Blockmodellen är en visuell problemlösningsstrategi som kommer från Singapore och som används för att lösa matematiska problemlösningsuppgifter. Modellen används i allt fler svenska skolor men trots det saknas svensk forskning inom området. Syftet med denna studie är att ge en bild av hur svenska mellanstadieelever, som är bekanta med Blockmodellen, använder den i problemlösning. Detta syfte besvaras med följande två forskningsfrågor: På vilka olika sätt använder elever i åk 5 Blockmodellen och vilka svårigheter framträder? På vilka olika sätt beskriver elever i åk 5 att Blockmodellen är till stöd för dem vid problemlösning? Studien genomförs genom att 36 elever i åk 5, som i skolans undervisning använder sig av Blockmodellen, gör ett skriftligt test med problemlösningsuppgifter och därefter intervjuas åtta av dessa elever individuellt. För att analysera materialet användes meningskategorisering som metod. Resultatet visar att de svårigheter som framträder är: svårigheter med att rita en modell, problem med översättning mellan modellen och beräkningar och att flera elever tror att de ska dividera med det totala antalet block för att räkna ut värdet av ett block. I studien framkommer det att elever beskriver att Blockmodellen hjälper dem att urskilja den viktiga informationen i textuppgiften och att de uppritade blocken hjälper dem att förstå problemet på ett bättre sätt. En slutsats som dragits av denna studie är att undervisningen behöver ge eleverna en djup förståelse för Blockmodellen för att de ska kunna använda den framgångsrikt i problemlösning. / The Bar Model is a visual problem-solving strategy that comes from Singapore and that is used to solve mathematical problem-solving tasks. The model is becoming more frequent in Swedish schools but despite this there is a lack of Swedish research in the area. The purpose of this study is to provide a picture of how Swedish intermediate school students, who are familiar with the Bar Model, use it in problem solving. This purpose is answered with the following two research questions: In what different ways do students in grade 5 use the Bar Model and what difficulties appear? In what different ways do students in grade 5 describe that the Bar Model is a support for them in problem solving? The study is conducted by having 36 students in grade 5, who use the Bar model in school teaching, conduct a written test with problem-solving tasks and then eight of these students are then interviewed individually. To analyze the material, sentence categorization was used as a method. The results show that the difficulties that emerge are: difficulties in drawing a model, problems with translation between the model and calculations and that several students think that they should divide by the total number of blocks to calculate the value of one block. The study shows that students describe that the Bar Model helps them to distinguish the important information in the text task and that the drawn blocks help them to understand the problem in a better way. A conclusion drawn from this study is that teaching needs to give students a deep understanding of the Bar Model for them to be able to use it successfully in problem solving.

Blockmodellen är en problemlösningsstrategi där block ritas upp för att visualisera information i problemlösningsuppgiften. Modellen har sitt ursprung i Singapore där den används flitigt. Idag används Blockmodellen även i många svenska klassrum trots att det knappt finns någon svensk forskning inom området. Syftet med denna studie är att ge en bild av hur undervisning med Blockmodellen kan se ut i svenska klassrum och hur elever i årskurs 4–6 använder modellen utifrån den undervisning som erbjuds. För att besvara syftet används observation som metod där lärare blir observerade i deras undervisning där de vägleder elever i att lösa problemlösningsuppgifter med hjälp av Blockmodellen. Det samlas även in elevlösningar som har gjorts under lektionerna. Materialet har analyserats utifrån en tematisk analys där olika teman har identifierats och sammanställts.  Resultatet visar att lärare bland annat vägleder elever på olika sätt genom problemlösningsuppgifter där de ska använda Blockmodellen som verktyg. Det sker genom lektionsupplägg, stödstrukturer och exempellösningar. Resultatet visar även att elever ställs inför vissa utmaningar när de använder Blockmodellen, vilket kan vara placering av information i blocken, exempellösningar, placering av frågetecknet och blockens storlek. En slutsats är att lärarens vägledning kan både stötta och hindra eleverna i användandet av Blockmodellen och en annan är att eleverna kan se modellen som en inlärd regel där de enbart ritar block utifrån en mall. / The Bar model is a problem-solving strategy where blocks are drawn to visualize information in the problem-solving task. The model originated in Singapore where it is widely used. Today, the Bar model is also used in many Swedish classrooms, even though there is hardly any Swedish research in the area. The purpose of this study is to give a picture of how teaching with the Bar model can look like in Swedish classrooms and how students in grades 4–6 use the model based on the teaching offered. To answer the purpose, observation is used as a method where teachers are observed in their teaching where they guide students in solving tasks using the Bar model. Student solutions that have been made during the lessons are also collected. The material has been analyzed based on a thematic analysis where themes have been identified and compiled. The result shows that teachers guide students in different ways through problem-solving tasks where they must use the Bar model as a tool. It is through lesson plans, scaffolding and example solutions. The result also shows that students are faced with certain challenges when using the Bar model, for example, placement of information in the blocks, example solutions, placement of the question mark and the size of the blocks. One conclusion is that the teacher's guidance can either support or hinder the students in using the Bar model. A second conclusion is that the students can see the model as a learned rule where they only draw blocks based on a template.

