Orientador: Vinício de Macedo Santos / Banca: Célia Maria Carolino Pires / Banca: José Carlos Miguel / Abstract: The rational numbers are shown as a subject that the students of the Elementary and High School have difficulties to learn. Some of these difficulties are due to the difference established between the daily use of the rational numbers by the student and the way it is taught at the school and, also for the ignorance, on the part of the school, of the multiplicity of their meanings. While the social use is centered in the decimal form, the school use lies more on the fractional form of the rational numbers. It is an undesirable separation that the school practices have accentuated through time. This study tried to characterize the existent dichotomization between it the use and the teaching of the Mathematics, starting from bibliographical research and of documental study that end up being responsible for damages in the students' learning.. This can be verified in the mistakes committed in the official tests (SARESP, SAEB...). It was sought to analyze how that separation has been reinforced in the official documents, by the pedagogic proposals and curricula. It was verified how the different documents and official publications deal with the rational numbers and the articulation among perspectives of the school use and the daily use of the rational numbers. That analysis made possible to understand different types of arguments and justifications for the teaching of the fractions, present in the official curricula, as well as explain the contents and the most appropriate methodologies of the conceptions presented in such documents. All this made possible to know part of the problems that happen with the teaching of fractions and their causes, and so, make suggestions on how these problems can be solved. Although the establishment of relationships between the social use and school use still doesn't happen in an effective way, it is recognized... (Complete abstract, click electronic address below) / Resumo: Os números racionais apresentam-se como conteúdo que os alunos do Ensino Fundamental e Médio têm dificuldades para aprender. Parte dessas dificuldades decorre da diferença instituída entre o uso cotidiano dos números racionais pelo aluno e a maneira como são ensinados na escola e, também pelo desconhecimento, por parte da escola, da multiplicidade dos significados dos racionais. Enquanto o uso social centra-se na forma decimal o uso escolar recai mais sobre a forma fracionária dos números racionais. É uma separação indesejável que as práticas escolares trataram de acentuar ao longo do tempo. A partir de pesquisa bibliográfica e de estudo documental procurou-se caracterizar, nesse trabalho, a dicotomização existente entre o uso e o ensino da Matemática, que acabam sendo responsáveis por prejuízos na aprendizagem dos alunos. Isto pode ser verificado nos erros que os alunos cometeram nas provas oficiais (SARESP, SAEB...). Procurou-se analisar como essa separação vem sendo reforçada nos documentos oficiais, por meio das propostas pedagógicas e curriculares. Verificaram-se como diferentes documentos e publicações oficiais abordam os números racionais e tratam da articulação entre a perspectivas do uso escolar e a do uso cotidiano dos números racionais. Essa análise possibilitou compreender diferentes tipos de argumentações e justificativas para o ensino das frações, presentes nos currículos oficiais, bem como explicitar os conteúdos e metodologias adequadas às concepções apresentadas em tais documentos. Tudo isso possibilitou conhecer parte dos problemas que ocorrem com o ensino de frações e suas causas e por isso sugerir propostas que sinalizam para a sua superação. Embora o estabelecimento de relações entre o uso social e uso escolar ainda não ocorra de maneira efetiva, reconhece-se que aquelas orientações... (Resumo completo, clicar acesso eletrônico abaixo) / Mestre

Made available in DSpace on 2016-04-27T16:58:28Z (GMT). No. of bitstreams: 1 Marcelo Eduardo Pereira.pdf: 4788851 bytes, checksum: f240246d61aa2e0ffdb09fe10593ffca (MD5) Previous issue date: 2007-10-19 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The purpose of this research is to analyze learning situations concerning argumentations and mathematical proofs combined with a computational tool and had been developed into the AprovaME project - Argumentação e Prova na Matemática Escolar (Argumentation and Proof in School Mathematics), particularly during the second phase. We are founded by researches that explore the assuming functions of proof as well as evaluate them in the school context, underneath various aspects and generality levels. Guided by the main results of this studies and the survey of proof conceptions made by the