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 5^th 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.

This work proposes the complete design cycle for several auxetic materials where the cycle consists of three steps (i) the design of the micro-architecture, (ii) the manufacturing of the material and (iii) the testing of the material. We use topology optimization via a level-set method and asymptotic homogenization to obtain periodic micro-architectured materials with a prescribed effective elasticity tensor and Poisson’s ratio. The space of admissible micro-architectural shapes that carries orthotropic material symmetry allows to attain shapes with an effective Poisson’s ratio below −1−1. Moreover, the specimens were manufactured using a commercial stereolithography Ember printer and are mechanically tested. The observed displacement and strain fields during tensile testing obtained by digital image correlation match the predictions from the finite element simulations and demonstrate the efficiency of the design cycle.

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

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

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

STEAM (Science, Technology, Engineering, Arts, and Mathematics) research emphasises interdisciplinary approaches to developing skills needed for the 21st century. In this paper, we explore three aesthetically rich mathematical activities involving dance and bodily performance (e.g., creative body postures) with 6-year-old students. Supported by an embodied perspective of task design and a Deweyan view of aesthetics, we argue that a high degree of bodily engagement and a high degree of task integration allow space for creativity and imagination that may involve aesthetic dimensions. Specifically, we argue that body postures, bodily rotation, rhythm, and fluency in composite sequences of movement may connect art and mathematics in a way that treats the students as active creators of aesthetic experiences, in contrast to other STEAM approaches that do not consider bodily performance as a way of connecting art and mathematics.

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

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

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

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

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

Content and Language Integrated Learning (CLIL) is a relatively new method for learning a foreign language. CLIL programs are growing in popularity and more and more schools are adopting it all around the world. There is still no special education for the teachers working with CLIL, though research suggests it may be an effective method for achieving good results.To get a realistic view of the work of a CLIL teacher, I have interviewed four different mathematic teachers who work at CLIL international schools in Spain with English as the foreign language. I have compared their methods and experiences with the main ideas of current research in this particular field.The survey shows that the teachers’ views about how CLIL is to be used do not completely correspond to that of the research. My conclusion is therefore that training in CLIL is needed.

With the ongoing digitization in today's society, which we also see in school operations, it is of great interest to examine its impact in the classroom. The aim of the present study was to investigate teachers' use and perception of interactive whiteboards in mathematics teaching. Furthermore, teachers' perception of their digital competence was also investigated. To fulfill the purpose of the study, eight qualitative interviews were conducted with teachers from four different schools active in grades f-6.

The results of the study showed that all teachers had a positive attitude towards the use of interactive whiteboards in mathematics teaching. The teachers used the board mainly in briefings where they had a perception that interactive whiteboards clarified, structured, and adapted the teaching. The teachers' main purpose for using the interactive whiteboard was to create engagement among the students.

A further conclusion of the study's results is that the digital competence of the teachers is of great importance for interactive whiteboards to be used as efficiently as possible. Furthermore, the results of the study showed that neither general nor subject-specific training within the interactive whiteboards was common. The teachers, on the other hand, expressed the desire to receive education. In conclusion, the results of the study showed that the teacher's self-interest and prioritization of time were decisive in the use of interactive whiteboards. In summary, this indicates that education within the interactive whiteboard is both desirable and necessary for the it to be used effectively in mathematics teaching. 

Digital teaching materials are becoming increasingly more common in the current education in mathematics. This means that the numbers of different digital teaching materials are increasing, which could possibly make it problematic for teachers to determine the quality of the content. Two digital materials that was created for grades 1-3 have been examined through a content analysis. The purpose of the study was to examine how subtraction including subtracting across the tens boundary within the number field 0-20 was represented and which useful strategies that students could use to solve the subtraction tasks. The empirics was collected using an analyzing tool that contained explicit definitions of the representations and the strategies that have been examined in the study. The results showed a relatively low spread of relevant representations connected to subtraction. An interesting discovery was that the type of tasks where two amounts can be compared to determine the difference was not represented at all in the digital materials that were analyzed. The most common representation appeared to be figures and symbols, while static and moving images with a distinct connection to the object of learning (subtraction) was not common at all. A general conclusion is that there is room for development when it comes to a wider variation of both representations and strategies in both of the digital materials. A wide representation of an object of learning, subtraction in the context of the study, creates an opportunity for the students to develop a deeper understanding for it. 

