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  • 101.
    Berggren, Jacob
    et al.
    Linnéuniversitetet, Fakulteten för teknik (FTK), Institutionen för matematik (MA).
    Landhammar, Anna
    Linnéuniversitetet, Fakulteten för teknik (FTK), Institutionen för matematik (MA).
    Johansson, Emma
    Linnéuniversitetet, Fakulteten för teknik (FTK), Institutionen för matematik (MA).
    ”Man kan ju skriva också” -Kommunikation, inte bara en muntlig förmåga: - En analys av kommunikationsförmågan inom matematiken2018Independent thesis Advanced level (professional degree), 240 hpOppgave
    Abstract [sv]

    Som ett led i att förbättra kvaliteten på matematikundervisning fokuserar den senaste versionen av läroplanen på förmågor som elever ska ges möjlighet att utveckla. Kommunikationsförmågan är en av dessa som nämns i styrdokumenten och som denna studie kommer att behandla. Våra erfarenheter, likväl som Skolinspektionen, har visat att alla lärare inte planerar och undervisar enligt styrdokumenten. Denna studie kommer därför att undersöka hur lärare definierar kommunikationsförmågan i matematikämnet samt vilka metoder och arbetssätt de låter eleverna använda för att utveckla denna förmåga.

    Genom intervjuer och observationer av matematiklektioner ska studien ge en bild av hur lärare i årskurs 4-6 tänker och arbetar för att utveckla just kommunikationsförmågan.

    Resultatet av detta visar att de intervjuade lärarnas definition av kommunikationsförmågan är relativt likvärdiga, det handlar om att prata matematik. Däremot skiljer det sig hur de väljer att arbeta med att låta eleverna utveckla förmågan. De metoder som väljs för att fokusera på just kommunikationsförmågan är främst problemlösning i par eller grupp, men också genomgångar och gemensamma diskussioner. Resultatet i denna studie visar även att få av lärarna nämner eller tillämpar kommunikation via skrift. Inte förrän de har tänkt igenom och analyserat förmågan på ett djupare plan beskriver de att även en skriftlig version är viktig att beakta.

  • 102.
    Bergling, Fredrik
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Tillämpad matematik och statistik.
    Applications of Quantiles In Portfolio Management2018Independent thesis Advanced level (degree of Master (Two Years)), 20 poäng / 30 hpOppgave
  • 103.
    Bergman Ärlebäck, Jonas
    et al.
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten.
    Doerr, Helen M.
    Syracuse University.
    At the core of modelling: Connecting, coordinating and integrating models2015Inngår i: Proceedings of the 9th Congress of European Research in Mathematics Education / [ed] K. Krainer & N. Vondrová, 2015, s. 802-808Konferansepaper (Fagfellevurdert)
    Abstract [en]

    This theoretical paper introduces the notions of connecting, coordinating and integrating models to analyse and reflect on how models are created and developed. We define, discuss and apply these constructs to some theoretical perspectives in the present modelling discourse. We draw on an example from a model application activity within a model development sequence to illustrate these constructs. Our hope is to spark a discussion that will enhance our understanding about the nature of mathematical modelling and the teaching and learning of, and through, modelling.

  • 104.
    Bergner, Julia
    et al.
    University of California, Riverside, USA.
    Hackney, Philip
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen. University of California, Riverside, USA.
    Group actions on Segal operads2014Inngår i: Israel Journal of Mathematics, ISSN 0021-2172, E-ISSN 1565-8511, Vol. 202, nr 1, s. 423-460Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We give a Quillen equivalence between model structures for simplicial operads, described via the theory of operads, and Segal operads, thought of as certain reduced dendroidal spaces. We then extend this result to give a Quillen equivalence between the model structures for simplicial operads equipped with a group action and the corresponding Segal operads.

  • 105.
    Bergner, Julia
    et al.
    University of California, Riverside, USA.
    Hackney, Philip
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen. University of California, Riverside, USA.
    Reedy categories which encode the notion of category actions2015Inngår i: Fundamenta Mathematicae, ISSN 0016-2736, E-ISSN 1730-6329, Vol. 228, s. 193-222Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We study a certain type of action of categories on categories and on operads. Using the structure of the categories Δ and Ω governing category and operad structures, respectively, we define categories which instead encode the structure of a category acting on a category, or a category acting on an operad. We prove that the former has the structure of an elegant Reedy category, whereas the latter has the structure of a generalized Reedy category. In particular, this approach gives a new way to regard group actions on categories and on operads.

  • 106.
    Bergqvist, Ewa
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik. Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Umeå forskningscentrum för matematikdidaktik (UFM).
    A verbal factor in the PISA 2003 mathematics items: Tentative analyses2009Inngår i: Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education: In Search for Theories in Mathematics Education, Thessaloniki, Greece: PME , 2009, s. 145-152Konferansepaper (Fagfellevurdert)
    Abstract [en]

    This study uses a statistical method to identify verbal items among mathematical items from PISA 2003. The verbal items are preliminary analysed and compared to the non-verbal items concerning number of text lines, response types, cognitive level, and competences measured. The results show that the verbal items, to a higher percentage than the non-verbal items, measures the reproduction competency, are straigh­t­forward, and of open constructed-response type. These results and proposed further analyses are discussed.

  • 107.
    Bergqvist, Ewa
    et al.
    Umeå universitet, Umeå forskningscentrum för matematikdidaktik (UFM).
    Dyrvold, Anneli
    Umeå universitet, Umeå forskningscentrum för matematikdidaktik (UFM).
    Österholm, Magnus
    Umeå universitet, Umeå forskningscentrum för matematikdidaktik (UFM).
    Relating vocabulary in mathematical tasks to aspects of reading and solving2012Inngår i: Evaluation and comparison of mathematical achievement: Dimensions and perspectives. Proceedings of MADIF 8, The Eighth Mathematics Education Research Seminar, Umeå, January 24-25, 2012 / [ed] Christer Bergsten, Eva Jablonka & Manya Raman, Linköping: SMDF , 2012, s. 61-70Konferansepaper (Fagfellevurdert)
    Abstract [en]

    This paper focuses on relationships between vocabulary in mathematical tasks and aspects of reading and solving these tasks. The paper contains a framework that highlights a number of different aspects of word difficulty as well as many issues to consider when planning and implementing empirical studies concerning vocabulary in tasks, where the aspect of common/uncommon words is one important part. The paper also presents an empirical method where corpora are used to investigate issues of vocabulary in mathematical tasks. The results from the empirical study show that there are connections between different types of vocabulary and task difficulty, but that they seem to be mainly an effect of the total number of words in a task.

