To Örebro University

Society is highly digitalized and digital learning resources have been given a large space at school in almost all grades. The purpose of using digital learning resources is to support students in their learning and development. Digital learning resources have another purpose, which is to facilitate the teacher's work. In this study, a systematic literature study will be done to investigate what previous research that has focused on digital learning resources in mathematics education will be compiled. digital learning resources in mathematics teaching. The purpose of this literature study is to investigate the use of digital learning resources during mathematics teaching and how it can promote student learning. The study will also considerthe use of digital learning resources based on the socio-cultural perspective. The results of the compiled research showed that digital learning resources of various kinds can promote students' learning in different ways. Among other things, it promotes motivation for learning, commitment, problem-solving skills, algebraic thinking, number calculations and pattern recognition. The results also show that a large part of the underlying aspects that can prevent students' learning from being promoted through digital learning resources is the teachers' lack of knowledge regarding the use of the material

I denna studie har undersökningarna Kortperiodisk Sysselsättningsstatistiks (KS) och Konjunkturstatistik över Vakansers (KV) befintliga granskningssystem utvärderats med avseende på hur effektivt det är. Processdata har framställts och analyserats. Resultaten tyder på att många av de inkomna blanketterna med misstänkt felaktiga uppgifter inte rättas upp, utan tvingas igenom trots att granskningssystemet ej accepterade uppgifterna. Det befintliga granskningssystemet har en högre träffsäkerhet avseende KS-undersökningen, men både KS och KV skulle kunnas granskas mer effektivt.

För att utvärdera det befintliga granskningssystemet ytterligare användes en poängfunktion. Till studien fanns tillgång till både helt ogranskat material och helt granskat material och dessa material användes i poängfunktionen. Det uppräknade ogranskade värdet för varje objekt jämfördes med det uppräknade granskade värdet och ställdes i relation till respektive skattade branschtotal. De poängsatta blanketterna rangordnades sedan. Därefter analyserades materialet för att försöka finna var det skulle vara lämpligt att sätta det tröskelvärde som skulle skilja det material som ”egentligen” skulle ha behövts granskas från det som kunde ha lämnats orört. Att sätta tröskelvärdet är svårt. Här gjordes det godtyckligt utifrån kriterierna att det fel som införs i skattningarna för att allt material inte granskas skulle hållas så lågt som möjligt samt att antalet blanketter som skulle behöva granskas manuellt av produktionsgruppen också skulle hållas så lågt som möjligt. Även här visade det sig att det befintliga granskningssystemet inte är så effektivt som önskas. När resultaten från denna del av utvärderingen analyserades upptäcktes problem som beror på blankettutformningen. Skulle blanketterna ses över och åtgärdas skulle det fel som införs för att allt material inte granskas kunna minskas avsevärt. Genom att minska det införda felet kan tröskelvärdet förmodligen sättas på en ny nivå vilket medför att omfattningen av granskningen skulle minska ytterligare.

Hur skulle då ett mer effektivt granskningssystem kunna se ut? I den här studien har valet fallit på att testa ”significance editing” på KS-undersökningen, det som på svenska kallas för effektgranskning. En poängfunktion användes även här, denna tilldelar de inkomna blanketterna varsin poäng och dessa poäng rangordnas därefter. Efter att poängen rangordnats bestäms en gräns, ett tröskelvärde, och de blanketter med en poäng som överstiger tröskelvärdet granskas och rättas upp av produktionsgruppen. De blanketter med en poäng som understiger det satta tröskelvärdet rättas inte upp, utan behåller sina originalvärden. Poängfunktionen jämför det inkomna ogranskade, uppräknade, värdet med ett uppräknat ”förväntat” värde och ställer denna differens i relation till den skattade branschtotalen. Svårigheten ligger ofta i att hitta ett bra förväntat värde och detta problem uppstår ideligen i urvalsundersökningar. Tanken med effektgranskning är att omfattningen av granskningen ska minska och den granskning som utförs ska ha effekt på slutresultatet.