Blockmodellen är en visuell metod som används inom matematik vid arbete med problemlösning och textuppgifter. Metoden utvecklades i Singapore på 1980-talet och har sedan dess visats vara framgångsrik inom matematik. Dess framgång har gjort att den spridits internationellt och även används i svenska skolor. Syftet med studien är att bidra med kunskap om hur elever i årskurs 2 som är bekanta med Blockmodellen väljer att lösa textuppgifter. Syftet uppfylls genom två frågeställningar som berör vilken metod eleverna använder, samt vad som påverkar deras val av metod. Studien genomfördes i två olika klasser med totalt 36 elever. Samtliga elever fick ett antal textuppgifter att lösa och utifrån elevlösningarna valdes tio elever ut för individuella intervjuer. Materialet analyserades utifrån tematisk analys. Studien visar att elever väljer att använda olika metoder för att lösa textuppgifter trots att de är bekanta med Blockmodellen. Vid de individuella intervjuerna framkom det att eleverna valde, enligt dem, den effektivaste metoden för att lösa uppgiften. Eleverna beskrev Blockmodellen som tidskrävande men samtidigt hjälpsam. Genom att använda blocken får de syn på vad som finns, vad som saknas och vad som söks i uppgiften. En slutsats är att elevernas val av metod varierar vilket tyder på att eleverna har olika preferenser när det gäller vilken metod de väljer att använda vid arbete med textuppgifter. / The Bar Model is a visual method used in mathematics when working with problem- solving and word problems. The method was developed in Singapore in the 1980s and has since been shown to be successful in mathematics. Its success has led to its international spread and is now used in Swedish schools as well. The aim of the study is to contribute knowledge about how students in second grade who are familiar with the Bar Model choose to solve word problems. The aim is answered through two questions that concern which method the students use and what influences their choice of method. The study was conducted in two different classes with a total of 36 students. All students were given several word problems to solve and based on the student solutions, ten of the students were selected for individual interviews. Thematic analysis was used to analyze the material. The study show that students choose to use different methods to solve word problems despite being familiar with the Bar Model. During the individual interviews, it emerged that the students chose what they considered to be the most effective method for problem solving. The student describer the Bar Model Bar Model as time-consuming but also helpful. By using the bars, they could see what is present, what is missing and what is being sought. One conclusion is that students' choice of method varies, indicating that they have different preferences regarding which method to use for solving word problems.

Made available in DSpace on 2016-06-02T20:02:57Z (GMT). No. of bitstreams: 1 6458.pdf: 5523001 bytes, checksum: 13a03e09a64c7737467c2e1eee8e86bc (MD5) Previous issue date: 2014-10-17 / The dissertation presents a proposal for teaching the content of fractions to a 7th grade classroom of basic education, based on a concrete didactical material, Fraction Case , and the Bar Model inspired in Singapore Mathematics, focused on the conceptual understanding along with basic operation skills. The motivation to this work was the recognition of great difficulty in learning with understanding of rational numbers, shown by the students of basic education across the grades. The learning of rational numbers is of fundamental importance to the accomplishment of the mathematics curriculum of basic education. We adopted the Fraction Case and the Bar Model because they actually offer the possibility of visualizing concrete representation of ideas that underline the subject of fractions. The proposed activities were inspired by the didactical ideas and activities in Baldin & Malagutti (2006), and they aim at the understanding of the concept of equivalent fractions together with the basic operations of addition, subtraction, multiplication and division of fractions, starting from the perspective of the part-whole relationship. We bring out the outcome resulted from the application of activities in three classrooms of 7th grade students of a private school of city of Bauru. / Apresentamos neste trabalho uma proposta de ensino do conteúdo de frações para uma turma de 7º ano, baseada na utilização de um material concreto intitulado Estojo das frações e do Modelo de Barras da Matemática de Singapura, e centrada na consolidação dos conceitos e das operações básicas. A realização deste trabalho foi motivada por uma constatação da enorme dificuldade de aprendizagem e compreensão dos números racionais apresentada por alunos de diversas séries do ensino fundamental. A aprendizagem dos números racionais é de fundamental importância para o desenvolvimento do conteúdo curricular da matemática do ensino fundamental. O Estojo de frações e o Modelo de Barras foram utilizados, pois oferecem a possibilidade de visualizar concretamente os conceitos relacionados ao tema de frações. As atividades propostas foram baseadas nas ideias didáticas e atividades do material de Baldin e Malagutti (2006), e buscam a compreensão do significado de frações equivalentes e das operações de adição, subtração, multiplicação e divisão de frações a partir do significado parte-todo. Trazemos também os resultados obtidos a partir da aplicação dessa proposta com três turmas de 7º ano de uma escola particular de Bauru.