  • 108.
    Bergqvist, Ewa
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Umeå forskningscentrum för matematikdidaktik (UFM). Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för naturvetenskapernas och matematikens didaktik.
    Dyrvold, Anneli
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Umeå forskningscentrum för matematikdidaktik (UFM). Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Österholm, Magnus
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Umeå forskningscentrum för matematikdidaktik (UFM). Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för naturvetenskapernas och matematikens didaktik.
    Relating vocabulary in mathematical tasks to aspects of reading and solving2012Inngår i: Evaluation and comparison of mathematical achievement: Dimensions and perspectives. Proceedings of MADIF 8, The Eighth Mathematics Education Research Seminar, Umeå, January 24-25, 2012 / [ed] Christer Bergsten, Eva Jablonka & Manya Raman, Linköping: SMDF , 2012, s. 61-70Konferansepaper (Fagfellevurdert)
    Abstract [en]

    This paper focuses on relationships between vocabulary in mathematical tasks and aspects of reading and solving these tasks. The paper contains a framework that highlights a number of different aspects of word difficulty as well as many issues to consider when planning and implementing empirical studies concerning vocabulary in tasks, where the aspect of common/uncommon words is one important part. The paper also presents an empirical method where corpora are used to investigate issues of vocabulary in mathematical tasks. The results from the empirical study show that there are connections between different types of vocabulary and task difficulty, but that they seem to be mainly an effect of the total number of words in a task.

  • 109.
    Bergqvist, Ewa
    et al.
    Umeå universitet, Umeå forskningscentrum för matematikdidaktik (UFM).
    Dyrvold, Anneli
    Umeå universitet, Umeå forskningscentrum för matematikdidaktik (UFM).
    Österholm, Magnus
    Umeå universitet, Umeå forskningscentrum för matematikdidaktik (UFM).
    Relating vocabulary in mathematical tasks to aspects of reading and solving2012Inngår i: Evaluation and comparison of mathematical achievement: Dimensions and perspectives. Proceedings of MADIF 8, The Eighth Mathematics Education Research Seminar, Umeå, January 24-25, 2012 / [ed] Christer Bergsten, Eva Jablonka & Manya Raman, Linköping: SMDF , 2012, s. 61-70Konferansepaper (Fagfellevurdert)
    Abstract [en]

    This paper focuses on relationships between vocabulary in mathematical tasks and aspects of reading and solving these tasks. The paper contains a framework that highlights a number of different aspects of word difficulty as well as many issues to consider when planning and implementing empirical studies concerning vocabulary in tasks, where the aspect of common/uncommon words is one important part. The paper also presents an empirical method where corpora are used to investigate issues of vocabulary in mathematical tasks. The results from the empirical study show that there are connections between different types of vocabulary and task difficulty, but that they seem to be mainly an effect of the total number of words in a task.

  • 110.
    Bergqvist, Ewa
    et al.
    Umeå universitet, Umeå forskningscentrum för matematikdidaktik (UFM).
    Theens, Frithjof
    Umeå universitet, Umeå forskningscentrum för matematikdidaktik (UFM).
    Österholm, Magnus
    Umeå universitet, Umeå forskningscentrum för matematikdidaktik (UFM).
    Relations between linguistic features and difficulty of PISA tasks in different languages2016Inngår i: Proceedings of the 40th Conference of the International Group for the Psychology of Mathematics Education / [ed] Csíkos, C., Rausch, A., & Szitányi, J., Szeged, Hungary: PME , 2016, s. 125-125Konferansepaper (Fagfellevurdert)
  • 111.
    Bergqvist, Ewa
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Umeå forskningscentrum för matematikdidaktik (UFM). Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för naturvetenskapernas och matematikens didaktik.
    Theens, Frithjof
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Umeå forskningscentrum för matematikdidaktik (UFM). Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för naturvetenskapernas och matematikens didaktik.
    Österholm, Magnus
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Umeå forskningscentrum för matematikdidaktik (UFM). Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för naturvetenskapernas och matematikens didaktik.
    Relations between linguistic features and difficulty of PISA tasks in different languages2016Inngår i: Proceedings of the 40th Conference of the International Group for the Psychology of Mathematics Education / [ed] Csíkos, C., Rausch, A., & Szitányi, J., Szeged, Hungary: PME , 2016, Vol. 1, s. 125-125Konferansepaper (Fagfellevurdert)
  • 112.
    Bergqvist, Ewa
    et al.
    Umeå universitet, Umeå forskningscentrum för matematikdidaktik (UFM).
    Österholm, Magnus
    Umeå universitet, Umeå forskningscentrum för matematikdidaktik (UFM).
    A theoretical model of the connection between the process of reading and the process of solving mathematical tasks2010Inngår i: Mathematics and mathematics education: Cultural and social dimensions. Proceedings of MADIF 7 / [ed] C. Bergsten, E. Jablonka & T. Wedege, Linköping, Sweden: Svensk förening för matematikdidaktisk forskning, SMDF , 2010, s. 47-57Konferansepaper (Fagfellevurdert)
    Abstract [en]

    In this paper we suggest a theoretical model of the connection between the process of reading and the process of solving mathematical tasks. The model takes into consideration different types of previous research about the relationship between reading and solving mathematical tasks, including research about traits of mathematical tasks (a linguistic perspective), about the reading process (a psychological perspective), and about behavior and reasoning when solving tasks (a mathematics education perspective). In contrast to other models, our model is not linear but cyclic, and considers behavior such as re-reading the task.

  • 113.
    Bergqvist, Ewa
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Umeå forskningscentrum för matematikdidaktik (UFM). Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Österholm, Magnus
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Umeå forskningscentrum för matematikdidaktik (UFM). Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för naturvetenskapernas och matematikens didaktik.
    A theoretical model of the connection between the process of reading and the process of solving mathematical tasks2010Inngår i: Mathematics and mathematics education: Cultural and social dimensions. Proceedings of MADIF 7 / [ed] C. Bergsten, E. Jablonka & T. Wedege, Linköping, Sweden: Svensk förening för matematikdidaktisk forskning, SMDF , 2010, s. 47-57Konferansepaper (Fagfellevurdert)
    Abstract [en]

    In this paper we suggest a theoretical model of the connection between the process of reading and the process of solving mathematical tasks. The model takes into consideration different types of previous research about the relationship between reading and solving mathematical tasks, including research about traits of mathematical tasks (a linguistic perspective), about the reading process (a psychological perspective), and about behavior and reasoning when solving tasks (a mathematics education perspective). In contrast to other models, our model is not linear but cyclic, and considers behavior such as re-reading the task.

  • 114.
    Bergqvist, Ewa
    et al.
    Umeå universitet, Umeå forskningscentrum för matematikdidaktik (UFM).
    Österholm, Magnus
    Umeå universitet, Umeå forskningscentrum för matematikdidaktik (UFM).
    Communicating mathematics or mathematical communication?: An analysis of competence frameworks2012Inngår i: Proceedings of the 36th Conference of the International Group for the Psychology of Mathematics Education: Vol. 2: opportunities to learn in mathematics education / [ed] Tai-Yih Tso, 2012, s. 67-74Konferansepaper (Fagfellevurdert)
    Abstract [en]

    In this study we analyse the communication competence included in two different frameworks of mathematical knowledge. The main purpose is to find out if mathematical communication is primarily described as communication of or about mathematics or if it is (also) described as a special type of communication. The results show that aspects of mathematics are mostly included as the content of communication in the frameworks but the use of different forms of representation is highlighted both in the frameworks and also in prior research as a potential cause for characterising mathematical communication differently than "ordinary" communication.