teenager students that had been accomplished during the first phase of the project, we prepared a sequence of activities intending to engage them throughout the stages of proving and to argue about the conditions of transiting between pragmatic and conceptual proofs. We search, into this context, to explore different functions proof, beyond the verification one, and to analyze the role of the Microsoft Excel tool in the students empirical work. The activities were applied in extra classes sessions for three pairs of volunteer students between 15-16 years old from a private school in Santos-SP. As result, it was verified that students interaction with the computer had dynamized the process of surveying conjectures and validating them. Also through the computer experience they were able to notice the manipulated objects properties which developed the production of justifications beyond empirical evidences. Therefore, within this proposal, the students had experienced the moments of proving and presented, by deductive reasoning, argumentations that show clearly the generality involved in the suggested tasks / A proposta deste trabalho é analisar situações de aprendizagem envolvendo argumentações e provas matemáticas, integrando uma ferramenta computacional, tendo sido desenvolvido no âmbito do projeto AProvaME Argumentação e Prova na Matemática Escolar, referindo-se, particularmente, à 2ª Fase deste projeto. Fundamentamo-nos em pesquisas que exploram as funções que uma prova pode assumir e as avaliam, no contexto escolar, sob vários aspectos e níveis de generalidade. À luz dos principais resultados desses estudos e do levantamento das concepções sobre prova de alunos adolescentes, realizado na 1a Fase do Projeto, elaboramos uma seqüência de atividades com o intuito de engajá-los nas várias etapas do processo de prova e discutir as condições de transição das provas pragmáticas para as conceituais. Buscamos, neste contexto, explorar outras funções da prova, além da função de verificação e avaliar o papel da ferramenta Microsoft Excel no trabalho empírico dos alunos. A seqüência foi aplicada, em sessões extraclasse, a três duplas de alunos de 15-16 anos de uma escola particular da cidade de Santos-SP, que participaram voluntariamente da experimentação. Como resultado, verificou-se que a interação dos alunos com o computador dinamizou o processo de produção de conjecturas e de validação experimental destas, bem como a observação de propriedades dos objetos manipulados, favorecendo a elaboração de justificativas que vão além das evidências empíricas. Assim, por meio desta abordagem, os alunos tiveram a oportunidade de vivenciar as etapas do processo de prova, apresentando, por meio de raciocínios dedutivos, argumentos que evidenciam a generalidade envolvida nas tarefas propostas

Bakgrund och syfte: Matematik är ett ämne som skapar lust och är stimulerande för vissa människor medan det för andra är ångestladdat, svårt och meningslöst (Skolverket, 2002). Decimaltal kan för vissa elever skapa problem (Mange, 1962). Syftet med studien är att ta reda på hur sex elever i skolår 6 utifrån en matematiklektion om beräkning av decimaltal, ser på sitt eget lärande och lektionsupplägget. Samt att ta reda på om eleverna själva ser sin egen förståelse utifrån lektionsinnehållet. Metod: Vi utgick från en kvalitativ tradition med en fenomenologisk ansats. För att få fram data till analysen utgick vi från tekniken stimulated recall. Det innebar att vi filmade en matematiklektion och där inspelningen sedan var utgångspunkten vid intervjuerna, som genomförs med sex utvalda elever. Vid intervjuerna fick eleverna se på filminspelningen och utifrån inspelningen fick eleverna kommentera vad de tänkte under den dokumenterade lektionen. När vi analyserade datan utgick vi från ett metakognitivt perspektiv. Resultat: Vi såg att eleverna kunde uttrycka vad de hade lärt sig och de berättade även vad som i lektionsupplägget underlättade eller hindrade deras lärande. Överlag tyckte eleverna att lärarens genomgång var för noggrann och för lätt. Eleverna menade att de redan innan lektionen kunde det som läraren skulle gå igenom, men vi kunde se brister i elevernas konceptuella kunskap av decimaltal. Diskussion: Elevernas konceptuella brister hindrade inte eleverna från att klara för- och eftertestet. Vi kunde inte urskilja om läraren var medveten om elevernas brister, dock tror vi oss veta att eleverna inte var medvetna om sina brister. Vi tror att läraren skulle kunna utmana sina elever genom att lägga störst vikt på den konceptuella kunskapen, istället för den procedurella kunskapen. / Background and purpose: Mathematics is a subject that creates desire and are stimulating for some people while for others it is difficult and pointless (Skolverket, 2002). Decimal numbers can for some students create problems (Mange, 1962). The study aims to find out how six pupils in school years 6 from a mathematics lesson, on the calculation of decimal numbers, look at their own learning, lesson structure, and find out whether the students themselves see their own understanding based on lesson content. Method: We started with a qualitative tradition with a phenomenological approach. In order to obtain data for analysis, we assumed stimulated recall. This means that we filmed a math lesson, where the recording then was the starting point of the interviews conducted with six selected pupils. At the interview the pupils were watching the movie and pupils comments based on the movie, on what they thought during the documented lesson. When we analyzed the data we started from a meta-cognitive perspective. Results: We can see that the students can express what they have learned and they said also what in the lesson structure that facilitated or hindered their learning. Overall, pupils felt that the teacher's review was thorough and easy. The pupils said even before the lesson that they had good knowledge about what the teacher would go through, but we could see the gaps in pupils' conceptual understanding of decimal numbers. Discussion: Students' conceptual deficiencies did not prevent the pupils to cope with pre-and post test. We cannot discern whether the teacher is aware of pupils' deficiencies, however, we think that we know that pupils are not aware of their shortcomings. We believe that the teacher should be able to challenge her pupils by putting most emphasis on the conceptual understanding rather than on procedural knowledge.

<p><strong>Bakgrund och syfte:</strong> Matematik är ett ämne som skapar lust och är stimulerande för vissa människor medan det för andra är ångestladdat, svårt och meningslöst (Skolverket, 2002). Decimaltal kan för vissa elever skapa problem (Mange, 1962). Syftet med studien är att ta reda på hur sex elever i skolår 6 utifrån en matematiklektion om beräkning av decimaltal, ser på sitt eget lärande och lektionsupplägget. Samt att ta reda på om eleverna själva ser sin egen förståelse utifrån lektionsinnehållet.</p><p><strong>Metod: </strong>Vi utgick från en kvalitativ tradition med en fenomenologisk ansats. För att få fram data till analysen utgick vi från tekniken stimulated recall. Det innebar att vi filmade en matematiklektion och där inspelningen sedan var utgångspunkten vid intervjuerna, som genomförs med sex utvalda elever. Vid intervjuerna fick eleverna se på filminspelningen och utifrån inspelningen fick eleverna kommentera vad de tänkte under den dokumenterade lektionen. När vi analyserade datan utgick vi från ett metakognitivt perspektiv.</p><p><strong>Resultat: </strong>Vi såg att eleverna kunde uttrycka vad de hade lärt sig och de berättade även vad som i lektionsupplägget underlättade eller hindrade deras lärande. Överlag tyckte eleverna att lärarens genomgång var för noggrann och för lätt. Eleverna menade att de redan innan lektionen kunde det som läraren skulle gå igenom, men vi kunde se brister i elevernas konceptuella kunskap av decimaltal.</p><p><strong>Diskussion:</strong> Elevernas konceptuella brister hindrade inte eleverna från att klara för- och eftertestet. Vi kunde inte urskilja om läraren var medveten om elevernas brister, dock tror vi oss veta att eleverna inte var medvetna om sina brister. Vi tror att läraren skulle kunna utmana sina elever genom att lägga störst vikt på den konceptuella kunskapen, istället för den procedurella kunskapen.</p> / <p><strong>Background and purpose: </strong>Mathematics is a subject that creates desire and are stimulating for some people while for others it is difficult and pointless (Skolverket, 2002).<strong> </strong>Decimal numbers can for some students create problems (Mange, 1962). The study aims to find out how six pupils in school years 6 from a mathematics lesson, on the calculation of decimal numbers, look at their own learning, lesson structure, and find out whether the students themselves see their own understanding based on lesson content.</p><p><strong>Method:</strong> We started with a qualitative tradition with a phenomenological approach. In order to obtain data for analysis, we assumed stimulated recall. This means that we filmed a math lesson, where the recording then was the starting point of the interviews conducted with six selected pupils. At the interview the pupils were watching the movie and pupils comments based on the movie, on what they thought during the documented lesson. When we analyzed the data we started from a meta-cognitive perspective.