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  • 201.
    Basarab-Horwath, Peter
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Lie algebras and classification of partial differential equations2004In: Symmetry in Nonlinear Mathematical Physics,2004, 2004Conference paper (Other academic)
  • 202.
    Basarab-Horwath, Peter
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Gungor, F.
    Istanbul Technical University, Turkey.
    Linearizability for third order evolution equations2017In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 58, no 8, article id 081507Article in journal (Refereed)
    Abstract [en]

    The problem of linearization for third order evolution equations is considered. Criteria for testing equations for linearity are presented. A class of linearizable equations depending on arbitrary functions is obtained by requiring presence of an infinite-dimensional symmetry group. Linearizing transformations for this class are found using symmetry structure and local conservation laws. A number of special cases as examples are discussed. Their transformation to equations within the same class by differential substitutions and connection with KdV and mKdV equations is also reviewed in this framework. Published by AIP Publishing.

  • 203.
    Basarab-Horwath, Peter
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Gungor, F
    Dogus University, Turkey .
    Lahno, V
    Pedag University, Ukraine .
    Symmetry Classification of Third-Order Nonlinear Evolution Equations. Part I: Semi-simple Algebras2013In: Acta Applicandae Mathematicae - An International Survey Journal on Applying Mathematics and Mathematical Applications, ISSN 0167-8019, E-ISSN 1572-9036, Vol. 124, no 1, p. 123-170Article in journal (Refereed)
    Abstract [en]

    We give a complete point-symmetry classification of all third-order evolution equations of the form u (t) =F(t,x,u,u (x) ,u (xx) )u (xxx) +G(t,x,u,u (x) ,u (xx) ) which admit semi-simple symmetry algebras and extensions of these semi-simple Lie algebras by solvable Lie algebras. The methods we employ are extensions and refinements of previous techniques which have been used in such classifications.

  • 204.
    Basarab-Horwath, Peter
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Güngör2, F.
    Istanbul Technical University, Turkey.
    Özemir, C.
    Istanbul Technical University, Turkey.
    Infinite-dimensional symmetries of a general class of variable coefficient evolution equations in 2+1 dimensions2013In: ISQS21, Institute of Physics Publishing (IOPP), 2013, Vol. 474Conference paper (Refereed)
    Abstract [en]

    We consider generalized KP-Burgers equations and attempt to identify subclasses admitting Virasoro or Kac-Moody type algebras as their symmetries. We give reductions to ODEs constructed from invariance requirement under these infinite-dimensional Lie symmetry algebras and integrate them in cases where it is possible. We also look at the conditions under which the equation passes the Painleve test and construct some exact solutions by truncation.

  • 205.
    Basarab-Horwath, Peter
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Lahno, V
    Group classification of the general quasi-linear wave equation: invance under low-dimensional Lie algebras2004Report (Other academic)
  • 206.
    Basarab-Horwath, Peter
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Lahno, V.
    Magda, O.
    The structure of Lie algebras and the classification problem for partial differential equations2004In: Proceedings of Institute of Mathematics of NAS of Ukraine.Mathematics and its Applications, Vol. 50, p. 40-46Article in journal (Refereed)
  • 207.
    Basarab-Horwath, Peter
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Lahno, V.
    Zhdanov, R.
    Classifying evolution equations2001In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 47, no 8, p. 5135-5144Article in journal (Refereed)
    Abstract [en]

    A Lie point symmetry classification of evolution equations in 1+1 time-space dimensions was presented. A combination of the standard Lie algorithm for point symmetry and the equivalence group of the given type of equation was used for the classification. For each canonical evolution the maximal symmetry algebra was calculated and related theorems were proved.

  • 208.
    Basarab-Horwath, Peter
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Lahno, V
    Linkoping Univ, S-58183 Linkoping, Sweden Pedag Univ, UA-314000 Poltava, Ukraine Inst Math, UA-252004 Kiev, Ukraine.
    Zhdanov, R
    Linkoping Univ, S-58183 Linkoping, Sweden Pedag Univ, UA-314000 Poltava, Ukraine Inst Math, UA-252004 Kiev, Ukraine.
    The structure of lie algebras and the classification problem for partial differential equations2001In: Acta Applicandae Mathematicae - An International Survey Journal on Applying Mathematics and Mathematical Applications, ISSN 0167-8019, E-ISSN 1572-9036, Vol. 69, no 1, p. 43-94Article in journal (Refereed)
    Abstract [en]

    The present paper solves completely the problem of the group classification of nonlinear heat-conductivity equations of the form u(t)=F(t,x,u,u(x))u(xx)+G(t,x,u,u(x)). We have proved, in particular, that the above class contains no nonlinear equations whose invariance algebra has dimension more than five. Furthermore, we have proved that there are two, thirty-four, thirty-five, and six inequivalent equations admitting one-, two-, three-, four- and five-dimensional Lie algebras, respectively. Since the procedure which we use relies heavily upon the theory of abstract Lie algebras of low dimension, we give a detailed account of the necessary facts. This material is dispersed in the literature and is not fully available in English. After this algebraic part we give a detailed description of the method and then we derive the forms of inequivalent invariant evolution equations, and compute the corresponding maximal symmetry algebras. The list of invariant equations obtained in this way contains (up to a local change of variables) all the previously-known invariant evolution equations belonging to the class of partial differential equations under study.

  • 209.
    Basarab-Horwath, Peter
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Zhdanov, R.Z.
    Institute of Mathematics, 3 Tereshchenkivska Street, 252004 Kyiv, Ukraine.
    Initial-value problems for evolutionary partial differential equations and higher-order conditional symmetries2001In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 42, no 1, p. 376-389Article in journal (Refereed)
    Abstract [en]

    We suggest a new approach to the problem of dimensional reduction of initial/ boundary value problems for evolution equations in one spatial variable. The approach is based on higher-order (generalized) conditional symmetries of the equations involved. It is shown that reducibility of an initial value problem for an evolution equation to a Cauchy problem for a system of ordinary differential equations can be fully characterized in terms of conditional symmetries which leave invariant the equation in question. We also give some examples of the solution of initial value problems for second- and third-order nonlinear differential equations by reduction by their conditional symmetries. We give a systematic classification of general second-order partial differential equations admitting second-order conditional symmetries, based on Lie's classification of invariant second-order ordinary differential equations. This yields five classes of principally new initial value problems for nonlinear evolution equations which admit no Lie symmetries and are reducible via second-order conditional symmetries. © 2001 American Institute of Physics.

  • 210.
    Bastani, Spencer
    et al.
    Linköping University, Department of Mathematics.
    Andersson, Olov
    Linköping University, Department of Mathematics.
    Stochastic Optimization in Dynamic Environments: with applications in e-commerce2007Independent thesis Advanced level (degree of Magister), 20 points / 30 hpStudent thesis
    Abstract [en]

    In this thesis we address the problem of how to construct an optimal algorithm for displaying banners (i.e advertisements shown on web sites). The optimization is based on the revenue each banner generates, with the aim of selecting those banners which maximize future total revenue. Banner optimality is of major importance in the e-commerce industry, in particular on web sites with heavy traffic. The 'micropayments' from showing banners add up to substantial profits due to the large volumes involved. We provide a broad, up-to-date and primarily theoretical treatment of this global optimization problem. Through a synthesis of mathematical modeling, statistical methodology and computer science we construct a stochastic 'planning algorithm'. The superiority of our algorithm is based on empirical analysis conducted by us on real internet-data at TradeDoubler AB, as well as test-results on a selection of stylized data-sets. The algorithm is flexible and adapts well to new environments.

