Beräkningar av subtraktionsuppgifter visar sig vara en utmaning för svenska elever i årskurs 3. Problematiken ligger i valet av beräkningsmetoder, där dessa inte är lika effektiva och framgångsrika som vid beräkningar av addition. Denna studie är inspirerad av en fenomenografisk ansats, som syftar till att redogöra för vilka uppfattningar elever i årskurs 3 har av subtraktion, samt vad som påverkar elevers metodval vid beräkningar av subtraktion. Studiens syfte besvaras med följande frågeställningar; hur uppfattar elever subtraktion? Vilka aspekter påverkar elever att välja en specifik metod vid beräkningar av olika subtraktionsuppgifter? Med utgångspunkt från semistrukturerade individuella intervjuer med 10 elever från två klasser i årskurs 3, visar det sig att samtliga elever inledningsvis uppfattar räknesättet subtraktion som en subtraktiv förändring, där något försvinner eller tas bort. Vid beräkning av sex olika subtraktionsuppgifter; tre textuppgifter och tre uppgifter med okända tal, visar eleverna en variation i att vara flexibel respektive anpassningsbar i metodvalet. Resultatet redovisar kategorier som representerar olika beräkningsmetoder. Inom respektive kategori lyfts elevers egna beräkningar inklusive deras motiveringar till valet av beräkningsmetod.  Standardalgoritmen visar sig vara den dominanta beräkningsmetoden, där ett fåtal elever använder standardalgoritmen vid samtliga uppgifter. Där valet av att använda standardalgoritmen som beräkningsmetod beror på att metoden är enklast att använda och att det går fort att räkna ut svaret med standardalgoritmen. I resultatet hävdar eleverna även att valet av metod beror till stor del på de ingående talens värde i den givna uppgiften. / Calculations of subtraction tasks has shown to be challenging for Swedish pupils in grade 3. The problem lies in the choice of calculation methods, where these methods aren´t as effective and successful as those methods for calculations of addition.  This study is inspired by a phenomenographic approach and aims to explain what perceptions pupils in grade 3 has of subtraction, and what affects pupils’ choice of methods in calculations of subtraction. The aim in the study focuses on the following questions; how do pupils perceive subtraction? Which aspects influence pupils to choose a specific method when calculating different subtraction tasks? Based on semi-structured individual interviews with 10 pupils from two classes in grade 3, it turns out that all pupils initially perceive subtraction as a subtractive change, where something disappears or is taken away. In the calculations of six different subtractions task; three text tasks and three tasks with unknown numbers, the pupils show a variation by being flexible and adaptable in their choice of method. The result reports categories that represents different calculation methods. In each category, pupils own calculations is mentioned including their motivations for the choice of calculation method. The standard written algorithm is shown to be the most dominant calculation method, where a few pupils use the standard algorithm in all tasks. The choice to use the standard algorithm as calculation method is due to the simplicity of the method and that it´s faster to calculate the answer with the standard algorithm. In the result, the pupils also claims that the choice of methods depends on the value of the including numbers in the given task.