  • 115.
    Bergqvist, Ewa
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Umeå forskningscentrum för matematikdidaktik (UFM). Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för naturvetenskapernas och matematikens didaktik.
    Österholm, Magnus
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Umeå forskningscentrum för matematikdidaktik (UFM). Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för naturvetenskapernas och matematikens didaktik. Monash University, Australia.
    Communicating mathematics or mathematical communication?: An analysis of competence frameworks2012Inngår i: Proceedings of the 36th Conference of the International Group for the Psychology of Mathematics Education: Vol. 2: opportunities to learn in mathematics education / [ed] Tai-Yih Tso, 2012, s. 67-74Konferansepaper (Fagfellevurdert)
    Abstract [en]

    In this study we analyse the communication competence included in two different frameworks of mathematical knowledge. The main purpose is to find out if mathematical communication is primarily described as communication of or about mathematics or if it is (also) described as a special type of communication. The results show that aspects of mathematics are mostly included as the content of communication in the frameworks but the use of different forms of representation is highlighted both in the frameworks and also in prior research as a potential cause for characterising mathematical communication differently than "ordinary" communication.

  • 116.
    Bergqvist, Ewa
    et al.
    Umeå universitet, Umeå forskningscentrum för matematikdidaktik (UFM).
    Österholm, Magnus
    Umeå universitet, Umeå forskningscentrum för matematikdidaktik (UFM).
    Språkbrukets roll i matematikundervisningen2014Inngår i: Nämnaren : tidskrift för matematikundervisning, ISSN 0348-2723, Vol. 2014, nr 1, s. 27-31Artikkel i tidsskrift (Annet (populærvitenskap, debatt, mm))
    Abstract [sv]

    Det språk vi använder oss av i matematikklassrummet kan fokuseras på många olika sätt. Språket är också nödvändigt att förhålla sig till vid utvecklingen av sitt matematiska tänkande. Författarna diskuterar här relationer mellan språk och lärande.

  • 117.
    Bergqvist, Ewa
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Umeå forskningscentrum för matematikdidaktik (UFM). Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för naturvetenskapernas och matematikens didaktik.
    Österholm, Magnus
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Umeå forskningscentrum för matematikdidaktik (UFM). Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för naturvetenskapernas och matematikens didaktik.
    Språkbrukets roll i matematikundervisningen2014Inngår i: Nämnaren : tidskrift för matematikundervisning, ISSN 0348-2723, Vol. 2014, nr 1, s. 27-31Artikkel i tidsskrift (Annet (populærvitenskap, debatt, mm))
    Abstract [sv]

    Det språk vi använder oss av i matematikklassrummet kan fokuseras på många olika sätt. Språket är också nödvändigt att förhålla sig till vid utvecklingen av sitt matematiska tänkande. Författarna diskuterar här relationer mellan språk och lärande.

  • 118.
    Bergström Lissmyr, Jannice
    Högskolan i Gävle, Akademin för teknik och miljö, Avdelningen för elektronik, matematik och naturvetenskap, Matematik.
    En kvalitativ studie om lärarens syn på lässvårigheters påverkan på matematikinlärningen: samt andra svårigheters påverkan2017Independent thesis Advanced level (degree of Master (One Year)), 20 poäng / 30 hpOppgave
    Abstract [sv]

    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.

  • 119.
    Bergwall, Andreas
    Örebro universitet, Institutionen för naturvetenskap och teknik.
    Conceptualizing reasoning-and-proving opportunities in textbook expositions: Cases from secondary calculus2017Inngår i: Proceedings of the Tenth Congress of the European Society for Research in Mathematics Education (CERME10, February 1-5, 2017) / [ed] Dooley, T., & Gueudet, G., Dublin, Ireland: European Society for Research in Mathematics Education , 2017, s. 91-98Konferansepaper (Fagfellevurdert)
    Abstract [en]

    Several recent textbook studies focus on opportunities to learn reasoning-and-proving. They typically investigate the extent to which justifications are general proofs and what opportunities exist for learning important elements of mathematical reasoning. In this paper, I discuss how a particular analytical framework for this might be refined. Based on an in-depth analysis of certain textbook passages in upper secondary calculus textbooks, I make an account for analytical issues encountered during this process and identify aspects of reasoning-and-proving in textbooks that might be missed unless the framework is refined. Among them there are characterizations of generality, use of different representations, logical and mathematical structure, and ordering of material and student activities. Finally, implications beyond textbook research are discussed.

  • 120.
    Bergwall, Andreas
    Örebro universitet, Institutionen för naturvetenskap och teknik.
    On a generality framework for proving tasks2015Inngår i: Proceedings of the Ninth Congress of the European Society for Research in Mathematics Education / [ed] Konrad Krainer; Nada Vondrova, Prague: Charles University in Prague, Faculty of Education and ERME , 2015, s. 86-92Konferansepaper (Fagfellevurdert)
    Abstract [en]

    In this paper I present an analytic framework for generality in textbook proving tasks that involve functions. The framework is discussed in relation to results obtained when analysing tasks in integral calculus. The results show that the frameworks’ categories are easily distinguishable if the functions are explicitly described. The results are also promising regarding the possibility to clarify differences between textbooks. The analysed sections exemplify that there is not necessarily a correlation between the number of general proving tasks and the opportunities for students to engage in reasoning about arbitrary functions. Limitations and possible refinements of the framework are also discussed.

  • 121.
    Bergwall, Andreas
    et al.
    Örebro universitet, Institutionen för naturvetenskap och teknik.
    Hemmi, Kirsti
    Åbo Akademi University, Åbo, Finland; Uppsala University, Uppsala, Sweden.
    The State of Proof in Finnish and Swedish Mathematics Textbooks: Capturing Differences in Approaches to Upper-Secondary Integral Calculus2017Inngår i: Mathematical Thinking and Learning, ISSN 1098-6065, E-ISSN 1532-7833, Vol. 19, nr 1, s. 1-18Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Students’ difficulties with proof, scholars’ calls for proof to be a consistent part of K-12 mathematics, and the extensive use of textbooks in mathematics classrooms motivate investigations on how proof-related items are addressed in mathematics textbooks. We contribute to textbook research by focusing on opportunities to learn proof-related reasoning in integral calculus, a key subject in transitioning from secondary to tertiary education. We analyze expository sections and nearly 2000 students’ exercises in the four most frequently used Finnish and Swedish textbook series. Results indicate that Finnish textbooks offer more opportunities for learning proof than do Swedish textbooks. Proofs are also more visible in Finnish text-books than in Swedish materials, but the tasks in the latter reflect a higher variation in nature of proof-related reasoning. Our results are compared with methodologically similar U.S. studies. Consequences for learning and transition to university mathematics, as well as directions for future research, are discussed.

