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• 1.
Clausthal University of Technology, Clausthal, Germany..
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. O. Ya, Usikov Institute for Radiophysics and Electronics of the National Academy of Sciences of Ukraine, Kharkiv, Ukraine.
Mathematical models for scattering and generation of plane wave packets on layered cubically polarisable structures2013In: Far East Journal of Applied Mathematics, ISSN 0972-0960, Vol. 81, no 1-2, p. 1-31Article in journal (Refereed)

The paper deals with different formulations of mathematical models for the analysis of processes of resonance scattering and generation of plane wave packets on isotropic, nonmagnetic, linearly polarised media with a nonlinear, layered dielectric structure of cubic polarisability. For each formulation, sufficient conditions for the existence and, partially, uniqueness of the corresponding solution are derived.

• 2.
Technische Universität Clausthal, Institut für Mathematik, Clausthal-Zellerfeld, Germany .
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics. O.Ya. Usikov Institute for Radiophysics and Electronics of the National Academy of Sciences of Ukraine, Kharkiv, Ukraine .
Eigenmodes of linearised problems of scattering and generation of oscillations on cubically polarisable layers2015In: Inverse Problems and Applications / [ed] Larisa Beilina, Springer-Verlag New York, 2015, Vol. 120, p. 67-80Conference paper (Refereed)

In the frequency domain, the resonant properties of nonlinear structures are determined by the proximity of the scattering/generation frequencies of the nonlinear structures to the complex eigenfrequencies of the corresponding homogeneous linear spectral problems with the induced nonlinear permeability of the medium. Here the case of cubically polarisable, canalising, and decanalising layers is considered.

• 3.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.
A study of a problem solving oriented lesson structure in mathematics in Japan2011In: Proceedings of the  seventh Congress  of  the  European  Society  for  Research  in Mathematics Education (CERME 7), Rzeszów, Poland.: Universitetsforlaget, 2011Conference paper (Refereed)

This paper presents and analyses “Mondaikaiketsu no jugyou” which translates to “the problem solving oriented – approach”.  It is a set of didactic techniques with the aim of motivating the students’ positive attitude toward engaging in mathematical activities and fostering mathematical thinking. As an analytical tool, The Anthropological Theory of Didactics (ATD) will be applied.

• 4.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.
Applying problem solving oriented lesson structure in Swedish lower secondary school: With a viewpoint of the anthropological theory of didactics (ATD)2011In: Proceedings of the 44th congress of Japanese mathematical education / [ed] Japanese society for research in mathematics education, Tokyo: Japanese society for research in mathematics education , 2011, p. 1051-1052Conference paper (Refereed)
• 5.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics. Linköpings universitet, Matematiska institutionen.
Designing Mathematics Lessons Using Japanese Problem Solving Oriented Lesson Structure: A Swedish case study2015Licentiate thesis, monograph (Other academic)

This licentiate thesis is concerned with applying the Japanese problem solving oriented (PSO) teaching approach to Swedish mathematics classrooms. The overall aim of my research project is to describe and investigate the viability of PSO as design tool for teaching mathematics. The PSO approach is a variation of a more general Japanese teaching paradigm referred to as “structured problem solving”. These teaching methods aim to stimulate the process of students’ mathematical thinking and have their focus on enhancing the students’ attitudes towards engaging in mathematical activities. The empirical data are collected using interviews, observations and video recordings over a period of nine months, following two Swedish lower secondary school classes. Chevallard’s anthropological framework is used to analyse which mathematical knowledge is exposed in the original Japanese lesson plans and in the lessons observed in the classrooms. In addition, Brousseau’s framework of learning mathematics is applied to analyse the perception of individual students and particular situations in the classroom.

The results show that the PSO based lesson plans induce a complex body of mathematical knowledge, where different areas of mathematics are linked. It is found that the discrepancy between the Japanese and Swedish curriculum cause some limitations for the adaptation of the lesson plans, especially in the area of Geometry. Four distinct aspects of the PSO approach supporting the teaching of mathematics are presented.

• 6.
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. Institutionen för pedagogik och didaktik, Göteborgs universitet. University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.
Application of Variation Theory in Teaching and Learning of Taylor Polynomials with GeoGebra2012In: Proceedings of the 12th International Congress on Mathematics Education. Seoul. Korea, Seoul. Korea, 2012, p. 3479-3488Conference paper (Refereed)

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.

• 7.
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.
Generating the patterns of variation with GeoGebra: the case of polynomial approximations2016In: 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)

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.

• 8.
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.
Improving undergraduate mathematics teaching2010In: The proceedings of MADIF7; The 7th Swedish Mathematics  Education Research Seminar, Stockholm, January 26-27, 2010., 2010Conference paper (Refereed)
• 9.
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.
The use of mathematics software in university mathematics teaching2011In: 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 (Refereed)

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.

• 10.
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. University of Eastern Finland.
The Learning Study Model and the Teaching of the Definite Integral Concept2010In: 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 (Refereed)

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.

• 11.
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. University of Eastern Finland, Finland.
Varied ways to teach the definite integral concept2013In: International Electronic Journal of Mathematics Education, ISSN 1306-3030, Vol. 8, no 2-3, p. 81-99Article in journal (Refereed)

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.

• 12.
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. University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.
Teaching Inverse Functions at Tertiary level2013In: 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 (Refereed)

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.

