Undersökningen beskriver den matematiska kommunikationen mellan speciallärare och matematiksvaga elever. Den empiriska datan består av två matematiklektioner från år 9 i grundskolan. Datan analyserades för att finna olika typer av kommunikation. Lärarna var i många delar medvetna i sitt sätt att kommunicera med eleverna och undervisningsmiljön kändes trygg. Under lektionerna observerades två parallella språk, ett vardagsanknutet och ett matematiskt språk. Det framkom att ett tydligt förhållningssätt beträffande matematiskt språk kan underlätta för elevens inlärning samt medvetandegöra läraren om den kommunikativa betydelsen i undervisningen med styrdokumentens mål i fokus. / The present investigation describes the mathematical communication between the special needs teacher and pupils with mathematical disabilities. The empirical data consists of two mathematics lessons from grade 9 in the compulsory school. The data was analyzed in order to find different kinds of communication. The teachers were in many ways aware of their way to communicate with pupils and the teaching environment felt confident. During the lessons two parallel languages were observed, one everyday related and one mathematical language. It was found that keeping a distinct approach to the mathematical language can facilitate student learning and make the teacher aware about teaching communicatively with steering documents goals in focus.

Syftet med denna empiriska studie är att undersöka hur lärare kan erbjuda möjligheter för 6-åringar att vidga sin förståelse av likhetstecknets innebörd. Undersökningen är inspirerad av Learning Study metoden vilket är en metod som har sin utgångspunkt i variationsteorin. Barnens initiala förmågor analyseras före planeringen av undervisningen och det ger en uppfattning om barnens kunskaper. Utifrån det planeras lektionerna och de aspekter som är de mest kritiska för barnen lyfts och på så sätt utvecklas deras kunskap. Vår slutsats är att det är viktigt att arbeta utifrån ett variationsinriktat lärandeperspektiv. Barnen lyckas bra i de uppgifter där de får laborera med konkret material. / The purpose of this empirical study is to examine how teachers can provide opportunities for children aged 6 to extend their understanding of the equal sign. This study is inspired by the Learning Study method which is a method that has its origins in variation theory.  By analyzing the children's initial abilities before planning our lessons, it gives us an idea of the children's knowledge. From there the lessons are planned and the aspects that are the most critical for the children are lifted and thereby develop their knowledge. Our conclusion is that it is important to work with variety oriented learning. The children are successful in tasks in which they are allowed to experiment with concrete materials.

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. / Språk och ämnesintegrerad inlärning (SPRINT) är ett relativt nytt sätt att lära sig språk på. SPRINT har växt sig allt starkare och fler och fler skolor satsar på denna metod världen över. Det finns ännu ingen specialiserad utbildning för lärare som arbetar med SPRINT -metoden, men många SPRINT - forskare menar att det vore nödvändigt med en sådan för att man skall uppnå goda resultat. För att göra mig en bild av hur SPRINT- lärare i verkligheten arbetar, har jag genomfört intervjuer med fyra matematiklärare som arbetar med SPRINT i Spanien på internationella skolor med engelska som inriktning. Jag har sedan jämfört dessa lärares metod och erfarenhet med vad forskningen säger.Undersökningen har visat att lärares uppfattning om hur SPRINT är menat att användas inte alltid stämmer överens med SPRINT – forskarnas.  Min slutsats blir därför att det skulle behövas en utbildning i denna metod.

The aim of this thesis is to try to make the mathematics content of textile handicraft visible. This is done through an attempt at integrated studies in the two subjects with 6th grade students. The main question is: In what way may integrated studies in mathematics and textile handicraft lead to making the mathematics content of textile handicraft visible to the students? This question has been divided into three sub-questions: 1. What knowledge do students have on the order of the millimeter, centimeter, decimeter and meter units of length?, 2. In what situations do students mention knowledge of measuring as being of use?, and 3. To what extent do students see a connection between mathematics and textile handicraft before and after an attempt at integrated studies in mathematics and textile handicraft? The methods of use in this thesis are questionnaires and interviews. The theoretical frame is constituted by the sociocultural theory. This theory focuses on the idea that thinking is closely connected to our activities, which makes it a suitable basis for the thesis. The results of the study show that overall the students’ knowledge of the order of the units of length in question is good. The students mention knowledge of measuring as being of use mostly in situations related to consumption, building and painting, and mathematics. No student mentions a connection between mathematics and textile handicraft in the questionnaires. However, the interviewees mentioned measuring as being of use in textile handicraft. The study gives no definite answer to the question of whether integrated studies can make the mathematics of textile handicraft visible. The fact that students in interview after integrated studies are able to discover a connection between the two subjects may indicate that the mathematics content of textile handicraft has been made visible.

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.

