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• 1.
Mälardalen University, School of Education, Culture and Communication.
Risk-Efficient Portfolios; Estimation Error In Essence2016Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis

This thesis primarily looks at estimation error problems and other related issues arising in connection with portfolio optimization. With some available assets, a portfolio program or optimizer seeks to distribute a fixed amount of capital among these available assets to optimize some cost function. In this regard, Markowitz portfolio selection basis defines the variance of the portfolio return to being that of the portfolio risk and tries to find an allocation that reduces or minimizes the risk subject to a target mean or expected return. Should the mean return vector and the covariance matrix of returns for the underlying assets be known, the Markowitz problem is said to have a closed-form solution.

In practice, however, an estimation is made from historical data for unknown expected returns and the covariance matrix of the returns, and this brings into the domain several problems such as estimation problems and renders the Markowitz theory impracticable in real-life portfolio applications. Estimators necessary to remedy these problems would be made bare to show how possible it is to tackle such issues.

In the concept demonstration sections, the analysis starts with the price data of 40 stocks and the S\&P index. The efficient frontier is introduced and used to show how the estimators take effect.

Finally, implementation is made possible using the R Programming Language to demonstrate the necessary concepts with the conclusion presented at the end.

Matematikverkstad: hur kan lärare och elever arbeta där?2010Student paper other, 10 credits / 15 HE creditsStudent thesis

The starting point of this work has been to get knowledge about teachers working in a mathematical workshop with the students. I also wanted to investigate what students think about math, both within the workshop and in traditional teaching.

To achieve this I chose to interview a teacher in charge within the framework of a mathematical workshop, and six students in the 5th grade. I also did a lesson observation with the purpose to see how well the teacher’s tutoring agrees with the results of the interview.

In this study I have discovered that the lessons in the mathematical workshop are planned after what the students are working with in the textbook. The teacher opens the lessons with the whole class and then splits it up in groups. In the interview the respondent conveys the importance of discussions among the students where they can explain their thoughts, especially for the weaker students that, thanks to the discussions are able to show their knowledge more. This is something I did not see during my lesson observation, however. The students also seemed to miss this when I spoke to them in the interview. They describe that they cannot see the connection between what they learn in the workshop and the textbook.

• 3.
Stockholm University, Faculty of Science, Department of Mathematics.
Hardy and spectral inequalities for a class of partial differential operators2014Doctoral thesis, monograph (Other academic)

This thesis is devoted to the study of Hardy and spectral inequalities for the Heisenberg and the Grushin operators. It consists of five chapters. In chapter 1 we present basic notions and summarize the main results of the thesis. In chapters 2-4 we deal with different types of Hardy inequalities for Laplace and Grushin operators with magnetic and non-magnetic fields. It was shown in an article by Laptev and Weidl that for some magnetic forms in two dimensions, the Hardy inequality holds in its classical form. More precisely, by considering the Aharonov-Bohm magnetic potential, we can improve the constant in the respective Hardy inequality. In chapter 2 we establish an Lp - Hardy inequality related to Laplacians with magnetic fields with Aharonov-Bohm vector potentials. In chapter 3 we introduce a suitable notion of a vector field for the Grushin sub-elliptic operator G and obtain an improvement of the Hardy inequality, which was previously obtained in the paper of N. Garofallo and E. Lanconelli. In chapter 4 we find an Lp version of the Hardy inequality obtained in chapter 2. Finally in chapter 5 we aim to find the CLR and Lieb-Thirringbninequalities for harmonic Grushin-type operators. As the Grushin operator is non-elliptic, these inequalities will not take their classical form.

• 4.
Linnaeus University, Board of Education Science.
Hur används den utökade resursen till matematikämnet i grundskolan?2014Independent thesis Basic level (university diploma), 10 credits / 15 HE creditsStudent thesis
• 5.
Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
Elevers möjlighet att utveckla matematiska förmågor utifrån läromedlet Pixel2012Student paper other, 10 credits / 15 HE creditsStudent thesis

Syftet med arbetet är att ta reda på i vilken utsträckning elever ges förutsättningar att utveckla de olika förmågor som beskrivs i kursplanen för ämnet matematik, LGR11, när undervisningen bygger på ett vanligt förekommande läromedel, Pixel. Metoden är kvalitativ och det har skett en textanalys på geometriavsnitten i läromedlet. De förmågor som eleverna når upp till är att använda sig av olika matematiska begrepp, samt att formulera och lösa problem. Eleverna har goda möjligheter att själva kunna välja olika sätt att lösa uppgifter. De får även kunskap att förstå och använda olika uttrycksformer. Eleverna uppmuntras inte med hjälp av detta läromedel, att föra egna matematiska diskussioner.

• 6.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
Matematik från grunden: sortering och klassificering med de yngsta barnen i förskolan2009Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis

Syftet med examensarbetet är att med hjälp av enkätintervjuer och observationer få mer kunskap om hur man arbetar med sortering och klassificering med de yngsta barnen i förskolan. Vi har intervjuat arbetslag som nyligen fått kompetensutveckling i matematik. I vår bakgrund har vi utifrån ett teoretiskt perspektiv förklarat sortering och klassificering, pedagogens roll samt den pedagogiska miljön samt hur dessa faktorer kan påverka barns matematiska förmåga.  Resultatet visar att det förekommer mycket sortering och klassificering i verksamheten på förskolor i vardagsmatematiken. Pedagogerna i undersökningen menar att de vill ha mer kunskap om matematiska begrepp för att på ett bättre sätt ge barnen kunskap i matematik. Undersökningen visar hur betydelsefull pedagogen är och denna roll är inte alltid pedagogerna ute i verksamheten medvetna om.