• 13.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.
University of Gävle, Faculty of Health and Occupational Studies, Department of Occupational and Public Health Sciences, Biology.
School-University Action Research: Impacts on Teaching Practices and Pupil Learning2017In: International Journal of Science and Mathematics Education, ISSN 1571-0068, E-ISSN 1573-1774, Vol. 15, no 2, p. 313-330Article in journal (Refereed)

The aim of this article is to describe a design and implementation of a schoolâuniversity action research project about teaching and learning biology and mathematics in primary school. Nine teachers in grades 1 to 6, in collaboration with two researchers, were using content representation (CoRe) in learning study (LS)-inspired cycle as pedagogical tools when planning, implementing, and reflecting on lessons and pupil learning. By using pre- and post-tests, the teachers acquired knowledge about pupil subject-specific knowledge and learning. Some examples are given on how the tools in the project influenced the teaching practices and pupil learning. This research design brought together university and school practitioners to work collaboratively in a professional learning community, which developed teaching and learning in biology and mathematics.

• 14.
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, Electronics. University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Physics.
MST education in Swedish preschool2013Conference paper (Refereed)

The present study is part of the EU-Fp7 project SECURE. The research question in the present study is: How is the current Swedish MST curriculum for preschool implemented by the teachers, and perceived by the 5-year old learners?

The introduction of a revised preschool curriculum in 2011 has given MST a prominent place in Swedish preschool, and may be intended by the authorities to change the weak positions and declining trends in the PISA and TIMSS surveys for Swedish school children. The new teaching task seems to be highly appreciated by the preschool teachers. However, the curriculum is quite ambitious, and a major concern is how the implementation will be done. The concept of ”emergent science” is discussed in order to induce positive attitudes towards MST.

In this study, semi-structured interviews with teachers in 15 Swedish preschools are the main source of information combined with some interviews with learners and photographic materials. The method for the analysis uses a category system of which results from some of the categories are reported.

The preschool teachers are aware of the new role as teachers rather than just child minders. The revised curriculum seems to increase the self-confidence for MST of the learners as well as the teachers. The teachers say that the learners are interested in MST and that it is a case of catching opportunities, which may lead to emergent science. The 5-year old learners confirm that they like the activities related to MST.

Working with hypotheses and storyline activities is shown as an example how emergent science can be promoted in 5-year old children. Activities connected to ”emerging science” have been common in Swedish preschools for some time, but after the implementation of the revised curriculum teachers also verbalize this for themselves and the learners.

• 15.
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.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Electronics. University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Electronics. University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Electronics. University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Physics. University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Physics. University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Physics.
MST-utbildning i den svenska förskolan2013Conference paper (Refereed)

Svenska elever hamnar efter i PISA- och TIMSS-undersökningarna. I matematik har matningarna visat på sjunkande kunskaper sedan början av 2000-talet och i naturvetenskap ligger de svenska eleverna ständigt under det internationella genomsnittet. Detta är alarmerande för landet och skälen bör utredas.Som en del i EU-projektet SECURE inom ramprogrammet Fp7 presenteras resultat från lokal forskning om genomförandet av MST-kursplanerna och attityder till ämnena bland lärare och elever. Med MST avser vi matematik, naturvetenskap och teknik. Intervjuer och enkäter har utförts bland lärare och elever i 47 klasser i 15 skolenheter i några mellansvenska kommuner. De åldersgrupper som studerats är elever från åldersgrupperna 5, 8, 11 resp. 13 år. Vi fokuserar här på förskolans verksamhet, d.v.s. arbetet med 5-åringar.Som en följd av införandet 2011 av en reviderad läroplan för förskolan, kan vi se att MST har fått en framträdande plats i den svenska förskolan som verkar vara mycket uppskattad av förskollärarna. I den reviderade läroplanen framgår att förskollärare ska ansvara för att arbetet i barngruppen genomförs så att barnen stimuleras och utmanas i sin matematiska utveckling, samt stimuleras och utmanas i sitt intresse för naturvetenskap och teknik. Exempel på hur de skall gå tillväga finns också i läroplanen. Detta kan i praktiken uppfattas som en informell tidigareläggning av den obligatoriska skolan i Sverige, eftersom förskolan inte är obligatorisk men utnyttjas av de allra flesta. Det skulle därmed kunna innebära en vändpunkt för resultaten i grundskolan och förhoppningsvis ge avtryck i kommande PISA- och TIMSS-undersökningar.De förskollärare som intervjuats är medvetna om vikten av att undervisa MST i förskolan. Detta tydliggörs av några uttalanden från intervjuade förskollärare. 'Det som är nytt nu är att man pratar om lärare och att vi utbildar barnen.' 'Det här lärandet är för livet. Det är inte bara för individen. Verksamheten på förskolan bidrar till hela samhällsutvecklingen.' Även föräldrarnas förväntningar på förskolan har förändrats. 'Föräldrarna uppskattar resultaten i matematik och ser förskolan som en skolform snarare än förvaring av barn.'I denna presentation ges exempel på hur MST-ämnena undervisas på ett integrerat sätt i förhållande till den praxis som råder i svenska förskolan, samt i förhållande till ålder och mognad hos barnen.