</p><p><strong>Results: </strong>We can see that the students can express what they have learned and they said also what in the lesson structure that facilitated or hindered their learning. Overall, pupils felt that the teacher's review was thorough and easy. The pupils said even before the lesson that they had good knowledge about what the teacher would go through, but we could see the gaps in pupils' conceptual understanding of decimal numbers.</p><p><strong>Discussion: </strong>Students' conceptual deficiencies did not prevent the pupils to cope with pre-and post test. We cannot discern whether the teacher is aware of pupils' deficiencies, however, we think that we know that pupils are not aware of their shortcomings. We believe that the teacher should be able to challenge her pupils by putting most emphasis on the conceptual understanding rather than on procedural knowledge.</p>

The aim of this study is to explore the adult immigrant students’ experience of maths in Sweden. I will present an understanding rather than an explanation on how second language adult students learn maths. It can be argued that people who study maths as adults in a new homeland and in a foreign language face particular challenges. At the same time research reports that people sometimes approach the subject in a more fruitful way as adults compared to their childhood experiences. I want to contribute to the general knowledge of the subject and furthermore provide improved understanding of how mathematics teachers can guide their students towards their goals.I have performed semi-structured qualitative research interviews. My informants are my own maths students on the basic level with incomplete grades in maths from secondary school, or they have failed in their maths studies in upper secondary school due to a low level of know-ledge. They are over 20 years of age and they are all immigrants and have arrived in Sweden as adults. I have used my students statements, written as narratives as the material which is to be interpreted and understood. Because of my use of my own students in the interview, I will not take into account their statements about the teacher’s role in my conclusion.I find that:1. The difficult experience of being forced to leave the home country, together with a wish to take revenge on the failures from their youth, can lead to a kind of struggle for decom-pensation that can be reflected in the participants' positive evaluation of their maths studies.2. Having a family is a great motivational help for studying regardless of the time it takes to take care of the same.3. The memories of previous failures with the incomprehensible, abstract mathematics characterise the students’ inception of the subject.4. It seems possible that adult students can understand themselves in a new way and redefine their relationship with maths and their own ability to study the subject.

Submitted by Raniere Barreto (raniere.barros@ufvjm.edu.br) on 2018-04-12T16:51:05Z No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) raphael_neves_matos.pdf: 4286914 bytes, checksum: 4faddab9001b8b035017adfd9a2d6d75 (MD5) / Approved for entry into archive by Rodrigo Martins Cruz (rodrigo.cruz@ufvjm.edu.br) on 2018-04-20T14:12:28Z (GMT) No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) raphael_neves_matos.pdf: 4286914 bytes, checksum: 4faddab9001b8b035017adfd9a2d6d75 (MD5) / Made available in DSpace on 2018-04-20T14:12:28Z (GMT). No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) raphael_neves_matos.pdf: 4286914 bytes, checksum: 4faddab9001b8b035017adfd9a2d6d75 (MD5) Previous issue date: 2017 / Este trabalho teve como objetivo principal apresentar uma contribui??o para o ensino aprendizagem dos n?meros racionais, destacando principalmente a rela??o entre d?zimas peri?dicas e progress?es geom?tricas. A metodologia utilizada permitiu a an?lise da abordagem e sequ?ncia did?tica dos t?picos D?zima peri?dica e Progress?o Geom?trica Infinita, contemplada nos livros did?ticos aprovados pelo Programa Nacional do Livro Did?tico. Nesta abordagem as fra??es e os n?meros decimais, especialmente os decimais infinitos e peri?dicos, e por consequ?ncia o c?lculo de sua fra??o geratriz, foram objetos de estudo centrais e instigadores dessa pesquisa. Realizou-se um estudo mais detalhado sobre a representa??o decimal dos n?meros racionais e analisando a compreens?o destes n?meros em n?vel fundamental e m?dio. Foi ainda proposto uma abordagem das maneiras mais usuais do c?lculo da fra??o geratriz, bem como, explorado a rela??o entre os decimais infinitos e peri?dicos e as progress?es geom?tricas. Durante o desenvolvimento deste trabalho, foi poss?vel perceber que h? mais de uma abordagem did?tica dos t?picos de ensino inerentes ao tema central analisado. O reconhecimento de que a parte decimal das d?zimas peri?dicas pode ser expressa como uma soma