Det var inte lätt att hitta ett bra förväntat värde på den tid som stod till förfogande. Två problem som snabbt upptäcktes var dels att i KS-undersökningen finns inte uträknade säsongs- eller trendfaktorer per variabel. Dessutom byttes en mycket stor del av urvalet ut till kvartal 2 (som denna studie har avgränsats till att behandla). Detta har fått till följd att cirka hälften av objekten i urvalet inte går att följa bakåt i tiden eftersom de inte ingått i urvalet tidigare. I studien har respektive stratums medelvärde använts som förväntat värde. Resultaten visar att det valda förväntade värdet inte skulle ha använts i praktiken, men det fungerar bra i syfte att illustrera hur det i praktiken skulle kunna gå till att införa en mer effektiv granskning.

Students who find mathematics difficult and complicated can lose their confidence in their abilities after the first years of school to cope with mathematical tasks and gradually give up all efforts to learn mathematics. This systematic literature study aims to shed light on the teacher's role in developing students' interest in mathematics. The goal is to point out important aspects that can strengthen the teacher's teaching to arouse and create a positive interest in the students in their teaching. The result is research on different perceptions from different continents in the world about essential aspects that teachers can work from to promote students' interest in the mathematical subject. The most central aspect focuses on the importance of the teacher's knowledge of the students. Half of the articles that form the basis of this study also shed light on the importance of the current mathematics classroom. Finally, there are themes about communication and creating self-confidence through feedback.

This study aims to alter students’ stereotypical views of incorrect answers and teachers’ feedback on them. By conducting a systematic qualitative literature review, this study demonstrates the role of verbal feedback and incorrect answers in students’ development of their learning abilities. It also aims to equip mathematics teachers with the required tools for creating a positive learning environment. The study’s findings are presented in two sections: The first section highlights the positive impact of errors and feedback on the mathematical performance of students. The second section provides tools for the effective delivery of positive feedback. The study concludes that, to create a positive learning environment, teachers should know how to use verbal feedback (response) positively and spark constructive discussions in mathematics classrooms. Thus, teachers create a positive classroom climate in which incorrect answers are welcome. 

This paper presents the computation of likelihood prediction with the generalized linear and mixed models. The method of likelihood prediction is briefy discussed and approximate formulae are provided to make easy computation of the likelihoodprediction with generalized linear models. For complicated prediction problems, simulation methods are suggested. An R add-in package is accompanied to carryout the computation of the predictive inference with the generalized linear and mixed models. The likelihood prediction is applied to the prediction of the credit defaults using a real data set. Results show that the predictive likelihood can be a useful tool to predict portfolio credit risk.

This thesis deals with developing and testing feasible computational procedures to facilitate the estimation of and carry out the prediction with the generalized linear mixed model (GLMM) with a scope of applying them to large data sets. The work of this thesis is motivated from an issue arising incredit risk modelling. We have access to a huge data set, consisting of about one million observations, on credit history obtained from two major Swedish banks. The principal research interest involved with the data analysis is to model the probability of credit defaults by incorporating the systematic dependencies among the default events. In order to model the dependent credit defaults we adopt the framework of GLMM which is apopular approach to model correlated binary data. However, existing computational procedures for GLMM did not offer us the flexibility to incorporate the desired correlation structure of defaults events.For the feasible estimation of the GLMM we propose two estimation techniques being the fixed effects (FE) approach and the two-step pseudolikelihood approach (2PL). The preciseness of the estimation techniques and their computational advantages are studied by Monte-Carlo simulations and by applying them to the credit risk modelling. Regarding the prediction issue, we show how to apply the likelihood principle to carryout prediction with GLMM. We also provide an R add-in package to facilitate the predictive inference for GLMM.

This paper presents a two-step pseudo likelihood estimation for generalized linear mixed models with the random effects being correlated between groups. The core idea is to deal with the random intractable integrals in  the likelihood function by multivariate Taylor's approximation. The accuracy of the estimation technique is assessed in a Monte-Carlo study: An application of it with binary response variable is presented using a real dara set on credit defaults from two Swedish banks. Thanks to   the use of two-step estimation technique, the proposed algorithm outperforms conventional likelihood algoritms in terms of computational time.

