Theorem of Calculus) during three lectures on an introductory course in calculus. We also discuss the challenges and possibilities of the LS model and the Variation Theory in the development of the teaching of tertiary mathematics in general. The experiment was carried out at aSwedish university. The data of the study consists of the documents of the observation of three lectures and the students’ answers to the pre- and post-tests of each lesson. The analysis of learning results revealed some critical aspects of the definite integral concept and patterns of variations that seem to be effective to a significant degree. For example, we found several possibilities to use GeoGebra to enrich students’ learning opportunities.

Open this publication in new window or tab >>Application of Variation Theory in Teaching and Learning of Taylor Polynomials with GeoGebra### Attorps, Iiris

University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.### Björk, Kjell

University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.### Lingefjärd, Thomas

### Radic, Mirko

University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_some",{id:"formSmash:j_idt184:3:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_otherAuthors",{id:"formSmash:j_idt184:3:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_otherAuthors",multiple:true}); 2012 (English)In: Proceedings of the 12th International Congress on Mathematics Education. Seoul. Korea, Seoul. Korea, 2012, p. 3479-3488Conference paper, Published paper (Refereed)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Seoul. Korea: , 2012
##### National Category

Didactics
##### Identifiers

urn:nbn:se:hig:diva-12901 (URN)
##### Conference

ICMI, the 12th International Congress on Mathematics Education. Seoul. Korea, July 8-15 2012
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Available from: 2012-09-16 Created: 2012-09-16 Last updated: 2019-09-27Bibliographically approved

Institutionen för pedagogik och didaktik, Göteborgs universitet.

We report on a teaching experiment regarding Taylor polynomial approximations at the level of university mathematics teaching. The experiment was carried by using the free dynamic mathematics software GeoGebra. A student group (n = 19) was taught Taylor polynomials with the assistance of GeoGebra while a control group (n = 18) was taught in a traditional way. The theoretical assumptions of this study rest on Variation theory. The data were gathered by doing a post test concerning Taylor polynomials. Our experiment revealed that the answers from the GeoGebra group in the post test were more varied compared with the results in the control group.

Open this publication in new window or tab >>The use of mathematics software in university mathematics teaching### Attorps, Iiris

University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.### Björk, Kjell

University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.### Radic, Mirko

University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_some",{id:"formSmash:j_idt184:4:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_otherAuthors",{id:"formSmash:j_idt184:4:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_otherAuthors",multiple:true}); 2011 (English)In: The proceedings of the Seventh Congress of the European Society for Research in Mathematics education. Poland. University of Rzeszow. / [ed] M. Pytland, E. Swoboda & T. Rowland, 2011, p. 2188-2197Conference paper, Published paper (Refereed)
##### Abstract [en]

##### Keywords

definite integral, geogebra, learning, variation theory
##### National Category

Didactics
##### Identifiers

urn:nbn:se:hig:diva-10020 (URN)
##### Conference

The Seventh Congress of the European Society for Research in Mathematics Education, Rzeszów, Poland, between 9-13 February 2011
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Available from: 2011-09-05 Created: 2011-09-05 Last updated: 2019-09-27Bibliographically approved

In this paper we report on a teaching experiment regarding the definite integral concept in university mathematics teaching. The experiment was carried out at a Swedish university by using the free dynamic mathematics software GeoGebra. In our theoretical framework we apply Variation Theory, originating in the phenomenographic research tradition. The data of this study consist of the lecture plan and the engineering students’ answers to pre and post tests. In the analysis of the data we applied statistical methods. The experiment revealed that by using GeoGebra it is possible to create learning opportunities of the definite integral concept that support the students’ learning.

Open this publication in new window or tab >>Improving undergraduate mathematics teaching### Attorps, Iiris

University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.### Björk, Kjell

University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.### Radic, Mirko

University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_5_j_idt188_some",{id:"formSmash:j_idt184:5:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_5_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_5_j_idt188_otherAuthors",{id:"formSmash:j_idt184:5:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_5_j_idt188_otherAuthors",multiple:true}); 2010 (English)In: The proceedings of MADIF7; The 7th Swedish Mathematics Education Research Seminar, Stockholm, January 26-27, 2010., 2010Conference paper, Published paper (Refereed)
##### Identifiers

urn:nbn:se:hig:diva-10022 (URN)
##### Conference

MADIF7, The 7th Swedish Mathematics Education Research Seminar, Stockholm 26-27 januari 2010
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Available from: 2011-09-05 Created: 2011-09-05 Last updated: 2019-09-27Bibliographically approved

Open this publication in new window or tab >>The Learning Study Model and the Teaching of the Definite Integral Concept### Attorps, Iiris

University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.### Björk, Kjell

University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.### Radic, Mirko

University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.### Tossavainen, Timo

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_some",{id:"formSmash:j_idt184:6:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_otherAuthors",{id:"formSmash:j_idt184:6:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_otherAuthors",multiple:true}); 2010 (English)In: Reports and Studies in Education, Humanities, and Theology / [ed] M. Asikainen, P. E. Hirvonen & K. Sormunen, Joensuu: University of Eastern Finland , 2010, p. 77-86Conference paper, Published paper (Refereed)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Joensuu: University of Eastern Finland, 2010
##### Keywords

definite integral, learning study, undergraduate students, variation theory
##### National Category

Didactics
##### Identifiers

urn:nbn:se:hig:diva-10023 (URN)
##### Conference

Matematiikan ja luonnontieteiden opetuksen tutkimuspäivät Joensuussa (Symposium of the Finnish Mathematics and Science Education Research Association, Joensuu) 22-23 oktober 2009
#####

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Available from: 2011-09-05 Created: 2011-09-05 Last updated: 2019-09-27Bibliographically approved

University of Eastern Finland.

In recent years, there have been several studies in mathematics education basing on the variation theory and the model of Learning Study that involves co-operation between teachers and researchers in an iterative process. Most of these studies have focused on the teaching and learning of elementary school mathematics rather than topics in advanced mathematics. In this paper, we discuss some challenges and possibilities of the Learning Study model and the variation theory when developing the teaching of mathematics at advanced levels. More precisely, we report on a series of teaching experiments on the definite integral concept. The experiments were carried out at a Swedish university. The data of this study consists of the documents on the observation of three lectures and the students’ answers to pre and post tests. Both engineering and teacher students participated. In the analysis of the data, we applied statistical methods. Although the series consisted only of three lectures, it revealed that the students’ understanding about certain – but not necessarily all – aspects of the definite integral concept can be enhanced by using the Learning Study model.