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• 1.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Concept definition and concept image: in the case of equations2007In: Ämnesdidaktik ur ett nationellt och internationellt perspektiv: rapport från Rikskonferensen i ämnesdidaktik, Kristianstad: Kristianstad University Press , 2007, p. 89-98Conference paper (Refereed)

The purpose of this study is to analyse what kind of conceptions secondary school teachers in mathematics have about equations and how these conceptions are related to the formal definition of the concept of equation. Data was gathered by interviews and questionnaires. Both newly graduated and experienced secondary school teachers were participated in this study. The phenomenographic research approach in order to analyse research outcomes was applied in the investigation. From a phenomenographic analysis of the interview transcripts I found that some patterns could be identified in them and the three qualitatively distinct categories of description about equations could be discerned among the teachers’ conceptions. The research results indicated that equations were apprehended as a procedure, as an answer and as a ‘rewritten’ expression.

• 2.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Definitions and Problem Solving: Problem Solving in Mathematics Education Proceedings of the ProMath meeting June 30- July 2, 2004 in Lahti, Finland2005Report (Other academic)

In this article student teachers’ conceptions of equations are discussed. Data was gathered from 75 student teachers by interviews and questionnaires. Both the quantitative and qualitative research methods were applied in the study. The research results indicate that some specific examples of concepts, which at first appear in teaching, and learning of mathematics are more central to the student teachers than other examples. The former examples are often called prototypes. The investigation shows that students tend to identify the concept equation with one of the prototypical examples and that the concept image develops from one unique prototypical example to include more examples of increasing distance from the prototypical example. The results also indicate that the student teachers’ concept images include erroneous conceptions of the concept equation.

• 3.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Matematikdidaktik för blivande 4-9-lärare2001Report (Other academic)
• 4.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Mathematics teachers' conceptions about concept learning in algebra2007In: Current state of research on mathematical beliefs XII: proceedings of the MAVI-7 [i.e. MAVI-12] Workshop, May 25-28, 2006, Helsinki: University of Helsinki , 2007, , p. 12Chapter in book (Other academic)

In this article I discuss the mathematics teachers’ conceptions about concept learning in algebra from school and university time. The study has a focus on the particular topic of equations. The data was gathered by interviews and questionnaires. Both newly graduated and experienced secondary school teachers were participated in this study. The phenomenographic research method in order to analyse research outcomes was applied in the investigation. The research results indicated that the teachers experience the learning of equations from school and university time on four qualitatively different ways. Learning was apprehended as doing routine problems, as memorizing and reproduction of rules and models, as doing applications and as interaction with other students.

• 5.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Mathematics teachers' conceptions about equations2006Doctoral thesis, monograph (Other scientific)

The aim of this study is to describe and to clarify the mathematics teachers’ subject matter and pedagogical content conceptions about equations. As the basis of these conceptions, the teachers’ experiences of the concept learning of equations from their own school time are described. The early research of conceptions has been concentrated on pupils’ conceptions of the topic as a contrast to scientific conceptions since the middle of the 1970s. Research of teachers’ conceptions of mathematics and mathematics teaching and learning has grown during the last decade. However, in these studies teachers’ conceptions of a specific content area in mathematics have not been investigated. In the theoretical background of the research, different traditions of school mathematics learning and teaching are treated. By using theories of experiential learning, it has been possible to study such learning situations and experiences, which may lead to the development of subjective conceptions of mathematical concepts. In order to understand difficulties concerning the concept formation in mathematics the theory of the concept image and the concept definition as well as the theory based on the duality of mathematical concepts have been studied. The acquired experiences from school time seem to lay the basis of both the teachers’ subject matter and pedagogical content-specific conceptions and decisions. Different components in teacher knowledge base together with current research both in teachers’ subject matter and pedagogical content knowledge are therefore presented at the end of the theoretical framework. By combining different kinds of methods like questionnaires, recorded interviews, videotape recording of six lessons in mathematics and observations the research empirical material was collected. In this investigation, five novice, five expert and 75 student teachers in mathematics participated. The preliminary investigation included 30 student teachers. In the study the phenomenographic approach is used in order to reveal differences between the teachers’ conceptions and experiences about equations. The research results indicate that equations are not apprehended as complete, static objects. Conceptions about equations reveal that equations are closely related to the symbols x and y and solving procedures. The teachers’ experiences of learning and teaching of mathematics may have formed their conceptions. The conceptions about equations seem to be based on the teachers’ experiences in arithmetic and their first impressions of learning the process of solving equations. The teachers apprehend equation teaching as a study of procedures rather than as a study of central ideas and concepts of algebra. Both aspects are however equally important at compulsory school, since the teaching of algebra should develop pupils’ ability both to use and to understand the basic algebraic concepts. Some of the teachers do not have a clear conception what the pupils should attain in algebra at compulsory school according to the specific goals in Swedish mathematics curriculum. The research results further show that both the expert and the novice teachers have various apprehensions of the pupils’ difficulties concerning equations.

