Giving effective instructional explanations is one of the most important teacher competences. Recent di dactic literature provides, however, little insight on teacher explanations. In our previous work we devel- oped guidelines for designing comprehensible explanations in the field of business (teacher) education, which are along general lines transferable to other subject areas and target audiences. In this article, we first compare our guidelines to the state of research in general and mathematics didactics. We then in vestigate their applicability to teaching operations research at university level, based on interviews with professors of the international operations research community.

Studiens syfte är att genom ett frågeformulär undersöka elevers kunskaper inom det matematiska området tal i bråkform. Vidare ämnar studien undersöka vad som särskiljer elevers kunskaper i årskurs 4, 5 och 6. Frågeformuläret är konstruerat utefter Blooms reviderade taxonomi och utförs i en årskurs 4, 5 och 6. Datan redovisas, analyseras och diskuteras utefter Blooms reviderade taxonomi för att besvara frågeställningen. I resultatet framkom det att elever i årskurs 4 visar större svårigheter gentemot årskurs 5 och 6 vid användning av procedurkunskaper för beräkning av bråktal samt för att beräkna antal med hjälp av bråk. Det framkom även att elever i årskurs 4 är förtrogna med att använda bilder som representationsformer medan elever i årskurs 6 är förtrogna med båda bilder och symboler som representationsformer. Elever i årskurs 5 verkar däremot befinna sig i någon slags övergångsfas då de vill använda symboler som representation men gör så med blandad framgång.

Arbetet handlar om matematikens kulturhistoria ur ett didaktiskt perspektiv i gymnasieskolans undervisning. Syftet är dels att undersöka hur lärare och elever ser på matematikens historia och dess inkluderande i matematikundervisningen, dels om det finns en diskrepans mellan olika läroboksförfattare hur man tolkar Skolverkets intention om att ge eleverna möjlighet att relatera matematiken till ett historiskt sammanhang. Baserat på några didaktikforskares olika synvinklar behandlas också frågor kring varför vi bör inkludera matematikens historia, vilka alternativa metoder som kan användas samt vilka eventuella hinder som kan finnas för detta. Dessutom ställs matematikhistorisk undervisning i relation till både teoretiska lärandeperspektiv och jämställdhet mellan män och kvinnor. Metoderna som använts i arbetet utgörs, dels av en kvalitativ enkätundersökning baserad på frågeformulär ställda till några lärare och elever, dels av en litteraturjämförelse mellan fyra av de vanligaste förekommande läroboksserierna i matematik på den svenska läromedelsmarknaden. Litteraturstudien baseras på både en kvantitativ och en hermeneutisk ansats där såväl mängden av historiskt innehåll liksom texternas innebörd studeras och uttolkas baserat på det sätt på vilket de framställts. Resultatet av studien visar att undervisning i matematikens historia prioriteras relativt olika bland lärarna och den ges i huvudsak låg prioritet då den ställs mot matematikundervisningens andra delar som aritmetik, algebra, geometri med flera. Vidare visar sig intresset vara begränsat hos merparten av eleverna. Ett något större intresse noteras däremot hos lärarna, som dock i relativt hög andel själva saknar utbildning i matematikens historia. Studien visar också att det finns betydande skillnader i hur läromedelsförfattarna ser på elevernas behov av matematikhistoriskt innehåll, inte minst mellan matematikkursernas a-, b- och c-spår. Även sättet på vilket det matematikhistoriska innehållet presenteras skiljer sig avsevärt mellan de studerade läroboksserierna. / This work is about the history of mathematics from a didactic perspective in the upper secondary school ’s education. The purpose is partly to investigate how teachers and students look at the history of mathematics’ incorporation in mathematics teaching and partly if there is a discrepancy between different school book writers how to interpret the Swedish National Agency for Education's intention to give students the opportunity to relate mathematics in a historical context. Based on the different viewpoints of some didactics researchers, issues about why we should include the history of mathematics, which alternative methods that can be used, and any possible barriers to doing so, are discussed. In addition, teaching the history of mathematics is viewed in relation to both theoretical learning perspectives and gender equality. The methods used in the work consist, partly of a qualitative survey based on questionnaires addressed to some teachers and students, and partly from a literature comparison between four of the most commonly used school book series in mathematics on the Swedish literature market. The literature study is based on both a quantitative and a hermeneutic approach where both the amount of historical content and the meaning of the texts are studied and interpreted based on the way in which they were presented. The result of the study shows that teaching history of mathematics is prioritized relatively different among the teachers, and it is given essentially low priority when it competes against subjects like arithmetics, algebra, geometry, and others. Furthermore, the interest is shown to be limited by most of the students. On the other hand, a somewhat greater interest is noted among the teachers, although in relatively high proportion they themselves lack education in the history of mathematics. The study also shows that there are significant differences in how the school book writers look at the students' need for mathematics-historical content, not least between the a-, b- and c-tracks of the mathematics courses. Even the way in which the mathematics-historical content is presented differs substantially between the studied textbook series.

