Multiplikation ingår i många delar av matematiken, vilket gör att goda kunskaper och förståelse för multiplikation ses vara av betydelse för eleverna. Samtidigt ses det inte helt oproblematiskt att lära in dessa kunskaper. Syftet med denna studie är att undersöka vad matematiklärare i årskurs 4–6 anser är svårt för elever att lära in vid undervisning av multiplikation samt vilka metoder och resurser dessa lärare använder för att utveckla elevernas multiplikativa tänkande. Undersökningen genomfördes genom kvalitativa intervjuer med matematiklärare, där insamlad data sedan analyserades utifrån Lithners (2008) ramverk. Resultatet visar att det finns flera svårigheter som kan uppstå vid lärande av multiplikation, men där den främsta svårigheten för eleverna är att lära sig multiplikationstabellens utantill. Vidare visar resultatet att lärarna aktivt arbetar med att eleverna ska förstå och kunna omsätta sina kunskaper i olika sammanhang, där samtidigt vissa metoder och resurser ses vara mindre gynnsamma för utveckling av det multiplikativa tänkandet. Denna studie visar att multiplikation innefattar olika typ av kunskap, vilka är viktiga för att eleverna ska kunna utvecklas inom skolans matematik. Studien ses även bidra till kunskaper om hur undervisningen i multiplikation kan utvecklas vidare. / Multiplication is included in many parts of mathematics, which means that good knowledge and understanding of multiplication is important for students. At the same time, learning multiplication is not unproblematic. The purpose of this study is to investigate what mathematics teachers in grade 4–6 consider difficult for students to learn in the teaching of multiplication, and what methods and resources these teachers use to develop students` multiplicative thinking. The survey was conducted through qualitative interviews with mathematics teachers, where collected data was analyzed based on Lithner’s (2008) framework. The results show several difficulties that can arise when learning multiplication, but where the main difficulty for the students consists of learning the multiplication tables by rote. The results also show that the teachers actively work with the students to understand and be able to transform their knowledge into different contexts. At the same time, certain methods and resources considered to be less favorable for the development of multiplicative thinking. This study shows that multiplication involves different types of knowledge, which are important for students to be able to develop within school mathematics. The study also contributes with knowledge about how teaching in multiplication can be further developed.

I detta arbete undersöks huruvida ett konceptuellt fokus på fysikundervisningen, samt en förståelseinriktad examinationsform, påverkar elevers konceptuella förståelse av fysik. Detta är intressant av två anledningar: 1) Skolans styrdokument efterfrågar (bl.a.) konceptuella förståelsekunskaper, och 2) Traditionell fysikundervisning är dålig på att skapa en korrekt, konceptuell förståelse av fysik. Det här arbetet granskar resultaten från en icketraditionell undervisning, där flervalsfrågeverktyget FCI använt före och efter undervisning ger en bild av elevernas konceptuella förståelseutveckling. Den undervisning som undersöks i detta arbete visade sig vara drygt 100 % effektivare än motsvarande traditionella undervisning vad gällde att förbättra resultaten på diagnosverktyget FCI. Även om elevantalet var väldigt litet (endast tio elever) är resultatet så markant att det nog ändå äger viss giltighet. Arbetet undersöker även elevers föreställningar om fysik. Ett mindre påtagligt resultat i kombination med det lilla elevantalet gör det svårt att säga något om huruvida elevernas föreställningar om fysik blev mer fördelaktiga än vid traditionell undervisning. Resultaten pekar på det, men det är långtifrån säkerställt. Slutligen undersökte arbetet också varaktigheten hos de kunskaper eleverna erhöll. Det visade sig att de fyra elever som skrev FCI efter ett uppehåll i fysikundervisningen på tio månader presterade nästan lika bra som då de skrev provet direkt efter avslutad undervisning. De hade inte tappat mer än 10 % av sina poäng.