• 6.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Ongoing research in mathematics education in school context2009In: Perspectives on mathematical knowledge: proceedings of MADIF 6 : the 6th Swedish Mathematics Education Research Seminar, Stockholm, January 29-30, 2008 / [ed] Christer Bergsten, Barbro Grevholm, Thomas Lingefjärd, Linköping: LiU-Tryck , 2009, p. 116-117Conference paper (Refereed)
• 7.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Secondary school teachers’ conceptions about algebra teaching2005Conference paper (Refereed)
• 8.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Secondary school teachers’ pedagogical content knowledge2004In: Proceedings of the Annual Meeting of the International Group for the Psychology of Mathematics Education (PME). Volume 1 / [ed] H. M.Jonsen & A. B. Fuglestad, Norge: University College , 2004Conference paper (Refereed)
• 9.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Teachers’ image of the "equation" concept2003In: CERME 3: Third Conference of the European Society for Research in Mathematics education, 2003Conference paper (Refereed)

The purpose of this study is to analyse what kind of conceptions secondaryteachers in mathematics have of the equation concept and what kind of experiencesgave rise to their own concept learning. Ten secondary school teachers ofmathematics participated in the study. Data was gathered by interviews andquestionnaires. The ‘phenomenographic’ research method was applied ininterpreting the results of the investigation. The results indicate that the teachers’conceptions of the equation concept differ from the formal definition of the equationconcept. At school they spent most of their time developing procedural skills insteadof mathematical understanding.

• 10.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Teachers’ knowledge about the equation concept2002In: International group for the psychology of mathematics education PME 26, 2002, p. 1-335Conference paper (Other academic)
• 11.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Uppfattningar hos lärare av ekvationsbegreppet2003Report (Other academic)

Syftet med min studie är att beskriva hur matematiklärare uppfattar begreppet ekvation. Min målsättning är också att undersöka lärarnas erfarenheter av ekvationsinlärning från grundskolan till universitetsnivån. Tio lärare från grundskolans högstadium har deltagit i undersökningen. Fem är nyutexaminerade lärare med mindre än ett års yrkeserfarenhet. Data i undersökningen samlades in genom enkäter och intervjuer. Undersökningsresultat analyserades genom fenomenografisk metod. Resultatet pekar på att lärarnas uppfattningar av ekvationsbegreppet avviker från den formella begreppsdefinitionen. De känner osäkerhet inför matematiska symboler, bokstavsuttryck och lösningsprocedurer. Deras skolerfarenheter visar att de har använt största delen av tiden till att utveckla algoritmiska färdigheter istället för matematisk förståelse.

• 12.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Area och omkrets i förskoleklass2008In: Nämnaren, ISSN 0348-2723, Vol. 35, no 1, p. 34-36Article in journal (Other academic)

Här beskrivs ett antal undervisningstillfällen kring begreppen area och omkrets i en förskoleklass. I planeringen har lärarna utgått från att barnen ska samarbeta, kommunicera med varandra, använda flera sinnen, reflektera över sitt och andras tänkande samt dokumentera och sätta egna ord på det de varit med om.

• 13.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Kellner, EvaUniversity of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för naturvetenskap.
Conceptions and beliefs in mathematics and science education including MAVI XIII2008Conference proceedings (editor) (Other academic)
• 14.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
University of Eastern Finland, Finland.
Is there always truth in equation?2009In: Nordic Research in Mathematics Education. Proceedings from NORMA 08 / [ed] C. Winsløw, Rotterdam: Sense Publishers , 2009, p. 143-149Conference paper (Refereed)

In this study we analyse conceptions that prospective teachers in mathematics have about equations and how these conceptions are related to the truth value (of mathematical statements). Our phenomenographic research reveals that there exists a large variety of fundamentally different conceptions about equations and that in students’ actual concept image of equation it is commonly assumed that a mathematical expression must possess the truth value ‘true’ in order to be an equation in spite of the fact that any consideration related to the truth value only rarely appears in concept definitions given by students.

• 15.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Is there equality in equation?2007In: European research in mathematics education 5: proceedings of the Fifth Congress of the European Society for Research in Mathematics education : Larnaca, Cyprus 22- 26 February 2007, 2007Conference paper (Refereed)
• 16.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
University of Eastern Finland, Finland.
On the equivalence relation in the students' concept image of equation2009In: Perspectives on Mathematical Knowledge: Proceedings of MADIF6, The 6th Swedish Mathematics Education Research Seminar Stockholm, January 29-30, 2008 / [ed] C. Bergsten, B. Grevholm & T. Lingefjärd, Linköping: LiU-Tryck , 2009, p. 118-119Conference paper (Refereed)
• 17.
University of Gävle, Department of Mathematics, Natural and Computer Sciences. University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
VARIATIONSTEORI I PRAKTIK: BESKRIVNING AV ÖVNINGSUNDERVISNING I FÖRSKOLEKLASSEN2009Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis

Examensarbete beskriver hur jag inom fenomenografins ramar provar Learning Study somkompetensutvecklingsverktyg. Det är ett undervisningsförsök i förskoleklassensmatematikundervisning. Med variationsteori som grund jag försöker ta reda på kritiska aspekter för att göra lärande möjligt. Genom att förintervjua barnen, utföra planerade lektioner med det avsedda lärandeobjektet och återintervjua om det erfarna lärandeobjektet, är min avsikt att komma till värdering om det iscensatta lärandeobjektet under videofilmad lektion. Resultat visar att Learning study är ett utmärkt instrument för lärares egen lärande och gör det möjligt att finna och målmedvetet fullgöra kritiska villkor för lärande.

• 18.
Department of Mathematics, Stockholm University, Stockholm, Sweden.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Geometric interplay between function subspaces and their rings of differential operators2006In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 359, no 5, p. 2075-2108Article in journal (Refereed)

We study, in the setting of algebraic varieties, finite-dimensional spaces of functions $V$ that are invariant under a ring $\Dc^V$ of differential operators, and give conditions under which $\Dc^V$ acts irreducibly. We show how this problem, originally formulated in physics \cite{kamran-Milson-Olver:invariant,turbiner:bochner}, is related to the study of principal parts bundles and Weierstrass points \cite{EGA4,laksov-thorup}, including a detailed study of Taylor expansions. Under some conditions it is possible to obtain $V$ and $\Dc^V$ as global sections of a line bundle and its ring of differential operators. We show that several of the published examples of $\Dc^V$ are of this type, and that there are many more - in particular, arising from toric varieties.

• 19. Danielson, Mats
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Structure information in decision trees and similar formalisms2007In: Proceedings of the Twentieth International Florida Artificial Intelligence Research Society Conference, Menlo Park, CA: AAAI Press , 2007, p. 62-67Conference paper (Refereed)

In attempting to address real-life decision problems, where uncertainty about input data prevails, some kind of representation of imprecise information is important and several have been proposed over the years. In particular, first-order representations of imprecision, such as sets of probability measures, upper and lower probabilities, and interval probabilities and utilities of various kinds, have been suggested for enabling a better representation of the input sentences. A common problem is, however, that pure interval analyses in many cases cannot discriminate sufficiently between the various strategies under consideration, which, needless to say, is a substantial problem in real-life decision making in agents as well as decision support tools. This is one reason prohibiting a more wide-spread use. In this article we demonstrate that in many situations, the discrimination can be made much clearer by using information inherent in the decision structure. It is discussed using

second-order probabilities which, even when they are implicit, add information when handling aggregations of imprecise representations, as is the case in decision trees and probabilistic networks. The important conclusion is that since structure carries information, the structure of the decision problem influences evaluations of all interval representations and is quantifiable.

• 20.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.

I denna uppsats står vardagsrelatering i fokus och här undersöks hur fyra läroböcker i

matematik, avsedda för elever i årskurs nio, gör kopplingar till elevers vardag. Syftet med

uppsatsen är att genom analys av läroböcker i matematik bringa klarhet i om och hur dessa

kopplar textuppgifter till elevens vardag. Som metod har en komparativ textanalys använts för

att urskilja de textuppgifter som har vardagsrelatering för en elev. Resultatet visar att ungefär

knyter an till elevens vardag och att dessa relaterar till eleven genom ett av följande områden,

fritid, vardag, skola eller handel. Läroböckerna utgörs av en acceptabel mängd textuppgifter

med vardagsanknytning för elever men det finns gott om utrymme för ännu fler. De

vardagsrelateringar som görs i läromedlen kan förbättras både i fråga om variation och vilka

räknesätt som används vid lösning av uppgifterna.

• 21.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Varför blir så många elever underkända i Matematik A?2009Independent thesis Basic level (professional degree), 20 credits / 30 HE creditsStudent thesis

Syftet med studien är att se varför så många elever blir underkända i Matematik A på gymnasiet. Vidare är syftet att ta reda på elevers och lärares syn på vad detta problem kan bero på.

Studien gjordes genom en enkätundersökning med elever och lärare och bearbetades statistisk som presenterades i tabeller och diagram.

Resultatet i studien visar att ca 20 % av eleverna får IG i Matematik A på gymnasiet. Enligt eleverna beror det på flera orsaker som t ex bristande arbetsro och dåliga lärare medan lärarna tycker att eleverna saknar förkunskaper och får för bra betyg från grundskolan.

En viktig slutsats är att det inte finns en entydig förklaring till varför eleverna misslyckas. För att komma tillrätta med problemet föreslås ett ökat samarbete mellan grundskolan och gymnasiet.