  • 122.
    Bernhoff, Niclas
    Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för matematik och datavetenskap.
    Boundary layers for discrete kinetic models: Multicomponent mixtures, polyatomic molecules, bimolecular reactions, and quantum kinetic equations2017Inngår i: Kinetic and Related Models, ISSN 1937-5093, E-ISSN 1937-5077, Vol. 10, nr 4, s. 925-955Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We consider some extensions of the classical discrete Boltzmann equation to the cases of multicomponent mixtures, polyatomic molecules (with a finite number of different internal energies), and chemical reactions, but also general discrete quantum kinetic Boltzmann-like equations; discrete versions of the Nordheim-Boltzmann (or Uehling-Uhlenbeck) equation for bosons and fermions and a kinetic equation for excitations in a Bose gas interacting witha Bose-Einstein condensate. In each case we have an H-theorem and so for the planar stationary half-space problem, we have convergence to an equilibrium distribution at infinity (or at least a manifold of equilibrium distributions). In particular, we consider the nonlinear half-space problem of condensation and evaporation for these discrete Boltzmann-like equations. We assume that the flow tends to a stationary point at infinity and that the outgoing flow is known at the wall, maybe also partly linearly depending on the incoming flow. We find that the systems we obtain are of similar structures as for the classical discrete Boltzmann equation (for single species), and that previously obtained results for the discrete Boltzmann equation can be applied after being generalized. Then the number of conditions on the assigned data at the wall needed for existence of a unique solution is found. The number of parameters to be specified in the boundary conditions depends on if we have subsonic or supersonic condensation or evaporation. All our results are valid for any finite number of velocities.

    Fulltekst tilgjengelig fra 2018-12-03 14:57
  • 123.
    Bernhoff, Niclas
    Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för matematik och datavetenskap (from 2013).
    Half-Space Problems for a Linearized Discrete Quantum Kinetic Equation2015Inngår i: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 159, nr 2, s. 358-379Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We study typical half-space problems of rarefied gas dynamics, including the problems of Milne and Kramer, for a general discrete model of a quantum kinetic equation for excitations in a Bose gas. In the discrete case the plane stationary quantum kinetic equation reduces to a system of ordinary differential equations. These systems are studied close to equilibrium and are proved to have the same structure as corresponding systems for the discrete Boltzmann equation. Then a classification of well-posed half-space problems for the homogeneous, as well as the inhomogeneous, linearized discrete kinetic equation can be made. The number of additional conditions that need to be imposed for well-posedness is given by some characteristic numbers. These characteristic numbers are calculated for discrete models axially symmetric with respect to the x-axis. When the characteristic numbers change is found in the discrete as well as the continuous case. As an illustration explicit solutions are found for a small-sized model.

  • 124.
    Bernhoff, Niclas
    et al.
    Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för matematik och datavetenskap (from 2013).
    Vinerean Bernhoff, Mirela
    Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för matematik och datavetenskap (from 2013).
    Discrete Velocity Models for Mixtures Without Nonphysical Collision Invariants2016Inngår i: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 165, nr 2, s. 434-453Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    An important aspect of constructing discrete velocity models (DVMs) for the Boltzmann equation is to obtain the right number of collision invariants. It is a well-known fact that DVMs can also have extra collision invariants, so called spurious collision invariants, in plus to the physical ones. A DVM with only physical collision invariants, and so without spurious ones, is called normal. For binary mixtures also the concept of supernormal DVMs was introduced, meaning that in addition to the DVM being normal, the restriction of the DVM to any single species also is normal. Here we introduce generalizations of this concept to DVMs for multicomponent mixtures. We also present some general algorithms for constructing such models and give some concrete examples of such constructions. One of our main results is that for any given number of species, and any given rational mass ratios we can construct a supernormal DVM. The DVMs are constructed in such a way that for half-space problems, as the Milne and Kramers problems, but also nonlinear ones, we obtain similar structures as for the classical discrete Boltzmann equation for one species, and therefore we can apply obtained results for the classical Boltzmann equation.

  • 125.
    Bertolino, Mattias
    et al.
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Tillämpad matematik och statistik.
    Jonnarth, Arvi
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Tillämpad matematik och statistik.
    Monte Carlo methods applied to tree-structured decision processes2017Independent thesis Basic level (degree of Bachelor), 10 poäng / 15 hpOppgave
  • 126.
    Bertse, Erik
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Tillämpad matematik och statistik.
    A simulational approach to parking2018Independent thesis Basic level (degree of Bachelor), 10 poäng / 15 hpOppgave
  • 127.
    Birky, Geoffrey
    et al.
    Georgetown University.
    Campbell, Connie M
    Millsaps College, USA.
    Raman, Manya
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Umeå forskningscentrum för matematikdidaktik (UFM). Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för naturvetenskapernas och matematikens didaktik.
    Sandefur, James
    Georgetown University.
    Somers, Kay
    Moravian College, USA.
    One problem, nine student-produced proofs2011Inngår i: The College Mathematics Journal, ISSN 0746-8342, E-ISSN 1931-1346, Vol. 42, nr 5, s. 355-360Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    The context of this paper is a project aimed at meeting the challenges of teaching an introduction-to-proof course at our own institutions. These courses are typically taken by second year students intending to major in mathematics or computer science and are taught in sections of 15 to 24 students. These students benefit from being given the freedom to explore different methods of proof without the presumption that they will use a certain method.  This is the story of how what could have been a routine, closed problem became an open one.

  • 128.
    Bisenius Sellgren, Kajsa
    Södertörns högskola, Lärarutbildningen.
    En studie om elevers val av metoder vid subtraktionsberäkningar2010Independent thesis Basic level (professional degree), 10 poäng / 15 hpOppgave
    Abstract [en]

    In this qualitative study of students’ methods of calculating subtractions, I have used interviews, subtraction exercises and analysis of teaching material. The purpose of my study was to explore which methods students in grade three uses when calculating subtractions. I also wanted to highlight which strategies the students use and their comprehension of the concept of subtraction. In the study, I also highlight the different pedagogical ideas on which the teaching material is based on and the students’ choice of methods.  The study shows that the students choose to use the methods”deduct” and “kind of number” independently. Further it also shows that the students choose to switch from the method “deduct” to “kind of number” when the numbers in the exercises are further up on the number axis. When asked, the students answered that subtraction means “minus” which they in turn explained as removing something, an explanation confirmed by the Swedish Academy dictionary. The students’ choice of methods and teaching material is based on different fundamental pedagogical view.