In this paper we consider the inverse problem of the permittivity determination of thin multisectional diaphragm in a rectangular waveguide. We perform a detailed analysis for one-, two- and three-sectional thin diaphragms. Numerical results are presented.

The mathematics teaching young children encounter affects not only the mathematics they are given the opportunity to learn but also their view of what mathematics is, how mathematics is taught and how they view their ability to learn mathematics. In this paper, an Educational Design Research study on problem solving and problem posing with Swedish six-year-olds will serve as an example when we elaborate on how children’s participation in education may affect their views of what it means to be taught and to learn mathematics. We do so by comparing drawings, of mathematics classrooms made of children from eight classes of which four participated in the study. The results indicate a reform-oriented and more diverse view of content and form in the drawings of the children participating in the intervention. 

This master thesis focuses on mathematics textbooks for preschool class. We have studied to what extent play is found in the textbooks and what kind of the support for implementation is offered to the teacher. We have focused on play which can contribute to the pupils’ concept and/or reasoning ability. 

Preschool class is now a compulsory school form. Preschool class is a class between preschool and primary school and is expected to function as a bridge between these two. There are many expectations for preschool class: implementing preschool pedagogy while introducing the primary school’s routines and teaching methods.

We have used content analysis as a method in this study. The method has been used in both a quantitative and a qualitative way. 

The results show that teacher guides offer play contributing to conceptual and/or reasoning. The results also show that the teacher gets support for implementing such play in the classroom. This offers opportunities for pupils to consolidate and develop their knowledge of mathematical concepts as well as their reasoning.

Further, the study shows that play in the pupils’ textbook largely is lacking. If the teacher chooses to only use the pupil’s textbooks, the consequence can be that the pupils are missing out on play in mathematical education.

Mathematical modelling is one of the competences students will have the possibility to develop in upper secondary school. However, it is not entirely obvious among teachers what modelling really means and it is questionable if the learning activities that are being used in relation to modelling education today really are effective. This study focuses on the students’ potential development within a series of lessons, a so-called modelling module, adapted for the course Mathematics 5 in upper secondary school. The students’ potential development is studied based on their involvement in the various sub-processes that are considered to be the modelling process as a hole. The result indicates that the modelling module involves students in all of these sub-processes. At the same time, the results show this is not self-evident and some of the subprocesses seem to be harder to involve the students in than others.

Som student på lärarutbildningen fick jag bland annat lära mig om olika inlärningsstilar och om vikten av varierade arbetsformer i undervisningen. Något förenklat menar vissa forskare att man kan skilja den teoretiskt lagde eleven från den mer praktiskt lagde, vilket betyder att en mix av teoretisk och praktisk/laborativ undervisning skapar bra förutsättningar för att så många som möjligt ska kunna ta del av den. Inte minst viktigt är detta i ämnet matematik och budskapet från Skolverket är det samma vad gäller detta ämne.

Denna uppsats handlar om matematikundervisningen på högstadiet, det vill säga grundskolans senare år, åk 7-9. Den är kopplad till de faktorer som styr vårt lärande och om elevernas förståelse för matematik gagnas av praktiska och laborativa övningar eller inte.

Då det i diskussioner om dagens matematikundervisning ofta framkommer att användandet av den praktiska/laborativa matematikundervisningen på högstadiet borde vara större, har jag i denna studie valt att undersöka om dess inverkan verkligen är så viktig som görs gällande.

Är tron på praktisk och laborativ matematikundervisning relevant eller överdriven?

Med hjälp av litteratur, samtal med lärare och pedagoger och en betygsjämförelse har jag försökt att hitta fakta som belyser den praktiska/laborativa undervisningens betydelse för studieresultatet i matematik. I studien har 1038 elever med betyg G i matematik deltagit och deras betyg i matematik, idrott och slöjd har jämförts och analyserats.

Resultatet visar att hälften av alla dessa elever uppvisar ett bättre betyg i idrott såväl som i slöjd. Detta resultat kan ses som en indikation på att elever med betyg G i matematik är mer praktiskt lagda och främjas av en praktiskt orienterad undervisning.