Den här studien undersöker gymnasieelevers begreppsförståelse inom trigonometri. Eleverna, som samtliga läser Ma 4, besvarade anonymt två frågeformulär där det ena bestod av en öppen fråga, medan det andra var upplagt som ett test med uppgifter att lösa. Den kvalitativa analysen baseras på Tall och Vinners (1981) teori om begreppsbild och Sfards (1991) uppdelning av begreppsförståelse i de tre processerna interiorisering, kondensering och reifiering. Resultatet indikerar att elever tenderar att använda cirkeltrigonometri före triangeltrigonometri för att lösa uppgifter, men att majoriteten av deras begreppsbilder består av både och. Det visade sig att radianer var det trigonometriska begrepp eleverna hade svårast för. Detta stämmer överens med tidigare forskning på området. Tre elever analyseras djupare med avseende på hur långt de har kommit i sin utveckling av begreppsförståelse inom trigonometri. Det ges även exempel på hur utvecklingsfaserna kan ge sig till uttryck hos enskilda elever. Såväl begreppsbild som vilka stadier eleverna befinner sig på i sin begreppsförståelse varierar stort. / This study investigates the conceptual understanding of high school students in a trigonometry context. The students, all of them taking the Swedish math course Ma 4, anonymously answered two questionnaires, one of them containing an open question, while the other was constructed like a test with problems to solve. The qualitative analysis is based on Tall & Vinner’s (1981) theory of concept image and Sfard’s (1991) division of conceptual understanding into the three processes interiorization, condensation and reification. The result indicates that students tend to use circle trigonometry over triangle trigonometry for problem solving, however the majority of their concept images consist of both. Radians seemed to be the trigonometrical concept that the students had most trouble understanding. This corresponds with earlier research of trigonometry. In the analysis three students are more deeply analysed, based on how far they have come in their development of trigonometric conceptual understanding. Furthermore there are concrete examples of how the phases of development can be presented among individual students. The concept image as well as the levels of the students’ conceptual understanding vary considerably.