infinita de parcelas que, a partir de certo ponto, descreve uma progress?o geom?trica infinita de raz?o compreendida entre zero e um, ? um ponto chave na proposta de interven??o apresentada para a sala de aula. Diante desse quadro, foi verificado a ordem atualmente seguida pelos professores do 1? Ano do Ensino M?dio, o que permitiu constatar que os conte?dos D?zimas Peri?dicas e Progress?es Geom?tricas Infinitas s?o tratados sem liga??o significativa e, diante disso, foi proposta uma altera??o na ordem de abordagem desses conte?dos no Ensino M?dio. Ao final foram propostas algumas sugest?es de atividades resolvidas e outras para serem desenvolvidas em sala de aula. / Disserta??o (Mestrado Profissional) ? Programa de P?s-Gradua??o Matem?tica, Universidade Federal dos Vales do Jequitinhonha e Mucuri, 2017. / The aim of this work was to present a contribution to the teaching of rational numbers, emphasizing mainly the relation between periodic tithe and geometric progression. The methodology used allowed the analysis of the approach and didactic sequence of the topics Periodic Dizima and Infinite Geometric Progression, contemplated in textbooks approved by the National Textbook Program. In this approach fractions and decimal numbers, especially the infinite and periodic decimals, and consequently the calculation of their generative fraction, were central objects and instigators of this research. A more detailed study on the decimal representation of rational numbers was carried out and the understanding of these numbers at the fundamental and medium levels was analyzed. It was also proposed an approach of the most usual ways of calculating the generative fraction, as well as exploring the relationship between infinite and periodic decimals and geometric progressions. During the development of this work, it was possible to perceive that there is more of a didactic approach of the teaching topics inherent to the central theme analyzed. The recognition that the decimal part of the periodic tithe can be expressed as an infinite sum of plots which, from a certain point, describes an infinite geometric progression of ratio between zero and one, is a key point in the proposal of intervention presented for the classroom. In view of this situation, we verified the order currently being followed by teachers of the 1? Year of High School, which allowed to verify that the Periodic Dictionaries and Infinite Geometric Progressions are treated without significant connection and, accordingly, a change was proposed in order to approach these contents in High School. At the end, some suggestions for solved activities and others to be developed in the classroom were proposed.

Made available in DSpace on 2014-06-11T19:24:21Z (GMT). No. of bitstreams: 0 Previous issue date: 2003Bitstream added on 2014-06-13T18:52:14Z : No. of bitstreams: 1 valera_ar_me_mar.pdf: 594283 bytes, checksum: 7fa747413b18f73739f058ca4ea1146e (MD5) / Os números racionais apresentam-se como conteúdo que os alunos do Ensino Fundamental e Médio têm dificuldades para aprender. Parte dessas dificuldades decorre da diferença instituída entre o uso cotidiano dos números racionais pelo aluno e a maneira como são ensinados na escola e, também pelo desconhecimento, por parte da escola, da multiplicidade dos significados dos racionais. Enquanto o uso social centra-se na forma decimal o uso escolar recai mais sobre a forma fracionária dos números racionais. É uma separação indesejável que as práticas escolares trataram de acentuar ao longo do tempo. A partir de pesquisa bibliográfica e de estudo documental procurou-se caracterizar, nesse trabalho, a dicotomização existente entre o uso e o ensino da Matemática, que acabam sendo responsáveis por prejuízos na aprendizagem dos alunos. Isto pode ser verificado nos erros que os alunos cometeram nas provas oficiais (SARESP, SAEB...). Procurou-se analisar como essa separação vem sendo reforçada nos documentos oficiais, por meio das propostas pedagógicas e curriculares. Verificaram-se como diferentes documentos e publicações oficiais abordam os números racionais e tratam da articulação entre a perspectivas do uso escolar e a do uso cotidiano dos números racionais. Essa análise possibilitou compreender diferentes tipos de argumentações e justificativas para o ensino das frações, presentes nos currículos oficiais, bem como explicitar os conteúdos e metodologias adequadas às concepções apresentadas em tais documentos. Tudo isso possibilitou conhecer parte dos problemas que ocorrem com o ensino de frações e suas causas e por isso sugerir propostas que sinalizam para a sua superação. Embora o estabelecimento de relações entre o uso social e uso escolar ainda não ocorra de maneira efetiva, reconhece-se que aquelas orientações...