• 7.
Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
Hur kan elevers kunskapsutveckling i matematik förbättras: Formativ bedömning i matematikundervisning2010Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis

Syftet med examensarbetet är att fördjupa vår kunskap om och förståelse för formativ bedömning i matematik. Vidare vill vi undersöka om den formativa bedömningen kan förbättra eleverna i klassens kunskapsutveckling i matematik. Vi har gjort en fallstudie i en klass som arbetar formativt för att undersöka hur det kan se ut i praktiken. Underlaget för det samlade materialet består av observation och intervju för att besvara våra två frågeställningar som följer. Hur kan en formativ bedömning se ut i praktiken? Hur kan en koppling mellan lärares och elevers uppfattningar om den pedagogiska verksamheten se ut? Vi har kommit fram till att den formativa bedömningen kan förbättra eleverna i klassens kunskapsutveckling i matematik. I den formativa bedömningen har vi sett vikten av mötet mellan lärare och elev. Att arbeta formativt är tidskrävande.

• 8.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
Analytical and Numerical Study of the Poincaré Map with Applications on the Computation of Periodic Orbits2015Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
• 9. Aleksanyan, Hayk
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
Applications of Fourier Analysis in Homogenization of the Dirichlet Problem: L-p Estimates2015In: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 215, no 1, 65-87 p.Article in journal (Refereed)

Let u(epsilon) be a solution to the system div(A(epsilon)(x)del u(epsilon)(x)) = 0 in D, u(epsilon)(x) = g(x, x/epsilon) on partial derivative D, where D subset of R-d (d >= 2), is a smooth uniformly convex domain, and g is 1-periodic in its second variable, and both A(epsilon) and g are sufficiently smooth. Our results in this paper are twofold. First we prove L-p convergence results for solutions of the above system and for the non-oscillating operator A(epsilon)(x) = A(x), with the following convergence rate for all 1 <= p < infinity parallel to u(epsilon) - u(0)parallel to (LP(D)) <= C-P {epsilon(1/2p), d = 2, (epsilon vertical bar ln epsilon vertical bar)(1/p), d = 3, epsilon(1/p), d >= 4, which we prove is (generically) sharp for d >= 4. Here u(0) is the solution to the averaging problem. Second, combining our method with the recent results due to Kenig, Lin and Shen (Commun Pure Appl Math 67(8): 1219-1262, 2014), we prove (for certain class of operators and when d >= 3) ||u(epsilon) - u(0)||(Lp(D)) <= C-p[epsilon(ln(1/epsilon))(2)](1/p) for both the oscillating operator and boundary data. For this case, we take A(epsilon) = A(x/epsilon), where A is 1-periodic as well. Some further applications of the method to the homogenization of the Neumann problem with oscillating boundary data are also considered.

• 10.
Izmir University, Izmir, Turkey. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
On the global solution of a fuzzy linear system2014In: Journal of Fuzzy Set Valued Analysis, ISSN 2193-4169, jfsva-00190Article in journal (Other academic)

The global solution of a fuzzy linear system contains the crisp vector solution of a real linear system. So discussion about the global solution of a $n\times n$ fuzzy linear system $A\tilde{x}=\tilde{b}$ with a fuzzy number vector $b$ in the right hand side and crisp a coefficient matrix $A$ is considered. The advantage of the paper is developing a new algorithm to find the solution of such system by considering a global solution based upon the concept of a convex fuzzy numbers. At first the existence and uniqueness of the solution are introduced and then the related theorems and properties about the solution are proved in details. Finally the method is illustrated by solving some numerical examples.

• 11.
Stockholm University, Faculty of Science, Department of Mathematics and Science Education, PRIM-gruppen.
På upptäcktsfärd i elevernas  värld av tal2007In: Matematikdidaktiska texter – Beprövad erfarenhet och vetenskaplig grund / [ed] Lena Alm ..., Stockholm: PRIM-gruppen, Institutionen för undervisningsprocesser, kommunikation och lärande, Lärarhögskolan i Stockholm , 2007, 43-55 p.Chapter in book (Other academic)
• 12. Almquist, Martin
KTH, School of Engineering Sciences (SCI), Aeronautical and Vehicle Engineering, Marcus Wallenberg Laboratory MWL.
Atmospheric Sound Propagation Over Large-Scale Irregular Terrain2014In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 61, no 2, 369-397 p.Article in journal (Refereed)

A benchmark problem on atmospheric sound propagation over irregular terrain has been solved using a stable fourth-order accurate finite difference approximation of a high-fidelity acoustic model. A comparison with the parabolic equation method and ray tracing methods is made. The results show that ray tracing methods can potentially be unreliable in the presence of irregular terrain.

• 13.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
Harmonic Lyapunov functions in the analysis of periodically switched systems2006In: PROCEEDINGS OF THE 45TH IEEE CONFERENCE ON DECISION AND CONTROL, VOLS 1-14, 2006, 2759-2764 p.Conference paper (Refereed)

The dynamic phasor model of a time-periodic system is used to derive a stability test involving a harmonic Lyapunov function. This reveals a new interpretation of the harmonic Lyapunov function with an appealing time-domain representation. Most importantly, it indicates that the ideas behind the harmonic Lyapunov equation can be generalized to include cyclic switching systems that have different pulse form in each period.