It is commonly agreed that the credit defaults are correlated. However, the structure and magnitude of such dependence is not yet fully understood. This paper contributes to the current understanding about the defaults comovement in the following way. Assuming that the industries provides the basis of defaults comovement it provides empirical evidence as to how such comovements can be modeled using correlated industry shocks. Generalized linear mixed model (GLMM) with correlated random effects is used to model the defaults comovement. It is also demonstrated as to how a GLMM with complex correlation structure can be estimated through a very simple way. Empirical evidences are drawn through analyzing quarterly individual borrower level credit history data obtained from two major Swedish banks between the period 1994 and 2000. The results show that, conditional on the borrower level accounting data and macro business cycle variables, the defaults are correlated both within and between industries but not over time (quarters). A discussion has also been presented as to how a GLMM for defaults correlation can be explained.

This paper presents the techniques of likelihood prediction for the generalized linear mixed models. Methods of likelihood prediction is explained through a series of examples; from a classical one to more complicated ones. The examples show, in simple cases, that the likelihood prediction (LP) coincides with already known best frequentist practice such as the best linear unbiased predictor. The paper outlines a way to deal with the covariate uncertainty while producing predictive inference. Using a Poisson error-in-variable general-ized linear model, it has been shown that in complicated cases LP produces better results than already know methods.

This article presents the techniques of likelihood prediction for the generalized linear mixed models. Methods of likelihood prediction are explained through a series of examples; from a classical one to more complicated ones. The examples show, in simple cases, that the likelihood prediction (LP) coincides with already known best frequentist practice such as the best linear unbiased predictor. This article outlines a way to deal with the covariate uncertainty while producing predictive inference. Using a Poisson errors-in-variable generalized linear model, it has been shown in certain cases that LP produces better results than already known methods.

In this paper, we discuss how a regression model, with a non-continuous response variable, which allows for dependency between observations, should be estimated when observations are clustered and measurements on the subjects are repeated. The cluster sizes are assumed to be large.We find that the conventional estimation technique suggested by the literature on generalized linear mixed models(GLMM) is slow and sometimes fails due to non-convergence and lack of memory on standard PCs.We suggest to estimate the random effects as fixed effects by generalized linear model and to derive the covariance matrix from these estimates.A simulation study shows that our proposal is feasible in terms of mean-square error and computation time.We recommend that our proposal be implemented in the software of GLMM techniques so that the estimation procedure can switch between the conventional technique and our proposal, depending on the size of the clusters.

Motivating cooperation is a study about the impact of cooperative learning on Swedish students’ motivation for the subject of mathematics. The study is based on previous research on the impact of cooperative learning methods on students’ motivation to mathematics and examines this in a Swedish context. Collection of data has taken place through observations and student-answered questionnaires about the students’ self-perceived attitude to mathematics after working in cooperative learning methods. The design of the questionnaire and the observation chart is based on the socio-cultural perspective and the concepts of motivation and commitment. Previous research indicates that students lose their motivation for mathematics during middle school and therefore the study has been limited to examining students in grade 5 and 6 in Swedish schools. The results show that students’ motivation for mathematics is promoted by cooperative learning methods. Through the categories students’ attitude and commitment to mathematics teaching, motivation-building teaching and social climate in teaching, the effect on students’ motivation to mathematics that were made visible during the study are explained. 

Pupils' achievement and knowledge in mathematics is a highly current issue that is discussed and compared both in Sweden and internationally. International measurements have for a long time pointed to a lower level of knowledge and in Sweden they discuss the reasons for these downward results. Researchers believe that pupils' attitude, adjustment and motivation to the subject of mathematics are one of the reasons when they are considered crucial to address the subject of mathematics. It also strives to answer the teaching methods that influence students' motivation. The Swedish curriculum for elementary school states that teaching should promote students' interest and the desire to learn, which represents a large part of the motivation of students. The choice of teaching method can thus influence students' motivation. In this systematic literature study, three different main categories of teaching methods appear in the results; Methods where the teacher is at the center, methods where technical aids are at the center and methods in which the student is at the center. The categories consist of articles that deal with different types of teaching methods, which in the result had an influence on student motivation.