O estudo de impressos direcionados para professores, publicados pelos órgãos oficiais de Educação, contendo sugestões sobre os modos de fazer em sala de aula, constituindo uma literatura cinzenta escolar, aparecem como um instrumento eficaz para o aprofundamento dos estudos da História da Educação Matemática no Brasil e das relações entre programas, conteúdos e práticas escolares. Diante disso, o objetivo da tese foi problematizar de que modo foram construídas as propostas de alterações metodológicas para o ensino do número nas séries iniciais do Ensino Fundamental, no período entre 1961 e 1979, de modo a tentar compreender como foram produzidas as representações de ensino moderno, fundamentadas no ideário do Movimento da Matemática Moderna (MMM), nas publicações das Secretarias de Educação de São Paulo. E também, os modos de produção desses modelos, ou seja, a transformação na representação didático-pedagógica do conceito de número, no período analisado, nas orientações publicadas. Ainda, mais especificamente, busca-se entender como ocorre a apropriação dos estudos de Zoltan Paul Dienes nesses impressos. Acredito que o estudo que explore o diálogo entre passado e presente, que procura compreender as condições que permitiram a produção das representações sobre como ensinar e aprender Matemática pode subsidiar as problematizações diárias sobre a prática e possíveis novas propostas. Isso implica seguir e procurar desvendar os processos de apropriação utilizados pelos elaboradores das publicações, além de procurar caracterizar e diferenciar o MMM no ensino primário. O período histórico da pesquisa foi determinado pelas fontes selecionadas, após um levantamento das publicações existentes (Implantação da escola municipal de oito anos, de 1969, e os quatro volumes do Manual de Detalhamento de Currículo de 1974, 1976, 1977 e 1979, na memória técnica documental do Município de São Paulo e no Arquivo Pessoal Lucília Bechara Sanchez. A opção por essas fontes relaciona-se ao reconhecimento do valor atribuído às publicações no subsídio de professores para as mudanças, num período de expansão e criação dos sistemas de ensino no Brasil, com transformações na estrutura, no funcionamento, nos programas e no currículo de Matemática, de acordo com as normativas impostas pelas LDB 4.024/1961 e LDB 5.672/1971. Para complementar a análise, foi necessário problematizar as dificuldades do trabalho com essa literatura cinzenta escolar como fonte. A escassez de pesquisas que as utilizam pode ser explicada pela profusão desses textos, que, apesar de emanados de um mesmo órgão público, têm fases diferentes, consoante com os grupos que os produziram. Na articulação das questões, fiz uso da abordagem da história cultural e me apoiei nos conceitos de representação, apropriação e estratégias, postas por Chartier (1991) e Certeau (1982). Concluí que, no período estudado, as publicações produzidas pelas Secretarias foram utilizadas como estratégia de reformulação curricular e divulgação para implementar as novas diretivas para o ensino de Aritmética na escola primária paulista, adequando-se às recomendações dos novos campos da psicologia e da didática. A pesquisa ainda assinalou que a apropriação das ideias de Zoltan Dienes, defendendo uma abordagem estruturalista para a Matemática, produziu grandes reformulações na didática da Matemática, ressignificando o quê, como e pra quem ensinar. / The study of papers targeted to teachers, published by the official agencies of Education, containing suggestions on how to do in the classroom, making up a blurry school literature, appear as an effective instrument for deepening the studies of Mathematics Education History in Brazil and the relationships between programs, contents and school practices. Given the importance of these papers, this thesis aims to discuss how the proposals of methodological changes to the teaching of numbers in the early grades of elementary school were built. I want to understand how the representations were made of \"modern education\" based on the ideals of MMM, in the publications of the Departments of Education of Sao Paulo and the ways of production of these models. What transformation does the didactic teaching go through on the concept of number in the analyzed period through the guidelines published to the teachers by the Education Department? And more specifically, what reveals the papers of the Education Departments in terms of ownership of the studies conducted by Zoltan Paul Dienes? I believe that the questioning about what it was like to teach arithmetic and the methods suggested for this teaching in the initial grades may contribute to the understanding of the process of learning mathematics and how this process influenced - and continues to influence - the teaching of mathematics in the current educational context. The historical period of the research was determined by selected sources, after a survey of the existing publications (Implementation of the eight year municipal school, of 1969, and the four volumes of the MDC, of 1974, 1976, 1977 and 1979) in the Document Technical Memory of São Paulo and APLB. The choice of these sources is related to the recognition of the value assigned to the publications in the supplies of teachers for changes in a period of expansion and creation of educational systems in Brazil, with changes in the structure, operations, programs and in the Mathematics curriculum in accordance with the regulations imposed by BDL 4024/61 and BDL 5672/71. The work also included the comparison between publications with LDB/61 and the LDB/71, considering the place where the production of the official papers was held and the everyday of the production process, their dynamics and backstage. In order to complement the analysis, it was necessary to discuss the difficulties of working with this blurry school literature as a source. In articulating the issues, I use the approach of cultural history and lean against the concepts of representation, ownership and strategies put by Chartier (1991) and Certeau (1982). I conclude that during the studied period, the publications produced by the Departments of Education were used as a strategy adapting to the recommendations of the new fields of psychology and didactics. The survey also noted that ownership of the ideas of Zoltan Dienes, advocating a structuralist approach to mathematics, produced major reformulations in mathematics didactics, giving new meaning to teaching, how to teach and for whom to teach.

Programmering som verktyg för lärande i matematik- En empirisk studie av elevers resonemangsförmåga i två olika undervisningsmiljöer För att förbereda grundskoleelever för denna allt mer digitaliserade värld, infördes programmering i kursplanen för matematik den första juli 2018. Syftet med den här studien var att undersöka om arbete i en programmeringsmiljö kan erbjuda nya möjligheter för lärande i matematik jämfört med en ouppkopplad lärmiljö. Detta undersöktes med hjälp av två för ändamålet särskilt designade lektioner, en där arbetet skedde ouppkopplat och en där arbetet skedde i en programmeringsmiljö. De deltagande eleverna arbetade parvis, och deras arbete observerades med hjälp av både fältanteckningar och ljudinspelningar. Elevernas lärande undersöktes genom att analysera deras matematiska resonemang vid de båda lektionstillfällena. Analysen skedde med hjälp av fyra analysfrågor, och resultatet visar tendenser till en skillnad i elevernas matematiska resonemang vid arbete i de två olika lärmiljöerna. På individnivå pekar resultatet på en variation i vilken av lärmiljöerna som var mest fördelaktig. På gruppnivå var det däremot fler elever som i större utsträckning följde varandras resonemang när de arbetade i programmeringsmiljön. Dessutom visade majoriteten av eleverna på en större uthållighet i att lösa uppgifterna när de arbetade med programmering. Vad dessa skillnader kan bero på diskuteras såväl i samband med studiens resultat som tidigare forskning. Slutsatsen lyder att programmering kan erbjuda elever nya sätt att lära matematik och därför bör användas som ett av flera verktyg i undervisningen. / Programming as a learning tool in mathematics - An empirical study of pupils’ mathematical reasoning in two different educational environments In order to prepare pupils for a more and more digitalised world, programming has been included in the Swedish curriculum for mathematics since July 1, 2018. The purpose of this study was to examine if working in a programming environment, in comparison to an unplugged environment, would offer pupils new opportunities for learning mathematics. This was examined by analysing the mathematical reasoning of the pupils during two different lessons; one where they worked without computers and one where they used computers and worked with block programming. The participating pupils worked in pairs, and the work and process of the pupils was observed and recorded by field notes and audio recordings. The learning opportunities was examined and the pupils' mathematical reasoning during both lessons was analysed. Four questions served as basis for the analysis, and the results showed a difference in the pupils’ mathematical reasoning in the two different learning environments. At an individual level, the results varied as regards which working environment was the most beneficial. At a group level, on the other hand, more of the pupils were able to follow each other’s mathematical reasoning when working in the programming environment. Furthermore, most of the pupils were more perseverant in solving the tasks when working in the programming environment. The possible cause of these differences is discussed in connection to the results of this study as well as to previous research. The conclusion is, a programming environment can offer the pupils new opportunities to learn and should be used as one of many ways to teach mathematics.