• 22.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för naturvetenskap.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för naturvetenskap. University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik. University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för naturvetenskap.
Lärarstudenters uppfattning om elevers begrepp om gas2007In: Resultatdialog 2007: forskning inom utbildningsvetenskap, Stockholm: Stockholm , 2007, p. 52-58Chapter in book (Other (popular science, discussion, etc.))
• 23.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för naturvetenskap.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för naturvetenskap. University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik. University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för naturvetenskap.
Lärarstudenters uppfattning om elevers begrepp om gas2007In: Ämnesdidaktik ur ett nationellt och internationellt perspektiv: rapport från Rikskonferensen i ämnesdidaktik 2006, Kristianstad: Kristianstad University Press , 2007, p. 133-142Chapter in book (Other (popular science, discussion, etc.))
• 24.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för naturvetenskap.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för naturvetenskap. University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik. University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för naturvetenskap. Göteborg University, Göteborg, Sweden.
Prospective teachers’ initial conceptions about pupils’ understanding of science and mathematics2008In: European Journal of Teacher Education, ISSN 0261-9768, E-ISSN 1469-5928, Vol. 31, no 3, p. 257-278Article in journal (Refereed)

Earlier reports have shown that prospective teachers' conceptions about teaching science to a high degree are resistant and do not change substantially during the teacher-training programme. In our investigation we elucidate the prospective teachers' initial conceptions about pupils' understanding of science and mathematics. We applied 'The Lesson Preparation Method' and used a phenomenographic approach in order to reveal the range of conceptions that the prospective teachers hold. A third of the prospective teachers did not consider pupils' conceptions when planning lessons. The rest of the 32 participants expressed awareness; some of the prospective teachers even referred to subject-specific teaching experience. Also regarding the prospective teachers' conceptions about pupils' knowledge and beliefs, as well as about pupils' difficulties, there was a significant diversity. By raising these issues about pedagogical content knowledge the prospective teachers' conceptions can be extended and developed during the education.

• 25.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Långdivision ur tiden2009Independent thesis Basic level (university diploma), 10 credits / 15 HE creditsStudent thesis

Syftet med uppsatsen är att undersöka vilka de bakomliggande orsakerna kan vara till att en algoritm för långdivision inte längre i någon större omfattning lärs ut i skolan och även hur långt avvecklingen av divisionsalgoritmen i det allmänna medvetandet har kommit.

Uppsatsen bygger dels på två enkäter riktade till grupperna lärare och allmänhet, ett divisionstest med elever i år ett på gymnasiet samt på litteraturstudier och tre telefonintervjuer med läromedelsförfattare. Vad som visat sig är att det är få människor under 40 år och med en bakgrund i den svenska grundskolan som behärskar en potent divisionsalgoritm. Det kan konstateras att kunskapen att utföra en standardalgoritm för division med flersiffrig nämnare i alla fall i denna undersökning inte längre finns bland eleverna i den svenska skolan och att den situationen bär skolöverstyrelsens och lärarkårens välsignelse. Det är dock en öppen fråga om utvecklingen dit föregåtts av genomtänkta beslut.

• 26.
Matematiska Institutionen, Umeå Universitet, Umeå, Sweden.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Orthogonal Latin Rectangles2008In: Combinatorics, probability & computing, ISSN 0963-5483, E-ISSN 1469-2163, Vol. 17, no 4, p. 519-536Article in journal (Refereed)

We use a greedy probabilistic method to prove that, for every ε > 0, every m × n Latin rectangle on n symbols has an orthogonal mate, where m = (1 − ε)n. That is, we show the existence of a second Latin rectangle such that no pair of the mn cells receives the same pair of symbols in the two rectangles.

• 27.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Department of Mathematics, Rutgers University, Piscataway, NJ, United States. Department of Mathematics, Rutgers University, Piscataway, NJ, United States.
Factors in random graphs2008In: Random structures & algorithms (Print), ISSN 1042-9832, E-ISSN 1098-2418, Vol. 33, no 1, p. 1-28Article in journal (Refereed)

Let H be a fixed graph on v vertices. For an n-vertex graph G with n divisible by v, an H-factor of G is a collection of n/v copies of H whose vertex sets partition V (G).

In this work, we consider the threshold thH(n) of the property that an Erds-Rényi random graph (on n points) contains an H-factor. Our results determine thH(n) for all strictly balanced H.

The method here extends with no difficulty to hypergraphs. As a corollary, we obtain the threshold for a perfect matching in random k-uniform hypergraph, solving the well-known Shamir's problem.

• 28.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Department of Mathematics, Uppsala University, Uppsala, Sweden.
Square summability of variations and convergence of the transfer operator2008In: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417, Vol. 28, no 4, p. 1145-1151Article in journal (Refereed)

In this paper we study the one-sided shift operator on a state space defined by a finite alphabet. Using a scheme developed by Walters [P. Walters. Trans. Amer. Math. Soc. 353(1) (2001), 327-347], we prove that the sequence of iterates of the transfer operator converges under square summability of variations of the g-function, a condition which gave uniqueness of a g-measure in our earlier work [A. Johansson and A. Öberg. Math. Res. Lett. 10(5-6) (2003), 587-601]. We also prove uniqueness of the so-called G-measures, introduced by Brown and Dooley [G. Brown and A. H. Dooley. Ergod. Th. & Dynam. Sys. 11 (1991), 279-307], under square summability of variations.