  • 211.
    Becevic, Semir
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    Klassrumsbedömning i matematik på gymnasieskolans nivå2011Licentiate thesis, monograph (Other academic)
    Abstract [en]

    Classroom Assessment in Mathematics at secondary school level is a licentiate study with the intention to approach and interpret teachers’ actions, thoughts and experiences of assessment in mathematics. Another intention of the study is to, through qualitative research on teachers’ experiences of the mathematical courses, bridge the gap between the target-related course system and teachers ́ practical work. Finally, the intent is to provide qualitative formative feedback of the course assessment regarding teachers ́ evaluation strategies. A more qualitative feedback from evaluation, analysis, discussion and reflection will undoubtedly lead to improved mathematics assessment.There are many other factors that also affect teachers’ activities, thoughts and decisions within the assessment area of which few are only concerned within the study. In the present study, interviews and focus-group discussions have been conducted with a group of secondary mathematics teachers. All interviews were tape recorded and detailed analysed in order to give an insight into teachers ́ thoughts and action and further, their experiences of assessing students ́ mathematic knowledge. The analytic and theoretical method of the grounded theory has gradually led to eight assessment strategies and a theory of how the teachers describe and reflect on assessment in mathematics at upper secondary school. Those strategies were: the intuitive, the awaited, the continuous, the equivalent, the target-related, the test-oriented, the instruction-connected and self-assessment strategy. Finally the theory that was grounded out of the data brought a number of concepts and their categorizations. In turn the concepts and the categorizations have facilitated the understanding of the teachers’ activities and thinking in mathematics assessment at upper secondary school.The study gives an important insight into teachers ́ assessment strategies and the reason for their actions, something which has implication for mathematics education in upper secondary school as well as teacher education practices.

  • 212.
    Bengtsson, Ingemar
    et al.
    Stockholms Universitet, AlbaNova, Fysikum, Stockholm, Sweden.
    Bruzda, Wojciech
    Instytut Fizyki im. Smoluchowskiego, Uniwersytet Jagielloński, Kraków, Poland.
    Ericsson, Åsa
    Stockholms Universitet, AlbaNova, Fysikum, Stockholm, Sweden.
    Larsson, Jan-Åke
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Tadej, Wojciech
    Wydział Matematyczno-Przyrodniczy, Szkoła Nauk Ścisłych, Universytet Kardynała Stefana Wyszyńskiego, Warszawa, Poland.
    Zyczkowski, Karol
    Instytut Fizyki im. Smoluchowskiego, Uniwersytet Jagielloński, Kraków, Poland and Centrum Fizyki Teoretycznej, Polska Akademia Nauk, Warszawa, Poland.
    Mutually unbiased bases and Hadamard matrices of order six2007In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 48, no 5, p. 052106-1-052106-21Article in journal (Refereed)
    Abstract [en]

    We report on a search for mutually unbiased bases (MUBs) in six dimensions. We find only triplets of MUBs, and thus do not come close to the theoretical upper bound 7. However, we point out that the natural habitat for sets of MUBs is the set of all complex Hadamard matrices of the given order, and we introduce a natural notion of distance between bases in Hilbert space. This allows us to draw a detailed map of where in the landscape the MUB triplets are situated. We use available tools, such as the theory of the discrete Fourier transform, to organize our results. Finally, we present some evidence for the conjecture that there exists a four dimensional family of complex Hadamard matrices of order 6. If this conjecture is true the landscape in which one may search for MUBs is much larger than previously thought.

  • 213.
    Bengtsson, Martin
    Linköping University, Department of Mathematics.
    Analytical Expressions for the Hawking Mass in slowly rotating Kerr and Kerr-Newman Space-times2007Independent thesis Advanced level (degree of Magister), 20 points / 30 hpStudent thesis
    Abstract [en]

    Penrose's inequality which relates the total mass of a space-time containing a black hole with the area of the event horizon, is a yet unproven condition that is required for the cosmic censorship hypothesis. It is believed that the inequality could be proved by using properties of the Hawking mass. This thesis gives analytical expressions for the Hawking mass in slowly rotating Kerr and Kerr-Newman space-times. It is also shown that the expressions are monotonically increasing, a result that does not contradict Penrose's inequality.

  • 214.
    Bennevall, Marcus
    Linköping University, Department of Mathematics.
    Creativity in Mathematics Curricula – An International Comparison between Singapore, Hong Kong, Sweden, and Norway2017Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    Studies have shown that creative mathematically founded reasoning (CMR) outperforms algorithmic reasoning (AR) in regards to retention and (re)construction of knowledge. This suggests that creativity should be encouraged in national high-school mathematics curricula. The aim of the present study is to compare how creativity is framed in different national high-school mathematics curricula, using the following definition: creativity is the characteristics of people, processes, and environments which lead to new and original products that are useful or otherwise attractive to an individual or a society. Utilizing content and discourse analysis, the present study thus contrasts how the high-school mathematics curricula of Singapore, Hong Kong, Sweden, and Norway handle and value creativity, and also examines which role creativity takes in each curricula.

    Findings suggest that Singapore’s curriculum emphasizes creativity the most, and frequently does so in relation to assessment. Hong Kong’s curriculum is found to emphasize creativity in diverse ways, often using words with connotations to playfulness. Analysis of Sweden’s curriculum indicates a relatively minute focus on creativity, tending to put it in a teacher-centered context. A feature of Norway’s curriculum is an increasing emphasis on creativity as courses approach tertiary education. This also suggests a rising value of creativity in its curriculum. A similar though not as pronounced trajectory is found also in Singapore’s curriculum. In the Asian and Norwegian curricula, creativity is expressed both as a means and an end, while in Sweden’s curriculum it is only seen as an end.

    The results are discussed in terms of potential reasons for the prominent national features, and the study also includes an evaluation of the aptness of the suggested definition of creativity, a review of the limitations of the study, as well as propositions for further research. Finally, two recommendations are given to the National Agency for Education in Sweden – Skolverket – based on the results of the study: 1) diversify the emphasis on creativity in the curriculum, and 2) ensure alignment between what teachers value and what Skolverket values with respect to creativity.

  • 215.
    Bennevall, Marcus
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Cultivating Creativity in the Mathematics Classroom using Open-ended Tasks: A Systematic Review2016Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    Creativity is an ever relevant concept in problem-solving. Indeed, one could argue that no problem is really problematic unless it requires creative thinking; such problems can be solved by simply applying known facts and employing suitable algorithms. Yet, that is exactly how many tasks in mathematics textbooks are structured today. The present study aims to find other tasks, which can infuse creativity in the mathematics classroom. Special attention is turned towards a class of tasks known as open-ended tasks because of their creative potential. A literature review spanning 70 sources yields 17 types of open-ended tasks, and these are subsequently exemplified, classified, analyzed, and discussed from a teacher’s perspective.