The purpose of this study is to observe some factors that may affect the understanding of subtraction with borrowing. The study is done in a foreign environment, in Kenya. Factors that will be looked into are: the classroom environment, the situation of learning in a second language. The study will also observe factors that can cause an erroneous answer and what those may look like. Within this area manipulatives will be mentioned. The study has used a qualitative as well as quantitative approach. The qualitative method has been accomplished through interviews and observations, the quantitative method through tests given to the learners. For the analysis of the observations, Cobb’s and Yackel’s model (1995) of the mathematical classroom has been used. The study is interpreted from a sociocultural perspective focusing interaction. The result in this study shows that the interaction in the classroom is important to the individual learning, perceptions of mathematics and the expectations on the individual. Learning in a second language can be a barrier. The use of manipulatives can work as a scaffold, but it can also hinder the learner to develop a deeper understanding. / Syftet med studien är att studera faktorer som kan påverka förståelsen av subtraktion med växling. Studien är gjord i en annan lärmiljö, i Kenya. Faktorer som kommer att uppmärksammas är lärmiljön i klassrummet och att inlärningen sker på ett för eleven andra språk. Studien kommer också att uppmärksamma faktorer som kan bidra till ett felaktigt svar och hur de kan se ut. Inom detta område kommer laborativt material nämnas. Studien har både ett kvalitativt och kvantitativt tillvägagångssätt. Den kvalitativa metoden har genomförts genom intervjuer och observationer, den kvantitativa metoden med hjälp av elevtester. Cobbs och Yackels (1995) modell över matematik klassrummet har använts som analysmaterial. Studien tolkas utifrån ett sociokulturellt perspektiv. Focus är på interaktionen.   Resultaten i studien visar att interaktionen I klassrummet är viktigt för det individuella lärandet, uppfattningen om matematik och förväntningar på individen. Undervisning på ett andra språk kan bli ett hinder. Användandet av laborativt material kan fungera som ett stöd, men kan också hindra utvecklandet av en djupare förståelse.

Research has shown mathematics education is highly dependent on teaching materials. Here mathematics textbooks have a dominant role in Swedish schools. When curriculum determines the content of textbooks, it promotes confidence in both schools and teachers. The purpose of this study is to analyze and compare textbooks using Basil Bernstein’s theoretical concepts of classification, frames and sequencing to investigate how content and structure are connected to the calculation methods of addition and subtraction. The following questions are posed: In what way have the calculation methods of addition and subtraction of the natural and rational numbers been presented in four selected textbooks for grade 4 in Swedish public schools? What are the similarities and differences in content and structure in these teaching materials? The study finds a variety and selection of material presented in the selected textbooks. Consequently, it is important for teachers to take in to consideration the content and structure of a textbook in mathematics to realize its potential in the classroom.

I den här studien undersöks om och vilken typ av lärarstöd avseende Mathematical Knowledge of Teaching (MKT) lärarhandledningar till Favorit matematik 3A och Rik matematik 3A. Vi analyserar vilket stöd som lärarhandledningarna ger till lärare i områdena addition och subtraktion för årskurs 3. Analysen utgår från MKT:s analysmodell med stöd av fyra utvalda kategorier: Common Content Knowledge, Subject Matter Knowledge, Knowledge of Content and Students, Knowledge of Content and Teaching och Knowledge of Content and Curriculum. En komparativ analys genomfördes med MKT som teoretiskt ramverk. Resultatet visar på att lärarhandledningarna innehåller stöd till läraren i addition och subtraktion och delar av MKT. Lärarhandledningarna ger stöd till lärarna vid planering av undervisning, begreppsanvändning i undervisning, olika undervisningsmetoder, nivåökning av kapitel och hur läraren ges kunskap om materialet. Lärarhandledningarnas struktur förmedlar stödet på ett varierande sätt vilket också synliggör likheter och skillnader om de delar av MKT som lärarhandledningarna inte innehåller. / This study examines whether and what type of teacher support regarding Mathematical knowledge of teaching (MKT) contains teacher guides for Favorite Mathematics 3A and Rich Mathematics 3A. We analyze the support of teacher guides provide to teachers in the areas of addition and subtraction for year 3. The analysis is based on MKT's analysis model with the support of four selected categories; Common Content Knowledge, Subject Matter Knowledge, Knowledge of Content and Students, Knowledge of Content and Teaching and Knowledge of Content and Curriculum. A comparative analysis was performed with MKT as the theoretical framework. The results prove that the teacher guides contain support for the teacher in addition and subtraction and parts of MKT. The teacher guides provide support to the teachers when planning teaching, use of concepts in teaching, different teaching methods, level increase of chapters and how the teacher is given knowledge of the material. The structure of the teacher guides the support in a varying way, which also highlights similarities and differences about the parts of MKT that the teacher guides do not contain.