We study a 2-functor that assigns to a bimodule category over a finite k-linear tensor category a k-linear abelian category. This 2-functor can be regarded as a category-valued tracefor 1-morphisms in the tricategory of finite tensor categories. It is defined by a universalproperty that is a categorification of Hochschild homology of bimodules over an algebra.We present several equivalent realizations of this 2-functor and show that it has a coherent cyclic invariance.Our results have applications to categories associated to circles in three-dimensional topological field theories with defects. This is made explicit for the subclass of Dijkgraaf-Wittentopological field theories.

Lyubashenko's construction associates representations of mapping class groups Map_{g,n} of Riemann surfaces of any genus g with any number n of holes to a factorizable ribbon category. We consider this construction as applied to the category of bimodules over a finite-dimensional factorizable ribbon Hopf algebra H. For any such Hopf algebra we find an invariant of Map_{g,n} for all values of g and n. More generally, we obtain such invariants for any pair (H,omega), where omega is a ribbon automorphism of H. Our results are motivated by the quest to understand correlation functions of bulk fields in two-dimensional conformal field theories with chiral algebras that are not necessarily semisimple, so-called logarithmic conformal field theories.

Abstract

By finding out the pupils' choice of strategies for problem solving, we as teachers, can get a better understanding of their way of thinking, and thus also help them develop in their learning of mathematics. In a classroom environment where several different strategies could be found, the pupils' learning is facilitated as they, with the help of classmates, find positive and negative attributes of different kinds of strategies. The pupils are, in other words, each other's resources, rather than competitors. The main point is that the classroom environment, as well as the role of the teacher in the classroom, are both key elements for good mathematics learning.

In order to increase the understanding of the pupils' choice of learning strategies, a survey was performed based on two school classes, one third grade and one six grade class at the same school. Both classes received the same mathematical problem to work with, and the strategies observed were analyzed and compared. The results show that the pupils' choice of strategies in the two grades did not differ significantly.

The Swedish National Agency (2002a) for Education emphasizes the importance of students developing their reasoning skills, which is one of the five competencies outlined in the mathematics curriculum. Despite the curriculum's emphasis, research shows that students often spend a lot of time working independently with textbooks during mathematics instruction. Therefore, our study examines how first-grade students reason when working on tasks designed to promote creative mathematical reasoning (CMR). Additionally, the study explores the opportunities and challenges associated with using such tasks. To answer the research questions, Lithner's (2008) categorization of different types of reasoning and Gray and Tall's (1994) theory of proceptual thinking were used as theoretical frameworks. Vygotsky's theory was also employed for a broader understanding of the study's results (Säljö, 2020). The study identified opportunities for using this approach as a method, provided it is adapted and supported appropriately. This can help students develop deeper mathematical insights and improve their reasoning skills. One challenge with this approach was finding a task that is challenging for all students despite differences in their prior knowledge. The method used to collect empirical data was "task-based interviews." Students engaged in mathematical activities and were asked to think aloud to reveal their thought processes. The results showed that although it was challenging to apply Lithner's (2008) framework to younger students, most were able to engage in CMR. 

This study investigates resource use by upper-secondary mathematics teachers in the context of collaborative planning. A thematic content analysis is conducted on audio-recorded teacher discussions in order to find out what resources are used by the teachers, how they are used, and for what reasons. The findings show that although teachers use a variety of resources to support their instructional enactment as well as instructional design, there is a difference in how they use different resources to support different planning practices. For instructional design, curriculum resources provide support for the mathematical content, while social resources, self-generated documents, and cognitive resources provide support for the design of instructional activities. Authority is given by teachers to curriculum resources, but conflicts of authority emerge in discussions, when teachers’ abilities to exert their agency are not supported by curriculum resources. We discuss the findings in relation to authority and resource use, as well as for established conceptualizations of resources. The findings bear practical implications for the design of curriculum resources.

The purpose of this study was to identify which thought's teachers had on the curriculum and its term to connect mathematics with reality, and how to combine this with concept of mathematics. The results presented in this report are based on interviews with seven teachers.