• 29.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Square summability of variations of g-functions and uniqueness of g-measures2003In: Mathematical Research Letters, ISSN 1073-2780, E-ISSN 1945-001X, Vol. 10, no 5, p. 587-601Article in journal (Refereed)

We prove uniqueness of $g$-measures for $g$-functions satisfying quadratic summability of variations. Our result is in contrast to the situation of, \eg, the one-dimensional Ising model with long-range interactions, since $\ell_1$-summability of variations is required for general potentials. We illustrate this difference with some examples. To prove our main result we use a product martingale argument. We also give conditions for uniqueness of general $G$-measures, \ie, the case for general potentials, based on our investigation of the probabilistic case involving $g$-functions.

• 30.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Department of Mathematics, Uppsala University, Uppsala, Sweden. Mathematics Institute, University of Warwick, Coventry, United Kingdom.
Countable state shifts and uniqueness of g-measures2007In: American Journal of Mathematics, ISSN 0002-9327, E-ISSN 1080-6377, Vol. 129, no 6, p. 1501-1511Article in journal (Refereed)

In this paper we present a new approach to studying $g$-measures which is based upon local absolute continuity. We extend an earlier result that square summability of variations of $g$ ensures uniqueness of $g$-measures. The first extension is to the case of countably many symbols. The second extension is to some cases where $g \geq 0$, relaxing the earlier requirement that $\inf g>0$.

• 31.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för naturvetenskap.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för naturvetenskap. University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik. University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för naturvetenskap.
Prospective teachers' initial conceptions of pupils' difficulties: holistic view of concepts in science and matematics2007In: ESERA 2007: European Science Education Research Association International Conference in Malmö, Sweden, August 21-25, 2007, 2007Conference paper (Refereed)
• 32.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Liftable derivations of generically separably algebraic morphisms of schemes2009In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 361, no 1, p. 495-523Article in journal (Refereed)

We consider dominant, generically algebraic (e. g. generically finite), and tamely ramified (if the characteristic is positive) morphisms pi : X/S -> Y/S of S-schemes, where Y, S are Noetherian and integral and X is a Krull scheme (e. g. normal Noetherian), and study the sheaf of tangent vector fields on Y that lift to tangent vector fields on X. We give an easily computable description of these vector fields using valuations along the critical locus. We apply this to answer the question when the liftable derivations can be defined by a tangency condition along the discriminant. In particular, if p is a blow-up of a coherent ideal I, we show that tangent vector fields that preserve the Ratliff-Rush ideal (equals [I(n+1) : I(n)] for high n) associated to I are liftable, and that all liftable tangent vector fields preserve the integral closure of I. We also generalise in positive characteristic Seidenberg's theorem that all tangent vector fields can be lifted to the normalisation, assuming tame ramification.

• 33.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Preservation of defect sub-schemes by the action of the tangent sheaf2005In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 203, no 1-3, p. 166-188Article in journal (Refereed)

Let X/S be a noetherian scheme with a coherent -module M, and TX/S be the relative tangent sheaf acting on M. We give constructive proofs that sub-schemes Y, with defining ideal IY, of points xX where or Mx is “bad”, are preserved by TX/S, making certain assumptions on X/S. Here bad means one of the following: is not normal; has high regularity defect; does not satisfy Serre's condition (Rn); has high complete intersection defect; is not Gorenstein; does not satisfy (Tn); does not satisfy (Gn); is not n-Gorenstein; Mx is not free; Mx has high Cohen–Macaulay defect; Mx does not satisfy Serre's condition (Sn); Mx has high type. Kodaira–Spencer kernels for syzygies are described, and we give a general form of the assertion that M is locally free in certain cases if it can be acted upon by TX/S.

• 34.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
MATERMINO: utformning och prövning av ett begreppsförklaringsspel i geometri för grundskolans senare år2009Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis

Syftet med denna studie var att undersöka hur ett pedagogiskt spel för grundskolans senare år som övar geometriska begrepp kan utformas samt huruvida det kan fungera som ett givande komplement till befintliga material för geometrisk begreppsinlärning. Spelet, som kom att kallas MATERMINO, testades av 33 elever (15 flickor och 18 pojkar) i skolår 6. Dessa elever samt tre lärare för grundskolans senare år bedömde sedan spelet; de förstnämnda i enkäter och de senare i intervjuer. MATERMINO bedömdes främst passa elever i skolår 6-8. Vidare visade studien att MATERMINO var omtyckt och roligt och att spelet tycktes leda till inlärning gällande geometriska begrepp. Eleverna presterade nämligen signifikant bättre på ett test direkt efter att ha spelat MATERMINO jämfört med ett identiskt test de hade gjort precis innan respektive spelomgång (medelförbättringen var 14 %). Slutsatsen blev därmed att MATERMINO kan fylla en viktig funktion som kompletterande undervisningsmaterial för begreppsinlärning i skolår 6-8.