• 16.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.
University of Eastern Finland.
Erfarenheter av ”Learning Studies” som ett pedagogiskt verktyg i skola och högskola2010In: Toisensa kohtaavat ainedidaktiikat [ Möte med ämnesdidaktik] / [ed] E. Ropo, H. Silfverberg & T. Soini, TampereTampereen yliopistopaino Oy - Juvenes Print, 2010, p. 25-33Conference paper (Refereed)

I detta dokument diskuterar vi pågående projekt i matematikdidaktik vid Högskolan i Gävle. Syftet med projektet är att utveckla och testa metoder för att förbättra lärande i matematik från förskolan till universitetsnivån. Dessutom är syftet att utveckla en forskningsmiljö för lärande i matematik som kan fungera som en plattform för att utveckla lärarnas kompetens i matematik. Den övergripande forskningsfrågan är: Hur påverkar lärarens sätt att hantera lärandeobjekt (t.ex. funktionsbegrepp, integralbegrepp etc.) i klassrummet elevens/studentens lärande? Vilka är de kritiska aspekterna för studenternas lärande? Som ett teoretiskt ramverk använder vi Learning Study modellen designad av Marton et al. (2004). Modellen bygger på variationsteorin (Marton et al. 2004), som härrör från den fenomenografiska forskningstraditionen och som introducerades i Sverige under senare delen av 1960-talet och i början av 1970-talet. Resultaten av studierna antyder bl.a. att det är betydligt enklare att hitta olika typer av konkreta men användbara representationer av lärandeobjektet i undervisningen av den elementära matematiken jämfört med undervisningen av den högre matematiken. Troligen beror det på att universitetsmatematik är mer abstrakt och bygger på den tidigare erhållna kunskapen.

• 17.
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.
Creating the patterns of variation with GeoGebra when teaching derivative graphs for first year engineering students2015In: International Journal of Engineering ,Science and Innovative Technology, ISSN 0949-149X, E-ISSN 2277-3754, Vol. 31, no 6, p. 1605-1612Article in journal (Refereed)

The present study investigates how technology assisted and designed teaching influences engineering students’ understanding of the connection between the graph of a function and its derivatives. An engineering student group (n = 27) was taught with the assistance of GeoGebra while a control group (n =20) was taught in a traditional way. The data of the study consist of the documents and photos of the observation of two lectures and the students’ answers to the pre and post tests. In our theoretical framework we discuss the distinction between conceptual and procedural knowledge. When creating the teaching sequences we applied variation theory. In the analysis of the students’ pre and post tests results we applied statistical methods. Our experiment revealed that the GeoGebra-assisted teaching design created more opportunities for students to grasp the connection between a function and its derivatives.

• 18. Buy this publication >>
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.
Variationsteoretiskt perspektiv på matematikundervisningen2016In: Ämnesdidaktiska utmaningar - inom matematik, naturvetenskap och teknik / [ed] Mikael Björling, Gävle: Gävle University Press , 2016, 1, p. 15-26Chapter in book (Other academic)
• 19.
Department of Mathematical Sciences, Chalmers University of Technology, Gothenburg University, Gothenburg, Sweden.
Department of Mathematical Sciences, Chalmers University of Technology, Gothenburg University, Gothenburg, Sweden. Institut de Mathematiques de Marseille, CNRS, UMR 7373, Ecole Centrale, Aix-Marseille University, Marseille, France. University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.
Preface for the session "recent Progress in Electromagnetic Field Theory and New Trends in Inverse Problems"2017In: AIP Conference Proceedings, ISSN 0094-243X, E-ISSN 1551-7616, Vol. 1863, article id 370001Article in journal (Refereed)
• 20.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics. Linköpings universitet, Matematiska institutionen.
Gymnasieelevers kommunikativa strategier i matematikklassrummet: En fallstudie av ett smågruppsarbete om derivata2014Licentiate thesis, monograph (Other academic)

This case study takes its focus on upper secondary school students’ work in small groups with a problem related to the derivative. The analysis aims to identify factors that promote or hinder an individual’s participation in and development of the mathematical communication in the classroom. The theoretical basis of the study is Anna Sfard’s commognitive framework, where learning mathematics is seen as participating in a mathematical discourse.

For more than a decade, reports about Swedish students’ decreasing levels of school mathematical knowledge have been put forward. Research points to various factors behind this development. The prevailing educational culture, where students largely work individually from the textbook, is seen as one explanation for the deterioration in the results, and that teaching does not give students the opportunity to develop all the required competencies in the curriculum. To achieve this, both research and the new Swedish curriculum from 2011 emphasize the importance of student communication in mathematics. In this perspective, there is a need to highlight the differences in student participation in the communication of mathematics in the classroom, particularly in the context of small group learning, and how this is assumed to influence students’ opportunities for learning.

The focus of the research is directed towards the participants’ contributions to the group’s mathematical discourse, i.e. when they communicate about mathematical objects or processes, and how these affect students’ opportunities and participation in the communication. Focus is also directed to the communication that involves participants in the group, what the students are doing and how they evaluate each other’s way to participate in the mathematical discourse in the classroom. This type of communication is in the framework referred to as subjectifying, and is assumed to affect the individual’s mathematical learning.

Data collection methods used are interviews, audio and video recordings, as well as “smart pens” to combine verbal and written communication. In the first step of the analysis, the mathematical discourse was studied regarding differences in the content of the participants’ utterances. The second step of analysis focused on the interaction flow of the group to understand more of the differences in each student’s participation and contribution to the communication.