The results showed that mathematics connected to reality can be interpreted from two basic principles. 1. To adapt the teaching to the pupils’ everyday life and by that facilitate learning here and now. 2. To prepare the pupils for their future so they feel secure and confidence in using mathematics. It was also based on these two approaches that the teachers' view of how the teaching of mathematics connected to reality became visible. For those who had the con-venience of that the students should learn here and now, was working with everyday math and to experiment with practical material current. The long-term approach, learning for life, as-sumed to work with problem solving in which interaction, discussion and argument were im-portant parts. Some of the teachers, who thought that facilitate learning here and now was how to connect mathematics to reality also thought about teaching, unconsciously or con-sciously, as learning for life.

En säker informationskanal med hög överföringskvalitet krävs i dessa dagar när informationsöverföringen ökar för varje år som går. Det finns olika sätt att skapa detta. Antingen genom att se till att överföringsmediet är av mycket hög kvalitet eller att skapa en skyddsmekanism som gör att de överföringsfel som kan uppstå kan detekteras och även korrigeras om man önskar detta. Denna uppsats handlar om detta, att kunna detektera och korrigera fel. Denna gren inom matematiken kallas kodningsteori.

Uppsatsen presenterar grunden för kodningsteorin, för att sedan presentera några vanligt förekommande kodningsalgoritmer, Hamming koder, BCH koder, Reed-Solomon. Jag går in på djupet av en av de absolut äldsta kodningsalgoritmerna, en kod som presenterades 1954 av David E. Muller, något senare presenterade en annan föregångare inom kodningsteori, Irving S. Reed, en avkodningsalgoritm för Mullers kod. Denna kod blev känd under namnet Reed-Muller kod.

Jag presenterar teorin bakom Reed-Muller kod och hur ett Reed-Muller kodord skapas med hjälp av teorin. Jag visar också hur man avkodar Reed-Muller kod med hjälp av olika algoritmer där Irving S. Reeds algoritm står i centrum. För att testa kodning och avkodning i simulerad verklighet används datorprogrammet Matlab. Slutligen presenteras hur kodnings- och avkodningsalgoritmer kan skapas med hjälp av grindnät.

The purpose of this study is to investigate students’ self-esteem regarding the subject of

mathematics and what affects their motivation in the subject. The applied method in

this study is a survey that generated both quantitative and qualitative data. The basis of

the survey and the analysis is the self-efficacy theory and the self-determination theory.

The results of my study indicates that the students’ need for competence, togetherness

and autonomy are not met to the degree that allows them to feel motivation in the

subject of matematics. The result can also be interpreted as that students’ self-esteem is

relatively low and in these cases has a negative impact on the students’ motivation in

the subject. In order for us teachers to be able to further motivate students in the subject

of matematics we have to ensure that these needs are fulfilled and that we help them to

increase their self-esteem in the subject.

Feedback in school is a relevant and important aspect of students' learning and knowledge development. In the school system, the digital opportunities have increased and with this, the digital teaching materials have also taken on a larger and more significant role in the school's operations. Feedback can take place in different ways that contribute to different types of processes in the students in their knowledge acquisition. In this study, we have analyzed three digital teaching materials that all market themselves with relevant concepts that are linked to feedback. The analysis has been based on four feedback levels: task, process, self-regulation and the individual level. We have tried to identify how well these teaching materials provide feedback at these levels for students' individual learning. Results showed that there are large differences between the different levels and how well the teaching aid provides the student with feedback. But the feedback structure between the different teaching aids is relatively similar.

Ground-based mechanized forestry requires the traversal of terrain by heavy machines. The routes they take are often called machine trails, and are created by removing trees from the trail and placing the logs outside it. Designing an optimal machine trail network is a complex locational problem that requires understanding how forestry machines can operate on the terrain as well as the trade-offs between various economic and ecological aspects. Machine trail designs are currently created manually based on intuitive decisions about the importance, correlations, and effects of many potentially conflicting aspects. Badly designed machine trail networks could result in costly operations and adverse environmental impacts. Therefore, this study was conducted to develop a holistic optimization framework for machine trail network design. Key economic and ecological objectives involved in designing machine trail networks for mechanized cut-to-length operations are presented, along with strategies for simultaneously addressing multiple objectives while accounting for the physical capabilities of forestry machines, the impact of slope, and operating costs. Ways of quantitatively formulating and combining these different aspects are demonstrated, together with examples showing how the optimal network design changes in response to various inputs.