Den vanligaste arbetsformen i matematikundervisningen är att använda sig av ett läromedel. Därav vårt syfte med denna studie, som är att analysera hur tals del-helhetsrelation behandlas i den inledande undervisningen av addition och subtraktion i de tre vanligaste förekommande läromedlen i Sverige. Utifrån delar av variationsteori har vi analyserat hur innehållet i läromedlen gör det möjligt för eleverna att se och använda sig av tals del-helhetsrelation samt vilka representationer som används i läromedlen för att synliggöra tals del-helhetsrelationer. Av resultatet framgår att det inte är givet att det läromedel som används tar upp de aspekter som forskning anser är kritiska för förståelsen av tals del-helhetsrelation samt användningen av del-helhetsrelationen i addition och subtraktion. Istället visar studien att det inte alltid är möjligt för eleverna att se och förstå del-helhetsrelationen genom de uppgifter som förekommer i läromedlen. Detta i sin tur kan leda till att eleverna istället lär sig att använda strategier som inte är generaliserbara och som också kan leda till svårigheter i matematik. Resultatet visar på att det är viktigt att lärare är medvetna om vilka aspekter som är kritiska i undervisningen, samt ser läromedlet som ett komplement till sin undervisning och inte som den viktigaste arbetsformen.

In order to help children become effective at addition and subtraction, it is important to provide them with an opportunity to investigate and discover the interconnectedness of the two operations. Fact families are one method teachers use to try and help children develop and understand how the operations relate to one another. This paper documents a strategy that was used with a seven year old boy to help him connect addition to subtraction. The strategy incorporated flash card tools to help him create logical problems to discover the mathematical relationship of fact families. With just a few trials, the child was able to create and explain problems that demonstrated the interconnectedness of fact families through addition and subtraction. The model was successful in helping the child advance his understanding. Additionally, it can be extended to more complex addition and subtraction problems as well as multiplication and division fact families.

In order to help children become effective at addition and subtraction, it is important to provide them with an opportunity to investigate and discover the interconnectedness of the two operations. Fact families are one method teachers use to try and help children develop and understand how the operations relate to one another. This paper documents a strategy that was used with a seven year old boy to help him connect addition to subtraction. The strategy incorporated flash card tools to help him create logical problems to discover the mathematical relationship of fact families. With just a few trials, the child was able to create and explain problems that demonstrated the interconnectedness of fact families through addition and subtraction. The model was successful in helping the child advance his understanding. Additionally, it can be extended to more complex addition and subtraction problems as well as multiplication and division fact families.