The results point to big differences regarding participation and content in student communication, both at group level and individual level. The development of students’ mathematical discourse benefits from the use of multiple mediators to represent the mathematical objects. When connections to a previously acquired discourse are offered, this leads to discursive advancements. Students were observed to have difficulties to interpret and use the formal mathematical symbolic language that would support their mathematizing. Students’ interpretation of the equality sign, the sign for inequality, and the symbol f´(x) on a process level, create obstacles to developing the mathematical discourse in the desired direction. The discourse about the participants and their own traits  (identification) constitutes about 10% of all utterances and are almost all negative reviews, frequently used in order to exclude or incorporate themselves or others from participating in the mathematizing activity.

This research study points to a need for more knowledge about how mathematics teachers can best organize work in small groups to increase student engagement and the quality of their mathematizing. The study also indicates the importance of mathematics teachers highlighting and varying the use of different mediators to represent the mathematical objects to learn. The case study also highlights the importance of building up a permissive environment in which students do not evaluate themselves and others, but instead dare to ask questions that will make them increasingly involved in the mathematical discourse. A need emerges for further research not only on the assessment between teacher and student, but also on the assessment that goes on in the classroom between the students, which can affect what roles they take or are assigned to in the classroom. This can be assumed to be of great importance to the way students communicate about mathematics with other students in the classroom, which is also likely to influence learning.

• 21.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.
En kvalitativ studie om lärarens syn på lässvårigheters påverkan på matematikinlärningen: samt andra svårigheters påverkan2017Independent thesis Advanced level (degree of Master (One Year)), 20 credits / 30 HE creditsStudent thesis

Syftet med den här studien är att undersöka hur lärare ser på lässvårigheters påverkan på matematikinlärningen i årskurserna 4–6. Metodvalet är inom det didaktiska forskningsområdet, arbetet består av en litteraturbakgrund, observationer och intervjuer med sex verksamma lärare och en specialpedagog för att undersöka deras syn på hur lässvårigheter kan påverka matematikinlärningen.

Resultatet visar att det kan finnas ett samband mellan lässvårigheter och matematiksvårigheter. Det som framkommer i arbetet är att det finns bakomliggande faktorer som påverkar läsinlärningen och matematikinlärningen. Det är flera delar som påverkar inlärningen, en del är den som är kopplad till de språkliga inlärningsförmågorna, som är en bristande fonologisk förmåga, läsförståelsen och ett svagt arbetsminne. När eleven har svårt med dessa faktorer påverkar det även möjligheten till inlärning i matematik. Förutom de här svårigheterna visar resultatet att de emotionella känslorna påverkar elevers förmåga att prestera. En av de emotionella känslorna som påverkar eleven är motivation och intresse för de som ska läras, om det inte finns motivation och intresse sker ingen inlärning. Det som forskare och lärare är absolut överens om är att ett dåligt självförtroende är det som påverkar eleven mest vid inlärningen. Ett dåligt självförtroende kan leda till blockeringar inte bara för läsinlärningen utan för övriga ämnen i skolan så som matematik.

• 22.
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, Electronics. University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Electronics.
Is technology an attractive school subject?: A pilot study of the subject in theory and practice2010In: Socio-cultural and Human Values in Science and Technology Education / [ed] Slavko Dolinšek, Terry Lyons, Ljubljana, Slovenia: IRI UL , 2010Conference paper (Refereed)

How does a teacher convey technological understanding through interactions with learners?

A case study is performed using a stimulated-recall methodology with one teacher in a limited Technology learning module in grade 8. Based on the official syllabus and local interpretations, what in fact occurs in the meeting with pupils is discussed regarding how technology is portrayed by the teacher.

An important result is what messages about technology the teacher is expecting the pupils to understand from each lesson. In order to get appropriate feedback, selected pupils are interviewed about their impressions of the lesson.

• 23.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.
Problemlösning bland yngre elever2018Independent thesis Advanced level (professional degree), 20 credits / 30 HE creditsStudent thesis

Syftet med uppsatsen är att undersöka vilka stragier som lärare i lågstadiet anser är mest lämpade för yngre elever, och vilka metoder lärarna använder sig av i sin undervisning. Studien har en kvalitativ ansats, och intervjuer och klassrumsobservationer har genomförts med två lärare. Lärarna hade varit yrkesverksamma i cirka 10 år men arbetade inte på samma skola. Studien har genomförts i en kommun i Mellansverige. Strategierna som eleverna använde sig mest av var att rita sina lösningar och att jobba med laborativa material. Undervisningen innehöll mycket diskussioner bland eleverna. Lärarna betonade att de ville att eleverna skulle reflektera i sina tankegångar och sätta ord på sina tankeprocesser.

• 24.
University of Gävle, Faculty of Education and Business Studies, Department of Culture Studies, Religious Studies and Educational Sciences, Curriculum studies. University of Gävle, Faculty of Education and Business Studies, Department of Culture Studies, Religious Studies and Educational Sciences, Curriculum studies.
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.
Teaching Mathematics in the Pre-school Context2012In: Journal of US-China Education Review B, ISSN 2161-6248, p. 1-16Article in journal (Other academic)

The aim of this study is to describe what kind of knowledge base is needed when pre-school teachers work goal-oriented with children’s mathematical learning. The question of this study aims to answer is: What kind of knowledge base do pre-school teachers need when they work with the object of learning in the pre-school context? Both as a theoretical and an analytical framework, we use variation theory. In all, four pre-school teachers are involved in the study. The research results indicate that it is important for pre-school teachers to have subject and pedagogical content knowledge in order to recognize children’s experiences and be aware of their expressions.