The aim of this Diploma thesis is focused on a design of the road bridge between Brodzany and Partizanske municipality, spaned the Valley. The design is processed in three preliminary designs. Selected variant is a continuos girder box with inclined walls, post-tensioned by bonded cables. Structural analysis includes the influences of construction by TDA method. The sctructure is assessed for temporary and permanent states. Design and check were carried out according to EC.

En utgångspunkt för denna studie har varit att flera svenska elever har svårigheter att lösa matematiska textuppgifter. I ett försök att utveckla elevers förmågor att lösa matematiska textuppgifter har Singaporemetoden blivit allt mer populär. I Singaporemetoden används blockmodellen för att visualisera textuppgifter. Syftet med följande uppsats är att, utifrån ett sociokulturellt perspektiv, synliggöra hur fem lärare uppfattar att blockmodellen kan stötta elever i att utveckla problemlösningsförmågan när de löser textuppgifter. Uppsatsen syftar även till att synliggöra vilka erfarenheter lärarna har av att arbeta med blockmodellen i matematikundervisningen för årskurserna 1–6. Studien utgår från en kvalitativ metod där syftet besvaras genom fem enskilda semistrukturerade intervjuer med grundlärare som undervisar utifrån blockmodellen i matematikämnet. Studien har utgångspunkt i Vygotskijs sociokulturella teori och har inspirerats av en tematisk metodanalys. Resultatet visar att lärare uppfattar att det finns såväl fördelar som nackdelar med att använda blockmodellen i matematikundervisningen. Lärarna uppfattar att blockmodellen främst är stöttande genom att modellen strukturerar textuppgifter och stödjer elever och lärare att ta sig an textuppgifter. Med stöd av blockmodellen visualiseras dessutom matematiska områden och elever ges möjlighet att utveckla matematisk förståelse. Modellen kan även hjälpa lärare att skapa samtalssituationer för att låta elever utvecklas i den proximala utvecklingszonen. I samtalen får elever möjlighet att utveckla problemlösnings-, begrepps-, resonemangs- och kommunikationsförmåga. Resultatet visar även att främsta utmaningen är att modellen innehåller många steg som kräver att rita och skriva vilket inte passar för alla elever. Blockmodellen ger dessutom upphov till utmaningar när det gäller att differentiera undervisningen, eftersom arbetet oftast sker gemensamt med uppgifter på en specifik svårighetsgrad. Slutsatsen är att blockmodellen kan stötta elever i att utveckla problemlösningsförmågan i textuppgifter i kombination med lärarens kompetens. / The premise for this study has been that swedish upper primary- and lower secondary pupils have been shown to have difficulties in solving textual tasks in mathematics. In an attempt to develop pupils’ abilities in solving these textual tasks, the Singapore model has become increasingly popular. The Singapore model utilizes what is known as the bar model, which is used to visualize the text within these tasks. The purpose of this study is to highlight how five teachers perceive that the bar model can support pupils in developing problem-solving skills in textual tasks, from a socio-cultural perspective. The study also aims to highlight the experiences the teachers have gained from working with the bar model in mathematics teaching for grades 1-6. The study's purpose has been answered through five individual, qualitative and semi-structured interviews, with primary teachers who teach mathematics with the support of the the bar model. The study uses Vygotskij’s sociocultural theory and has been further inspired by a thematic method analysis to present and understand the results. The results show that teachers perceive both advantages and disadvantages in using the bar model in mathematics teaching. The teachers expressed that the bar model is primarily supportive in that the model structures textual tasks and supports pupils and teachers when approaching textual tasks. With the support of the bar model, abstract areas are also visualized and the pupils gain an increased mathematical understanding. The model can also contribute to creating conversational situations in order for pupils to develop in regards to the proximal development zone. In these conversations, pupils are given the opportunity to develop problem-solving, conceptual, reasoning and communicational skills. The results also show that the main challenge is that the model contains many elements that require drawing and writing, which are not suitable for all pupils. The bar model also gives rise to challenges in regard to differentiating the teaching, since the work usually takes place jointly with tasks at a specific degree of difficulty. The conclusion is that the bar model can support pupils in developing problem-solving skills in textual tasks in combination with teacher competence.