• 14.
Linnaeus University, Faculty of Engineering and Technology, Department of Mathematics Education.
Muntlig bedömning: Med fokus på elevers representationer2013Independent thesis Advanced level (professional degree), 20 credits / 30 HE creditsStudent thesis

Muntlig bedömning är, från och med höstterminen 2011, något som alla matematiklärare inom gymnasieskolan kommer att få genomföra då en muntlig del läggs som obligatorisk för alla elever inom årskurs 1. Syftet med den här studien är därför att konstruera kriterier som lärare kan använda som hjälp vid muntlig bedömning. Fokus kommer att ligga på elevers representationsformer och transformationer däremellan utifrån ett sociokulturellt perspektiv. Elevgrupper filmades då de löste en problemlösningsuppgift och analyserades utifrån ett begreppsligt ramverk. Resultatet visar att elevers transformationer mellan representationer visar både missuppfattningar och förståelse beroende på hur dessa transformationer ter sig. Elever som obehindrat lyckas vandra mellan olika representationer visar på god förståelse för det matematiska innehållet. Även elevers förmåga att skapa sig ett multimodalt arbetssätt visar sig, för läraren, vara värd att uppmärksamma.

• 15.
KTH, School of Electrical Engineering (EES), Electric Power Systems.
Texas Tech University.
The Feedback Control of Glucose: On the road to type II diabetes2006In: Proceedings of the 45th IEEE Conference on Decision & Control, 2006, 685-690 p.Conference paper (Refereed)

This paper develops a mathematical model for the feedback control of glucose regulation in the healthy human being and is based on the work of Sorensen (1985). The proposed model serves as a starting point for modeling type H diabetes. Four agents - glucose and the three hormones insulin, glucagon, and incretins - are assumed to have an effect on glucose metabolism. By letting compartments represent anatomical organs, the model has a close resemblance to a real human body. Mass balance equations that account for blood flows, exchange between compartments, and metabolic sinks and sources are written, and these result in simultaneous differential equations that are solved numerically. The metabolic sinks and sources - removing or adding glucose, insulin, glucagon, and incretins - describe physiological processes in the body. These processes function as feedback control systems and have nonlinear behaviors. The results of simulations performed for three different clinical test types indicate that the model is successful in simulating intravenous glucose, oral glucose, and meals containing mainly carbohydrates.

• 16.
Dalarna University, School of Education and Humanities, Educational Work.
Att uttrycka matematik: En observationsstudie av vilka uttrycksformer och matematiska aktiviteter som uppmuntras och används i förskolan2013Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis

Syftet med studien är att fördjupa kunskaperna om vilka matematiska aktiviteter ochuttrycksformer pedagogerna i förskolan använder sig av i planerade matematikaktiviteter.Utgångspunkt för undersökningen var uttrycksformerna som finns beskrivna i förskolansläroplan samt sex historiskt och kulturellt grundade matematiska aktiviteter.Syftet konkretiseras i följande frågeställningar: Vilka matematiska aktiviteter och vilka uttrycksformer uppmuntras och stöds avpedagogerna i planerade matematikaktiviteter? Vilka likheter och skillnader finns beroende på barnens ålder?För att få svar på dessa frågor genomfördes strukturerade observationer av planeradematematikaktiviteter på tre olika förskoleavdelningar. Barnens åldrar var 1-5 år och varjeavdelning observerades vid ett tillfälle under 45 minuter. Tekniker som användes vidgenomförandet av observationerna var papper, penna, kamera och videokamera.Studiens resultat visar att samtal och rörelse är de vanligast förekommande uttrycksformerna iplanerade matematikaktiviteter. Ingen pedagog använde sig av uttrycksformerna lek/drama ellerdans. Av de matematiska aktiviteterna var lokalisera och mäta de mest förekommande, följt avkonstruera hos de äldsta barnen.En slutsats som kan dras av studien är att det behövs en ökad medvetenhet och kunskap hospedagogerna om hur förskolan kan använda alla de uttrycksformer som beskrivs i Lpfö 98/2010för att stimulera och utveckla barnens matematiklärande.

• 17. Anco, S.
Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
Solutions of gasdynamic equations associated with classical and new conservation laws2015In: Applied Mathematics and Computation, ISSN 0096-3003, E-ISSN 1873-5649, Vol. 268, 52-58 p.Article in journal (Refereed)

Exact solutions of the one-dimensional gasdynamic equations are constructed by applying the method of conservation laws to all point-wise conserved vectors of the equations under consideration. Â© 2015 Elsevier Inc.

• 18. Andersen, Jorgen Ellegaard
Reducibility of quantum representations of mapping class groups2010In: Letters in Mathematical Physics, ISSN 0377-9017, E-ISSN 1573-0530, Vol. 91, 215-239 p.Article in journal (Refereed)

In this paper we provide a general condition for the reducibility of the Reshetikhin–Turaev quantum representations of the mapping class groups. Namely, for any modular tensor category with a special symmetric Frobenius algebra with a non-trivial genus one partition function, we prove that the quantum representations of all the mapping class groups built from the modular tensor category are reducible. In particular, for SU(N) we get reducibility for certain levels and ranks. For the quantum SU(2) Reshetikhin–Turaev theory we construct a decomposition for all even levels. We conjecture this decomposition is a complete decomposition into irreducible representations for high enough levels.

• 19.
Chalmers .
University of Borås, Faculty of Textiles, Engineering and Business.
Approximate solution of a Thomas-Fermi model equation for bulk self-gravitating stellar objects in two dimensions2017In: European journal of physics, ISSN 0143-0807, E-ISSN 1361-6404, Vol. 38, 1-8 p., 015406Article in journal (Refereed)

Direct variational methods are used to find simple approximate solutions of the Thomas–Fermi equations describing the properties of self-gravitating radially symmetric stellar objects both in the non-relativistic and ultra-relativistic cases. The approximate solutions are compared and shown to be in good agreement with exact and numerically obtained solutions.