Vårt samhälle blir mer och mer digitaliserat, vilket också visar sig i skolans värld. Med digitaliseringens framfart blir det vanligare med digitala läromedel som kompletterar eller ersätter de analoga. Tekniken kan bidra till och möjliggöra en omvandling av undervisningen samt elevernas inlärningsprocess. Vad det gäller problemlösning i matematikundervisning kan det digitala läromedlet bidra till att vidga det sociala rummet, för att elever ska kunna kommunicera matematik med andra. Det skulle även kunna innebära att eleverna ges möjlighet att resonera och argumentera för valda strategier genom att spela in ljud och film. Det kan också innebära att problemlösningsuppgifter presenteras genom korta animationer istället för abstrakta meningar och symboler. Vi upptäckte på vår verksamhetsförlagda utbildning (VFU) att tekniken inte utnyttjade dessa fördelar när eleverna arbetade med problemlösning i digitala läromedel. Utvecklingen av elevernas problemlösningsförmåga hämmades snarare än utvecklades av tekniken. Genom att analysera läromedel som riktar sig till elever i årskurs tre, besvaras frågan om vår observation var en tillfällighet eller inte. För att ta reda på studiens syfte har ett analysverktyg skapats utifrån teorier av Choppin m.fl. (2014), Taflin (2007) och Hughes m.fl. (2006).

Studien syftar till att bidra med mer kunskap om hur digitala läromedel kan påverka utvecklingen av elevernas problemlösningsförmåga i årskurs tre. Med hjälp av en kvantitativ datainsamling och en kvalitativ analys har studiens syfte besvarats. Resultatet visar att elevernas möjligheter vid problemlösningsuppgifter varierar mellan de analyserade läromedlen. Vidare visar resultatet att samtliga läromedel följer en förutbestämd progression oavsett elevernas kunskapsnivå och behov. Det visade sig också att det endast är genom användning av lärarhandledning en differentiering av innehållet kan ske. En slutsats är därmed att digitala läromedel har utvecklingspotential vad det gäller att utnyttja teknikens fördelar, för att bidra till en progression av elevers problemlösningsförmåga. 

If students are to build knowledge, it is important to connect disciplinary knowledge to students’ everyday knowledge when teaching. In this study, the relationship between disciplinary- and everyday knowledge in subject-teaching is analysed, based on the semantic dimensions used in Legitimation Code Theory. Un-packing disciplinary concepts through concrete examples implies a shift towards a context-dependent everyday knowledge, while re-packing them entails a shift towards disciplinary abstractions and context-independent generalizations. Over time these shifts constitute so-called semantic waves argued to facilitate students´ knowledge building. Earlier research suggest that the form of these semantic waves can differ between school-subjects. Therefore, this article examines semantic waves in two contrasting subjects – Mathematics and Social Science Education, aiming at a better understanding of how and why semantic waves differ. The results reveal that the forms of semantic waves differ between the two subjects. In Mathematics, when teaching different geometrical concepts, the semantic shifts through un-packing and re-packing were frequent and evenly distributed. When teaching about pricing and household in Social Science, un-packing activities dominated, making shifts biased towards everyday knowledge. These differences are discussed in terms of semantic waves constituting a different pulse when making knowledge-building possible in Mathematics and Social Science.

In recent years, significant efforts have been made to increase access to digital resources in schools, leading to a rise in both availability and usage of digital educational games. This study investigates occurrences and highlights differences of feedback among the five most commonly used digital educational games in elementary school mathematics. This was accomplished through content analysis based on a framework comprising of five different types and four levels of feedback. 

The results indicate that all types and levels of feedback occur to varying extents in the five digital educational games. Across all games, feedback addressing past performance, encouraging feedback, and result-oriented feedback were present. Feedback correcting incorrect answers was found in four out of five games, with two of them offering this feedback as optional and two automatically correcting the answer. The least prevalent feedback type was forward-looking feedback, which was found in one of the five games.