Syftet med studien är att synliggöra lärares erfarenheter av undervisningsutformning när de lär ut problemlösning genom blockmodellen på mellanstadiet. Tidigare forskning har visat att blockmodellen är en framgångsrik problemlösningsmetod med både för- och nackdelar. Studien anses relevant då det inte finns mycket forskning om blockmodellen i svenska sammanhang. För att samla in data användes individuella semistrukturerade intervjuer och materialet analyserades genom en tematisk analys. Fem lärare som är verksamma på mellanstadiet intervjuades med fokus på vilka hinder och anpassningar de erfarit i sin undervisning med blockmodellen. Studiens teoretiska utgångspunkt är sociokulturell teori med fokus på hur elever tar till sig kulturella redskap. Resultatet visar fyra teman som kunde identifieras i lärarnas utsagor. I resultatets två första teman hanteras hinder som lärarna beskriver i samband med problemlösningsundervisning med blockmodellen. I resultatets två sista teman hanteras anpassningar som lärarna beskriver i samband med problemlösningsundervisning med blockmodellen. Samtliga lärare har en positiv inställning till blockmodellen som problemlösningsmetod men de är överens om flera utmaningar som behöver hanteras när undervisningen med metoden utformas. / The purpose of this study is to make teachers’ experiences of instruction design when teaching problem solving with the bar model in middle school visible. Previous research has shown that the bar model is a successful method for mathematical problem solving with both pros and cons. The study is considered relevant since there has not been much research conducted on the bar model in Sweden. Individual semi-structured interviews were conducted in order to collect data which later was analysed through a thematic analysis. Five active middle school teachers were interviewed with focus on the obstacles they have experienced and the adaptations they have implemented while teaching with the bar model. The theoretical framework used in this study is sociocultural theory with focus directed at how pupils assimilate cultural tools. The result shows four themes that could be identified from the teachers’ responses. The teachers’ descriptions of obstacles connected to teaching using the bar model are compiled in the first two themes. In the two following themes teachers’ descriptions of adaptations are compiled. All the teachers share a positive attitude towards using the bar model when teaching mathematical problem solving. They are however in agreement about several challenges that needs to be handled when designing instruction for problem solving with the bar model.

Blockmodellen är en matematisk modell som ger visuellt stöd vid textuppgifter. Den har fått stor spridning i Singapore. Modellen har visat sig skapa goda resultat för elevers matematikkunskaper. Syftet med studien är att ta reda på vilka utmaningar och möjligheter som lärare kan stöta på när de arbetar med blockmodellen, men även hur detta arbete utformas. Detta gjordes genom semistrukturerade intervjuer, där svenska lärare i grundskolan intervjuades om deras erfarenheter. Totalt deltog tio lärare, där alla undervisade om blockmodellen i matematik på mellanstadiet. Analysverktygen som användes var en tematisk analys. Med hjälp av CPA-teorin och scaffolding skapades en infallsvinkel på studien. CPA-teorin utgår från att symboler skapar representationer inom exempelvis problemlösning.  Resultatet visade att pedagogerna i den svenska skolan inte ser några större utmaningar med blockmodellen samt att det är ett bra verktyg för alla elever, speciellt för elever med lässvårigheter. Lärarna upplevde att det är en gynnsam metod för elever att lära sig tidigt i grundskolan, för att underlätta elevernas vidare matematikutveckling. Resultatet visade också att lärarna anser att man bör arbeta kollegialt med blockmodellen, både att eleverna arbetar tillsammans, att läraren stöttar eleverna och att lärarna själva arbetar kollegialt med kunniga kollegor. / The Bar Model is a mathematical method that gives visual support. It is spread across Singapore. The model has been shown to produce positive results in student’s mathematical skills. The purpose of this study is to explore the challenges and opportunities that teachers may encounter when working with The Bar Model, but also as how this work is designed. This was done through semi-structured interviews, where Swedish elementary school teachers were interviewed about their experiences. A total of ten teachers participated, all of whom teached according to The Bar Model in mathematics in middle school mathematics. The analytical tool used included a thematic analysis. An analytical approach was created using the CPA-theory and scaffolding. The CPA theory assumes that symbols create representations within, for example, problem solving. The results showed that the teachers in the Swedish school system didn’t perceive any major challenges with The Bar Model, and that it is an effective tool for all students, especially those with reading difficulties. The teachers found it to be a beneficial method for students to learn early in elementary school, to simplify their further mathematical development. The results also indicated that teachers believe collaborative work is important with The Bar Model, both in terms of students working together and teachers supporting students, as well as teachers collaborating with skilled colleagues.