     

  • 129.
    Björk, Annika
    et al.
    Högskolan i Halmstad, Sektionen för lärarutbildning (LUT).
    Nilsson, Sara
    Högskolan i Halmstad, Sektionen för lärarutbildning (LUT).
    ”Det finns jättemycket luckor, att de inte förstår”: –      En studie om barns och pedagogers kunskap i antalsuppfattning.2012Independent thesis Basic level (degree of Bachelor), 10 poäng / 15 hpOppgave
    Abstract [sv]

    Matematik finns överallt men trots det har vi genom studien fått fram att ”det finns jättemycket luckor, att de inte förstår”.  Detta citat är hämtat ifrån en av våra intervjuer och speglar vårt resultat. Studien är kvalitativ och vi har intervjuat pedagoger samt låtit barn, i förskola och årskurs 1, utföra matematikuppgifter. I denna studie har vi fokuserat på grunderna i matematik. Syftet med studien har varit att utröna kunskaper inom antalsuppfattning hos barn. Vi har även belyst pedagogernas kunskaper inom antalsuppfattning samt hur de arbetar med detta. I studien har vi influerats av Piagets och Vygotskijs tankar om hur barn lär. Resultaten i vår studie har analyserats med inspiration av hermeneutiken. Resultaten visar att det finns okunskap hos både barn och pedagoger gällande antalsuppfattning. Det gäller bland annat barn som helt saknar kunskaper inom detta samt att pedagoger uppfattar att subitizing, se stycke 3.4., och ett till ett-principen, se stycke 3.5., är detsamma som antalsuppfattning. Pedagoger behöver djupare förståelse för de matematiska begreppen inom den grundläggande matematiken. Det i sin tur kan medföra ett tydligare matematiskt språk vilket kan underlätta kommunikationen sinsemellan. I sin tur kan det möjliggöra för pedagoger att tillsammans lättare kunna utveckla barns matematiska färdigheter som de behöver i sin vardag som vuxna samhällsmedborgare. 

  • 130.
    Björkqvist, Filip
    Linnéuniversitetet, Fakulteten för teknik (FTK), Institutionen för matematik (MA).
    Läromedelsanalys gällande textbaserade uppgifter i matematik2018Independent thesis Advanced level (degree of Master (One Year)), 10 poäng / 15 hpOppgave
    Abstract [sv]

    Syftet med studien har varit att utveckla en kunskap om vilket innehåll olika läromedel kan ha gällande textbaserade uppgifter i ämnet matematik. Arbetet är baserat på en läromedelanalys på två olika läromedel i årskurs tre som har analyserats och granskats. Förhoppningen har varit att genom arbetet upptäcka vilka kritiska aspekter de två olika läromedlen kan innehålla för eleverna när de ska lösa textbaserade uppgifter. Studien har utgått från variationsteorin när resultatet och analysen har gjorts. I resultatet har det framkommit att läroböckerna kan innehålla kritiska aspekter för eleverna, vilket kan göra att det försvårar för eleverna att utveckla sina kunskaper gällande att lösa textbaserade uppgifter i matematik. I diskussionsdelen diskuteras det om de två olika läromedlen ger eleverna möjlighet att utveckla sin förmåga att lösa textbaserade uppgifter eller om läromedlets framställning av textbaserade uppgifter inte tar hänsyn till alla kritiska aspekter när eleverna ska lösa textbaserade uppgifter.

  • 131.
    Björkström, Angela
    Mälardalens högskola, Akademin för utbildning, kultur och kommunikation.
    Is it all in their heads?: A study of the strategies used in mental arithmetic by Swedish pupils in their last years of the obligatory school and in the upper secondary school2008Independent thesis Basic level (degree of Bachelor), 10 poäng / 15 hpOppgave
    Abstract [en]

    Competence in mental arithmetic is recognised by many as essential to be active participants in the fast flowing, high technological society we live in today.  Many have noticed pupils’ unwillingness to set their calculators aside and practice this aspect of mathematics when possible.  Furthermore, some studies show that pupils’ ability to compute mentally deteriorates as they pass through the school system.  Through testing classes in a Swedish obligatory school and an upper secondary school, the aim of this thesis is to see if the goals set by The National [Swedish] Agency for Education regarding mental arithmetic, are being fulfilled.  Through using questionnaires to collect the strategies and ideas of the pupils, a wide range of problematic mathematical misconceptions became evident.  These are highlighted since they are important aspects teachers should be aware of.  The results of this study show that the obligatory school classes are far from reaching the goals set for them whereas the upper secondary classes show good results.  Furthermore, there is an apparent improvement in their progression, resulting in a fulfilment the official goals.  Many pupils however, seem reluctant to rely on their mental arithmetic capabilities and resort to algorithmic strategies.  Other problems to emerge are in carrying out table calculations and in a lack of number sense when deeming if the answers are reasonable.   

  • 132.
    Blanck, Andreas
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för fysik.
    American Option Price Approximation for Real-Time Clearing2018Independent thesis Advanced level (professional degree), 20 poäng / 30 hpOppgave
    Abstract [en]

    American-style options are contracts traded on financial markets. These are derivatives of some underlying security or securities that in contrast to European-style options allow their holders to exercise at any point before the contracts expire. However, this advantage aggravates the mathematical formulation of an option's value considerably, explaining why essentially no exact closed-formed pricing formulas exist. Numerous price approximation methods are although available, but their possible areas of application as well as performance, measured by speed and accuracy, differ. A clearing house offering real-time solutions are especially dependent on fast pricing methods to calculate portfolio risk, where accuracy is assumed to be an important factor to guarantee low-discrepancy estimations. Conversely, overly biased risk estimates may worsen a clearing house's ability to manage great losses, endangering the stability of a financial market it operates.

    The purpose of this project was to find methods with optimal performance and to investigate if price approximation errors induce biases in option portfolios' risk estimates. Regarding performance, a Quasi-Monte Carlo least squares method was found suitable for at least one type of exotic option. Yet none of the analyzed closed-form approximation methods could be assessed as optimal because of their varying strengths, where although the Binomial Tree model performed most consistently. Moreover, the answer to which method entails the best risk estimates remains inconclusive since only one set of parameters was used due to heavy calculations. A larger study involving a broader range of parameter values must therefore be performed in order to answer this reliably. However, it was revealed that large errors in risk estimates are avoided only if American standard options are priced with any of the analyzed methods and not when a faster European formula is employed. Furthermore, those that were analyzed can yield rather different risk estimates, implying that relatively large errors may arise if an inadequate method is applied.

  • 133.
    Blom, Veronica
    et al.
    Mälardalens högskola, Akademin för utbildning, kultur och kommunikation.
    Gelius-Lundberg, Ulrika
    Mälardalens högskola, Akademin för utbildning, kultur och kommunikation.
    Samband mellan de fyra räknesätten- en läromedelsanalys2012Independent thesis Basic level (university diploma), 10 poäng / 15 hpOppgave
    Abstract [sv]

    Sambandet mellan de fyra räknesätten är en del i utvecklingen av den matematiska förståelsen. Syftet med denna studie är att analysera hur ett antal olika läromedel i matematik riktade till årskurs 1-3 tar upp dessa samband. Med läromedel menas då läroböcker, samt tillhörande lärarhandledningar. Resultatet visar att flertalet av de analyserade läromedlen inte tar upp alla samband, medan ett fåtal tar upp alla samband men i olika stor omfattning. Dessutom visar analysen hur viktig lärarhandledningen är för att ge eleverna fler övningstillfällen i att träna de fyra sambanden, eftersom en stor del av dessa tillfällen förekommer i lärarhandledningen och inte i läroböckerna. Lärare som endast följer ett enskilt läromedel i sin undervisning kan alltså gå miste om mycket när det gäller sambandet mellan de fyra räknesätten.