Fractions are an area of mathematics that causes major problems for the school's students, especially the younger students in grades 4-6. Since it is difficult to understand fractions as well as to teach fractions, it is good if mathematics teachers have knowledge of a way that can simplify the understanding of the abstract through concretization. The purpose of the study is to examine research on how concretization and laboratory work create understanding of numbers in fractional form among students in primary school grades 4-6. To achieve the aim, a question has specified how does concretization and laboratory work make it easier for students in primary school grades 4-6 to understand numbers in fraction form? What forms of laboratory work are proposed in the previous research? The results of the study show that through concretization the pupils can gain a deeper understanding of fraction concepts. The teacher's use of different types of laboratory materials has contributed to making it easier for the pupils to do fraction calculations.

Realized covariance matrices are often constructed under the assumption that richness of intra-day return data is greater than the portfolio size, resulting in nonsingular matrix measures. However, when for example the portfolio size is large, assets suffer from illiquidity issues, or market microstructure noise deters sampling on very high frequencies, this relation is not guaranteed. Under these common conditions, realized covariance matrices may obtain as singular by construction. Motivated by this situation, we introduce the Singular Conditional Autoregressive Wishart (SCAW) model to capture the temporal dynamics of time series of singular realized covariance matrices, extending the rich literature on econometric Wishart time series models to the singular case. This model is furthermore developed by covariance targeting adapted to matrices and a sector wise BEKK-specification, allowing excellent scalability to large and extremely large portfolio sizes. Finally, the model is estimated to a 20-year long time series containing 50 stocks and to a 10-year long time series containing 300 stocks, and evaluated using out-of-sample forecast accuracy. It outperforms the benchmark models with high statistical significance and the parsimonious specifications perform better than the baseline SCAW model, while using considerably less parameters.

This paper investigates the problem of calibrating sensors affected by (i) heteroskedastic measurement noise and (ii) a polynomial bias, describing a systematic distortion of the measured quantity. First, a set of increasingly complex statistical models for the measurement process was proposed. Then, for each model the authors design a Bayesian parameters estimation method handling heteroskedasticity and capable to exploit prior information about the model parameters. The Bayesian problem is solved using MCMC methods and reconstructing the unknown parameters posterior in sampled form. The authors then test the proposed techniques on a practically relevant case study, the calibration of Light Detection and Ranging (Lidar) sensor, and evaluate the different proposed procedures using both artificial and field data.

Att använda lärarhandledningar som stöd har länge haft en betydande roll i finsk matematikundervisning, och på senare år har flera finska läromedel översatts till svenska (Hoelgaard, 2015). Syftet med den här studien var att bidra med ökad förståelse för vilken typ av stöd, i form av aktiviteter, en lärarhandledning som översatts från finska till svenska ger. Aktiviteter för nio lektioner ur tre lärarhandledningar för årskurs 1–3 från läromedlet ’Favorit matematik’ har studerats utifrån en kvalitativ textanalys, med kvantitativa inslag. Genom en kvalitativ intervju med två matematiklärare i årskurs 1–3 som använder de aktuella lärarhandledningarna har aktiviteterna studerats utifrån vilka aktiviteter lärarna använder, inte använder samt vilka aktiviteter de önskar fanns i lärarhandledningarna. Resultatet av textanalysen visar att det i det analyserade materialet erbjuds 102 aktiviteter. Elever arbetar enskilt och förvärvar kunskap i nästan en tredjedel av dem, men ett brett urval av aktiviteter där elever arbetar tillsammans och lär sig genom deltagande erbjuds också. Lärarna upplever dock att det finns ett riktat fokus mot enskilt arbete i elevboken, vilket också blir synligt genom textanalysen.

Using teacher guides as support for education is common and has been common in Finnish mathematics education for a long time. During the last few years Finnish curriculum materials have been translated into Swedish (Hoelgaard, 2015). The aim of this study was to contribute with knowledge about what kind of support, in form of activities, teacher guides provide. Activities for nine lessons in three teacher guides, for education in year 1-3 in Swedish compulsory school, from the curriculum material ‘Favorit matematik’ have been analysed with a qualitative text analysis with quantitative elements. Which activities teachers choose to use, not to use and what activities they wish were offered have been studied by interviewing two teachers in year 1-3, whom uses these teacher guides. The result from the text analysis show that 102 activities are offered in the analysed material. Almost one third of them are activities where students work on their own acquiring knowledge, but a wide selection of activities where student work together and knowledge comes from participating, are also offered in the teacher guides. Though, the teachers experience that there is a focus on activities where student work on their own with the textbook, which is confirmed by the text analysis.