Purpose: The current thesis assignment aims to explain, understand and follow the strategies the young children employ into a conception of numbers. This thesis is based on a study performed by Doverborg and Pramling Samuelson. The following research questions serve to refine the stated goal: How mathematical thinking regarding conception of numbers occurs within younger children and which strategies they use? How is the interaction between the teacher and the younger children? Method: The study is organized as a qualitative study and was conducted in a pre-school with a group of four children. The children that have participated within my study are between two and three years old. They were filmed in four exercises. Then the films were transcribed. The method used is an analytic approach intentional analysis which allows studying the children´s strategies according to the established questions and also it makes it possible to follow how they think regarding mathematics. Conclusion: My study group shows in the final results that they in their strategies made use of one or more principles in their conception of numbers. / Syfte: Bidra till en bättre förståelse för små barns tänkande kring matematik. Jag utgår ifrån Doverborg och Pramling Samuelsons studie som behandlar antalsuppfattning i matematik hos de yngsta barnen. Följande frågeställningar syftar har undersökts: Hur sker de yngsta barnens matematiska tänkande omkring antalsuppfattning och vilka strategier de använder sig av? Hur sker samspelet mellan pedagogen och de yngre barnen? Metod: Studien är upplagd som en kvalitativ studie och genomfördes i en förskola med en grupp på fyra barn. De barn som har deltagit i min studie är mellan två och tre år gamla. De filmades i fyra övningar varefter filmsekvenserna transkriberades. Metoden som använts är en analytisk metod intentionell analys som gör det möjligt att studera barnens strategier i enlighet med fastställda frågor och även gör det möjligt att följa hur de tänker om matematik. Slutsats: Min undersökningsgrupp visar i slutresultatet att de i sina strategier använde sig av en eller flera principer i antalsuppfattning, nämligen principen om ett till ett korespondens, principen om den stabila ordningen, kardinalprincipen, och principen om godtycklig ordning. Barnen kan också urskilja grupper av två eller tre föremål, det vill säga subitizing.

The aim of this study has been to investigate students’ understanding of fractions when they occur in the practical subject home economics. In order to investigate this, the learning study method was used. Learning study can be understood as a method to gain deeper understanding of what is learnt by students during a lesson to increase learning outcome. The variation theory framework has been utilized to analyze the results of this study. The object of learning was chosen to be addition of fractions bigger than one half when they appear in home economics. The results indicate that in order for the learners to experience the object of learning it requires that they discern some critical aspects; to distinguish between quarters and four parts, the relationship between the part and the whole in fractions, varied presentations of the same fraction along with an understanding of the meaning of the concept to double.   Moreover the students have carried out a math test which tested the students’ theoretical understanding of fractions to see whether they can transfer the knowledge from one learning context to another. The result evinces that most of the learners show a good understanding of fractions in a theoretical context as well as a practical. The learning study has increased the students learning according to the object of learning

Syftet med denna rapport är att få mer kunskap om hur digitala läromedel kan främja elevers kunskapsbildning i matematikundervisningen. Denna kvalitativa c-uppsats omfattar två delar. Den första delen består i att utveckla ett verktyg för att kunna analysera datorprogram gjorda för matematikundervisning. Detta verktyg baseras på tidigare analysverktyg, litteratur samt egna erfarenheter. Den andra delen består i att analysera tre olika programvaror med hjälp av verktyget för att se hur det fungerar samt se på vilka sätt programvarorna främjar matematikundervisningen. Resultatet visar att verktyget fungerar bra och att de testade programmen är mycket olika och att alla har sina egna styrkor för att underlätta matematikundervisningen. Alla program har självklart även svagheter vilka främst består i att de inte tillräckligt tagit tillvara på de möjligheter det digitala mediet ger programmen.

Type 2 diabetes is a global disease which affects an increasing number of peopleevery year. At the heart of the disease lies insulin resistance in the target tissues,primarily fat and muscle. The insulin resistance is caused by the failure of a complexsignalling network, and several mechanistic hypotheses for this failure havebeen proposed. Herein, we evaluate a hypothesis that revolves around the proteinmammalian target of rapamycin (mTOR) and its feedback signals to insulin receptorsubstrate-1 (IRS1). In particular, we have re-examined this hypothesis andrelevant biological data using a mathematical modelling approach. During the course of modelling we gained several important insights. For instance,the model was unable to reproduce the relation between the EC50-valuesin the dose-response curves for IRS1 and its serine residue 312 (Ser-312). Thisimplies that the presented hypothesis, where the phosphorylation of Ser-312 liesdownstream of the tyrosine phosphorylation of IRS1, is inconsistent with the provideddata, and that the hypothesis or the data might be incorrect. Similarly, wealso realized that in order to fully account for the information in the dose-responsedata, time curves needed to be incorporated into the model. A preliminary model is presented, which explains most of the data-sets, butstill is unable to describe all the details in the data. The originally proposed hypothesisas an explanation to the given data has been revised, and our analysisserves to exemplify that an evaluation of a mechanistic hypothesis by mere biochemicalreasoning often misses out on important details, and/or leads to incorrectconclusions. A model-based approach, on the other hand, can efficiently pin-pointsuch weaknesses, and if combined with a comprehensive understanding of biologicalvariation and generation of experimental data, mathematical modelling canprove to be a method of great potential in the search for mechanistic explanationsto the cause of insulin resistance in type 2 diabetics.

<p>The work is focused on the Value at Risk and the Expected Shortfallcalculation. We assume the returns to be based on two pillars - the white noise and the stochastic volatility. We assume that the white noise follows the NIG distribution and the volatility is modeled using the nGARCH, NIG-GARCH, tGARCH and the non-parametric method. We apply the models into the stocks of three Banks of the Nordic market. We consider the daily and the intraday returns with the frequencies 5, 10, 20 and 30 minutes. We calculate the one step ahead VaR and ES for the daily and the intraday data. We use the Kupiec test and the Markov test to assess the correctness of the models. We also provide a new concept of improving the daily VaR calculation by using the high frequency returns. The results show that the intraday data can be used to the one step ahead VaR and the ES calculation. The comparison of the VaR for the end of the following trading day calculated on the basis of the daily returns and the one computed using the high frequency returns shows that using the intraday data can improve the VaR outcomes.</p>