  • 216.
    Berg, Jens
    et al.
    Uppsala University, Department of Information Technology.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    A stable and dual consistent boundary treatment using finite differences on summation-by-parts form2012In: European Congress on Computational Methods in Applied Sciences and Engineering, Vienna University of Technology , 2012Conference paper (Other academic)
    Abstract [en]

    This paper is concerned with computing very high order accurate linear functionals from a numerical solution of a time-dependent partial differential equation (PDE). Based on finite differences on summation-by-parts form, together with a weak implementation of the boundary conditions, we show how to construct suitable boundary conditions for the PDE such that the continuous problem is well-posed and the discrete problem is stable and spatially dual consistent. These two features result in a superconvergent functional, in the sense that the order of accuracy of the functional is provably higher than that of the solution.

  • 217.
    Berg, Jens
    et al.
    Uppsala University, Sweden.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Duality based boundary conditions and dual consistent finite difference discretizations of the Navier–Stokes and Euler equations2014In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 259, p. 135-153Article in journal (Refereed)
    Abstract [en]

    In this paper we derive new farfield boundary conditions for the time-dependent Navier–Stokes and Euler equations in two space dimensions. The new boundary conditions are derived by simultaneously considering well-posedess of both the primal and dual problems. We moreover require that the boundary conditions for the primal and dual Navier–Stokes equations converge to well-posed boundary conditions for the primal and dual Euler equations.

    We perform computations with a high-order finite difference scheme on summation-by-parts form with the new boundary conditions imposed weakly by the simultaneous approximation term. We prove that the scheme is both energy stable and dual consistent and show numerically that both linear and non-linear integral functionals become superconvergent.

  • 218.
    Berg, Jens
    et al.
    Uppsala University, Department of Information Technology.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Duality based boundary treatment for the Euler and Navier-Stokes equations2013In: AIAA Aerospace Sciences - Fluid Sciences Event, 2013, p. 1-19Conference paper (Other academic)
    Abstract [en]

    In this paper we construct well-posed boundary conditions for the compressible Euler and Navier-Stokes equations in two space dimensions. When also considering the dual equations, we show how to construct the boundary conditions so that both the primal and dual problems are well-posed. By considering the primal and dual problems simultaneously, we construct energy stable and dual consistent finite difference schemes on summation-by-  parts form with weak imposition of the boundary conditions.

    According to linear theory, the stable and dual consistent discretization can be used to compute linear integral functionals from the solution at a superconvergent rate. Here we evaluate numerically the superconvergence property for the non-linear Euler and Navier{ Stokes equations with linear and non-linear integral functionals.

  • 219.
    Berg, Jens
    et al.
    Uppsala University, Department of Information Technology, SE-751 05, Uppsala, Sweden.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    On the impact of boundary conditions on dual consistent finite difference discretizations2013In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 236, p. 41-55Article in journal (Refereed)
    Abstract [en]

    In this paper we derive well-posed boundary conditions for a linear incompletely parabolic system of equations, which can be viewed as a model problem for the compressible Navier{Stokes equations. We show a general procedure for the construction of the boundary conditions such that both the primal and dual equations are wellposed.

    The form of the boundary conditions is chosen such that reduction to rst order form with its complications can be avoided.

    The primal equation is discretized using finite difference operators on summation-by-parts form with weak boundary conditions. It is shown that the discretization can be made energy stable, and that energy stability is sufficient for dual consistency.

    Since reduction to rst order form can be avoided, the discretization is significantly simpler compared to a discretization using Dirichlet boundary conditions.

    We compare the new boundary conditions with standard Dirichlet boundary conditions in terms of rate of convergence, errors and discrete spectra. It is shown that the scheme with the new boundary conditions is not only far simpler, but also has smaller errors, error bounded properties, and highly optimizable eigenvalues, while maintaining all desirable properties of a dual consistent discretization.

  • 220.
    Berg, Jens
    et al.
    Division of Scientific Computing, Department of Information Technology, Uppsala University, Sweden.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Spectral analysis of the continuous and discretized heat and advection equation on single and multiple domains2012In: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 62, no 11, p. 1620-1638Article in journal (Refereed)
    Abstract [en]

    In this paper we study the heat and advectionequation in single and multipledomains. The equations are discretized using a second order accurate finite difference method on Summation-By-Parts form with weak boundary and interface conditions. We derive analytic expressions for the spectrum of the continuous problem and for their corresponding discretization matrices.

    It is shown how the spectrum of the singledomain operator is contained in the multi domain operator spectrum when artificial interfaces are introduced. The interface treatments are posed as a function of one parameter, and the impact on the spectrum and discretization error is investigated as a function of this parameter. Finally we briefly discuss the generalization to higher order accurate schemes.

  • 221.
    Berg, Jens
    et al.
    Uppsala University, Department of Information Technology.
    Nordström, Jan
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
    Stable Robin solid wall boundary conditions for the Navier-Stokes equations2011In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 230, no 19, p. 7519-7532Article in journal (Refereed)
    Abstract [en]

    In this paper we prove stability of Robin solid wall boundary conditions for the compressible Navier–Stokes equations. Applications include the no-slip boundary conditions with prescribed temperature or temperature gradient and the first order slip-flow boundary conditions. The formulation is uniform and the transitions between different boundary conditions are done by a change of parameters. We give different sharp energy estimates depending on the choice of parameters.

    The discretization is done using finite differences on Summation-By-Parts form with weak boundary conditions using the Simultaneous Approximation Term. We verify convergence by the method of manufactured solutions and show computations of flows ranging from no-slip to almost full slip.

  • 222.
    Berg, Jens
    et al.
    Uppsala University, Department of Information Technology, Sweden.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Superconvergent functional output for time-dependent problems using finite differences on summation-by-parts form2012In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 231, no 20, p. 6846-6860Article in journal (Refereed)
    Abstract [en]

    Finitedifference operators satisfying the summation-by-parts (SBP) rules can be used to obtain high order accurate, energy stable schemes for time-dependent partial differential equations, when the boundary conditions are imposed weakly by the simultaneous approximation term (SAT).

    In general, an SBP-SAT discretization is accurate of order p + 1 with an internal accuracy of 2p and a boundary accuracy of p. Despite this, it is shown in this paper that any linear functional computed from the time-dependent solution, will be accurate of order 2p when the boundary terms are imposed in a stable and dual consistent way.

    The method does not involve the solution of the dual equations, and superconvergent functionals are obtained at no extra computational cost. Four representative model problems are analyzed in terms of convergence and errors, and it is shown in a systematic way how to derive schemes which gives superconvergentfunctionaloutputs.

  • 223.
    Bergelin, Victor
    Linköping University, Department of Mathematics. Linköping University, Faculty of Science & Engineering.
    Human Activity Recognition and Behavioral Prediction using Wearable Sensors and Deep Learning2017Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    When moving into a more connected world together with machines, a mutual understanding will be very important. With the increased availability in wear- able sensors, a better understanding of human needs is suggested. The Dart- mouth Research study at the Psychiatric Research Center has examined the viability of detecting and further on predicting human behaviour and complex tasks. The field of smoking detection was challenged by using the Q-sensor by Affectiva as a prototype. Further more, this study implemented a framework for future research on the basis for developing a low cost, connected, device with Thayer Engineering School at Dartmouth College. With 3 days of data from 10 subjects smoking sessions was detected with just under 90% accuracy using the Conditional Random Field algorithm. However, predicting smoking with Electrodermal Momentary Assessment (EMA) remains an unanswered ques- tion. Hopefully a tool has been provided as a platform for better understanding of habits and behaviour. 