Thema dieser Dissertation ist die Renormierung von perturbativer skalarer Quantenfeldtheorie bei großer Schleifenzahl. Der Hauptteil der Arbeit ist dem Einfluss von Renormierungsbedingungen auf renormierte Greenfunktionen gewidmet. Zunächst studieren wir Dyson-Schwinger-Gleichungen und die Renormierungsgruppe, inklusive der Gegenterme in dimensionaler Regularisierung. Anhand zahlreicher Beispiele illustrieren wir die verschiedenen Größen. Alsdann diskutieren wir, welche Freiheitsgrade ein Renormierungsschema hat und wie diese mit den Gegentermen und den renormierten Greenfunktionen zusammenhängen. Für ungekoppelte Dyson-Schwinger-Gleichungen stellen wir fest, dass alle Renormierungsschemata bis auf eine Verschiebung des Renormierungspunktes äquivalent sind. Die Verschiebung zwischen kinematischer Renormierung und Minimaler Subtraktion ist eine Funktion der Kopplung und des Regularisierungsparameters. Wir leiten eine neuartige Formel für den Fall einer linearen Dyson-Schwinger Gleichung vom Propagatortyp her, um die Verschiebung direkt aus der Mellintransformation des Integrationskerns zu berechnen. Schließlich berechnen wir obige Verschiebung störungstheoretisch für drei beispielhafte nichtlineare Dyson-Schwinger-Gleichungen und untersuchen das asymptotische Verhalten der Reihenkoeffizienten. Ein zweites Thema der vorliegenden Arbeit sind Diffeomorphismen der Feldvariable in einer Quantenfeldtheorie. Wir präsentieren eine Störungstheorie des Diffeomorphismusfeldes im Impulsraum und verifizieren, dass der Diffeomorphismus keinen Einfluss auf messbare Größen hat. Weiterhin untersuchen wir die Divergenzen des Diffeomorphismusfeldes und stellen fest, dass die Divergenzen Wardidentitäten erfüllen, die die Abwesenheit dieser Terme von der S-Matrix ausdrücken. Trotz der Wardidentitäten bleiben unendlich viele Divergenzen unbestimmt. Den Abschluss bildet ein Kommentar über die numerische Quadratur von Periodenintegralen. / This thesis concerns the renormalization of perturbative quantum field theory. More precisely, we examine scalar quantum fields at high loop order. The bulk of the thesis is devoted to the influence of renormalization conditions on the renormalized Green functions. Firstly, we perform a detailed review of Dyson-Schwinger equations and the renormalization group, including the counterterms in dimensional regularization. Using numerous examples, we illustrate how the various quantities are computable in a concrete case and which relations they satisfy. Secondly, we discuss which degrees of freedom are present in a renormalization scheme, and how they are related to counterterms and renormalized Green functions. We establish that, in the case of an un-coupled Dyson-Schwinger equation, all renormalization schemes are equivalent up to a shift in the renormalization point. The shift between kinematic renormalization and Minimal Subtraction is a function of the coupling and the regularization parameter. We derive a novel formula for the case of a linear propagator-type Dyson-Schwinger equation to compute the shift directly from the Mellin transform of the kernel. Thirdly, we compute the shift perturbatively for three examples of non-linear Dyson-Schwinger equations and examine the asymptotic growth of series coefficients. A second, smaller topic of the present thesis are diffeomorphisms of the field variable in a quantum field theory. We present the perturbation theory of the diffeomorphism field in momentum space and find that the diffeomorphism has no influence on measurable quantities. Moreover, we study the divergences in the diffeomorphism field and establish that they satisfy Ward identities, which ensure their absence from the S-matrix. Nevertheless, the Ward identities leave infinitely many divergences unspecified and the diffeomorphism theory is perturbatively unrenormalizable. Finally, we remark on a third topic, the numerical quadrature of Feynman periods.

The purpose of this study is to examine teachers, in the elementary grades 2 and 3, view on pupils ' understanding of regrouping in subtraction algorithms and the use of concrete materials to facilitate pupils' understanding of regrouping in subtraction algorithms. The study emanates from teachers who teach, or have taught, in regrouping in subtraction algorithms in the elementary grades of 2 and 3. With the use of a semi-structured interview, we were able to gain an understanding of teachers' opinions and experience about pupils' understanding of regrouping in subtraction algorithms and their use of concrete materials to facilitate pupils' understanding.  The teachers in our study agree on the fact that pupils can have difficulties with regrouping in subtraction algorithms. The study shows that the teachers´ opinions and experience indicate that concrete materials can facilitate pupils' understanding of mathematics, but they also show different opinions and experiences for whom the opportunities exist. Furthermore, the results of the study show that the teachers who participated in the study can also see limitations in using concrete materials in their mathematics teaching, such as the amount of material that is available, that the concrete material invites to play and more. Most of the teachers teach with concrete material for regrouping in subtraction algorithms to facilitate the pupils understanding and to provide clarity for the pupils, as seen from the results of the study. Furthermore, the results show that there are teachers who do not teach with concrete materials for regrouping in subtraction algorithms.