One of the challenges with predictive modeling is how to quantify the reliability of the models' predictions on new objects. In this work we give an introduction to conformal prediction, a framework that sits on top of traditional machine learning algorithms and which outputs valid confidence estimates to predictions from QSAR models in the form of prediction intervals that are specific to each predicted object. For regression, a prediction interval consists of an upper and a lower bound. For classification, a prediction interval is a set that contains none, one, or many of the potential classes. The size of the prediction interval is affected by a user-specified confidence/significance level, and by the nonconformity of the predicted object; i.e., the strangeness as defined by a nonconformity function. Conformal prediction provides a rigorous and mathematically proven framework for in silico modeling with guarantees on error rates as well as a consistent handling of the models' applicability domain intrinsically linked to the underlying machine learning model. Apart from introducing the concepts and types of conformal prediction, we also provide an example application for modeling ABC transporters using conformal prediction, as well as a discussion on general implications for drug discovery.

In this paper two-dimensional systems of dierential equations are consideredtogether with their stabilization by a hybrid feedback control. A stabilizinghybrid control for an arbitrary controlled system that belongs to acertain category within two-dimensional systems is constructed as a result ofthis study and some stabilization proprieties of the system with the obtainedhybrid control are presented.

Minskade anslag gör att den årliga undersökningen om våra levnadsförhållanden hotas att halveras. Det kan drabba redan svaga grupper som äldre, invandrare och ensamstående föräldrar.

We review and extend the results of [1] that gives a condition for reducibility of quantum representations of mapping class groups constructed from Reshetikhin-Turaev type topological quantum field theories based on modular categories. This criterion is derived using methods developed to describe rational conformal field theories, making use of Frobenius algebras and their representations in modular categories. Given a modular category C, a rational conformal field theory can be constructed from a Frobenius algebra A in C. We show that if C contains a symmetric special Frobenius algebra A such that the torus partition function Z(A) of the corresponding conformal field theory is non-trivial, implying reducibility of the genus 1 representation of the modular group, then the representation of the genus g mapping class group constructed from C is reducible for every g\geq 1. We also extend the number of examples where we can show reducibility significantly by establishing the existence of algebras with the required properties using methods developed by Fuchs, Runkel and Schweigert. As a result we show that the quantum representations are reducible in the SU(N) case, N>2, for all levels k\in \mathbb{N}. The SU(2) case was treated explicitly in [1], showing reducibility for even levels k\geq 4.

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.

I denna uppsats om analytisk fortsättning så ges en intuitiv inblick hur denna process går till. Av analytisk fortsättning får vi ett sätt att utöka bortom den domän där funktionen från början är definierad. En fortsättning kan fås på flera sätt. Det är framförallt en metod som presenteras i uppsatsen, nämligen analytisk fortsättning längs kurvor. Genom att studera homotopier så får vi ett sätt att se på hur dessa kurvor förhåller sig till varandra. Homotopier ger oss också viktiga definitioner som enkelt sammanhängande område, vilket kommer ha stor betydelse när vi ser på en analytisk fortsättning. När vi studerar hur en analytisk fortsättning påverkas av olika områden så fås en viktig sats, nämligen Monodromisatsen som ger oss ett sätt att bygga analytiska funktioner.

We disprove some power sum conjectures of Turan that would have implied the density hypothesis of the Riemann zeta-function if true.

We prove a variant of the Mergelyan approximation theorem that allows us to approximate functions that are analytic and nonvanishing in the interior of a compact set K with connected complement, and whose interior is a Jordan domain, with nonvanishing polynomials. This result was proved earlier by the author in the case of a compact set K without interior points, and independently by Gauthier for this case and the case of strictly starlike compact sets. We apply this result on the Voronin universality theorem for compact sets K, where the usual condition that the function is nonvanishing on the boundary can be removed. We conjecture that this version of Mergelyan's theorem might be true for a general set K with connected complement and show that this conjecture is equivalent to a corresponding conjecture on Voronin Universality.