• 35.
Department of Neuroscience, Rehabilitation Medicine, Uppsala University, Uppsala, Sweden; Centre for Research and Development, Uppsala University/County Council of Gavleborg, Uppsala, Sweden.
Department of Medicine, Clinical Epidemiology Unit, Karolinska University Hospital, Stockholm, Sweden. Centre for Research and Development, Uppsala University/County Council of Gavleborg, Uppsala, Sweden. University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik. Department of Neuroscience, Rehabilitation Medicine, Uppsala University, Uppsala, Sweden.
Prevalence and structure of symptoms at 3 months after mild traumatic brain injury in a national cohort2009In: Brain Injury, ISSN 0269-9052, E-ISSN 1362-301X, Vol. 23, no 3, p. 213-219Article in journal (Refereed)

Objectives: To describe symptom prevalence and structure after mild traumatic brain injury (MTBI) in a population-based cohort. Methods: Symptoms data were collected at 3 months post-MTBI by use of the Rivermead Post-concussion Symptoms Questionnaire (RPQ) at follow-up of 2602 patients attending 39 Swedish hospitals. Spearmans rank correlation analysis was used to explore correlations between symptoms and structural equation modelling (SEM) was performed by use of several fit indices to explore if data were compatible with one or more factors. Results: Questionnaires were received from 2523 (97%) patients with a mean age of 31 years (median 22, range 6-96). A majority of the respondents (56%) reported no remaining injury related symptoms, 24% reported three or more symptoms and 10% reported seven or more symptoms. All symptoms exhibited strong positive inter-relations and SEM provided strong support for a single or two factor solution. Fit indices were only slightly weaker for three and four factor solutions. Conclusions: A significant minority of patients reported multiple symptoms to persist at 3 months after MTBI. The observed structure of symptoms according to RPQ demonstrates a common factor for all symptoms, but also sub-groups of symptoms as previously suggested.

• 36.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Tillvägagångssätt vid lösning av algebraiska problem.2009Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis

Denna undersökning är en fallstudie som syftar till att ta reda på hur elever går tillväga när de löser ett algebraiskt problem. Syftet är att sätta sig in i elevernas tankar och sätt att lösa problem och genom ökad förståelse kunna förklara för dem på ett sätt de förstår och kan relatera till.

Metoden som använts är enkät och intervju, och studiegruppen är en klass i årskurs nio.

Eleverna är inte så vana vid att kombinera olika räknesätt i ett och samma tal. De har lättare att se algebraiska uttryck som uttryck för substantiv eller fasta siffror än de har för att se uttrycket som en variabel matematisk formel. De är heller inte vana vid att med ord beskriva vad de gör, därför löser de ofta talen rutinmässigt utan att reflektera över hur.

Det är viktigt att det ingår varierad problemlösning i undervisningen så att eleverna övar sig på både praktisk matematik samt olika matematiska områden. Eleverna tycker det är roligt att göra annat än enbart räkna i läroboken. För att ta uppmuntra deras intresse för matematik och få dem att utvecklas är det därför viktigt med olika sorters matematik..

• 37.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Samlingens Matematik2009Independent thesis Basic level (professional degree), 10 credits / 15 HE creditsStudent thesis

Syftet med den här uppsatsen är att undersöka vilka delar av matematiken som vanligtvis ingår i de olika delarna av samlingen på förskolan. Den tittar även närmare på hur medvetna pedagogerna är om ämnet matematik i samlingen samt hur det fungerar som inlärningstillfälle för barnen. Två avdelningar på olika förskolor har medverkat i undersökningen. Metoden som används för att få fram resultatet är enkätundersökningar bland pedagoger samt videoobservationer av samlingen. Pedagogerna hade olika kunskaper om ämnet matematik och endast hälften hänvisade till läroplanen på frågan om de hade en medveten målsättning inom ämnet matematik i samlingen. Matematiken fanns genomgående i samlingen och om det används på rätt sätt så kan det vara ett bra tillfälle för barnens utveckling och lärande.

• 38.
Chydenius Institut, Jyväskylä Universitet, Finland.
Chydenius Institut, Jyväskylä Universitet, Finland. University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Finnish and Swedish children’s spontaneous mathematical  expressions inspired by pictures2009In: The Proceedings of the conference MAVI-15. Ongoing research on beliefs in mathematics education. / [ed] F. Furinghetti and M. Morselli, 2009, p. 131-137Conference paper (Refereed)

Every child in Finland and Sweden has possibility to get pre-school education, in separate pre-school classes, and after that nine years compulsory primary school education. Concerning their math skills development, it is important that children have opportunities to express their own marks and meanings. In this paper we study six to eight year old Finnish (n=100) and Swedish (n=100) children’s spontaneous expressions measured by a pictorial test. In our cross-cultural comparative study we found some similarities and some differences. Both Finnish and Swedish children expressed most numbers and exercises which they use to practice in their early school years. Swedish children expressed significantly more numbers and exercises than the Finnish ones, whereas Finnish children expressed more word problems and mental models. Our cross-cultural interpretation is mainly based on different emphasis in Finnish and Swedish core curriculums. Finnish children follow an early school education that emphasises more learning skills and social development than academic skills learning. Swedish curriculum emphasises more subject-specific knowledge in mathematics.