• 25.
University of Gävle, Faculty of Education and Business Studies, Department of Educational sciences, Educational science. University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.
Lärares val av metod i matematikundervisningen: Med fokus på elever med svenska som andraspråk2018Independent thesis Advanced level (professional degree), 20 credits / 30 HE creditsStudent thesis

Detta arbete har som huvudsyfte att analysera några erfarna lärares olika metoder och val av material i sin matematikundervisning, och hur de motiverar sina val för att genomföra en likvärdig undervisning för alla elever - med fokus på elever med svenska som andraspråk. I detta arbete har jag valt att intervjua fyra lärare från två olika skolor. Då syftet med detta arbete var att analysera lärares val av metoder vid matematikundervisningen utgick jag från lärarnas egna perspektiv. Med inspiration från grounded theory har en kvalitativ dataanalys genomförts från intervjuerna. Både intervjuerna och litteraturgenomgången visar att det inte finns något rätt eller fel vid val av undervisningsmetod, men att vissa metoder ibland fungerar bättre än andra. Det gäller att hitta det sätt som fungerar i den grupp man jobbar i för tillfället. De egenskaper som verkar lysa igenom som några av de viktigare egenskaperna hos en lärare är att vara flexibel.

• 26.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.
Automatisera multiplikationstabellerna - är det nödvändigt?: Matematiklärares uppfattning om automatisering av multiplikationstabellerna samt framgångsrika automatiseringsmetoder.2017Independent thesis Advanced level (professional degree), 20 credits / 30 HE creditsStudent thesis

Syftet med arbetet är att skapa en bild av matematiklärares uppfattning om automatisering av multiplikationstabellerna samt att redogöra för både för- och nackdelar med automatiserad kunskap. Syftet är även att belysa och diskutera olika automatiseringsmetoder som på olika sätt främjar eller hindrar elevernas matematikutveckling. Det gjordes en digital enkätundersökning där urvalet utgörs av 333 frivilliga matematiklärare representerade från olika undervisningsnivåer. Resultatet visar att matematiklärare anser att det är nödvändigt att eleverna automatiserar multiplikationstabellerna för att få en chans att utvecklas inom matematiken på bästa sätt, men att det kan ske på olika sätt, med olika metoder. Den mest fördelaktiga metoden verkar vara att kombinera färdighetsträning med att lära ut beräkningsstrategier för att kunskapen ska bestå.

• 27.
O. Ya. Usikov Institute for Radiophysics and Electronics of the NAS of Ukraine, Kharkiv, Ukraine.
O. Ya. Usikov Institute for Radiophysics and Electronics of the NAS of Ukraine, Kharkiv, Ukraine. O. Ya. Usikov Institute for Radiophysics and Electronics of the NAS of Ukraine, Kharkiv, Ukraine. Lomonosov Moscow State University, Russia. Lomonosov Moscow State University, Russia. University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.
Reconstructing complex permittivity of local inhomogeneites in a radio-transparent dielectric matrix located in a waveguide2016In: 2016 22nd International Conference on Applied Electromagnetics and Communications (ICECOM), 2016Conference paper (Refereed)

The results are presented of the numerical solution to the forward problem of the electromagnetic wave scattering by inhomogeneous dielectric bodies in a waveguide and inverse problem of determining parameters of dielectric inclusions from the values of the transmission coefficient of the scattered electromagnetic wave. A comparison with the results of mesurements and analysis of the achieved threshold computational accuracy levels enable one to estimate and validate the simulation results and formulate the ways of improving the model and codes.

• 28.
O. Ya. Usikov Institute for Radiophysics and Electronics of the NAS of Ukraine, Kharkiv, Ukraine.
O. Ya. Usikov Institute for Radiophysics and Electronics of the NAS of Ukraine, Kharkiv, Ukraine. O. Ya. Usikov Institute for Radiophysics and Electronics of the NAS of Ukraine, Kharkiv, Ukraine. 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, Electronics.
Combined system of the microstrip antennas with different frequencies2016In: 2016 22nd International Conference on Applied Electromagnetics and Communications (ICECOM), 2016Conference paper (Refereed)

The combined system with microstrip antennas (f1=1.85GHz; f2=3.7GHz, f3=1.75GHz, f4= 3.5GHz) is presented. The measurements of the prototype characteristics have shown that elevation angle of peak directivity of the antennas is oriented to zenith. Simulated S11 at stated frequencies with the following minimum values of S11 = -34dB, f=1.85GHz; S11=-21dB, f=3.7GHz; S11 = -22dB, f=1.75GHz; S11=-19dB, f=3.5GHz are observed.

• 29.
O.Ya. Usikov Institute for Radiophysics and Electronics, National Academy of Sciences of Ukraine, Kharkiv, Ukraine.
O.Ya. Usikov Institute for Radiophysics and Electronics, National Academy of Sciences of Ukraine, Kharkiv, Ukraine. O.Ya. Usikov Institute for Radiophysics and Electronics, National Academy of Sciences of Ukraine, Kharkiv, Ukraine. 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. Amity University, Delhi, India. Penza State University, Penza, Russian Federation.
Determination of effective permittivity of metamaterial antenna cells2016In: 2016 10th European Conference on Antennas and Propagation: EuCAP 2016, Institute of Electrical and Electronics Engineers (IEEE), 2016, article id 7481369Conference paper (Refereed)

A numerical-analytical method of reconstructing complex frequency-dependent permittivities is proposed. Measurements of the transmission coefficient for three typical metamaterial antenna units are performed and the frequency-dependent effective complex permittivity of the units is reconstructed in a wide frequency range.