  • 224.
    Bergholm, Marie
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    Gymnasieelevers kommunikativa strategier i matematikklassrummet: En fallstudie av ett smågruppsarbete om derivata2014Licentiate thesis, monograph (Other academic)
    Abstract [en]

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

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

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

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

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

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

  • 225.
    Bergman, Johan
    et al.
    Pepto Systems AB, Stockholm, Sweden.
    Flisberg, Patrik
    Linköping University, Department of Mathematics, Optimization . Linköping University, The Institute of Technology.
    Rönnqvist, Mikael
    Linköping University, Department of Mathematics, Optimization . Linköping University, The Institute of Technology.
    Roll cutting at paper mills2002In: Proceedings of the Control Systems 2002, 2002, p. 159-163Conference paper (Other academic)
    Abstract [en]

    Roll cutting has attracted a great deal of research attention. However, most of the development of models and methods in academic research has not considered some key industrial requirements. These are practical production aspects that often create non-tractable optimization problems. In this paper, we study two important problems and show how these can be solved efficiently in an industrial setting. These are operative roll cutting where the number of reels and different patterns should be minimized and real-time cutting, when defects and quality restrictions on ordered rolls are included.

  • 226.
    Bergman, Tobias
    Linköping University, Department of Mathematics. Linköping University, Faculty of Science & Engineering.
    Läxa i matematik: Vad säger forskningen?: Vilka faktorer påverkar effekten av läxa i matematik?2019Independent thesis Basic level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [sv]

    Denna litteraturstudie undersöker vilka faktorer som påverkar effekten som läxa har på elevers kunskaper i  matematik. Studien visar att läxa är ett komplext fenomen som påverkas av många faktorer och även varierar beroende  på ämne. På grund av komplexiteten begränsas studien till att undersöka effekten av läxa på individnivå. Studien identifierar ett antal faktorer. Dessa faktorer visar sig ha en stark koppling till varandra och är svåra att skilja från varandra. Därför grupperas de i följande tre kategorier: lärarens syfte med läxan, läxans konstruktion samt elevens förutsättning att genomföra läxan. Studien visar även att läxa varken är strikt positiv eller negativ. Faktorer i alla kategorierna kan medföra att  elevens kunskaper påverkas positivt, negativt eller inte alls. Den viktigaste slutsatsen av studien är därför att man som  lärare bör planera sin läxa noggrant, ha ett explicit syfte och att ge tydlig och välgenomtänkt återkoppling

  • 227.
    Bergman Ärlebäck, Jonas
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Conformal Einstein spaces and Bach tensor generalization in n dimensions2004Licentiate thesis, monograph (Other academic)
    Abstract [en]

    In this thesis we investigate necessary and su±cient conditions for an n-dimensional space, n ≥ 4, to be locally conformal to an Einstein space. After reviewing the classical results derived in tensors we consider the four-dimensional spinor result of Kozameh, Newman and Tod. The involvement of the four-dimensional Bach tensor (which is divergence-free and conformally well-behaved) in their result motivates a search for an n-dimensional generalization of the Bach tensor Bab with the same properties. We strengthen a theorem due to Belfagón and Jaén and give a basis (Uab, V ab and Wab) for all n-dimensional symmetric, divergence-free 2-index tensors quadratic in the Riemann curvature tensor. We discover the simple relationship Bab = 1/2Uab + 1/6V ab and show that the Bach tensor is the unique tensor with these properties in four dimensions. Unfortunately we have to conclude, in general that there is no direct analogue in higher dimension with all these properties.

    Nevertheless, we are able to generalize the our-dimensional results due to Kozameh, Newman and Tod to n dimensions. We show that a generic space is conformal to an Einstein space if and only if there exists a vector field satisfying two conditions. The explicit use of dimensionally dependent identities (some of which are newly derived in this thesis) is also exploited in order to make the two conditions as simple as possible; explicit examples are given in five and six dimensions using these tensor identities. For n dimensions, we define the tensors babc and Bab, and we show that their vanishing is a conformal invariant property which guarantees that the space with non-degenerate Weyl tensor is a conformal Einstein space.

  • 228.
    Bergman Ärlebäck, Jonas
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    Conformal Einstein spaces and Bach tensor generalizations in n dimensions2004Licentiate thesis, monograph (Other academic)
    Abstract [en]

    In this thesis we investigate necessary and sufficient conditions for an n-dimensional space, n ≥ 4, to be locally conformal to an Einstein space. After reviewing the classical results derived in tensors we consider the four-dimensional spinor result of Kozameh, Newman and Tod. The involvement of the four-dimensional Bach tensor (which is divergence-free and conformally well-behaved) in their result motivates a search for an n-dimensional generalization of the Bach tensor Bab with the same properties. We strengthen a theorem due to Belfagón and Jaen and give a basis (Uab = Vab and Wab) for all n-dimensional symmetric, divergence-free 2-index tensors quadratic in the Riemann curvature tensor. We discover the simple relationship Bab = ½Uab + ⅙Vab and show that the Bach tensor is the unique tensor with these properties in four dimensions. Unfortunately we have to conclude, in general that there is no direct analogue in higher dimension with all these properties.

    Nevertheless, we are able to generalize the four-dimensional results due to Kozameh, Newman and Tod to n dimensions. We show that a generic space is conformal to an Einstein space if and only if there exists a vector field satisfying two conditions. The explicit use of dimensionally dependent identities (some of which are newly derived in this thesis) is also exploited in order to make the two conditions as simple as possible; explicit examples are given in five and six dimensions using these tensor identities.

    For n dimensions, we define the tensors 𝔟abc and 𝔅ab, and we show that their vanishing is a conformal invariant property which guarantees that the space with non-degenerate Weyl tensor is a conformal Einstein space.

  • 229.
    Bergman Ärlebäck, Jonas
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Designing, implementing and evaluating mathematical modelling modules at the upper secondary level2009Report (Other academic)
    Abstract [en]

    This study provides an account of how it is possible to work with mathematical models and modelling at the Swedish upper secondary level. It shows how a researcher and two teachers, departing from what is written in the mathematics curriculum document about mathematical models and modelling and the teachers’ mathematics teaching practices, collaborate to design and develop two so called modelling modules. These two modules designed for, and integrated in, the Mathematics C and Mathematics D course respectively, were implemented in the teachers’ classes and the whole process was evaluated. The students’ experiences of working with the modules were also investigated.

    The conceptual framework used to study this process of design, implementation, and evaluation is built up of design-based research methodology, cultural historical activity theory (CHAT), and co-learning agreement between the researcher and the participants.

    The teachers as well as the students expressed that they enjoyed working on the modules and found the experience positive and rewarding. However, some doubt concerning the learning outcome on behalf of the students were raised among the teachers, and the wish to incorporate more scheduled time into the modules was expressed by the students.