The theme of this graduation thesis is the education of homothety using CABRI II Plus. It includes interactive files on CD and methodical instruction how to use them. It is also very useful tool for self-education.

O estudo de impressos direcionados para professores, publicados pelos órgãos oficiais de Educação, contendo sugestões sobre os modos de fazer em sala de aula, constituindo uma literatura cinzenta escolar, aparecem como um instrumento eficaz para o aprofundamento dos estudos da História da Educação Matemática no Brasil e das relações entre programas, conteúdos e práticas escolares. Diante disso, o objetivo da tese foi problematizar de que modo foram construídas as propostas de alterações metodológicas para o ensino do número nas séries iniciais do Ensino Fundamental, no período entre 1961 e 1979, de modo a tentar compreender como foram produzidas as representações de ensino moderno, fundamentadas no ideário do Movimento da Matemática Moderna (MMM), nas publicações das Secretarias de Educação de São Paulo. E também, os modos de produção desses modelos, ou seja, a transformação na representação didático-pedagógica do conceito de número, no período analisado, nas orientações publicadas. Ainda, mais especificamente, busca-se entender como ocorre a apropriação dos estudos de Zoltan Paul Dienes nesses impressos. Acredito que o estudo que explore o diálogo entre passado e presente, que procura compreender as condições que permitiram a produção das representações sobre como ensinar e aprender Matemática pode subsidiar as problematizações diárias sobre a prática e possíveis novas propostas. Isso implica seguir e procurar desvendar os processos de apropriação utilizados pelos elaboradores das publicações, além de procurar caracterizar e diferenciar o MMM no ensino primário. O período histórico da pesquisa foi determinado pelas fontes selecionadas, após um levantamento das publicações existentes (Implantação da escola municipal de oito anos, de 1969, e os quatro volumes do Manual de Detalhamento de Currículo de 1974, 1976, 1977 e 1979, na memória técnica documental do Município de São Paulo e no Arquivo Pessoal Lucília Bechara Sanchez. A opção por essas fontes relaciona-se ao reconhecimento do valor atribuído às publicações no subsídio de professores para as mudanças, num período de expansão e criação dos sistemas de ensino no Brasil, com transformações na estrutura, no funcionamento, nos programas e no currículo de Matemática, de acordo com as normativas impostas pelas LDB 4.024/1961 e LDB 5.672/1971. Para complementar a análise, foi necessário problematizar as dificuldades do trabalho com essa literatura cinzenta escolar como fonte. A escassez de pesquisas que as utilizam pode ser explicada pela profusão desses textos, que, apesar de emanados de um mesmo órgão público, têm fases diferentes, consoante com os grupos que os produziram. Na articulação das questões, fiz uso da abordagem da história cultural e me apoiei nos conceitos de representação, apropriação e estratégias, postas por Chartier (1991) e Certeau (1982). Concluí que, no período estudado, as publicações produzidas pelas Secretarias foram utilizadas como estratégia de reformulação curricular e divulgação para implementar as novas diretivas para o ensino de Aritmética na escola primária paulista, adequando-se às recomendações dos novos campos da psicologia e da didática. A pesquisa ainda assinalou que a apropriação das ideias de Zoltan Dienes, defendendo uma abordagem estruturalista para a Matemática, produziu grandes reformulações na didática da Matemática, ressignificando o quê, como e pra quem ensinar. / The study of papers targeted to teachers, published by the official agencies of Education, containing suggestions on how to do in the classroom, making up a blurry school literature, appear as an effective instrument for deepening the studies of Mathematics Education History in Brazil and the relationships between programs, contents and school practices. Given the importance of these papers, this thesis aims to discuss how the proposals of methodological changes to the teaching of numbers in the early grades of elementary school were built. I want to understand how the representations were made of \"modern education\" based on the ideals of MMM, in the publications of the Departments of Education of Sao Paulo and the ways of production of these models. What transformation does the didactic teaching go through on the concept of number in the analyzed period through the guidelines published to the teachers by the Education Department? And more specifically, what reveals the papers of the Education Departments in terms of ownership of the studies conducted by Zoltan Paul Dienes? I believe that the questioning about what it was like to teach arithmetic and the methods suggested for this teaching in the initial grades may contribute to the understanding of the process of learning mathematics and how this process influenced - and continues to influence - the teaching of mathematics in the current educational context. The historical period of the research was determined by selected sources, after a survey of the existing publications (Implementation of the eight year municipal school, of 1969, and the four volumes of the MDC, of 1974, 1976, 1977 and 1979) in the Document Technical Memory of São Paulo and APLB. The choice of these sources is related to the recognition of the value assigned to the publications in the supplies of teachers for changes in a period of expansion and creation of educational systems in Brazil, with changes in the structure, operations, programs and in the Mathematics curriculum in accordance with the regulations imposed by BDL 4024/61 and BDL 5672/71. The work also included the comparison between publications with LDB/61 and the LDB/71, considering the place where the production of the official papers was held and the everyday of the production process, their dynamics and backstage. In order to complement the analysis, it was necessary to discuss the difficulties of working with this blurry school literature as a source. In articulating the issues, I use the approach of cultural history and lean against the concepts of representation, ownership and strategies put by Chartier (1991) and Certeau (1982). I conclude that during the studied period, the publications produced by the Departments of Education were used as a strategy adapting to the recommendations of the new fields of psychology and didactics. The survey also noted that ownership of the ideas of Zoltan Dienes, advocating a structuralist approach to mathematics, produced major reformulations in mathematics didactics, giving new meaning to teaching, how to teach and for whom to teach.