The aim of this study is to investigate how elementary school teachers use conceptual understanding in the teaching of the variable concept, namely, how these teachers connect the variable concept with other areas of mathematics, such as algebraic expressions.  The empirical data was obtained by qualitative methods comprising interviews with five mathematic elementary school teachers. In addition three observations in three classrooms were made; one observation is in grade 4 and the other two are in grade 6.The theoretical framework is based on Kilpatrick et al. (2001) theories: conceptual understanding, strategic competence and procedural fluency. Furthermore the theory of representations of Bergsten et. al (1997) and Persson (2010) were used in the theoretical framework. The results of this study show that none of the five participating teachers connects the variable concept to other areas of mathematics, such as algebraic expressions. However, two of them mention the variable concept in equations, problem-solving, algebraic expressions and shapes, but without explaining the meaning of the variable concept or how that concept is used and integrated into these mentioned areas of mathematics. Furthermore the study shows that the other three participating teachers mention the variable only to one to two areas of mathematics. These areas are problem- solving and shapes. These teachers do not explain how the variable concept is used in the mentioned areas of mathematics, but they focus on how the calculation will be performed to solve the data they used. The participating teachers do not use any representations to explain the variable concept and how it’s connected to the other areas of mathematics. In conclusion, the variable concept was not explained and its connection to other mathematical areas was not addressed, which due to the absence of usage of conceptual understanding in teaching the variable concept.

The aim of this study is to investigate how elementary school teachers use conceptual understanding in the teaching of the variable concept, namely, how these teachers connect the variable concept with other areas of mathematics, such as algebraic expressions.  The empirical data was obtained by qualitative methods comprising interviews with five mathematic elementary school teachers. In addition three observations in three classrooms were made; one observation is in grade 4 and the other two are in grade 6.The theoretical framework is based on Kilpatrick et al. (2001) theories: conceptual understanding, strategic competence and procedural fluency. Furthermore the theory of representations of Bergsten et. al (1997) and Persson (2010) were used in the theoretical framework. The results of this study show that none of the five participating teachers connects the variable concept to other areas of mathematics, such as algebraic expressions. However, two of them mention the variable concept in equations, problem-solving, algebraic expressions and shapes, but without explaining the meaning of the variable concept or how that concept is used and integrated into these mentioned areas of mathematics. Furthermore the study shows that the other three participating teachers mention the variable only to one to two areas of mathematics. These areas are problem- solving and shapes. These teachers do not explain how the variable concept is used in the mentioned areas of mathematics, but they focus on how the calculation will be performed to solve the data they used. The participating teachers do not use any representations to explain the variable concept and how it’s connected to the other areas of mathematics. In conclusion, the variable concept was not explained and its connection to other mathematical areas was not addressed, which due to the absence of usage of conceptual understanding in teaching the variable concept.

Svårigheter med beräkningar och problemlösning i kemi är ett vanligt förekommande fenomen. Men vilka orsaker kan ligga bakom dessa svårigheter, och vad kan lärare göra för att motverka dem? Syftet med studien är att undersöka dessa frågor. Genom en analys av relevant litteratur identifieras fyra faktorer som påverkar elevers svårigheter med beräkningar och problemlösning i kemi: läsförmåga, matematisk förmåga, överföring och konceptuell förståelse. Vidare visar litteraturanalysen att lärare kan motverka elevers svårigheter genom att förbättra elevers läsförmåga (lära ut lässtrategier och låta elever förklara texter för sig själva eller en vän) och överföringsförmåga (genom att medvetandegöra matematiken i kemin för sig själva och eleverna samt en god kommunikation med matematikläraren), samt fokusera på förståelse för centrala begrepp. Slutligen bör lärare vara uppmärksamma på vilken eller vilka faktorer som verkar bära huvudansvaret för enskilda elevers svårigheter med beräkningar och problemlösning i kemi, för att på så sätt kunna erbjuda individanpassat stöd.