In this thesis, the possibility of detection of fault location in the cable distribution systems by using traveling waves due to fault and circuit breaker operations is investigated. Waveforms originated from both actions and fault steady state are separately analyzed. During such switching actions, high frequency variations which are absent in the steady state conditions, take place. In order to simulate high frequency changes properly, system elements are modeled accordingly. In other words, frequency dependent models are introduced, and they are used in Electro-Magnetic Transients Program (EMTP). Since the characteristics of waveforms are different for separately analyzed portions, different fault locating algorithms with their limitations are introduced.

Made available in DSpace on 2016-06-02T20:02:57Z (GMT). No. of bitstreams: 1 6507.pdf: 2675622 bytes, checksum: 1fa2e1a89f27433883070d1a6ede575e (MD5) Previous issue date: 2014-10-17 / The difficulties in learning and teaching of algebra can be detected in the school cycle 4 (8th and 9th grades) of the Elementary School II and throughout High School, such difficulties being present in all Brazilian schools. These difficulties arise from an institutional failure, in others words, in the transition from arithmetic to algebra, in the phase of pre-algebra which occurs at cycle 3 ( 6th and 7th grades) of the Elementary School II. When this transition is unsatisfactory this compromises the subsequent studies making the students feel not motivated in learning the content of algebra. Therefore, in this research project we planned and executed 6 (six) activities based on the methodology of Problem Solving based on the phases proposed by George Polya, along with the methodology of the Bar Model from Singapore Mathematics. The activities were carried out in seventh grade classrooms of Elementary School II of Instituto Educacional Estilo , Campinas, SP. The results of this dissertation suggests to teachers of Elementary School II didactical sequences of activities that they can use and enjoy in classroom practices, so that they can improve also their teaching and learning, contributing to the development of the students. With the objective of achieving a satisfactory transition from arithmetic to algebra, the activities were developed and based on problems solving, and then analyzed critically using the Problem Solving steps. After 6 (six) activities, we applied a diagnostic evaluation in order to analyze the results and to check if the activities contributed to a meaningful learning of algebra. The dissertation presents a theoretical study about teaching and learning algebra as well as a study on the methodologies of Problem Solving in classroom practice and Bar Model from Singapore Mathematics. / As dificuldades na aprendizagem e no ensino da álgebra podem ser constatadas no ciclo 4 (8º Ano e 9º Ano) do Ensino Fundamental II e também em todo o Ensino Médio, tais dificuldades estão presentes em todas as escolas brasileiras. Essas dificuldades são decorrentes de uma falha na introdução, ou seja, na transição da aritmética para a álgebra, a pré-álgebra que ocorre no final do ciclo 3 (6º Ano e 7º Ano) do Ensino Fundamental II, já que feita de maneira não satisfatória pode comprometer as aulas seguintes fazendo com que os alunos se sintam desmotivados a aprenderem o conteúdo de álgebra. Deste modo foram planejadas e executadas 6 (seis) atividades utilizando a metodologia de Resolução de Problemas seguindo as etapas de George Polya, juntamente com a metodologia do Modelo de Barras segundo a Filosofia da Matemática de Singapura. As atividades foram aplicadas em duas turmas do sétimo ano do Ensino Fundamental II, no colégio Instituto Educacional Estilo, Campinas, SP. O trabalho desenvolvido nesta dissertação proporciona aos professores do Ensino Fundamental II e Ensino Médio uma sequência didática, que podem utilizar e aproveitar em suas aulas de forma que possam também melhorar em suas práticas de ensino e aprendizagens, de maneira a contribuir para o desenvolvimento de seus alunos. Com o objetivo de realizar uma transição satisfatória da aritmética para álgebra, as atividades foram elaboradas e baseadas na resolução de problemas, e depois analisadas criticamente por meio das etapas de resolução. Após as 6 (seis) atividades, aplicamos uma avaliação diagnóstica de forma a analisar os resultados para verificar se as atividades contribuíram com significado para uma aprendizagem da álgebra. O trabalho apresenta um estudo teórico sobre o ensino e aprendizagem da álgebra e também apresenta um estudo sobre as metodologias desenvolvidas no trabalho, Resolução de Problemas e Modelo de Barras segundo a Filosofia da Matemática de Singapura.