    It is concluded that the participating teachers could had difficulties in expressing a clear conception of the notions of mathematical models or modelling and no affirmative conclusions could be drawn that the participation of the teachers changed their attitudes towards mathematical models and modelling. The designing process was often was restrained by constraints originating from the local school context, and working with modelling highlights many systemic tensions in the established school practice. Also, meta-results in form of suggestions of how to resolve different kinds of tensions in order to improve the study design are reported.

  • 230.
    Bergman Ärlebäck, Jonas
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology. Linköping University, Faculty of Educational Sciences.
    Ett modell- och modelleringsperspektiv på lärande och undervisning2014In: Nämnaren : tidskrift för matematikundervisning, ISSN 0348-2723, no 4, p. 39-45Article in journal (Other (popular science, discussion, etc.))
    Abstract [sv]

    I denna artikel diskuterar och exemplifierar författaren tankeavslöjande aktiviteter som stöttar elever att utveckla och förfina sin förståelse av matematiska modeller. Ett modell- och modelleringsperspektiv används också i ett försök att fånga och beskriva kritiska aspekter av lärande och undervisning.

  • 231.
    Bergman Ärlebäck, Jonas
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Exploring the solving process of group solving realistic Fermi problems from the perspective of the Anthropological theory of didactics2011In: Proceedings of the Seventh Conference of European Research in Mathematics Education (CERME 7) / [ed] M. Pytlak, T. Rowland, & W Swoboda, 2011, p. 1010-1020Conference paper (Refereed)
    Abstract [en]

    This paper reports on a first attempt to use the notions of ‘Research and Study Course’ (RSC) and ‘praxeologies’ within the Anthropological Theory of the Didactics (ATD) to analyse groups of students engaged in the mathematical activity of solving realistic Fermi problems. By considering so called realistic Fermi problem as a generating question in a RSC the groups’ derived sub-questions are identified and the praxeologies developed to address these are discussed.

  • 232.
    Bergman Ärlebäck, Jonas
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Matematisk modellering i svenska gymnasieskolans kursplaner i matematik 1965-20002009Report (Other academic)
    Abstract [en]

    Denna rapport studerar hur begreppen matematisk modell och matematisk modellering behandlats i kursplanerna i matematik för motsvarande svenska gymnasieskolans naturvetenskapliga program mellan åren 1965 och 2000. En översikt över sex kursplaners olika form och innehåll ges och tre olika analyser på tre olika nivåer genomförs; en innehållsanalys, en kvalitativ analys, och en analys som beaktar relationen mellan begreppen modellering, tillämpningar och problemlösning. Analysen visar att matematiska modeller och modellering förekommit implicit sedan kursplanen från 1965, men att begreppen främst i de två senaste kursplanerna lyfts fram explicit och fått en central position i kursplanerna. I den nuvarande kursplanen Gy2000 framgår att matematiska modeller kan tolkas både som ett undervisningsmål i sig och som ett didaktiskt hjälpmedel att lära matematik.

  • 233.
    Bergman Ärlebäck, Jonas
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology. Linköping University, Faculty of Educational Sciences.
    Matematiska modeller och modellering - vad är det?2013In: Nämnaren : tidskrift för matematikundervisning, ISSN 0348-2723, no 3, p. 21-26Article in journal (Other (popular science, discussion, etc.))
    Abstract [sv]

    Vad är en matematisk modell? Vad innebär modellering? Denna artikel tar sig an dessa frågor, problematiserar bregreppen och undersöker hur de förhåller sig till angränsade begrepp som tillämpningar och problemlösning.

  • 234.
    Bergman Ärlebäck, Jonas
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Mathematical modelling in upper secondary mathematics education in Sweden2010Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    The aim of this thesis is to investigate and enhance our understanding of the notions of mathematical models and modelling at the Swedish upper secondary school level. Focus is on how mathematical models and modelling are viewed by the different actors in the school system, and what characterises the collaborative process of a didactician and a group of teachers engaged in designing and developing, implementing and evaluating teaching modules (so called modelling modules) exposing students to mathematical modelling in line with the present mathematics curriculum. The thesis consists of five papers and reports, along with a summary introduction, addressing both theoretical and empirical aspects of mathematical modelling.

    The thesis uses both qualitative and quantitative methods and draws partly on design-based research methodology and cultural-historical activity theory (CHAT). The results of the thesis are presented using the structure of the three curriculum levels of the intended, potentially implemented, and attained curriculum respectively.

    The results show that since 1965 and to the present day, gradually more and more explicit emphasis has been put on mathematical models and modelling in the syllabuses at this school level. However, no explicit definitions of these notions are provided but described only implicitly, opening up for a diversity of interpretations.

    From the collaborative work case study it is concluded that the participating teachers could not express a clear conception of the notions mathematical models or modelling, that the designing process often was restrained by constraints originating from the local school context, and that working with modelling highlights many systemic tensions in the established school practice. In addition, meta-results in form of suggestions of how to resolve different kinds of tensions in order to improve the study design are reported.

    In a questionnaire study with 381 participating students it is concluded that only one out of four students stated that they had heard about or used mathematical models or modelling in their education before, and the expressed overall attitudes towards working with mathematical modelling as represented in the test items were negative. Students’ modelling proficiency was positively affected by the students’ grade, last taken mathematics course, and if they thought the problems in the tests were easy or interesting. In addition empirical findings indicate that so-called realistic Fermi problems given to students working in groups inherently evoke modelling activities.

    List of papers
    1. Towards understanding teachers’ beliefs and affects about mathematical modelling
    Open this publication in new window or tab >>Towards understanding teachers’ beliefs and affects about mathematical modelling
    2009 (English)Conference paper, Published paper (Other academic)
    Abstract [en]

    Work in progress on a framework aiming at capturing teachers’ beliefs about mathematical models and modelling is presented. It is suggested that the belief structure of mathematical models and modelling as perceived by teachers fruitfully might be explored as partly constituted of the teachers’ beliefs about the real world, the nature of mathematics, school mathematics, and applying and applications of mathematics. Some aspects of the suggested framework are explored using two case study interviews. It is found that the two teachers do not have any well formed beliefs about mathematical models and modelling, and that the interpreted beliefs structure of the teachers contain inconsistencies which are made explicit within the framework. The empiric findings also suggest some modifications of the framework.

    National Category
    Pedagogy
    Identifiers
    urn:nbn:se:liu:diva-54316 (URN)
    Conference
    CERME 6 - Sixth Conference of European Research in Mathematics Education Lyon, France, January 28 - February1
    Available from: 2010-03-09 Created: 2010-03-09 Last updated: 2018-09-01
    2. On the use of realistic Fermi problems for introducingmathematical modelling in school
    Open this publication in new window or tab >>On the use of realistic Fermi problems for introducingmathematical modelling in school
    2009 (English)In: The Montana Mathematics Enthusiast, ISSN 1551-3440, E-ISSN 1551-3440, Vol. 6, no 3, p. 331-364Article in journal (Refereed) Published
    Abstract [en]

    In this paper an analytical tool referred to as the MAD (Modelling Activity Diagram) framework adapted from Schoenfeld’s parsing protocol coding scheme is used to address the issues of how to introduce mathematical modelling to upper secondary students. The work of three groups of students engaged in solving so called realistic Fermi problems were analysed using this framework, and it was observed that the processes involved in a typical mathematical modelling cycle were richly represented in the groups’ solving processes. The importance of the social interactions within the groups was noted, as well as the extensive use of extra-mathematical knowledge used by the students during the problem solving session.