• 20.
Jönköping University, School of Engineering, JTH, Mathematics.
Linnaeus university, Vaxjö, Sweden. Jönköping University, School of Engineering, JTH, Physics.
Fourier methods for harmonic scalar waves in general waveguides2016In: Journal of Engineering Mathematics, ISSN 0022-0833, E-ISSN 1573-2703, Vol. 98, no 1, 21-38 p.Article in journal (Refereed)

A set of semi-analytic techniques based on Fourier analysis is used to solve wave-scattering problems in variously shaped waveguides with varying normal admittance boundary conditions. Key components are the newly developed conformal mapping methods, wave splitting, Fourier series expansions in eigenfunctions to non-normal operators, the building block method or the cascade technique, Dirichlet-to-Neumann operators, and reformulation in terms of stable differential equations for reflection and transmission matrices. For an example, the results show good correspondence with a finite element method solution to the same problem in the low- and medium-frequency domains. The Fourier method complements finite element analysis as a waveguide simulation tool. For inverse engineering involving tuning of straight waveguide parts joining complicated waveguide elements, the Fourier method is an attractive alternative including time aspects. The prime motivation for the Fourier method is its added physical understanding primarily at low frequencies.

• 21.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
Validated Enclosures of Rotation Numbers2015Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
• 22.
WorldLight.com AB, Sweden.
Mälardalen University, School of Education, Culture and Communication.
The mathematics of internet search engines2008In: Acta Applicandae Mathematicae - An International Survey Journal on Applying Mathematics and Mathematical Applications, ISSN 0167-8019, E-ISSN 1572-9036, Vol. 104, no 2, 211-242 p.Article in journal (Refereed)

This article presents a survey of techniques for ranking results in search engines, with emphasis on link-based ranking methods and the PageRank algorithm. The problem of selecting, in relation to a user search query, the most relevant documents from an unstructured source such as the WWW is discussed in detail. The need for extending classical information retrieval techniques such as boolean searching and vector space models with link-based ranking methods is demonstrated. The PageRank algorithm is introduced, and its numerical and spectral properties are discussed. The article concludes with an alternative means of computing PageRank, along with some example applications of this new method.

• 23.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
On the Theorem of Uniform Recovery of Random Sampling Matrices2014In: IEEE Transactions on Information Theory, ISSN 0018-9448, Vol. 60, no 3, 1700-1710 p.Article in journal (Refereed)

We consider two theorems from the theory of compressive sensing. Mainly a theorem concerning uniform recovery of random sampling matrices, where the number of samples needed in order to recover an s-sparse signal from linear measurements (with high probability) is known to be m greater than or similar to s(ln s)(3) ln N. We present new and improved constants together with what we consider to be a more explicit proof. A proof that also allows for a slightly larger class of m x N-matrices, by considering what is called effective sparsity. We also present a condition on the so-called restricted isometry constants, delta s, ensuring sparse recovery via l(1)-minimization. We show that delta(2s) < 4/root 41 is sufficient and that this can be improved further to almost allow for a sufficient condition of the type delta(2s) < 2/3.

• 24.
Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
Att arbeta med begreppsinlärning: Tillsammans med elever med annat modersmål än svenska2012Independent thesis Basic level (university diploma), 10 credits / 15 HE creditsStudent thesis

Elever med annat modersmål än svenska besitter oftast inte den språkliga kompetens som förväntas av dem när de når skolåldern. Detta blir problematiskt vid begreppsinlärningen i matematik. Syftet med denna studie är att beskriva hur verksamma lärare arbetar med begreppsinlärning i klasser som har elever med annat modersmål än svenska. I studien har verksamma lärare intervjuats och observerats med fokus på begreppsinlärning och på elever med annat modersmål än svenska. Resultatet av studien visar att de verksamma lärarna är överens om att i klasser med elever med annat modersmål än svenska går det inte utgå ifrån att alla elever förstår begrepp genom ord. Istället behöver dessa elever arbeta med flera olika sinnen för att skapa förståelse.

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
Robustness to strategic uncertainty2014In: Games and Economic Behavior, ISSN 0899-8256, E-ISSN 1090-2473, Vol. 85, no 1, 272-288 p.Article in journal (Refereed)

We introduce a criterion for robustness to strategic uncertainty in games with continuum strategy sets. We model a player's uncertainty about another player's strategy as an atomless probability distribution over that player's strategy set. We call a strategy profile robust to strategic uncertainty if it is the limit, as uncertainty vanishes, of some sequence of strategy profiles in which every player's strategy is optimal under his or her uncertainty about the others. When payoff functions are continuous we show that our criterion is a refinement of Nash equilibrium and we also give sufficient conditions for existence of a robust strategy profile. In addition, we apply the criterion to Bertrand games with convex costs, a class of games with discontinuous payoff functions and a continuum of Nash equilibria. We show that it then selects a unique Nash equilibrium, in agreement with some recent experimental findings.