The master thesis is concerning about opening to decimal numbers at primary school and the interactive textbook called Cyril I created in the Imagine Logo programme. It contains theory about computer-aided teaching, decimal numbers, manual for the programme and the results of experiment which proved the textbook in practice. Important component of master thesis is the programme itself, which works both as a textbook and a workbook, and which is directed at completion opening to decimal numbers and his exercises at the same time.

Made available in DSpace on 2018-06-27T19:13:13Z (GMT). No. of bitstreams: 2 Livia Da Cas Pereira.pdf: 1967233 bytes, checksum: 85697e48fc48568430f859d209bc3879 (MD5) Livia Da Cas Pereira.pdf.jpg: 2867 bytes, checksum: 0701e2556802f3b56ef0d6a23247fa10 (MD5) Previous issue date: 2011-08-16 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The theme of this dissertation is the teaching and learning on the operations with decimal-numbers by the resolution of problems in Primary School. The main purpose is to evaluate if the method of resolution of problems contributes for a better understanding of operations involving decimal numbers. To develop this research of qualitative type, the following instruments to collect data have been used: participants were observed and their results recorded in a class diary which included all the incidents occurred in class. A diagnostic test as well as documents used in the resolution of problems which were applied to twenty students from the sixth year (5° grade) from primary school of a private school, located in the city of Santa Maria. The diagnostic test have been elaborated based on questions from SAERS (a system to evaluate the performance of the students in schools of Rio Grande do Sul), from the year of 2007, which involved decimal number problems of the sixth year (5° grade) from primary school and aimed to identify the most frequent doubts presented by the students who the researcher have works with. Between theses doubts the ones which stood out were: difficulties of interpretation of problem s statements; lack of attention; when the minuend is smaller than the subtrahend in subtraction and the position of the comma. From this diagnostic, hypothetic problems were set to be solved by the method of Resolution of Problems and according to it, it s been possible to conclude that the application of the method of Resolution of Problems has been satisfactory, since it allowed the students to develop a collective and collaborative job. Besides, it allowed the researcher to develop their own knowledge with a higher level of autonomy. It also made possible to the researcher to identify gaps in children s knowledge previously achieved, as the four mathematical operations and then, fulfill them. / Esta dissertação tem como tema ensino e aprendizagem das operações com números decimais através da Resolução de problemas no Ensino Fundamental. Seu objetivo é avaliar se o método de Resolução de problemas contribui para um melhor entendimento das operações com números decimais. Para a realização desta pesquisa de caráter qualitativo foram utilizados como instrumentos de coleta de dados: observação participante registrada por meio de um diário de aula onde foram relatados todos os acontecimentos ocorridos em classe e um teste diagnóstico, bem como documentos produzidos nas resoluções dos problemas, aplicados a vinte alunos do 6º ano (5ª série) do Ensino Fundamental de um colégio privado, localizado na cidade de Santa Maria. O teste diagnóstico foi elaborado a partir das questões do SAERS, do ano de 2007, que envolvem o conteúdo de Números Decimais do 6º ano do Ensino Fundamental e teve como objetivo diagnosticar as dúvidas mais frequentes apresentadas pelos alunos com os quais a pesquisadora trabalhou. Entre estas se destacaram: problemas de interpretação do enunciado do problema; relativas à falta de atenção; na subtração quando o minuendo é maior que o subtraendo; posicionamento da vírgula. A partir desse diagnóstico, elaboraram-se situações-problema para serem trabalhadas por meio do método de Resolução de problemas. Através desta pesquisa pode-se concluir que a aplicação do método de Resolução de problemas foi válida, uma vez que possibilitou aos alunos a realização de um trabalho coletivo e colaborativo, além de desenvolver, nos mesmos, uma maior autonomia na construção de seu próprio conhecimento. Também proporcionou a pesquisadora o diagnóstico de lacunas existentes em relação a aprendizados anteriores como, por exemplo, as quatro operações, possibilitando saná-las.