In einer initialen Studie verglichen wir das XQi Revolution, welches auf indirektem CsI (Cäsium Iodit) /a: Si (amorphes Silizium) basiert mit einem direkten Digitalröntgengerät a: SE (amorphes Selen) an einem CDRAD-Phantom bei vier unterschiedlichen Eintrittsdosen und an einem TRG-Phantom bei zwei unterschiedlichen Eintrittsdosen. Mittels des berechneten Bildqualitätsfaktors des CDRAD-Phantoms konnten wir zeigen, daß das indirekte im Vergleich zum direkten System bei niedrigeren Dosen eine bessere Detailerkennungsrate aufweist. Ein positiver Trend läßt sich auch beim TRG-Phantom darstellen. In einer weiteren Studie untersuchten wir anhand von Patientenbildern die Bildqualität des Dual Energy Systems bei zwei unterschiedlichen Dosisniveaus, der Standarddosis sowie einer doppelten Dosis, was einem Speed-Äquivalent von 400/1000 bzw. 200/500 entspricht. Bei hoher Dosis konnten wir eine signifikante Reduktion des Rauschens im Knochen- und Weichteilbild feststellen, gleichzeitig nahmen die Störungen durch Bewegungsartefakte signifikant zu. Im Anschluß verglichen wir die Erkennbarkeit verkalkter Lungenpathologien im Standard P/A Bild mit zusätzlichem Einsatz von Dual Energy. Als Goldstandard erfolgte der sichere Nachweis der Pathologien im CT. Bei zusätzlichem Einsatz von Dual Energy konnten wir eine signifikante Steigerung der Sensitivität erkennen. Dieses Ergebnis wurde durch den Qualitätsfaktor, der die Bildeigenschaften kumulativ beschreibt, bestätigt. Weiterhin untersuchten wir analog die Erkennbarkeit von nichtverkalkten Lungenrundherden. Auch bei diesen Pathologien ließ sich ein positiver Trend der Sensitivität und Spezifität bei zusätzlichem Einsatz von Dual Energy erkennen. Zusätzlich stieg die durchschnittliche Entscheidungssicherheit der Gutachter signifikant an. Damit bietet die Dual Energy Subtraktionstechnik eine wertvolle Ergänzung in der Diagnostik verkalkter und nichtverkalkter Lungenpathologien eine wertvolle Ergänzung zum Standardröntgen. / First study assess and quantify the image quality at two dose levels for an amorphous Silicon (a:Si) Cesium Iodide (CsI) flat panel system compared with a direct amorphous Selenium (a:Se) digital radiography system. Image quality of a:Si flat panel digital radiography proved to be superior to a:Se drum digital radiography using low-dose settings. Second study assess the image quality of subtracted soft tissue and bone images of a CsIdetector-based dual-energy system for chest radiography at varying dose levels. Radiation dose did not significantly influence the perception of dual-energy image quality. Next study assess the value of dual-energy chest radiography obtained using a cesium iodide flat-panel detector in addition to standard posteroanterior chest radiography for the detection of calcified chest abnormalities. When dual-energy images were added, sensitivity increased significantly. Brunner and Langer’s test revealed a highly significant difference between posteroanterior chest radiography and dual-energy imaging in the detection of calcified chest abnormalities. Dual-energy images added to standard posteroanterior chest radiographs significantly improve the detection of calcified chest lesions. Last study compare the sensitivity and specificity of digital chest radiography alone with digital chest radiography combined with dual-energy chest radiography in the detection of small non-calcified pulmonary nodules. Standard and dual-energy radiographs were obtained with a flat-panel digital chest system. The increase of nodule detection overall as well as for different size categories was significant. The increase of the confidence level rating was also significant. Dual energy added to standard posteroanterior chest radiography significantly improves the sensitivity, specificity, and confidence in detection of small non-calcified pulmonary nodules. Dual-energy subtraction has the potential to become a future routine application in chest radiography.