• 26.
Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
Flemish mathematics teaching: Bourbaki meets RME?2014In: Proceedings of the 8th British Congress of Mathematics Education 2014 / [ed] Sue Pope, 2014, 9-16 p.Conference paper (Refereed)

The Programme of International Student Assessment (PISA) and Trends in International Mathematics and Science Study (TIMSS) create much international interest in those countries perceived as high achieving. One such system, rarely acknowledged, is Flanders, the Dutch-speaking region of Belgium. In this paper I present the results of focused analyses of four sequences of video-taped mathematics lessons taught to students aged 10 to 14 years. These confirmed a mathematics education tradition drawing on two well-known curricular movements. The first presents mathematics as a Bourbakian set of interconnected concepts. The second exploits realistic problems in its presentation of mathematics.

• 27.
University of Cambridge, UK.
The curricular importance of mathematics: a comparison of English and Hungarian teachers' espoused beliefs2007In: Journal of Curriculum Studies, ISSN 0022-0272, E-ISSN 1366-5839, Vol. 39, no 3, 317-338 p.Article in journal (Refereed)

This paper reports an interview study of 45 English and 10 Hungarian teachers of mathematics. The semi‐structured interviews focused on the teachers’ professional life‐histories and invited them to discuss their beliefs about the necessary subject content for the teaching and learning of mathematics. Substantial differences emerged between the two cohorts, which accord with well‐defined national perspectives on education in general and mathematics education in particular. They reflect, at national rather than individual levels, the expectations of the curricular frameworks within which teachers operate. English teachers tended to view mathematics as applicable number and the means by which learners are prepared for a world beyond school. Hungarian teachers privileged mathematics as problem‐solving and logical thinking.

• 28.
University of Cambridge, UK.
Mathematics teacher typologies or nationally located patterns of behaviour?2007In: International Journal of Educational Research, ISSN 0883-0355, E-ISSN 1873-538X, Vol. 46, no 5, 306-318 p.Article in journal (Refereed)

This paper reports on a small-scale EU-funded study of the teaching of mathematics in the age range 10–14 in Flanders, England, Hungary, and Spain. Drawing on video recordings of sequences of lessons taught on standard topics, and exploiting a coding schedule developed from live observations in each country, the inferable learning objectives of, and the didactic strategies teachers employ during, each of a lesson's episodes are examined. Two analyses were undertaken. The first, despite some common ground, indicated that teachers in each country behave in ways similar to teachers from the same country that distinguishes them from teachers elsewhere. The second identified typologies of teacher behaviour that were international in their teacher composition and challenged the robustness of the national script.

• 29.
University of Cambridge, UK.
Negotiating meaning in cross-national studies of mathematics teaching: kissing frogs to find princes2007In: Comparative Education, ISSN 0305-0068, E-ISSN 1360-0486, Vol. 43, no 4, 489-509 p.Article in journal (Refereed)

This paper outlines the iterative processes by which a multinational team of researchers developed a low‐inference framework for the analysis of video recordings of mathematics lessons drawn from Flemish Belgium, England, Finland, Hungary and Spain. Located within a theoretical framework concerning learning as the negotiation of meaning, we discuss problems of linguistic and conceptual equivalence and the manner by which they were resolved. Significantly, when compared with the time‐stamped codes of projects like the TIMSS video studies, we argue that the unit of analysis adopted, the episode, allowed for the distinctive patterns of a lesson to be retained for comparison with others. Also, we suggest that the framework’s generic, though subject‐focused, codes are amenable to adaptation to other curriculum areas, thus providing an opportunity for the comparative study of subjects not normally associated with work of this nature.

• 30.
Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
Mathematics, PISA, and culture: An unpredictable relationship2015In: Journal of educational change, ISSN 1389-2843, E-ISSN 1573-1812, Vol. 16, no 3, 251-280 p.Article in journal (Refereed)

Recent studies have indicated, particularly in the European context, that students’ mathematical successes on international tests of student achievement may not be attributable to the quality of classroom instruction, although, as is shown, this is unlikely to be the case in Flanders, the autonomous Dutch-speaking region of Belgium. Flemish students’ mathematics performance on such tests have placed them at the head of the European rankings, warranting Flanders as a site of research interest that has been largely ignored by the international community. In this paper, drawing on analyses of four sequences of five lessons, taught by teachers construed locally as competent, I explore the nature of Flemish mathematics teaching. Framed by anecdotal reports that it reflects the structuralism of the now largely abandoned Bourbakian new mathematics movement humanised by the Dutch tradition of realistic mathematics education, the analyses focus on examining not only the extent to which these traditions are manifested in Flemish classrooms but the ways in which they interact. The dominant tradition seems to be that of mathematical structuralism mediated by teachers’ use of realistic problems; a tradition not unlikely to underpin Flemish students’ repeated successes. The results are discussed in relation to research highlighting the significance on students’ achievement of the broader cultural milieu in which they and their teachers operate.

• 31.
University of Cambridge, UK.
Comparative studies of mathematics teachers’ observable learning objectives: validating low inference codes2009In: Educational Studies in Mathematics, ISSN 0013-1954, E-ISSN 1573-0816, Vol. 71, no 2, 97-122 p.Article in journal (Refereed)

Videotape is an increasingly used tool in cross-national studies of mathematics teaching. However, the means by which videotaped lessons are coded and analysed remains an underdeveloped area with scholars adopting substantially different approaches to the task. In this paper we present an approach based on generic descriptors of mathematics learning objectives. Exploiting live observations in five European countries, the descriptors were developed in a bottom-up recursive manner for application to videotaped lessons from four of these countries, Belgium (Flanders), England, Hungary and Spain. The analyses showed not only that the descriptors were consistently operationalised but also that they facilitated the identification of both similarities and differences in the ways in which teachers conceptualise and present mathematics that resonated with the available literature. In so doing we make both methodological and theoretical contributions to comparative mathematics research in general and debates concerning the national mathematics teaching script in particular.