    Place, publisher, year, edition, pages
    Montana Council of Teachers of Mathematics & Information Age Publishing, 2009
    Keywords
    Fermi problems, Fermi estimates, Modelling cycles, Mathematical modelling, Modelling activity diagram
    National Category
    Mathematics
    Identifiers
    urn:nbn:se:liu:diva-54664 (URN)
    Available from: 2010-03-30 Created: 2010-03-30 Last updated: 2017-12-12
    3. First results from a study investigating Swedish upper secondary students’ mathematical modelling competencies
    Open this publication in new window or tab >>First results from a study investigating Swedish upper secondary students’ mathematical modelling competencies
    2009 (English)Conference paper, Published paper (Other academic)
    Abstract [en]

    This paper reports on the first results from a study investigating Swedish upper secondary students’ (11th – 12th grade) mathematical modelling competency. Using non-parametric statistical methods the data from 381 students are analysed and the students’ modelling competency is described in terms of seven subcompetencies. Possible factors affecting the students’ mathematical competency such as attitudes toward modelling, previous experiences, last taken mathematics course, grade, class and gender were also investigated.

    National Category
    Mathematics
    Identifiers
    urn:nbn:se:liu:diva-54317 (URN)
    Conference
    14th International Conference on the Teaching of Mathematical Modelling and Applications (ICTMA 14), University of Hamburg, 27th to 31st July, Germany
    Available from: 2010-03-09 Created: 2010-03-09 Last updated: 2014-06-23Bibliographically approved
    4. Matematisk modellering i svenska gymnasieskolans kursplaner i matematik 1965-2000
    Open this publication in new window or tab >>Matematisk modellering i svenska gymnasieskolans kursplaner i matematik 1965-2000
    2009 (Swedish)Report (Other academic)
    Abstract [en]

    Denna rapport studerar hur begreppen matematisk modell och matematisk modellering behandlats i kursplanerna i matematik för motsvarande svenska gymnasieskolans naturvetenskapliga program mellan åren 1965 och 2000. En översikt över sex kursplaners olika form och innehåll ges och tre olika analyser på tre olika nivåer genomförs; en innehållsanalys, en kvalitativ analys, och en analys som beaktar relationen mellan begreppen modellering, tillämpningar och problemlösning. Analysen visar att matematiska modeller och modellering förekommit implicit sedan kursplanen från 1965, men att begreppen främst i de två senaste kursplanerna lyfts fram explicit och fått en central position i kursplanerna. I den nuvarande kursplanen Gy2000 framgår att matematiska modeller kan tolkas både som ett undervisningsmål i sig och som ett didaktiskt hjälpmedel att lära matematik.

    Place, publisher, year, edition, pages
    Linköping University: Linköping Universtity Electronic Press, 2009. p. 93
    Series
    LiTH-MAT-R, ISSN 0348-2960 ; 2009:8
    National Category
    Mathematics Computational Mathematics
    Identifiers
    urn:nbn:se:liu:diva-54267 (URN)LiTH-MAT-R-2009-8 (ISRN)
    Available from: 2010-03-05 Created: 2010-03-05 Last updated: 2014-06-23
    5. Designing, implementing and evaluating mathematical modelling modules at the upper secondary level
    Open this publication in new window or tab >>Designing, implementing and evaluating mathematical modelling modules at the upper secondary level
    2009 (English)Report (Other academic)
    Abstract [en]

    This study provides an account of how it is possible to work with mathematical models and modelling at the Swedish upper secondary level. It shows how a researcher and two teachers, departing from what is written in the mathematics curriculum document about mathematical models and modelling and the teachers’ mathematics teaching practices, collaborate to design and develop two so called modelling modules. These two modules designed for, and integrated in, the Mathematics C and Mathematics D course respectively, were implemented in the teachers’ classes and the whole process was evaluated. The students’ experiences of working with the modules were also investigated.

    The conceptual framework used to study this process of design, implementation, and evaluation is built up of design-based research methodology, cultural historical activity theory (CHAT), and co-learning agreement between the researcher and the participants.

    The teachers as well as the students expressed that they enjoyed working on the modules and found the experience positive and rewarding. However, some doubt concerning the learning outcome on behalf of the students were raised among the teachers, and the wish to incorporate more scheduled time into the modules was expressed by the students.

    It is concluded that the participating teachers could had difficulties in expressing a clear conception of the notions of mathematical models or modelling and no affirmative conclusions could be drawn that the participation of the teachers changed their attitudes towards mathematical models and modelling. The designing process was often was restrained by constraints originating from the local school context, and working with modelling highlights many systemic tensions in the established school practice. Also, meta-results in form of suggestions of how to resolve different kinds of tensions in order to improve the study design are reported.

    Place, publisher, year, edition, pages
    Linköping University: Linköping Universtiy Electronic Press, 2009. p. 219
    National Category
    Mathematics Computational Mathematics
    Identifiers
    urn:nbn:se:liu:diva-54269 (URN)LiTH-MAT-R-2009-9 (Local ID)LiTH-MAT-R-2009-9 (Archive number)LiTH-MAT-R-2009-9 (OAI)
    Available from: 2010-03-05 Created: 2010-03-05 Last updated: 2014-06-23
  • 235.
    Bergman Ärlebäck, Jonas
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    On the use of realistic Fermi problems for introducingmathematical modelling in school2009In: The Montana Mathematics Enthusiast, ISSN 1551-3440, E-ISSN 1551-3440, Vol. 6, no 3, p. 331-364Article in journal (Refereed)
    Abstract [en]

    In this paper an analytical tool referred to as the MAD (Modelling Activity Diagram) framework adapted from Schoenfeld’s parsing protocol coding scheme is used to address the issues of how to introduce mathematical modelling to upper secondary students. The work of three groups of students engaged in solving so called realistic Fermi problems were analysed using this framework, and it was observed that the processes involved in a typical mathematical modelling cycle were richly represented in the groups’ solving processes. The importance of the social interactions within the groups was noted, as well as the extensive use of extra-mathematical knowledge used by the students during the problem solving session.

  • 236.
    Bergman Ärlebäck, Jonas
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Towards understanding teachers’ beliefs and affects about mathematical modelling2009Conference paper (Other academic)
    Abstract [en]

    Work in progress on a framework aiming at capturing teachers’ beliefs about mathematical models and modelling is presented. It is suggested that the belief structure of mathematical models and modelling as perceived by teachers fruitfully might be explored as partly constituted of the teachers’ beliefs about the real world, the nature of mathematics, school mathematics, and applying and applications of mathematics. Some aspects of the suggested framework are explored using two case study interviews. It is found that the two teachers do not have any well formed beliefs about mathematical models and modelling, and that the interpreted beliefs structure of the teachers contain inconsistencies which are made explicit within the framework. The empiric findings also suggest some modifications of the framework.