We use the social force model to study the behaviour of two crowds of pedestrians in a bi-directional flow. There are two main goals of the project. The first goal is to study the effects of perception anisotropy on lane formation of the two crowds. The perception anisotropy models the fact that people actually do not perceive their surroundings equally well in all directions, i.e they have a field of vision. The second goal is to develop pedestrian viscosity indices to characterize the motion of the crowds. Our concepts of viscosity indices estimate how fluid the motion of the crowds are. We develop two viscosity indices. A lane viscosity index which gives information of the flow on large timescales, and a space-dependent viscosity index which can pinpoint where in space the motion is less fluid. Our results indicate that there is a small correlation between the perception anisotropy and the lane formation of the two crowds. The two viscosity indices work as intended but more refinement is needed to cope with simultaneous space-time changes.

Children are able to learn mathematics in many different ways. To study and learn mathematics out of doors are not so common in today's nine-year compulsory school, but can work as a perfect complement to the usual teaching methods. The purpose of this essay is to make researches into advantages and disadvantages with outdoor mathematics. The conclusions of this essay are based on results from six structured interviews with teachers at a school in the west of Sweden, as well as literature and research. All of the teachers that have been interviewed in this essay, have experience of working with mathematics out of doors. Outdoor mathematics helps to put mathematics in a concrete form. The outdoor environment gives children a good chance to use all of their senses, which can confirm old and new knowledge, at the same time as they get the opportunity to use their whole bodies. Disadvantages with outdoor mathematics are often related to shortage of staff and increased planning from the teacher's side. In the end, the attitudes of the teacher are the major obstacle when it comes to outdoor mathematics.

Uppsatsen börjar med definitionen av en grupp och fortsätter med grundläggande begrepp och egenskaper hos grupper såsom undergrupper, homomorfismer, isomorfismer, normala undergrupper, faktorgrupper och permutationsgrupper. Sedan behandlas konjugerade undergrupper, konjugatklasser och klassekvationen med vars hjälp det visas att en grupp med ordning $p^{2}$ är abelsk. Vidare behandlas direkta och semidirekta produkter av grupper. Det konstateras att omvändningen till Lagranges sats i allmänhet inte gäller men att Sylows satser finns istället. Med hjälp av den tidigare teorin visas sedan att de enda enkla grupperna med ordning mellan 1 och 60 är av primtalsordning. Det visas också vilka olika grupper som finns med ordning upp till och med 15. Uppsatsen avslutas med en introduktion av datorprogrammet GAP.

We study the size of sets containing sums of two integer cubes such that their representation is unique and also fit between two consecutive integer cubes. We will try to write algorithms that efficiently calculate the size of these sets and also implement these algorithms in Python^TM.

Although we will fail to find a non-iterative algorithm, we will find different ways of approximating the size of these sets. We will also find that techniques used in our failed algorithms can be used to calculate the number of integer lattice points inside a circle.

Mathematics is a subject that contains many different concepts that are ultimately important for students to have knowledge of in order to cope with the subject, but also everyday situations. The teaching of mathematics often means that the students get to meet mathematics by working through the mathematics textbook, where an alternative could be to let the students meet mathematics and its concepts through play. Play is especially important as a teaching method in preschool class, because preschool class is expected to use the playful pedagogy from preschool to prepare students for the school years to come. The purpose of this study is therefore to gain more knowledge in how preschool class teachers can promote students´ concept development in mathematics, by using play as a teaching method. The following research questions are addressed: 

·       What kinds of games are presented by teachers on social media to develop students´ mathematical conceptual ability in mathematics? 

·       What aspects of the teacher´s role are displayed on social media when it comes to students´ mathematical concept development through play?

Posts on social media were collected, to see how active teachers in preschool class work with play in the subject of mathematics. The prevailing pandemic that took place at the time the work was written, influenced the choice of method.

Social media is a collective name for a number of different platforms and for this work, 4 different platforms were selected to see how teachers use play in their mathematics lessons. To help analyzing the various posts, an analysis protocol was developed with various aspects around, among other things, the teacher´s role in the play. The results show that play in many posts is used in concept development. The results also show that teachers let their students work with different pedagogically materials and play to attach many of all the concepts of mathematics. 