• 32.
University of Cambridge, UK.
Learning from others: Can PISA and TIMSS really inform curriculum development in mathematics?2012In: Mathematical Gazette, ISSN 0025-5572, Vol. 96, no 537, 386-407 p.Article in journal (Other academic)

One of the problems that will vex any President of the Mathematical Association is the topic of the address with which he or she closes his or her year of office. This occupied me, on and off, for more than a year. In my case, in addition to my desire to acknowledge the honour of the invitation made to me, I was deeply conscious of the fact that I would be the 100th individual to serve as President. I dabbled with some pet themes, typically concerning the lack of genuine problem-solving or proof in English school mathematics, before concluding that the most sensible thing would be to talk on the topic about which I know most. My research interests are in comparative mathematics education. I have been fortunate, over the last twenty years or so, to have been able to visit and videotape mathematics classrooms in several European countries. In so doing I have had my understanding of mathematics teaching transformed in ways that led, almost inevitably, to the theme of both this talk and the conference which brought my Presidency to an end: Learning from Others.

• 33.
University of Cambridge, UK.
The importance of acknowledging the cultural dimension in mathematics teaching and learning research2010In: Acta Didactica Napocensia, ISSN 2065-1430, Vol. 3, no 2, 3-16 p.Article in journal (Refereed)

In this paper, which is in four parts, I make a plea to those involved in research into mathematics teaching and learning of the need to acknowledge, however their work is framed, that it will be located in a culture, not always visible to a reader, that should be made explicit. In the first part I examine three key models of culture and their significance for education. In the second I further highlight the impact of culture on what children are expected by critiquing various models of curriculum. The third part examines how culture informs the particularities of four European mathematics curricula, while the fourth part explores culturally located differences in mathematics teaching. In so doing a plea to researchers is framed: Culture permeates all aspects of educational endeavour and should be acknowledged more explicitly than it is.

• 34.
University of Cambridge, UK.
Mathematics Teachers’ Didactic Strategies: Examining the Comparative Potential of Low Inference Generic Descriptors2009In: Comparative Education Review, ISSN 0010-4086, E-ISSN 1545-701X, Vol. 53, no 4, 559-581 p.Article in journal (Refereed)
• 35.
Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
The Emperor’s new clothes: PISA, TIMSS and Finnish mathematics2014In: Spaces for learning: past, present and future: Proceedings of the FMSERA 30th annual symposium in Vaasa, November 6-8, 2013 / [ed] Ann-Sofi Röj-Lindberg, Lars Burman, Berit Kurtén-Finnäs, Karin Linnanmäki, Åbo Akademi University , 2014, 43-65 p.Chapter in book (Refereed)

For nearly fifteen years, due to repeated successes on the Programme of International Student Assessment (PISA), Finnish education in general and mathematics education in particular have been construed internationally as benchmarks. In what is essentially a review paper I consider how the Finns explain their students’ repeated PISA successes before contrasting these explanations with observational evidence indicating that typical classroom practice is unlikely to account for such successes. In addition, I examine the relative failure of Finnish students on the Trends in International Mathematics and Science Study (TIMSS), particularly with respect to algebra and geometry, and highlight the extent to which Finnish students may be inadequately prepared for higher study of mathematics. I close by indicating that continued interest in Finland as a source of excellence in mathematics teaching may be misguided and that other European systems, like Flanders, may provide better warranted research locations for those interested in transferable insights.

• 36.
Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
Practice to inspire: Mathematics teaching in one Hungarian grade one classroom2014In: "Do you think it's all the same?": Proceedings of the Julianna Szendrei Memorial Conference / [ed] Judit Szitanyi, 2014, 63-78 p.Conference paper (Other academic)

In this paper I introduce and categorise the concept of foundational number sense (FONS). Broadly described as the number related competences expected of grade one children, which research has shown to be necessary for the later study of mathematics, FONS is operationalised as an eight dimensional framework for analysing the number-related opportunities teachers present to their students. Drawing on data from a case study of exemplary teaching in grade one classrooms, I analyse one teacher’s, Klara’s, practice against the framework to show not only that she provides some profound opportunities for her students to learn but does so in ways that reflect the long-standing Hungarian tradition of mathematics as a problem solving discipline taught in collaborative and socially dynamic ways.

• 37.
University of Cambridge, UK.
The teaching of linear equations: Comparing effective teachers from three high achieving European countries2011In: Proceedings of the Seventh Congress of the European Society for Research in Mathematics Education / [ed] Marta Pytlak, Tim Rowland, Ewa Swoboda, Rzeszów: University of Rzeszów , 2011, , 1555-1564 p.1555-1564 p.Conference paper (Refereed)

On various international tests of achievement Finnish, Flemish and Hungarian students have been amongst the more successful in Europe. Linear equations, a topic students traditionally find difficult, is a key topic in the transition from mathematics as inductive and concrete to deductive and abstract. This paper, by means of an analysis of video-taped lessons taught by case study teachers, one from each of Finland, Flanders and Hungary, examines comparatively how teachers defined locally as effective construct opportunities for their students to learn the mathematics of linear equations. The findings show that all three teachers acted in ways contrary to received research wisdom, exploiting the balance scale as the key metaphor for inducting students into the solution processes of algebraic equations.