• 39.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Dyskalkyli: en litteraturstudie2009Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis

Syftet med detta arbete är att få veta mer om dyskalkyli. Jag vill veta vad dyskalkyli är, hur det visar sig, hur lärare kan arbeta med dessa elever och om det finns någon koppling mellan dyslexi och dyskalkyli.

För att få svar på dessa frågor har jag gjort en litteraturstudie.

Dyskalkyli betyder specifika matematiksvårigheter. Det är inte ovanligt att dyskalkyliker får svårt med exempelvis planering och tidsuppfattning. Det är viktigt med en noggrann utvärdering av eleven för att den ska få rätt hjälp. Dessa elever behöver en bra struktur på undervisningen som ska ligga på en nivå som passar eleven. Det är viktigt med tillgång till laborativt material. När det gäller kopplingen mellan diagnoserna, har elever med dyslexi en sämre fonologisk förmåga än andra. Om barnet har ett svagt arbetsminne kommer den inte ihåg texten som den nyss läst. I matematiken märks detta genom att eleven har problem med huvudräkning.

• 40.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Distribution of expected utility in second-order decision analysis2007Licentiate thesis, comprehensive summary (Other academic)

In decision analysis maximising the expected utility is an often used approach in choosing the optimal alternative. But when probabilities and utilities are vague or imprecise expected utility is fraught with complications. Studying second-order effects on decision analysis casts light on the importance of the structure of decision problems, pointing out some pitfalls in decision making and suggesting an easy to implement and easy to understand method of comparing decision alternatives. The topic of this thesis is such second-order effects of decision analysis, particularly with regards to expected utility and interval-bound probabilities. Explicit expressions for the second-order distributions inherent in interval-bound probabilities in general and likewise for distributions of expected utility for small decision problems are produced. By investigating these distributions the phenomenon of warping, that is concentration of belief, is studied.

• 41.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Some second order effects on interval based probabilities2006In: Proceedings of the Nineteenth International Florida Artificial Intelligence Research Society Conference, Menlo Park, CA: AAAI Press , 2006, p. 848-853Conference paper (Refereed)

In real-life decision analysis, the probabilities and values of consequences are in general vague and imprecise. One way to model imprecise probabilities is to represent a probability with the interval between the lowest possible and the highest possible probability, respectively. However, there are disadvantages with this approach, one being that when an event has several possible outcomes, the distributions of belief in the different probabilities

are heavily concentrated to their centers of mass, meaning that much of the information of the original intervals are lost. Representing an imprecise probability with the distribution’s center of mass therefore in practice gives much the same result as using an interval, but a single number instead of an interval is computationally easier and avoids problems such as overlapping intervals. Using this, we demonstrate why second-order

calculations can add information when handling imprecise representations, as is the case of decision trees or probabilistic networks. We suggest a measure of belief density for such intervals. We also demonstrate important properties when operating on general distributions. The results herein apply also to approaches which do not explicitly deal with second-order distributions, instead

using only first-order concepts such as upper and lower bounds.

• 42.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Deptartment of Computer and Systems Sciences, Stockholm University and TH, Kista, Sweden. Deptartment of Computer and Systems Sciences, Stockholm University and TH, Kista, Sweden.
Warp effects on calculating interval probabilities2009In: International Journal of Approximate Reasoning, ISSN 0888-613X, E-ISSN 1873-4731, Vol. 50, no 9, p. 1360-1368Article in journal (Refereed)

In real-life decision analysis, the probabilities and utilities of consequences are in general vague and imprecise. One way to model imprecise probabilities is to represent a probability with the interval between the lowest possible and the highest possible probability, respectively. However, there are disadvantages with this approach; one being that when an event has several possible outcomes, the distributions of belief in the different probabilities are heavily concentrated toward their centres of mass, meaning that much of the information of the original intervals are lost. Representing an imprecise probability with the distribution’s centre of mass therefore in practice gives much the same result as using an interval, but a single number instead of an interval is computationally easier and avoids problems such as overlapping intervals. We demonstrate why second-order calculations add information when handling imprecise representations, as is the case of decision trees or probabilistic networks. We suggest a measure of belief density for such intervals. We also discuss properties applicable to general distributions. The results herein apply also to approaches which do not explicitly deal with second-order distributions, instead using only first-order concepts such as upper and lower bounds.