  • 237.
    Bergman Ärlebäck, Jonas
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Using a Models and Modeling Perspective (MMP) to frame and combine research, practice- and teachers’ professional development2017In: Proceedings of the Tenth Congress of European Research in Mathematics Education  / [ed] T. Dooley, & G. Gueudet, Ireland, Dublin, 2017Conference paper (Refereed)
    Abstract [en]

    This paper describes and discusses the framing of, and experiences from, a project that combine research, practice- and teachers’ professional development based on the tenets of the Models andModeling Perspective on teaching and learning (MMP). Besides providing a general description of the methodological considerations in the project design, the paper describes how the accumulated results and experiences in the research literature on so-called model eliciting activities are used to inform the design, implementation and evaluation of activities aiming at introducing functions to grade 8 students. The focus of the paper is on the implantation and aim to showcase how the teacher in question realized the offered perspective and tools in practice.

  • 238.
    Bergman Ärlebäck, Jonas
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Albarracín, Lluís
    Serra Húnter Fellow, Universitat Autònoma de Barcelona, Spain.
    Developing a classification scheme of definitions of Fermi problems in education from a modelling perspective2017In: Proceedings of the Tenth Congress of European Research in Mathematics Education  / [ed] T. Dooley, & G. Gueudet, Ireland, Dublin, 2017, p. 884-891Conference paper (Refereed)
    Abstract [en]

    In this paper we use a modelling perspective to analyse three descriptions and definitions of so- called Fermi problems found in the literature. We discuss how the three definitions align with, and what they potentially have to offer to, realistic or applied modelling, contextual modelling,educational modelling (either a didactical or conceptual), socio-critical modelling, epistemological or theoretical modelling, and cognitive modelling. Our findings show that the definitions share some similarities, but for the most part are formulated in lose terms. From a modelling perspective, we found that the conceptualisation of Fermi problem we studied foremost and directly align with contextual modelling and both strands of educational modelling. We also discuss the seemly incompatibility between Fermi problems and the other modelling perspectives, and suggest new lines of research on Fermi problems in particular, and on conceptualizing modelling in general.

  • 239.
    Bergman Ärlebäck, Jonas
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Bergsten, Christer
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Educational Sciences.
    On the Use of Realistic Fermi Problems in Introducing Mathematical. Modelling in Upper Secondary Mathematics2010In: Modeling Students' Mathematical Modeling Competencies, ICTMA 13 / [ed] Richard Lesh, Peter L. Galbrait, Christopher R. Haines and Andrew Hurford, Springer US , 2010, p. 597-609Conference paper (Refereed)
    Abstract [en]

    This paper reports on a study aiming to investigate the possible potential of using Fermi problems to introduce mathematical modelling to Swedish upper secondary school students. Three groups of students- work on realistic Fermi problems were analysed using an analytic tool, referred to as the -modelling activity diagram-, adapted and developed by the author. It was observed that the processes involved in a mathematical modelling cycle were richly represented in the problem solving processes of the groups. The students also frequently used their personal extra-mathematical knowledge in the solving process in three different ways; a creative, a verifying and a social way.

  • 240.
    Bergman Ärlebäck, Jonas
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Blomberg, Per
    Örebro University.
    Nilsson, Per
    Linnaeus University, Sweden.
    An instructional design perspective on data-modelling for learning statistics and modelling2015In: Development of mathematics teaching: Design, Scale, Effects : Proceedings of MADIF 9, The Ninth mathematics Education Research Seminar, Umeå, 2015, p. 37-46Conference paper (Refereed)
    Abstract [en]

    This theoretical and methodological paper discusses the emerging theoretical framing and methodological considerations in our efforts to develop a theoretical approach supporting instructional design for teaching and learning statistics and mathematical modelling. From an instructional design point of view aligned with the goals in governing curricula documents and real classroom constraints, we argue for the integration of the models and modelling perspective on teaching and learning mathematics with a data-modelling approach to facilitate students’ learning statistics and mathematical modelling. An application of the framework is given and future research discussed.

  • 241.
    Bergman Ärlebäck, Jonas
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Claesson, Marcus
    Kungsgårdsgymnasiet Norrköping.
    Hällgren, Malin
    Birgittaskolan vuxenutbildning Linköping.
    Lindh, Sofia
    Hagagymnasiet Norrköping.
    Lundell, Mia
    Djäkneparksskolan Norrköping.
    Nordling, Mattias
    Anders Ljungstedts gymnasium Linköping.
    IPAs - ihopparningsaktiviteter: aktiviteter med låg tröskel som tränar elevers begrepps-, procedur-/ metod- och resonemangsförmåga2017Report (Other academic)
    Abstract [sv]

    Syftet med denna rapport är att på ett beskrivande sätt delge hur arbetet i projektet om IPAs som genomförts HT2014 – VT2015 gått till och vilka erfarenheter och slutsatser som arbetet resulterat i. Den tänkte läsaren är framför allt kommunernas matematiklärare, men vi tror även att andra lärare och skolpersonal kan finna innehållet intressant och användbart. Det är vår förhoppning att rapporten med tillhörande aktiviteter kan inspirera till reflektion och en mer varierad undervisning. När pronomen som vi och oss används i rapporten åsyftats alla som aktivt ingått i IPAprojektets forskande grupp, vilken har bestått av såväl deltagande lärare som forskare:

    Marcus Claesson (Kungsgårdsgymnasiet, Norrköping), Malin Hällgren (Birgittaskolans vuxenutbildning, Linköping), Sofia Lindh (Hagagymnasiet, Norrköping), Mia Lundell (Djäkneparksskolan, Norrköping), Mattias Nordling (Anders Ljungstedska gymnasiet, Linköping) och Jonas Bergman Ärlebäck (LiU). Huvudansvaret för författandet av rapporten har Jonas haft som medverkande forskare i projektet. De medforskande lärarna har bidragit med analyser, erfarenheter och exempel från sina klassrum, utvecklat, testat och formulerat aktiviteterna som utgjort navet i projektet, samt läst och kommenterat rapporten under dess framställning.

    Rapporten följer samma logik och uppbyggnad som de rapporter skrivna inom det övergripande samarbetsprojektet Matematikdidaktik för bättre matematikkunskaper av det arbete och den forskning som bedrivits av Lisa Björklund Boistrup (Björklund Boistrup et al. 2013a; 2013b; 2013c; 2014a; 2014b). Vissa delar av texten använder därför, eller bygger till viss del på, Lisas texter, och när så är fallet tydliggörs detta med fotnoter. Andra delar av texten, så som exempelvis beskrivningen av de teoretiska utgångspunkterna, är i stort identiskt i samtliga delprojektsrapporter för de projekt som Jonas medverkat i som forskare (Ärlebäck et. al. 2017a, 2017b, 2017c, 2017d). Rapporten är dock skriven för att kunna läsas helt fristående från de andra rapporterna författade i projektet och förutsätter med andra ord ingen tidigare kunskap eller kännedom om projektet och de perspektiv som används. De läsare som redan tagit del av arbetet inom samarbetsprojektet Matematikdidaktik för bättre matematikkunskaper på olika sätt kommer här och var att känna igen sig, men vi hoppas att de delar som är unikt specifika för just detta delprojekt ska vara intressant och inspirerande.

  • 242.
    Bergman Ärlebäck, Jonas
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Doerr, Helen M.
    Syracuse University.
    At the core of modelling: Connecting, coordinating and integrating models2015In: Proceedings of the 9th Congress of European Research in Mathematics Education / [ed] K. Krainer & N. Vondrová, 2015, p. 802-808Conference paper (Refereed)
    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.