• 38.
University of Cambridge, UK.
Finnish mathematics teaching: a case of uniquely implicit didactics2011In: Fourth Conference on Research in Mathematics Education: Mathematics Teaching Matters / [ed] Thérèse Dooley, Dolores Corcoran, Miriam Ryan, Drumcondra, Ireland: St. Patrick’s College , 2011, 3-18 p.Conference paper (Refereed)

This paper reports on a qualitative analysis of video-taped mathematics lessons taught by four case study teachers, defined locally as effective, in a provincial university city in Finland. The aim was to examine how teachers conceptualise and present mathematics to their learners and, in so doing, understand the relationship between Finnish mathematics teaching practices, as reflected in case study lessons, and Finnish success on successive PISA assessments. Analysed by means of the process of constant comparison, the data yielded two key characteristics of case study classrooms. Firstly, irrespective of their intended learning outcome, teachers exploited a series of implicit didactic strategies focused on encouraging students to infer meaning. Secondly, three culturally located activities were identified that appeared complementary to this sense of the implicit. These were the systemic encouragement of students to make notes, teachers’ exploitation of the confident child and the assumed collaboration of parents.

• 39.
Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
European mathematics curricula and classroom practices2014In: Masterclass in mathematics education: international perspectives on teaching and learning / [ed] Paul Andrews, Tim Rowland, London: Bloomsbury Academic, 2014, 179-190 p.Chapter in book (Refereed)
• 40.
Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
Understanding the Cultural Construction of School Mathematics2016In: Mathematical Cultures: The London Meetings 2012-2014 / [ed] Brendan Larvor, Springer, 2016, 9-23 p.Chapter in book (Refereed)

In this chapter, I show how culture underpins all aspects of school mathematics, whether it be the curriculum specified by the system, the development of the textbooks that teachers may or may not be compelled to use, the ways teachers teach, the classroom interactions privileged by the system or the beliefs, attitudes and aspirations of teachers, students and parents. To do this, however, I will describe the nature of culture and its educational manifestation.

• 41.
Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
Instrument adaptation in cross-cultural studies of students' mathematics-related beliefs: Learning from healthcare research2015In: Compare, ISSN 0305-7925, E-ISSN 1469-3623, Vol. 45, no 4, 545-567 p.Article in journal (Refereed)

Much comparative research into education-related beliefs has exploitedquestionnaires developed in one culture for use in another. This hasbeen particularly the case in mathematics education, the focus of thispaper. In so doing, researchers have tended to assume that translationalone is sufficient to warrant a reliable and valid instrument forcross-cultural research, prompting concerns that a number of necessaryequivalences are unlikely to have been addressed. In this paper, we considerthe nature of these equivalences before examining the literature ofa different field, healthcare research, to synthesise an approach to instrumentadaptation that is pragmatic but rigorous. Finally, we demonstratehow this pragmatic approach, incorporating extensive cognitive interviews,enabled us to adapt and refine a mathematics-related beliefsquestionnaire, developed in Flanders, for use with students aged 14–15in England and Spain. Analyses indicate that the instrument so developedis multidimensional, reliable and cross-culturally valid. Someimplications are discussed.

• 42.
Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
Rowland, Tim
Masterclass in mathematics education: international perspectives on teaching and learning2014Collection (editor) (Refereed)
• 43.
Mälardalen University, Sweden.
Mälardalen University, Sweden. Mälardalen University, Sweden. Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
PISA, TIMSS and Finnish mathematics teaching: an enigma in search of an explanation2014In: Educational Studies in Mathematics, ISSN 0013-1954, E-ISSN 1573-0816, Vol. 87, no 1, 7-26 p.Article in journal (Refereed)

Finnish students’ success on all three content domains of each of the four cycles ofthe OECD’s Programme for International Student Assessment (PISA) has created muchinternational interest. It has also prompted Finnish academics to offer systemic explanationstypically linked to the structural qualities of Finnish schooling and teacher education. Lesswell-known has been the modest mathematics performance of Finnish grade 8 students on thetwo Trends in International Mathematics and Science Study (TIMSS) in which Finland hasparticipated, which, when compared with its PISA successes, has created something of anenigma. In this paper, we attempt to shed light on this enigma through analyses of Finnishmathematics classroom practice that draw on two extant data sets—interviews with Finnishteacher educators and video-recordings of sequences of lessons taught on standard topics. Dueto the international interest in Finnish PISA success, the analyses focus primarily on theresonance between classroom practice and the mathematical literacy component of the PISAassessment framework. The analyses indicate that Finnish mathematics didactics are morelikely to explain the modest TIMSS achievements than PISA successes and allude to severalfactors thought to be unique to the Finns, which, unrelated to mathematics teaching practices,may be contributory to the repeated Finnish PISA successes. Some implications for policyborrowingare discussed.

• 44.
Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
Identifying opportunities for grade one children to acquire foundational number sense: Developing a framework for cross cultural classroom analyses2015In: Early Childhood Education Journal, ISSN 1082-3301, E-ISSN 1573-1707, Vol. 43, no 4, 257-267 p.Article in journal (Refereed)

It is known that an appropriately developedfoundational number sense (FONS), or the ability tooperate flexibly with number and quantity, is a powerfulpredictor of young children’s later mathematical achievement.However, until now not only has FONS been definitionallyelusive but instruments for identifyingopportunities for children to acquire its various componentshave been missing from the classroom observationtools available. In this paper, drawing on a constant comparisonanalysis of appropriate literature, we outline thedevelopment of an eight dimensional FONS framework.We then show, by applying this framework to three culturallydiverse European grade one lessons, one English,one Hungarian and one Swedish, that it is both straightforwardlyoperationalised and amenable to cross culturalanalyses of classroom practice. Some implications arediscussed.