  • 134.
    Blomqvist, Charlotta
    et al.
    Berghemsskolan, Umeå.
    Gade, Sharada
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Umeå forskningscentrum för matematikdidaktik (UFM). Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för naturvetenskapernas och matematikens didaktik.
    Att kommunicera om likamedtecknet2013Inngår i: Nämnaren : tidskrift för matematikundervisning, ISSN 0348-2723, nr 4, s. 39-42Artikkel i tidsskrift (Annet (populærvitenskap, debatt, mm))
    Abstract [sv]

    I denna artikel beskriver en lärare och en forskare sitt arbete med att utveckla den muntliga kommunikationsförmågan inom matematik i en fjärdeklass. Aktionsforskning utgör bakgrunden för arbetet och likamedtecknet står i fokus när eleverna utmanas i sitt matematiska tänkande med hjälp av lappar.

  • 135.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Potapenko, Irina
    Russia.
    Monte Carlo methods and their analysis for Coulomb collisions in multicomponent plasmas2013Inngår i: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 246, s. 123-144Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    A general approach to Monte Carlo methods for Coulomb collisions is proposed. Its key idea is an approximation of Landau-Fokker-Planck equations by Boltzmann equations of quasi-Maxwellian kind. It means that the total collision frequency for the corresponding Boltzmann equation does not depend on the velocities. This allows to make the simulation process very simple since the collision pairs can be chosen arbitrarily, without restriction. It is shown that this approach includes the well-known methods of Takizuka and Abe (1977) [12] and Nanbu (1997) as particular cases, and generalizes the approach of Bobylev and Nanbu (2000). The numerical scheme of this paper is simpler than the schemes by Takizuka and Abe [12] and by Nanbu. We derive it for the general case of multicomponent plasmas and show some numerical tests for the two-component (electrons and ions) case. An optimal choice of parameters for speeding up the computations is also discussed. It is also proved that the order of approximation is not worse than O(root epsilon), where epsilon is a parameter of approximation being equivalent to the time step Delta t in earlier methods. A similar estimate is obtained for the methods of Takizuka and Abe and Nanbu. (C) 2013 Elsevier Inc. All rights reserved.

  • 136.
    Bodin, Kenneth
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Högpresterande beräkningscentrum norr (HPC2N).
    Lacoursière, Claude
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Högpresterande beräkningscentrum norr (HPC2N).
    Nilsson, Martin
    Servin, Martin
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för fysik.
    Constraint based particle fluids on GPGPU2011Konferansepaper (Annet vitenskapelig)
    Abstract [en]

    We present a fluid simulation method adapted for stream parallelism on general purpose graphics processingunits (GPGPU). In this method the equations of Navier and Stokes are discretized using particles and kernelfunctions as in Smoothed Particle Hydrodynamics (SPH), but rather than using penalty methods or solving for a divergence free velocity field, incompressibility is enforced using holonomic kinematic constraints [1]. We useone constraint for each smoothed particle stating that the local density should be kept constant. Other constraintsare used for boundary conditions and multiphysics coupling. We also present a viscosity model in which theshear rate at each pseudo particle is constrained to satisfy a given constitutive law. The computation of theconstraint forces, namely, the pressure and the stresses, requires the solution system of linear equations whichhave a sparse, saddle point structure. These are solved using the Uzawa method of preconditioned conjugate gradients (CG) applied directly to the symmetric indefinite matrix. The overall simulation method has its rootsin a discrete variational principle and the SPOOK time stepping scheme for constrained mechanical systems [2].The SPOOK method is second order accurate on the positions and constraints violations, and is stable at largetime-steps, thus often allowing several orders of magnitude larger timesteps in our method compared to intraditional SPH methods. The numerical implementation on GPGPU that is the main result of this paper consistsof the following components: particle neighbour searches based on spatial decomposition; summation of kernel densities; construction of Jacobians representing the constraints on the density, boundary conditions, viscosityand multiphysics couplings; a Uzawa CG solver for the system of linear equations; and finally, discrete timestepping of velocities and positions. The CG solver is particularly suitable for stream computing since it is basedon matrix-vector multiplications. The sparse system data is stored in a compressed matrix format and the algorithms operating on this data on GPGPU are implemented in CUDA and OpenCL. Our simulation resultsinclude performance measurements, and validation of the method for benchmark problems. We achieve up totwo orders of magnitude speed-up from the GPGPU over traditional processors and together with the increased timestep efficiency of our method we arrive at interactive performance for systems with up to two million fluidparticles representing an incompressible fluid.

  • 137.
    Bonner, Richard
    Mälardalens högskola, Akademin för utbildning, kultur och kommunikation.
    Operators that coerce the surjectivity of convolution2013Rapport (Annet vitenskapelig)
    Abstract [en]

    Considered are operators that leave the set of non-invertible (in the sense of Ehrenpreis) distributions stable. They simultaneously generalise the operation of convolution by a distribution with compact support and the operation of multiplication by a real analytic function; they are here called pseudo-convolutions since they also generalise pseudo-differential operators. (It is shown that the elliptic real analytic pseudo-differential operators leave both the non-invertible and the invertible distributions invariant.) But when the condition of real-analyticity is relaxed, such operators may map a non-invertible distribution to one invertible -- given that the invertibility in both cases concerns the same function space. By varying the space, however, one can measure the 'loss of non-invertibily' that a non-analytic perturbation may introduce. This phenomenon is here studied using the Beurling classes of functions and measuring the regularity of operator symbols in the Denjoy-Carleman sense; the Gevrey case turns out particularly simple.

  • 138.
    Bonner, Richard F.
    et al.
    Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, Utbildningsvetenskap och Matematik.
    Mamchych, Tetyana I.
    RFB Consulting, Sweden.
    Classifying Households by the (Sobolev) Norms of their Electricity Consumption2014Inngår i: Energy Procedia, ISSN 1876-6102, E-ISSN 1876-6102, Vol. 61, s. 1870-1873Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Numerical time series, but especially periodic such, are characterized up to pertinent symmetries by families of norms. The electricity consumption by a household, recorded daily during a month’s time, say, may then be encoded in a sequence of numbers; for example, as follows: the mean daily consumption, the mean daily variation of the consumption, the variation of the variation, the variation of the variation of the variation, etc. Now, replacing each of these numbers by the digits 0, 1, or 2, to say that a number is “low”, “medium”, or “high”, in relation to a collection of households, one naturally partitions the collection by the strings of these three digits; the household labeled 102   has then medium daily consumption, low daily variation, but high variation of variation, etc. We generally discuss this innocent idea and examine it in three ways: by way of toy examples, through its mathematical model (in detail presented elsewhere) and by accordingly classifying some actual electricity consumption data.