  • 243.
    Bergman Ärlebäck, Jonas
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Doerr, Helen M.
    Department of Mathematics, Syracuse University, Syracuse, NY, USA.
    Moving beyond a single modelling activity2015In: Mathematical modelling in educational research and practice: Cultural, social and cognitive influences Part III / [ed] Gloria Ann Stillman, Werner Blum and Maria Salett Biembengut, asdf: Springer, 2015, p. 293-303Chapter in book (Refereed)
    Abstract [en]

    In this theoretical chapter, we draw on a models and modelling perspective on teaching and learning to elaborate on the components of model development sequences using the lens of variation theory. We give empirical examples of how model exploration activities and model application activities can be described and understood from a variation theory perspective. The chapter concludes by presenting tentative principles for the design of such activities within a model development sequence.

  • 244.
    Bergman Ärlebäck, Jonas
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Doerr, Helen M.
    Syracuse University.
    Preserving students’ independence by encouraging students self-evaluation2015In: Nordic research in mathematics education – Proceedings of NORMA14, Turku: The Finnish Research Association for Subject Didactics , 2015, p. 257-266Conference paper (Refereed)
  • 245.
    Bergman Ärlebäck, Jonas
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Doerr, Helen M.
    Department of Mathematics, Syracuse University, Syracuse, USA.
    Students’ interpretations and reasoning about phenomena with negative rates of change throughout a model development sequence2018In: ZDM - the International Journal on Mathematics Education, ISSN 1863-9690, E-ISSN 1863-9704, Vol. 50, no 1-2, p. 187-200Article in journal (Refereed)
    Abstract [en]

    In this article, we examine how a sequence of modeling activities supported the development of students’ interpretations and reasoning about phenomena with nega- tive average rates of change in different physical phenomena. Research has shown that creating and interpreting models of changing physical phenomena is difficult, even for university level students. Furthermore, students’ reasoning about mod- els of phenomena with negative rates of change has received little attention in the research literature. In this study, 35 students preparing to study engineering participated in a 6-week instructional unit on average rate of change that used a sequence of modeling activities. Using an analysis of the students’ work, our results show that the sequence of mod- eling activities was effective for nearly all students in reason- ing about motion with negative rates along a straight path. Almost all students were successful in constructing graphs of changing phenomena and their associated rate graphs in the contexts of motion, light dispersion and a discharging capacitor. Some students encountered new difficulties in interpreting and reasoning with negative rates in the con- texts of light dispersion, and new graphical representations emerged in students’ work in the context of the discharg- ing capacitor with its underlying exponential structure. The results suggest that sequences of modeling activities offer a structured approach for the instruction of advanced math- ematical content.

  • 246.
    Bergman Ärlebäck, Jonas
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology. Linköping University, Faculty of Educational Sciences.
    Doerr, Helen M.
    Syracuse University, NY, USA .
    O'Neil, AnnMarie H.
    Syracuse University, NY, USA .
    A Modeling Perspective on Interpreting Rates of Change in Context2013In: Mathematical Thinking and Learning, ISSN 1098-6065, E-ISSN 1532-7833, Vol. 15, no 4, p. 314-336Article in journal (Refereed)
    Abstract [en]

    Functions provide powerful tools for describing change, but research has shown that students find difficulty in using functions to create and interpret models of changing phenomena. In this study, we drew on a models and modeling perspective to design an instructional approach to develop students abilities to describe and interpret rates of change in the context of exponential decay. In this article, we elaborate the characteristics of the model development sequence and we examine how students interpreted and described non-constant rates of change in context. We provide evidence for how a focus on the context made visible students reasoning about rates of change, including difficulties related to the use of language when describing changes in the negative direction. We argue that context and the use of language, forefronted in a modeling approach, should play an important role in supporting the development of students reasoning about changing phenomena.

  • 247.
    Bergman Ärlebäck, Jonas
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Edgar, Brian
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Herberthson, Magnus
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    The Bach tensor and other divergence-free tensors2005In: International Journal of Geometric Methods in Modern Physics (IJGMMP), ISSN 0219-8878, Vol. 2, no 1, p. 13-21Article in journal (Refereed)
    Abstract [en]

    In four dimensions, we prove that the Bach tensor is the only symmetric divergence-free 2-tensor which is also quadratic in Riemann and has good conformal behavior. In n > 4 dimensions, we prove that there are no symmetric divergence-free 2-tensors which are also quadratic in Riemann and have good conformal behavior, nor are there any symmetric divergence-free 2-tensors which are concomitants of the metric tensor gab together with its first two derivatives, and have good conformal behavior.

  • 248.
    Bergman Ärlebäck, Jonas
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Frejd, Peter
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    First results from a study investigating Swedish upper secondary students' mathematical modelling competencies2010In: EIMI 2010 Conference: Educational interfaces between mathematics and industry : proceedings / [ed] Araújo, Adérito; Fernandes, António; Azevedo, Assis; Rodrigues, José Francisco, Lissabon: Centro Internacional de Matemática , 2010, p. 63-74Conference paper (Refereed)
  • 249.
    Bergman Ärlebäck, Jonas
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology. Linköping University, Faculty of Educational Sciences.
    Frejd, Peter
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Modelling from the Perspective of Commognition – An emerging framework2013In: Teaching mathematical modelling: Connecting to research and Practice / [ed] Gloria Ann Stillman, Gabriele Kaiser, Werner Blum, Jill P. Brown, Springer Netherlands, 2013, p. 47-56Chapter in book (Refereed)
    Abstract [en]

    This bookprovides readers with an overview of recent international research and developments in the teaching and learning of modelling and applications from a variety of theoretical and practical perspectives. There is a strong focus on pedagogical issues for teaching and learning of modelling as well as research into teaching and practice. The teaching of applications of mathematics and mathematical modelling from the early years through primary and secondary school and at tertiary level is rising in prominence in many parts of the world commensurate with an ever-increasing usage of mathematics in business, the environment, industry and everyday life. The authors are all members of the International Community of Teachers of Mathematical Modelling and Applications and important researchers in mathematics education and mathematics. The book will be of interest to teachers, practitioners and researchers in universities, polytechnics, teacher education, curriculum and policy.

  • 250.
    Bergman Ärlebäck, Jonas
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Frejd, Peter
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    The bottleneck problem in modelling revisitedManuscript (preprint) (Other academic)
    Abstract [en]

    In this paper Sfard's theory of commognition is used to investigate how two students work collaboratively to develop mathematical models in a natural school setting. The students' communication is analysed from both an insider and an outsider perspective to recreate the construction of their developed models in terms of realization trees. The analysis shows how the students draw on different discourses and that the models evolved through activities of recognition and negotiation. In particular, the students formulated their mathematical models using already known mathematical models. However, these latter models were tacit models in that they were not explicitly mentioned or discussed during the developmental process. To facilitate students’ capability to productively engage in mathematical modelling, the result of this study points to the need renegotiate the learning-teaching agreement so that it incorporates activities offering the students opportunities to practice negotiation and recognition.

2345678 201 - 250 of 2483
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