This study serves as a response to a development need where difficulties regarding students' ability to solve mathematical problems were observed in two schools. The aim of this study has been to investigate the effects of using cooperative learning when teaching mathematics for students’ mathematics development. Specifically, it examines students’ conceptual and procedural knowledge in problem solving with a focus on proportionality. To study this, the following three questions were posed:  What differences are identified in the students' post-tests regarding their procedural and conceptual knowledge in problem solving with a focus on proportionality? What basic cooperative principles are made visible in the student groups during the  problem-solving lessons? How can cooperative learning contribute to students' mathematics development with a  focus on conceptual and procedural knowledge in the field of problem solving?  To answer these questions, qualitative text analyses have been carried out with the help of an analysis tool based on theories of conceptual and procedural knowledge. The texts have consisted of eight selected students' pre- and post-tests consisting of problem-solving tasks with a focus on proportionality. Between the pre- and post-tests, a total of four cooperative lessons have been completed and observed based on a structured observation schedule founded on the five basic principles of cooperative learning. The results of the study show that there has been a development in seven out of eight students regarding their conceptual and procedural knowledge. During the cooperative lessons, all five basic principles for cooperative learning could be observed, although to varying degrees in the different working groups. Based on this, it can be interpreted as reasonable that there are certain correlations between specific students' mathematics development and cooperative learning. Therefor it is plausible that cooperative learning can be assumed to have a positive effect on students’ mathematics development. However, it is important to keep in mind that cooperative learning does not automatically contribute to the mathematical development of students, nor is it a form of work that suits all students.

Empirisk forskning har visat att det finns tydliga kopplingar mellan intresse, attityder ochstudieframgångar. Enkätundersökningen som föreligger denna rapport ämnade att mäta hur intressetför fysik och attityder till fysik och fysikundervisningen skiljer sig åt mellan de olika årskurserna pågymnasiet. För att kunna mäta attityderna har enkätverktyget Maryland Physics Expectations(MPEX) Survey använts. Enkäten har besvarats av 605 respondenter från teknik- ochnaturvetenskapsprogrammet på två gymnasiumskolor i Mellansverige. Intresset för fysik är lågt påde undersökta skolorna och det tenderar att bli lägre med åren. Överlag är det fler ofördelaktiga svarhos de olika attitydsdimensionerna i årskurs 3 än i årskurs 1. Koncept är den dimension som har flestofördelaktiga svar både i tvåan och i trean. För att öka den konceptuella förståelsen hosgymnasiestudenterna bör konceptuell förståelse få en större del av undervisningen. Att examinerakonceptuell förståelse på hemläxor och prov är även det att föredra. / Empirical research has shown that there are clear links between the interests, attitudes, and studentsuccess. The aim of the survey, which is the foundation of this report, was to measure how theinterest in physics and attitudes towards physics and physics education differs between the differentyears in upper secondary school. Maryland Physics Expectations (MPEX) Survey has been used tomeasure the attitudes. The questionnaire was answered by 605 respondents from technology andnatural science program from two upper secondary schools in central Sweden. Interest in physics islow on the investigated schools and it tends to become lower through the ages. Overall, there aremore unfavorable responses of the different attitude dimensions in third grade than in first grade. Concept is the dimension that has the most unfavorable response in both the second and third grade.In order to increase the conceptual understanding of upper secondary school students, shouldconceptual understanding be offered a greater part of the teaching. Examining conceptualunderstanding in homework assignments and tests are also preferable.

The following study aims to examine the special mathematical knowledge needed in order to teach mathematics. Furthermore, the study attempts to explore how teachers’ views on the knowledge needed in order to teach mathematics affects their student’s opportunities to develop their conceptual understanding. Qualitative and quantitative empirical data was attained by observations and complementary interviews. A total of three teachers, all working at the same school, was observed and interviewed. The study used Ball, Thames & Phelps (2008) practice-based theory of mathematical knowledge for teaching, MKT, as its theoretical framework when analyzing the empirical data. The result of the observations displays that math teachers tend to use common content knowledge far more than specialized content knowledge during their lessons. The outcome of this also study reveals that there is a tendency among teachers to interfuse mathematical concepts with terminology. Conceptual understanding is equated with the use of correct terminology. The students are not exposed to the underlying ideas of the mathematical concepts. The study also concludes that there seems to be a sectioning between the mathematical content taught in grade 4-6 from the rest of the content being taught in elementary school, with a low number of connections being made between mathematical topics and concepts included in the curriculum.