The internal structure of adhesive tapes determines the effective mechanicalproperties. This holds true especially for blended systems, here consisting ofacrylate and rubber phases. In this note, we propose a lattice-based modelto study numerically the formation of internal morphologies within a fourcomponent mixture (of discrete particles) where the solvent components evaporate. Mimicking numerically the interaction between rubber, acrylate, andtwo different types of solvents, relevant for the technology of adhesive tapes,we aim to obtain realistic distributions of rubber ball-shaped morphologies—they play a key role in the overall functionality of those special adhesives.Our model incorporates the evaporation of both solvents and allows for tuningthe strength of two essentially different solvent–solute interactions and of thetemperature of the system.

Epigenetics is coming to the fore as a key process which underpins health. In particular emerging experimental evidence has associated alterations to DNA methylation status with healthspan and aging. Mammalian DNA methylation status is maintained by an intricate array of biochemical and molecular processes. It can be argued changes to these fundamental cellular processes ultimately drive the formation of aberrant DNA methylation patterns, which are a hallmark of diseases, such as cancer, Alzheimer’s disease and cardiovascular disease. In recent years mathematical models have been used as effective tools to help advance our understanding of the dynamics which underpin DNA methylation. In this paper we present linear and nonlinear models which encapsulate the dynamics of the molecular mechanisms which define DNA methylation. Applying a recently developed Bayesian algorithm for parameter estimation and model selection, we are able to estimate distributions of parameters which include nominal parameter values. Using limited noisy observations, the method also identified which methylation model the observations originated from, signaling that our method has practical applications in identifying what models best match the biological data for DNA methylation.

National and international studies, such as PISA and TIMSS, which are part of the national evaluation system, are carried out continuosly. Skolverket, in collaboration with researchers, are responsible for the implementation, evaluation, and analysis of the studies. National results show an increasing percentage of pupils who do not reach the goals in mathematics. The last completed international studies PISA 2009 and TIMSS 2007 show an impaired trend of knowledge, in international comparison. Moreover, studies also show that the differences between schools and also within the same school in Sweden have increased. In summary, the analysis of the studies shows that the Swedish students' math skills have been declining over time. The purpose of this report is to examine possible causes, essential factors that are affecting and future actions in order to achieve better results. Completed national and international studies and analysis show a range of factors that are affecting Swedish students´ declining mathematic skills. However, this report clarifies the big complexity that is revealed when results are being analyzed. This is due to a multitude of factors of influence of underlying causes. Nevertheless, results and analysis show that teachers’ skills are a strong factor which affects and influences the progression and accumulation of knowledge.

The new technologies, in particular social media in Web 2.0, enable rapid change in people’s behaviour, which needs to be considered in research on teacher empowerment and teacher professional development and growth. In this chapter we discuss how teachers in an informal, yet structured, way use social media to expand their professional learning communities beyond the local school context in Sweden. This is an example of how a new behaviour is emerging among teachers that changes the opportunities and the frames for professional development and growth. Through teachers’ engagement in social media, such as Facebook, extended professional learning communities arise and teachers’ professional development and growth become evident. Global levels influence local levels: teachers from different schools engage in structured discussions related to everyday practice, such as issues of learning goals in pre-school or topics related to a specific course in upper secondary school. The teachers’ arena for professional development and growth has changed, which means that the context of teacher empowerment is rapidly changing too. Consequently, the chapter includes theoretical reflections on professional learning communities in a Web 2.0 world and how this phenomenon may affect our approach to enhancing teachers’ professional development.

Teachers worldwide are using social media as a professional development resource. In studying social media as ‘a place’ for teachers’ professional development, we investigated large Facebook groups with themes connected to teaching and learning in compulsory schools. The interaction in these groups was analysed within the framework of systemic functional grammar. In order to reveal knowledge known and shared by teachers as a community, we have also used Shulman’s (1987) framework. Most posts received responses and this response is in line with the expected response pattern. The speech functions ‘Questions’ and ‘Offers’ were most common. Further, most posts addressed subject specific knowledge. The multi-theoretical approach used when researching mathematics teachers’ professional development in self-organized online groups showed that these large Facebook groups facilitated professional learning.