• 30.
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, Electronics. University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Electronics. University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Electronics. University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Electronics.
An analysis of Kautz-Volterra models for modeling block structure nonlinear systems2010Report (Other academic)

Kautz-Volterra (KV) models of some nonlinear systems were analyzed. The relationsbetween the true pole of the analyzed systems and the optimal pole of the KV modelswere analyzed. The properties of nonlinear systems depend on the order of thesubsystems, since nonlinear operators do not commute. Wiener (H-N) andHammerstein (N-H) systems were analyzed.

• 31.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.
Institutionen för matematik och matematisk statistik, Umeå univeristet. Institutionen för matematik och matematisk statistik, Umeå univeristet.
Factors of r-partite graphs and bounds for the strong chromatic number2010In: Ars combinatoria, ISSN 0381-7032, Vol. 95, p. 277-287Article in journal (Refereed)

We give an optimal degree condition for a tripartite graph to have a spanning subgraph consisting of complete graphs of order 3. This result is used to give an upper bound of 2 Delta for the strong chromatic number of n vertex graphs with Delta >= n/6.

• 32.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.
Department of Mathematics, University of Bristol, Bristol, United Kingdom. Department of Mathematics, Uppsala Universitet, Uppsala, Sweden. Mathematics Institute, University of Warwick, Coventry, United Kingdom.
Multifractal analysis of non-uniformly hyperbolic systems2010In: Israel Journal of Mathematics, ISSN 0021-2172, E-ISSN 1565-8511, Vol. 177, no 1, p. 125-144Article in journal (Refereed)

We prove a multifractal formalism for Birkhoff averages of continuous functions in the case of some non-uniformly hyperbolic maps, which includes interval examples such as the Manneville-Pomeau map.

• 33.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics. Mathematics Department, Uppsala University, Uppsala, Sweden.
Institut für Informatik, Universität Leipzig, Leipzig, Germany . Institut für Informatik, Universität Leipzig, Leipzig, Germany . Mathematics Department, Uppsala University, Uppsala, Sweden .
Tuning positive feedback for signal detection in noisy dynamic environments2012In: Journal of Theoretical Biology, ISSN 0022-5193, E-ISSN 1095-8541, Vol. 309, p. 88-95Article in journal (Refereed)

Learning from previous actions is a key feature of decision-making. Diverse biological systems, from neuronal assemblies to insect societies, use a combination of positive feedback and forgetting of stored memories to process and respond to input signals. Here we look how these systems deal with a dynamic two-armed bandit problem of detecting a very weak signal in the presence of a high degree of noise. We show that by tuning the form of positive feedback and the decay rate to appropriate values, a single tracking variable can effectively detect dynamic inputs even in the presence of a large degree of noise. In particular, we show that when tuned appropriately a simple positive feedback algorithm is Fisher efficient, in that it can track changes in a signal on a time of order L(h)= (vertical bar h vertical bar/sigma)(-2), where vertical bar h vertical bar is the magnitude of the signal and sigma the magnitude of the noise.

• 34.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.
School of Engineering and Applied Sciences, Harvard University.
A Slime Mold Solver for Linear Programming Problems2012In: How the World Computes: Turing Centenary Conference and 8th Conference on Computability in Europe, CiE 2012, Cambridge, UK, June 18-23, 2012. Proceedings / [ed] S. Barry Cooper , Anuj Dawar and Benedikt Löwe, Springer, 2012, p. 344-354Conference paper (Refereed)

Physarum polycephalum (true slime mold) has recently emerged as a fascinating example of biological computation through morphogenesis. Despite being a single cell organism, experiments have observed that through its growth process, the Physarum is able to solve various minimum cost flow problems. This paper analyzes a mathematical model of the Physarum growth dynamics. We show how to encode general linear programming (LP) problems as instances of the Physarum. We prove that under the growth dynamics, the Physarum is guaranteed to converge to the optimal solution of the LP. We further derive an efficient discrete algorithm based on the Physarum model, and experimentally verify its performance on assignment problems.

• 35.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.
Department of Mathematics, Uppsala University, Uppsala, Sweden. Mathematics Institute, University of Warwick, Coventry, UK.
Ergodic Theory of Kusuoka Measures2017In: Journal of Fractal Geometry, ISSN 2308-1317, Vol. 4, no 2, p. 185-214Article in journal (Refereed)

In the analysis on self-similar fractal sets, the Kusuoka measure plays an important role. Here we investigate the Kusuoka measure from an ergodic theoretic viewpoint, seen as an invariant measure on a symbolic space. Our investigation shows that the Kusuoka measure generalizes Bernoulli measures and their properties to higher dimensions of an underlying finite dimensional vector space. Our main result is that the transfer operator on functions has a spectral gap when restricted to a certain Banach space that contains the Hölder continuous functions, as well as the highly discontinuous g" role="presentation">g-function associated to the Kusuoka measure. As a consequence, we obtain exponential decay of correlations. In addition, we provide some explicit rates of convergence for a family of generalized Sierpinski gaskets.

• 36.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.
Uppsala University. University of Warwick.
Phase transitions in long-range Ising models and an optimal condition for factors of g-measures2017In: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417Article in journal (Refereed)

We weaken the assumption of summable variations in a paper by Verbitskiy [On factors of -measures. Indag. Math. (N.S.) 22 (2011), 315–329] to a weaker condition, Berbee’s condition, in order for a one-block factor (a single-site renormalization) of the full shift space on finitely many symbols to have a -measure with a continuous -function. But we also prove by means of a counterexample that this condition is (within constants) optimal. The counterexample is based on the second of our main results, where we prove that there is a critical inverse temperature in a one-sided long-range Ising model which is at most eight times the critical inverse temperature for the (two-sided) Ising model with long-range interactions.