• 43.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Shifted Dirichlet Distributions as Second-Order Probability Distributions that Factors into Marginals2009In: Proceedings of the Sixth International Symposium on Imprecise Probability: Theories and Applications, 2009, p. 405-410Conference paper (Refereed)

In classic decision theory it is assumed that a decision-maker can assign precise numerical values corresponding to the true value of each consequence, as well as precise numerical probabilities for their occurrences. In attempting to address real-life problems, where uncertainty in the input data prevails, some kind of representation of imprecise information is important. Second-order distributions, probability distributions over probabilities, is one way to achieve such a representation. However, it is hard to intuitively understand statements in a multi-dimensional space and user statements must be provided more locally. But the information-theoretic interplay between joint and marginal distributions may give rise to unwanted effects on the global level. We consider this problem in a setting of second-order probability distributions and find a family of distributions that normalised over the probability simplex equals its own product of marginals. For such distributions, there is no flow of information between the joint distributions and the marginal distributions other than the trivial fact that the variables belong to the probability simplex. marginal distributions may give rise to unwanted effects on the global level.

• 44.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Some properties of aggregated distributions over expected values2008In: MICAI 2008: Advances in Artificial Intelligence: 7th Mexican International Conference on Artificial Intelligence, Atizapán de Zaragoza, Mexico, October 27-31, 2008, Proceedings, Berlin, Heidelberg: Springer , 2008, p. 699-709Conference paper (Refereed)

Software agents and humans alike face severe difficulties in making decisions in uncertain contexts. One approach is to formalise the decision situation by means of decision theory, i.e. probabilities and utilities leading to the principle of maximising the expected utility. Expected utility is here considered as a stochastic variable; under the assumption that all utility values are equally likely, and that each vector of probability values is equally likely, the probability distribution of expected utility is calculated for two, three, and four possible outcomes. The effect of these probability distributions concentrating around the middle value is explored and its significance for making decisions.

• 45.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för naturvetenskap.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för naturvetenskap. University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för naturvetenskap. University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Developing transformative pedagogical content knowledge in science and mathematics teacher education2005Report (Other (popular science, discussion, etc.))
• 46.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Different views:  Teacher and engineering students on the concept of function2008In: Perspectives on Mathematical Knowledge: Proceedings of MADIF 6 / [ed] Christer Bergsten, Barbro Grevholm, Thomas Lingefjärd, Stockholm: SMDF , 2008, p. 97-106Conference paper (Other academic)

This study analyses what kind of conceptions teacher students and engineering students have about the function concept, and how these conceptions differ between the two groups. The study was conducted through questionnaires, and 34 students at a Swedish university participated. The function concepts of the students have been classified according to modified versions of models presented by Vinner & Dreyfus, Sfard and DeMarois & Tall. The study shows that the students primarily have operational concepts of function, with only a couple of students having a structural function concept. The study also shows distinct differences between prospective compulsory school teachers and engineering students, where the former have less developed functional concepts.

• 47.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
The many faces of function - students' understanding of the function concept2007In: Conceptions and Beliefs in Mathematics and Science Education including MAVI XIII / [ed] Iiris Attorps, Eva Kellner, Gävle: University of Gävle , 2007, p. 87-97Conference paper (Other academic)

This study analyses what kind of conceptions teacher students and engineering students have about the function concept, and how these conceptions differ between the two groups. The study was conducted through questionnaires, and 34 students at a Swedish university participated. The function concepts of the students have been classified according to modified versions of models presented by Vinner & Dreyfus, Sfard and DeMarois & Tall. The result of the study shows that the students primarily have operational concepts of function, with only a couple of students having anything resembling a structural function concept. The study also shows distinct differences between prospective compulsory school teachers and engineering students, where the former have less developed functional concepts.

• 48.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Analyticity theorems for parameter-dependent currents1991In: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 69, no 2, p. 179-198Article in journal (Refereed)

Plurisubharmonic functions of two groups of complex variables (x1,...,x(n)) and (a1,...,a(m)) are considered; their partial functions are defined by f(a)(x) = f(x, a). We discuss analyticity theorems for the level sets associated to Lelong numbers of the parameter-dependent currents dd(c)f(a).

• 49.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
The finite part of singular integrals in several complex variables1993In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 337, no 2, p. 771-793Article in journal (Refereed)

A divergent integral can sometimes be handled by assigning to it as its value the finite part in the sense of Hadamard. This is done by expanding the integral over the complement of a symmetric neighborhood of a singularity in powers of the radius, and throwing away the negative powers. In this paper the finite part of a singular integral of Cauchy type is defined, and this is then used to describe the boundary behavior of derivatives of a Cauchy-type integral. The finite part of a singular integral of Bochner-Martinelli type is studied, and an extension of the Plemelj jump formulas is shown to hold.

• 50.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Rehabilitering av långtidssjukskrivna: En studie av fyra projekt i Gävleborgs län 1992-19951999Report (Other academic)
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