  • 139.
    Bonner, Richard
    et al.
    Mälardalens högskola, Akademin för utbildning, kultur och kommunikation.
    Mamchych, T.
    Malchuk, I.
    On the problem of mining the Web – for a curriculum2008Konferansepaper (Fagfellevurdert)
  • 140.
    Bortolin, Gianantonio
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
    Gutman, Per-Olof
    A new algorithm for variable selection2006Inngår i: PROCEEDINGS OF THE 45TH IEEE CONFERENCE ON DECISION AND CONTROL, VOLS 1-14, 2006, s. 1309-1314Konferansepaper (Fagfellevurdert)
    Abstract [en]

    A new method for variable selection and estimation called Iteratively Scaled Ridge Regression, ISRR, is proposed. The method is an iterative algorithm based on ridge regression. Simulation studies show that ISRR shares the properties of both subset selection and ridge regression. It selects an optimal subset of the regressor variables and is robust to small changes in the data set. The ISRR algorithm was primarily developed for linear models, but is quite simple and general and can easily be extended to more general linear and nonlinear models.

  • 141.
    Bouhuis, Krijn
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Differentiation in teaching mathematics: An observational study of eight primary school teachers in Sweden2013Independent thesis Basic level (professional degree), 10 poäng / 15 hpOppgave
    Abstract [en]

    According to the Swedish curriculum for the primary school, the teachers have to meet the different conditions and demands of all the pupils. A way of meeting the different conditions and demands of all the pupils is differentiation. The aim of this study was to contribute to the discussion about differentiation in the classroom on the subject of mathematics in Swedish primary school.

    Differentiation means that the teacher adapts his teaching to the different learning styles, interests and levels of the pupils. The main question of this exam paper is: How do teachers in Swedish primary school differentiate during their mathematics lessons? In order to get an answer to this question eight teachers were observed during their mathematics lessons. An observation instrument, newly developed by the University of Utrecht, was used for this.

    The main parts of the observed lessons were whole class instruction and working independently, alone or in groups. The teachers’ instructions varied from concrete acts, the use of pictures and schedules to formal tasks. The teachers used the different principles of how a child is learning mathematics in their instructions as well. For example: understanding the concept and development of strategies. Another way of differentiation is to give the pupils freedom to choose their own working style while working individually. This kind of differentiation was not observed often during the eight lessons.

    This study shows that teachers in Swedish primary school are differentiating. However the differentiation can be improved and implemented more often. Furthermore it is unclear if the teachers differentiate consciously, which would mean that their level of consciousness could probably be increased.

  • 142.
    Bounadi, Monir
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
    The Foundations of Graph Pebbling2015Independent thesis Advanced level (degree of Master (One Year)), 240 hpOppgave
    Abstract [en]

    Graph pebbling modeling started as a method for solving a combinatorialnumber theory conjecture by Erdös and Lemke. Using thismethod, Chung proved the conjecture in 1989. Since then, the literaturehas grown considerably. Several variations and possible applicationshave been discussed, in graph theory, computer science andnetwork optimization.

    The main focus in graph pebbling is graphs, mathematical structuresmodeling binary relations between vertices. To every vertex insome graph we assign a number of pebbles. If two pebbles are movedacross an edge joining two distinct vertices, one pebble arrives andone pebble is lost. This is called a pebbling step.

    The basic question in graph pebbling asks if one may from a givendistribution of pebbles on a set of vertices move to another distributionon the same set via a series of pebbling steps.

    In this Master’s thesis we approach the above question using twomodels: a deterministic, which includes the notion of a pebblingnumber, and a probabilistic, which includes the notion of a threshold.

    For both these models we clarify earlier proofs, and provide newproofs, of foundational theorems in graph pebbling. These resultsconstitute the backbone for our discussion on recent research, whichconcentrates on generalizing and extending central notions in graphpebbling, for example the generalized idea of a pebbling number:the pi-pebbling function. Simultaneously, a corollary to the so calledcover pebbling theorem is derived. This corollary lets us prove established,and newly found, theorems.

    Regarding applications in graph pebbling, we argue that one shouldgeneralize existing results, and incorporate directed graphs into abigger part of the theory. We suggest how this can be done.

  • 143.
    Brandberg, Anna-Lena
    et al.
    Linnéuniversitetet, Fakultetsnämnden för naturvetenskap och teknik, Institutionen för datavetenskap, fysik och matematik, DFM.
    Wennerström, Maria
    Linnéuniversitetet, Fakultetsnämnden för naturvetenskap och teknik, Institutionen för datavetenskap, fysik och matematik, DFM.
    Speciallärares kommunikation med matematiksvaga elever2010Independent thesis Advanced level (professional degree), 10 poäng / 15 hpOppgave
    Abstract [sv]

    Undersökningen beskriver den matematiska kommunikationen mellan speciallärare och matematiksvaga elever. Den empiriska datan består av två matematiklektioner från år 9 i grundskolan. Datan analyserades för att finna olika typer av kommunikation.

    Lärarna var i många delar medvetna i sitt sätt att kommunicera med eleverna och undervisningsmiljön kändes trygg. Under lektionerna observerades två parallella språk, ett vardagsanknutet och ett matematiskt språk. Det framkom att ett tydligt förhållningssätt beträffande matematiskt språk kan underlätta för elevens inlärning samt medvetandegöra läraren om den kommunikativa betydelsen i undervisningen med styrdokumentens mål i fokus.

     

  • 144.
    Brattström, Gudrun
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen. Matematik.
    Leonardo da Pisa: Inger Christensen och Fibonacci2008Inngår i: Lyrikvännen, ISSN 0460-0762, Vol. 55, nr 6, s. 33-38Artikkel i tidsskrift (Annet (populærvitenskap, debatt, mm))
    Abstract [sv]

    Den danska poeten Inger Christensen byggde sin diktsamling Alfabet på Fibonaccis talserie. Artikeln är en kort historisk essä över Fibonacci, och beskriver även hur talserien är konstruerad.

  • 145.
    Britton, Tom
    et al.
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
    Deijfen, Maria
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
    Liljeros, Fredrik
    Stockholms universitet, Samhällsvetenskapliga fakulteten, Sociologiska institutionen.
    A weighted configuration model and inhomogeneous epidemics2011Inngår i: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 145, s. 1368-1384Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    A random graph model with prescribed degree distribution and degree dependent

    edge weights is introduced. Each vertex is independently equipped with a random number

    of half-edges and each half-edge is assigned an integer valued weight according to a distribution

    that is allowed to depend on the degree of its vertex. Half-edges with the same weight

    are then paired randomly to create edges. An expression for the threshold for the appearance

    of a giant component in the resulting graph is derived using results on multi-type branching

    processes. The same technique also gives an expression for the basic reproduction number

    for an epidemic on the graph where the probability that a certain edge is used for transmission

    is a function of the edge weight (reflecting how closely ‘connected’ the corresponding

    vertices are). It is demonstrated that, if vertices with large degree tend to have large (small)

    weights on their edges and if the transmission probability increases with the edge weight,

    then it is easier (harder) for the epidemic to take off compared to a randomized epidemic

    with the same degree and weight distribution. A recipe for calculating the probability of a

    large outbreak in the epidemic and the size of such an outbreak is also given. Finally, the

    model is fitted to three empirical weighted networks of importance for the spread of contagious

    diseases and it is shown that

    R0

    can be substantially over- or underestimated if the

    correlation between degree and weight is not taken into account.