• 37.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics. Uppsala University, Department of Mathematics.
Uppsala University, Department of Mathematics. University of Warwick, Mathematics Institute.
Unique Bernoulli g-measures2012In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 14, no 5, p. 1599-1615Article in journal (Refereed)

We improve and subsume the conditions of Johansson and Öberg and Berbee for uniqueness of a g-measure, i.e., a stationary distribution for chains with complete connections. In addition, we prove that these unique g-measures have Bernoulli natural extensions. We also conclude that we have convergence in the Wasserstein metric of the iterates of the adjoint transfer operator to the g-measure.

• 38.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.
Sannolikhet till vinst: Ett undervisningsmaterial åk 92018Independent thesis Advanced level (professional degree), 20 credits / 30 HE creditsStudent thesis
• 39. Buy this publication >>
University of Gävle, Faculty of Health and Occupational Studies, Department of Occupational and Public Health Sciences, Biology.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.
Kollegialt lärande: aktionsforskning i biologi- och matematikundervisning2016In: Ämnesdidaktiska utmaningar - inom matematik, naturvetenskap och teknik / [ed] Mikael Björling, Gävle: Gävle University Press , 2016, 1, p. 61-74Chapter in book (Other academic)
• 40.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Biology. University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Biology.
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.
Kollegialt lärande om progression i biologi- och matematikundervisning2014In: Nämnaren : tidskrift för matematikundervisning, ISSN 0348-2723, Vol. 41, no 4, p. 3-8Article in journal (Other (popular science, discussion, etc.))

Kan man utveckla undervisningen i biologi och matematik samtidigt? Ett lärarlag samarbetade med två forskare för att undersöka och svara på frågan. De undervisningsområden som valdes ut handlar om ekosystem och övergången mellan aritmetik och algebra.

• 41.
University of Gävle, Faculty of Health and Occupational Studies, Department of Occupational and Public Health Sciences, Biology.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.
Primary school teachers' concerns and needs in biology and mathematics teaching2015In: NorDiNa: Nordic Studies in Science Education, ISSN 1504-4556, E-ISSN 1894-1257, Vol. 11, no 3, p. 282-292Article in journal (Refereed)

The aim of this paper is to provide insights into nine primary school teachers’ concerns and instructional needs in biology and mathematics, grades 1 to 6. By using Content Representation, combined with Learning Study in an action research project, teachers were encouraged to reflect on their conceptions, processes of instructing and pupil learning. From concerns articulated by teachers three instructional needs emerged: (i) to make subject progression, especially in biology, and pupil learning more visible, (ii) to develop mathematics teaching in order to change pupils’ views of the subject, and (iii) to develop teachers’ subject matter knowledge and teaching in an ongoing collaborative process.  The paper argues that in order to stimulate teacher professional development it is important to make teacher concerns and thereby needs explicit. They help teachers to identify and reflect on relations between teacher subject matter knowledge, pedagogical content knowledge and pupil learning.

• 42.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Biology. University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Biology.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Biology. University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Biology. 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. University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.
Prospective teachers' initial conceptions about pupils difficulties in science and mathematics: a potential resource in teacher education2011In: International Journal of Science and Mathematics Education, ISSN 1571-0068, E-ISSN 1573-1774, Vol. 9, no 4, p. 843-866Article in journal (Refereed)

A crucial issue for prospective teachers (PTs) in their education is to develop pedagogical content knowledge (PCK; i.e. how to make a topic comprehensible to pupils). However, research has shown that PTs may have tacit ideas about teaching that act as filters preventing consideration of unfamiliar and discrepant ideas. These ideas must be elicited and taken into consideration in order to be modified. Therefore, PTs’ explicit conceptions may constitute a valuable resource in teacher education. The aim of this study was to investigate PTs’ ideas about pupils’ difficulties, at a topic-specific level, upon beginning the teacher education programme. The “Lesson Preparation Method” was used in four case studies to elicit the conceptions of 32 PTs regarding pupils’ difficulties in four specific science and mathematics topics: plant growth, gases, equations and heat and temperature. In all four topic groups (n = 5 – 11), there was a variety of initial conceptions about pupils’ difficulties, which were categorised into two to five topic-specific categories. Although, initially, PTs may not have expressed any notions about pupils’ difficulties, conceptions were elicited by using the Lesson Preparation Method. Furthermore, we found that the initial ideas corresponded with earlier research on pupils’ difficulties, which could provide a potential resource when creating a scaffolding context in teacher education programmes where PCK development is stimulated.

• 43.
Riga Technical University, Latvia.
Riga Technical University, Latvia. University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.
Accelerated boundary integral method for solving the problem of scattering by multiple multilayered circular cylindrical posts in a rectangular waveguide2017In: 2017 Progress In Electromagnetics Research Symposium - Fall (PIERS - FALL), Electromagnetics Academy , 2017, p. 3263-3271Conference paper (Refereed)

An accelerated boundary integral method for the analysis of scattering of the dominant mode by multiple multi-layered full-height circular cylindrical posts in a rectangular waveguide is presented. After some transformations the surface integral equation is converted to a system of equations whose entries can be evaluated analytically yielding Schlömilch series. Slow convergence of these series is accelerated using the Ewald technique. The proposed method gives results that are comparable in terms of accuracy with other approaches, including those incorporated in solvers HFSS and CST Studio and outperforms them in terms of computation time, especially for posts with large electrical sizes. The efficiency of the proposed method is confirmed by the examples of H-plane cylinder bandpass filters design.