La question de l’enseignement du nombre à l’école maternelle est abordée à travers les savoirs et les connaissances mathématiques spécifiques du professeur. Cette thèse s’inscrit dans le cadre de la théorie des situations didactiques. Une modélisation des savoirs mathématiques spécifiques pour l’enseignement du nombre à l’école maternelle est proposée afin d’étudier l’enseignement du nombre de dix enseignants. Le travail est complété par une micro formation relative à des savoirs spécifiques pour la quantité. La thèse étudie l’impact de nouveaux savoirs sur les connaissances mathématiques spécifiques de cinq professeurs. Enfin, les résultats de la recherche sont interprétés par rapport à la formation des enseignants / Teaching numbers in preschool is addressed though the specific mathematical knowledge of the teacher himself. This thesis work comes within theoretical didactical situations. A modeling system for specific mathematical knowledge is suggested in order to study the way ten teachers teach numbers. This work is completed by a short training course related to quantity knowledge. This thesis work analyses the impact of these new specific fields of knowledge for five amongst ten of these teachers. The final result of the research work is highlighted by teachers training.

La thèse étudie l'évolution sous contraintes du milieu que se donnent les élèves dans les cadres des premiers moments d'une activité d'étude et de recherche sur les équations du premier degré à une inconnue en classe de 4°. Le dispositif d'observation - analyses mathématique et didactique a priori soutenant la proposition d'activité, mise en place de groupes au sein des classes, films des interactions au sein du processus de recherche - recourt à une méthodologie de type clinique. Elle autorise une analyse micro-didactique qui, utilisant les récents développements de la théorie Anthropologique du didactique, conclut à la différenciation des milieux, des temporalités et des fonctions au sein des groupes observés et un enrichissement du concept de mémoire didactique. / The thesis studies the evolution under constraints of the milieu that pupils give themselves in the frames of the first moments of an activity of study and research on single-unknown equations of the first degree in class of 4 °.The observation device - a priori mathematical and didactical analyzes supporting the activity proposal, setting up groups within the classes, films of the interactions within the research process - resorts to a methodology of the clinical type. It allows a micro-didactic analysis which, using the most recent developments of the anthropological theory of didactics, concludes with the differentiation of the environments, the temporalities and the functions within the groups observed and an enrichment of the concept of didactic memory.