We consider an equidistributed concatenation sequence of pseudorandom rational numbers generated from the primes by an inversive congruential method. In particular, we determine the sharp convergence rate for the star discrepancy of said sequence. Our arguments are based on well-known discrepancy estimates for inversive congruential pseudorandom numbers together with asymptotic formulae involving prime numbers.

The Swedish national agency of education has written reports about what the first-teacher do in general. The purpose with this study is to exemplify what the first-teacher in mathematics do and what it possibly can result in, both for colleagues and pupils. By making qualitative interviews with seven first-teachers in mathematics the study illustrates how they work and what outcome they experience. The results show that the first-teacher assignments differs, for example, one teacher does nothing above her regular teaching in her assignment more than having a title. Another first-teacher actively work with screening to assess pupils’ knowledge in mathematics, enables teaching of practical mathematics and is responsible for different projects in developing competence for colleagues. The first-teachers work is done under different conditions, such as time assigned to the mission, number of pupils at the school they work at and other on-going projects. Something that is crucial for the first-teacher efforts is, in part, how committed principal and school leadersare, and what demands they make, partly how passionate and skilful the first-teacher is herself.

Cancer is a complex illness that affects millions of people every year. Amongst the most frequently encountered variants of this illness are solid tumours. The growth of solid tumours depends on a large number of factors such as oxygen concentration, cell reproduction, cell movement, cell death, and vascular environment. The aim of this thesis is to provide further insight in the interconnections between these factors by means of numerical simulations.

We present a multiscale model for tumor growth by coupling a microscopic, agent-based model for normal and tumor cells with macroscopic mean-field models for oxygen and extracellular concentrations.

We assume the cell movement to be dominated by Brownian motion. The temporal and spatial evolution of the oxygen concentration is governed by a reaction-diffusion equation that mimics a balance law.To complement this macroscopic oxygen evolution with microscopic information, we propose a lattice-free approach that extends the vascular distribution of oxygen. We employ a Markov chain to estimate the sprout probability of new vessels. The extension of the new vessels is modeled by enhancing the agent-based cell model with chemotactic sensitivity.

Our results include finite-volume discretizations of the resulting partial differential equations and suitable approaches to approximate the stochastic differential equations governing the agent-based motion.

We provide a simulation framework that evaluates the effect of the various parameters on, for instance, the spread of oxygen. We also show results of numerical experiments where we allow new vessels to sprout, i.e. we explore angiogenesis. In the case of a static vasculature, we simulate the full multiscale model using a coupled stochastic/deterministic discretization approach which is able to reduce variance at least for a chosen computable indicator, leading to improved efficiency and a potential increased reliability on models of this type.

Half the world’s river volume is affected by flow alterations and/or fragmentation, a figure that is likely to increase with the current global surge in hydropower development. At the same time, freshwater biodiversity is in rapid decline. In fluvial ecosystems, streamflow is a master variable, shaping riverine species’ habitat over space and time. Atlantic salmon (Salmo salar) and brown trout (S. trutta) are examples of species that need fluvial habitats for reproduction and juvenile rearing, and whose flow needs can come in conflict with hydropower production objectives. This necessitates tools for predicting the effect of fluvial habitat alteration on fish production.

In this thesis, I applied the individual-based model inSTREAM to simulate salmon and trout in the Gullspång Rapids, a residual flow stretch of the Gullspång River, Sweden, over a ten-year period. InSTREAM uses sub-daily time steps to simulate individual fish interacting with their biotic and physical environment, and tracks the fitness consequences of their main actions: habitat and activity selection. For inSTREAM input, I had to describe key habitat features, create a 2D hydraulic model of the rapids, model river temperature, and gather data on salmon and trout eco-physiology and life history characteristics in the Gullspång River. I ran simulation experiments varying either flow input, the number of spawners, spawning gravel, shelter and boulder availability, or temperature. Flow alterations had limited benefit. Only the scenario where the current minimum flow was set three times higher yielded increased production, and only for trout. Trout dominated salmon in competition, and production was density dependent. The model predicted that the only way to increase production of both species at current spawner numbers was to add instream structures offering more energetically profitable feeding positions.