• 44.
Riga Technical University, Latvia.
Riga Technical University, Latvia. University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.
Maximum-sensitivity method for minimizing uncertainty in the measurements of permittivity of a cylindrical dielectric sample in a rectangular waveguide2017In: 2017 Progress In Electromagnetics Research Symposium - Fall (PIERS - FALL), Electromagnetics Academy , 2017, p. 570-578Conference paper (Refereed)

A novel and efficient methodology is proposed for reducing measurement uncertainty associated with the dielectric constant of low-loss materials. The experimental setup consists of two H-plane cylindrical rods located in a rectangular waveguide. One of the rods has the unknown dielectric constant to be extracted, while the other dielectric rod whose dielectric constant is known in advance and serves as a tool for altering the shape of the dependence of the absolute value of the reflection coefficient on the dielectric constant in such a way that the resulting curve exhibits more rapid variations in the range of possible values of the dielectric constant of the post being characterized. The distance between the rods, radius of the optimizing rod, and position offset of the optimizing post serve as optimization parameters to be adjusted so that measurement uncertainty could be reduced which is confirmed by the results of numerical modeling. However, in the vicinity of the resonances, the method cannot be applied. © 2018 Electromagnetics Academy. All rights reserved.

• 45.
Institute of Information Technologies, Moscow Technological University (MIREA), 86 Vernadsky Avenue, Moscow, Russian Federation.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.
Complex waves in a dielectric rod and goubau line2017In: 2017 International Conference on Electromagnetics in Advanced Applications (ICEAA), Institute of Electrical and Electronics Engineers Inc. , 2017, p. 963-966, article id 8065417Conference paper (Refereed)

Existence of complex TM and TE waves in a dielectric waveguide of circular cross section and a Goubau line is proved by analyzing functional properties of the dispersion equations (DEs) using the theory of functions of several complex variables and validating the existence of complex roots of DE. The method proceeds from the analysis of general setting involving hybrid nonsymmetric azimuthally dependent real and complex waves.

• 46. Kuzmina, E.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.
Higher-order surface modes in the Goubau line2014In: Progress in Electromagnetics Research Symposium, Electromagnetics Academy , 2014, p. 2605-2609Conference paper (Refereed)

Existence of the higher-order surface waves of the Goubau line is proved and their structure is analyzed. An efficient computational approach is proposed based on numerical solution to initial-value problems obtained by the parameter differentiation. Several applications and further research directions are discussed.

• 47.
Moscow State Institute of Radio Engineering, Electronics and Automation, Technical University, Moscow, Russian Federation.
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.
Waves in a lossy Goubau line2016In: 2016 10th European Conference on Antennas and Propagation: EuCAP 2016, Institute of Electrical and Electronics Engineers (IEEE), 2016, article id 7481368Conference paper (Refereed)

We consider a homogeneous Goubau line (GL) with a lossy cover. We analyze the dispersion equation for symmetric waves with respect to the problem parameters, find radially symmetric complex waves, and examine their behavior. We show that longitudinal wavenumbers of complex waves are regular perturbations of the propagation constants of eigenwaves of lossless GL, weakly depend om the imaginary part of permittivity of the cover, and that attenuation in GL is low at higher losses.

• 48.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.
D-modules with finite support are semi-simple2014In: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 52, no 2, p. 291-299Article in journal (Refereed)

Let (R,m,k) be a regular local k-algebra satisfying the weak Jacobian criterion, and such that kR/k is an algebraic field extension. Let D be the ring of k-linear differential operators of R. We give an explicit decomposition of the DR-module D/Dm^(n+1) as a direct sum of simple modules, all isomorphic to D/Dm, where certain “Pochhammer” differential operators are used to describe generators of the simple components.

• 49.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.
Purity of Branch and Critical locus2013In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 379, p. 156-178Article in journal (Refereed)

To a dominant morphism $X/S \to Y/S$ of N\oe therian integral $S$-schemes one has the inclusion $C_{X/Y}\subset B_{X/Y}$ of the critical locus in the branch locus of $X/Y$. Starting from the notion of locally complete intersection morphisms, we give conditions on the modules of relative differentials $\Omega_{X/Y}$, $\Omega_{X/S}$, and $\Omega_{Y/S}$ that imply bounds on the codimensions of $C_{X/Y}$ and $B_{X/Y}$. These bounds generalise to a wider class of morphisms the classical purity results for finite morphisms by Zariski-Nagata-Auslander, and Faltings and Grothendieck, and van der Waerden's purity for birational morphisms.

• 50.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.
The Zariski-Lipman conjecture for complete intersections2011In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 337, no 1, p. 169-180Article in journal (Refereed)

The tangential branch locus BX/Yt∩BX/Y is the subset of points in the branch locus where the sheaf of relative vector fields TX/Y fails to be locally free. It was conjectured by Zariski and Lipman that if V/k is a variety over a field k of characteristic 0 and BV/kt=∅, then V/k is smooth (= regular). We prove this conjecture when V/k is a locally complete intersection. We prove also that BV/kt=∅ implies codimXBV/k≤1 in positive characteristic, if V/k is the fibre of a flat morphism satisfying generic smoothness.

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