• 45.
Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
Foundational number sense: A framework for analysing early number-related teaching2014In: Proceedings of MADIF 9, The Ninth mathematics Education Research Seminar, 2014Conference paper (Refereed)

In this paper, by means of an extensive review of the literature, we discuss the development of a framework for analysing the opportunities, both implicit and explicit, that grade one students receive for acquiring those number-related understandings necessary for later mathematical achievement but which do not occur without formal instruction. The framework, which we have called foundational number sense, currently comprises seven interrelated components, although additional components may exist. Each component, as warranted by earlier research, is known to underpin later mathematical understanding and, when viewed collectively, addresses a definitional gap in the literature.

• 46.
University of Cambridge, UK.
The University of Northampton, UK.
Teaching linear equations: Case studies from Finland, Flanders and Hungary2012In: Journal of Mathematical Behavior, ISSN 0732-3123, E-ISSN 1873-8028, Vol. 31, no 4, 476-488 p.Article in journal (Refereed)

In this paper we compare how three teachers, one from each of Finland, Flanders and Hungary, introduce linear equations to grade 8 students. Five successive lessons were videotaped and analysed qualitatively to determine how teachers, each of whom was defined against local criteria as effective, addressed various literature-derived equations-related problems. The analyses showed all four sequences passing through four phases that we have called definition, activation, exposition and consolidation. However, within each phase were similarities and differences. For example, all three constructed their exposition around algebraic equations and, in so doing, addressed concerns relating to students’ procedural perspectives on the equals sign. All three teachers invoked the balance as an embodiment for teaching solution strategies to algebraic equations, confident that the failure of intuitive strategies necessitated a didactical intervention. Major differences lay in the extent to which the balance was sustained and teachers’ variable use of realistic word problems.

• 47.
University of Cambridge, UK.
University of Northampton, UK.
Comparative studies of mathematics teaching: does the means of analysis determine the outcome?2013In: ZDM - the International Journal on Mathematics Education, ISSN 1863-9690, E-ISSN 1863-9704, Vol. 45, no 1, 133-144 p.Article in journal (Refereed)

This paper addresses four questions concerning the influence of culture on mathematics teachers’ professional practice. Firstly, drawing on categorical data yielded by the application of low inference coding schedule to video recordings of sequences of lessons taught by case study teachers on four common topics in England, Flanders, Hungary and Spain, we undertook an exploratory factor analysis to examine the ways in which such coded variables interact. This process yielded five factors, each of which was interpretable against the literature and highlighted the extent to which dichotomisations of mathematics teaching as reform or traditional are not necessarily helpful, not least because all project teachers exhibited characteristics of both. Secondly, factors scores were analysed by nationality to reveal culturally located practices resonant with the available literature. Thirdly, cluster analyses yielded four well-defined cross-cultural clusters of episodes, each indicative of particular didactical perspectives that appeared to challenge the exclusivity of these culturally located practices. Finally, the key methodological finding was that the manner in which data are analysed influences greatly the outcomes of comparative mathematics research.

• 48.
Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
Stockholm University, Faculty of Science, Department of Mathematics and Science Education. Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
Developing foundational number sense: Number line examples from Poland and Russia2015In: Proceedings of the Ninth Congress of the European Society for Research in Mathematics Education / [ed] Konrad Krainer, Nad'a Vondrová, 2015, 1681-1687 p.Conference paper (Refereed)

For a variety of reasons children start school with differing number-related skills, leading to differences in later mathematics achievement. Such differences prompt the question, what number-related experiences are necessary if the first year of school is to prepare children appropriately for their learning of mathematics? In this paper we discuss the development of an eight dimensional framework, foundational number sense (FoNS), that characterises those learning experiences. We then demonstrate the framework's analytical efficacy by evaluating episodes from two sequences of lessons, one Polish and one Russian, focused on the use of the number line. The results show that the FoNS framework is cross-culturally sensitive, simply operationalised and analytically powerful.

• 49.
Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
Analysing the relationship between the problem-solving-related beliefs, competence and teaching of three Cypriot primary teachers2015In: Journal of Mathematics Teacher Education, ISSN 1386-4416, E-ISSN 1573-1820, Vol. 18, no 4, 299-325 p.Article in journal (Refereed)

In this article, we analyse the problem-solving-related beliefs, competence and classroom practice of three Cypriot upper-primary teachers. Data derived from semi-structured interviews focused on teachers’ beliefs about the nature of mathematical problems, problem-solving, and their competence as both problem-solvers and teachers of problem-solving; clinical interviews during which teachers solved a context-free geometrical problem, and observations of a lesson during which teachers introduced that problem to students of grade six. Analyses, structured by a framework derived from key problem-solving literature, indicated firstly, that the framework was an effective tool, sensitive to variation within and across the data from teachers, and secondly, that all participants, in largely explicable ways, exhibited consistency and inconsistency in the ways in which their beliefs, competence and practice interacted. Some implications for further research are discussed.

• 50.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
Adaptive node distribution for on-line trajectory planning2006In: ICAS-Secretariat - 25th Congress of the International Council of the Aeronautical Sciences 2006, Curran Associates, Inc., 2006, 3150-3157 p.Conference paper (Refereed)

Direct methods for trajectory optimization are traditionally based on a priori temporal dis- cretization and collocation methods. In this work, the problem of node distribution is for- mulated as an optimization problem, which is to be included in the underlying non-linear mathematical programming problem (NLP). The benefits of utilizing the suggested method for on-line trajectory optimization are illustrated by a missile guidance example.

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