We use an estimate for character sums over finite fields of Katz to solve open problems of Montgomery and Turan. Let h >= 2 be an integer. We prove that inf(vertical bar zk vertical bar=1) max(nu=1,...,n)(h) vertical bar Sigma(n)(k=1) Z(k)(nu)vertical bar <= (h - 1 + o(1)root n. This gives the right order of magnitude for the quantity and improves on a bound of Erdos-Renyi by a factor of the order root logn.

We improve a recent universality theorem for the Riemann zeta-function in short intervals due to Antanas Laurincikas with respect to the length of these intervals. Moreover, we prove that the shifts can even have exponential growth. This research was initiated by two questions proposed by Laurin & ccaron;ikas in a problem session of a recent workshop on universality.

We consider the problem of approximating a function having no zeros on the interior of a set by polynomials having no zeros on the entire set.

We study universality properties of the Epstein zeta function E-n(L,s) for lattices L of large dimension n and suitable regions of complex numbers s. Our main result is that, as n -> infinity, E-n(L,s) is universal in the right half of the critical strip as L varies over all n-dimensional lattices L. The proof uses a novel combination of an approximation result for Dirichlet polynomials, a recent result on the distribution of lengths of lattice vectors in a random lattice of large dimension and a strong uniform estimate for the error term in the generalized circle problem. Using the same approach we also prove that, as n -> infinity, E-n(L-1,s) - E-n(L-2,s) is universal in the full half-plane to the right of the critical line as E-n(L,s) varies over all pairs of n-dimensional lattices. Finally, we prove a more classical universality result for E-n(L,s) in the s-variable valid for almost all lattices L of dimension n. As part of the proof we obtain a strong bound of E-n(L,s) on the critical line that is subconvex for n >= 5 and almost all n-dimensional lattices L.

This paper proposes several resampling algorithms suitable for error component models and evaluates them in the context of bootstrap testing. In short, all the algorithms work well and lead to tests with correct or close to correct size. There is thus little or no reason not to use the bootstrap with error component models.

Both from previous research and from the media the documentation form of education plan receives a lot of criticism in terms of both content and design. An education plan must be written if a student is suspected not to achieve current knowledge requirements. It will therefore be important in a subject like mathematics which has a relatively high percentage of students who fails to achieve the knowledge requirements. Therefore it is important that the process with the education plan fulfils its purpose. This systematic literature study aims to identify factors that may increase the effectiveness of education plan process in mathematics. The results and conclusions of this study put forward several different factors that must interact in order to make the educational plan process in mathematics successful. Factors that this study concludes that might have an affect are for example, pupils' and parents' participation and the headmasters’ commitment and volition to participate.

Academic year 2013/2014 almost every tenth student in grades 6 lacks passing grade in mathematics. Mathematics is one of the three subjects that a student needs to pass in order to later in their schooling get high school eligibility. The school has a statutory mandate to provide all students with opportunities to achieve acceptable knowledge requirements and if this is not done, a special support should be deployed and documented in an education plan. Education plans must constantly endure a lot of criticism because they do often lack in quality.

Through a qualitative text analysis 15 education plans will be reviewed for the purpose of developing knowledge about how the work with special support in math are projected and processed. The study shows that there is a link between education plan and knowledge requirement in 22 of the 29 necessary descriptions in 11 of the 15 education plans. In four of the 15 education plans a description of the student's special needs based on the Mathematical Syllabuses core content and given abilities is missing entirely. Of the 38 measures in the education plans there are six with distinct character and the remaining 32 are of a diffuse character. A clear majority of the measures do not fulfill the criteria of evaluability and concreteness specified in Skolverkets general advice. The study shows that 26 of the 38 measures are related to current research on what is considered to be favorable measures for students in mathematics difficulties. Only two of the 15 education plans shows a clear connection between the students written needs and measures.

The assumption of approximate normality of a pivotal used for constructing confidence intervals or tests often is violated by a substantial correlation between the point estimator at hand and the estimator of its variance. This can be caused by underlying skewness of data, generating both bias and skewness of the pivotal and leading to, for example, confidence intervals with poorer coverage properties, especially for one-sided intervals. We present in a general setting a simple, yet effective procedure that takes into account this correlation and leads to alternative adjusted intervals with more appealing coverage properties. We provide examples, including a simulation study.