  • 146.
    Brunosson, Albina
    Högskolan Kristianstad, Sektionen för Lärarutbildning.
    Kökets bråkstakar: elevers förståelse för bråktal i hem- och konsumentkunskap2010Independent thesis Advanced level (professional degree), 20 poäng / 30 hpOppgave
    Abstract [en]

    The aim of this study has been to investigate students’ understanding of fractions when they occur in the practical subject home economics. In order to investigate this, the learning study method was used. Learning study can be understood as a method to gain deeper understanding of what is learnt by students during a lesson to increase learning outcome.

    The variation theory framework has been utilized to analyze the results of this study. The object of learning was chosen to be addition of fractions bigger than one half when they appear in home economics. The results indicate that in order for the learners to experience the object of learning it requires that they discern some critical aspects; to distinguish between quarters and four parts, the relationship between the part and the whole in fractions, varied presentations of the same fraction along with an understanding of the meaning of the concept to double.

     

    Moreover the students have carried out a math test which tested the students’ theoretical understanding of fractions to see whether they can transfer the knowledge from one learning context to another. The result evinces that most of the learners show a good understanding of fractions in a theoretical context as well as a practical. The learning study has increased the students learning according to the object of learning

  • 147.
    Bruér, Rebecka
    et al.
    Högskolan Kristianstad, Sektionen för lärande och miljö.
    Wiklund, Jenny
    Högskolan Kristianstad, Sektionen för lärande och miljö.
    Elevers möjligheter att lära bråk2017Independent thesis Advanced level (professional degree), 10 poäng / 15 hpOppgave
    Abstract [sv]

    Syftet med studien är att undersöka hur elever i årskurs 6 uttrycker sina kunskaper om olika aspekter av bråk, jämförelser och division samt vilka resonemang och strategier de då använder. Syftet är även att undersöka vilka möjligheter eleverna upplever de fått att lära sig detta samt vilka möjligheter lärarna upplever att de har gett eleverna att lära sig, eftersom vi vill veta orsaker till att använda strategier valdes och vilka samband det kan finnas mellan elevers kunskaper om bråks olika aspekter och deras kunskaper om jämförelser och division med bråk. Studien är förankrad i teoretiska modeller om hur bråk kan förstås, teorier om lärande samt tidigare forskning med liknande frågeställningar. Det tillvägagångssätt som valts är flermetodsforskning som omfattar kvantitativ metod för insamling och analys av en skriftlig diagnos och kvalitativ metod för insamling och tematisk analys av intervjuer med elever och lärare. Resultatet visar att eleverna hade tillräckliga kunskaper totalt sett om bråks aspekter, jämförelser och divisioner men otillräckliga kunskaper om framför allt aspekten bråk som kvot vilket medförde otillräckliga kunskaper om innehållsdivision.

  • 148.
    Brännström, Katarina
    Linnéuniversitetet, Fakultetsnämnden för naturvetenskap och teknik, Institutionen för datavetenskap, fysik och matematik, DFM.
    Variationsrika uppgifter med syfte att utveckla matematiska förmågor: En analys av komplexa tal inom gymnasiekursen Matematik 42012Independent thesis Basic level (professional degree), 12 poäng / 18 hpOppgave
    Abstract [sv]

    Det finns ett behov av att kombinera förmågor och variation i undervisningen i högre kurser på gymnasiet då det uppfattas som att kurserna blir mer abstrakta, ensidiga och läroboksbundna. Syftet är att höja elevernas måluppfyllelse. Genom litteraturstudie av matematisk förmåga och variationsteori tillverkas eller väljs ut exempeluppgifter som sedan analyseras på vilket sätt de kan variera. Det visar sig att begreppsförmågan varieras mest följt av problemlösningsförmågan och båda har ovanliga uppgifter som den mest varierbara uppgiftstypen. När det gäller verklighetsnära uppgifter är det viktigt att de är anpassade till elevernas förkunskaper för att erbjuda meningsfull variation.

  • 149.
    Brännström, Åke
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Carlsson, Linus
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Rossberg, Axel
    School of Biological Sciences, Queen’s University Belfast, Belfast, BT9 7BL UK.
    Rigorous conditions for food-web intervality in high-dimensional trophic niche spaces2011Inngår i: Journal of Mathematical Biology, ISSN 0303-6812, E-ISSN 1432-1416, Vol. 63, nr 3, s. 575-592Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Food webs represent trophic (feeding) interactions in ecosystems. Since the late 1970s, it has been recognized that food-webs have a surprisingly close relationship to interval graphs. One interpretation of food-web intervality is that trophic niche space is low-dimensional, meaning that the trophic character of a species can be expressed by a single or at most a few quantitative traits. In a companion paper we demonstrated, by simulating a minimal food-web model, that food webs are also expected to be interval when niche-space is high-dimensional. Here we characterize the fundamental mechanisms underlying this phenomenon by proving a set of rigorous conditions for food-web intervality in high-dimensional niche spaces. Our results apply to a large class of food-web models, including the special case previously studied numerically.

  • 150.
    Brännström, Åke
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Gross, Thilo
    Max-Planck-Institute for Physics of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany.
    Blasius, Bernd
    Institute for Chemistry and Biology of Marine Environment, Oldenburg University, 26111 Oldenburg, Germany.
    Dieckmann, Ulf
    Evolution and Ecology Program, International Institute for Applied Systems Analysis (IIASA), Schlossplatz 1, 2361 Laxenburg, Austria.
    Consequences of fluctuating group size for the evolution of cooperation2011Inngår i: Journal of Mathematical Biology, ISSN 0303-6812, E-ISSN 1432-1416, Vol. 63, nr 2, s. 263-281Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Studies of cooperation have traditionally focused on discrete games such as the well-known prisoner’s dilemma, in which players choose between two pure strategies: cooperation and defection. Increasingly, however, cooperation is being studied in continuous games that feature a continuum of strategies determining the level of cooperative investment. For the continuous snowdrift game, it has been shown that a gradually evolving monomorphic population may undergo evolutionary branching, resulting in the emergence of a defector strategy that coexists with a cooperator strategy. This phenomenon has been dubbed the ‘tragedy of the commune’. Here we study the effects of fluctuating group size on the tragedy of the commune and derive analytical conditions for evolutionary branching. Our results show that the effects of fluctuating group size on evolutionary dynamics critically depend on the structure of payoff functions. For games with additively separable benefits and costs, fluctuations in group size make evolutionary branching less likely, and sufficiently large fluctuations in group size can always turn an evolutionary branching point into a locally evolutionarily stable strategy. For games with multiplicatively separable benefits and costs, fluctuations in group size can either prevent or induce the tragedy of the commune. For games with general interactions between benefits and costs, we derive a general classification scheme based on second derivatives of the payoff function, to elucidate when fluctuations in group size help or hinder cooperation.

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