Lärare som undervisar i matematik förväntas kunna mer avancerad matematik än vad de undervisar om. Men formell matematikkunskap anses inte vara tillräckligt för att lärare ska kunna undervisa så att ämnesinnehållet blir begripligt för eleverna, de behöver även pedagogical content knowledge (PCK). Begreppet belyser en speciell form av ämneskunskap för undervisning och skiljer sig från den matematikkunskap som används av andra välutbildade vuxna. Det har föreslagits att olika arrangemang av kollegialt och praktikbaserat lärande kan utveckla lärares PCK. Ett exempel på ett sådant arrangemang är learning study. Den här avhandlingen handlar om den kunskap om lärande och undervisning i matematik som studiens lärare utvecklar då de deltar i learning studies och utforskar sin praktik utifrån ett variationsteoretiskt perspektiv. Det yttersta syftet med en learning study är att utveckla elevernas lärande om specifika lärandeobjekt, genom att undersöka vad som kan vara kritiskt för elevernas lärande. I ett samarbetsprojekt med fyra högstadielärare genomfördes två learning studies i matematik, under ett år. Lärargruppen undersökte vad eleverna behöver lära för att de ska förstå i) varför en kvot kan vara större än talet i täljaren och ii) olika representationer av konstanterna k och m i räta linjens ekvation. Under learning study-arrangemangets olika steg samlades studiens empiri in och denna består av filmade lektioner, inspelade möten där lärargruppen planerade och analyserade undervisning och elevers lärande, skriftliga elevtest samt elevintervjuer. Studien har en variationsteoretisk utgångspunkt, vilket innebär att lärande förklaras ske när en person ser något på ett nytt och mer kvalitativt sätt, genom att personen urskiljer aspekter som han/hon inte tidigare har urskilt. Studien visar de två lärandeobjektens kritiska aspekter samt hur de kritiska aspekterna gradvis förändrades och specificerades. Förändringen var ett resultat av att lärargruppen fick syn på avgörande detaljer om på vilket sätt eleverna förstod ämnesinnehållet samt hur skilda sätt att förstå kunde användas i undervisningen för att utveckla elevernas lärande. Där av titeln att få syn på avgörande skillnader. Denna form av utvecklad kunskap om lärandeobjektet kan ses som ett bidrag om PCK och vad det kan vara. / It is a common view that teachers need more than formal content knowledge to teach and to make the content comprehensible to others. They also need pedagogical content knowledge, or PCK (Shulman, 1986). It has been suggested that different teacher collaboration approaches may support teachers’ development of PCK (Chapman, 2013, Davis &amp; Renert, 2014; Steele &amp; Rogers, 2012). This thesis aims to provide insights into the kind of knowledge about teaching and learning mathematics that teachers develop through their participation in a specific collaboration approach called learning study. Four teachers of mathematics and their 74 students (aged 15−16 years) participated in two learning studies over the course of one year. The foremost aim of a learning study is to enhance student learning about specific objects of learning and to identify what is critical for the students’ learning (Marton &amp; Tsui, 2004). The objects of learningin the two learning studies were to understand that dividing with a denominator between 0 and 1 gives a quotient larger than the numerator and to understand different representations of the constants b and m in the equation of the straight line. During the two learning studies data were collected from 8 video-recorded lessons, 2 written student tests, student interviews, and 14 audio-recorded sessions in which the teachers and I (PhD student) planned, analysed and revised teaching and student learning. The analysis was based on variation theory (Marton &amp; Tsui, 2004) and focused on what participants considered to be critical aspects of the objects of learning and on the components embedded in that knowledge. The result shows the identified critical aspects of the two objects of learning and, furthermore, how the teachers’ knowledge about those critical aspects gradually changed and became more refined and specified in relation to their students’ understanding. The thesis provides an insight into the value of the teachers’ enhanced knowledge of the object of learning, in relation to how PCK can be understood.