Sedan Lgr11 infördes i skolan har förmågan att kommunicera och resonera i matematik genom samtal och diskussioner fått en ny plats i undervisningen. I kunskapskraven står det att ska elevernas förmåga att muntligt resonera och kommunicera i matematik ska bedömas i årskurs 6 vilket innebär att lärarna behöver ge eleverna möjlighet att träna dessa förmågor. Syftet med denna studie är att analysera vilket stöd lärarna erbjuds genom lärarhandledningar för att kunna utveckla elevernas förmåga att samtala med och om matematik samt att föra matematiska resonemang. En innehållsanalys av lärarhandledningar visar att det finns stöd för lärarna att utveckla elevernas förmåga att samtala med och om matematik samt att föra matematiska resonemang, men att stödet varierar i stor utsträckning mellan de olika lärarhandledningarna. Innehållsanalysen är baserad på Shulmans ramverk om vilka kunskaper en lärare behöver ha för att undervisa, MKT-ramverket från Ball et al. som kopplar Shulmans ramverk till matematik samt Niss och Højgaard-Jensens ramverk om vilka kunskaper en matematiklärare behöver ha. Analysen visar att samtliga lärarhandledningar erbjuder stöd till lärare i sina ämnesdidaktiska kunskaper i form av aktiviteter som kan användas för att utveckla elevernas muntliga förmåga. Däremot varierar antalet förekomster av stöd för lärarna till sina ämnesdidaktiska- samt ämneskunskaper. / When the new curriculum was introduced in Sweden, 2011, the ability to communicate and reason in mathematics through discussions was put forward. The students’ ability to orally reason and communicate in mathematics is to be assessed in grade 6, which means that teachers need to give students opportunities to practice these skills. The purpose of the study is to analyze what kind of support teacher guides offer the teacher to help them develop the students’ ability to discuss with and about mathematics as well as to apply mathematical reasoning. A content analysis of teacher guides showed that there is support for teachers to develop the students’ ability to discuss with and about mathematics and to apply mathematical reasoning, but that the support varies to a great extend amongst teacher guides. The content analysis is based on Shulman’s knowledge base framework describing what a teacher needs to know in order to teach, the MKT-framework from Ball and colleagues that connects Shulman’s framework to mathematics and Niss och Højgaard-Jensen’s framework about which knowledge a mathematics teacher needs for teaching. The analysis shows that all analyzed teacher guides offer support in pedagogical content knowledge consisting of activities that can be used to develop the students’ orally ability. However, the amount of support varies concerning both pedagogical content knowledge and content knowledge.

Studier har visat att det i den traditionella undervisningen i matematik råder brist på förutsättningar för eleverna att utveckla sin resonemangsförmåga på ett betydande sätt (Sidenvall, Lithner och Jäder, 2015; Tranbeck, 2010; Jäder, 2015). Denna brist har identifierats som en av anledningarna till att elever har det svårt på matematikutbildningar på högskolenivå (Lithner, 2011). Detta har lett till att alternativa arbetssätt har undersökts med avseende på hur de främjar elevernas användning av resonemangsförmågan (Liljekvist, 2014; Brunström, 2015). Syftet med denna studie är att undersöka hur gymnasieelevers resonemang ser ut och vad de innehåller när de spelar spel med matematiskt innehåll från gymnasiets matematikkurser. En förståelse för detta kan bland annat möjliggöra en utvärdering av denna typ av spels relevans inom matematikundervisningen på gymnasialnivå. I denna studie studeras två gymnasieelevgruppers resonemang när de spelar ett kortspel som har ett matematiskt innehåll. Två strukturerade innehållsanalyser tillämpas i denna studie. Den första tillämpar kategorier som skapats utifrån tre olika ramverk för resonemang medan den andra utgår ifrån kategorier för ingenjörsmässiga förmågor som valts ur Conceive Design Implement Operate (CDIO) syllabus. Resultatet visar att ett flertal olika former av resonemang och ingenjörsmässiga förmågor förekommer bland elevernas resonemang. Det vanligaste är att eleverna i sina resonemang tillämpar kunskaper om generella matematiska regler och samband på ett imitativt sätt och resonemanget behandlar även samspelet mellan två variabler. Andelen kreativa resonemang var dock fortfarande högre i denna studie än i tidigare genomförda studier. Elevernas resonemang visar vidare främst på ingenjörsmässiga förmågor relaterat till initiativförmåga och beslutsfattning under osäkerhet. Utifrån resultatet görs bedömningen att det spel som studeras i denna studie är att betraktas som relevant för tillämpning inom matematikundervisningen, med några begränsningar. Mer generellt visar resultatet att när spel av den typ som studerats i denna studie används visar eleverna upp resonemangsförmågan på ett nyanserat sätt och även ingenjörsmässiga förmågor framkommer. / Studies have shown that in the traditional forms of mathematics education there is a lack of opportunity for students to improve their reasoning ability in a meaningful way (Sidenvall, Lithner och Jäder, 2015; Tranbeck, 2010; Jäder, 2015). This flaw has been identified as one reason for why students have troubles following the mathematics education at university level (Lithner, 2011). This has led to alternative methods being studied in regard to how they provide opportunities for the students to utilise their reasoning ability (Liljekvist, 2014; Brunström, 2015). The purpose of this study is to examine what the students’ reasoning looks like and its content when they play a game with mathematical content in high school mathematics courses. Understanding this can, among other things, enable an assessment of how relevant this type of games is for use in high school mathematics education. In this study, two groups of high school students reasoning are observed when they play a card game that has mathematical content. Two structured content analyses are used in this study. The first one applies categories that have been created through the combination of three other frameworks, while the second one applies categories of engineering skills that have been selected from Conceive Design Implement Operate (CDIO) syllabus. The result shows that many different forms of reasoning and engineering skills were present in the students’ reasoning. It is most common that the students’ reasoning incorporates knowledge of general mathematical rules and relations in an imitative way that also includes the interaction between two variables. The proportion of creative reasoning was higher than those found in previous studies. Furthermore, the students’ reasoning mostly reveals engineering skills related to taking initiative and decision making under uncertainty. With regards to the result the assessment is that the game that was studied in this study can be regarded as relevant for use in mathematics education, with some limitations. More generally, the result shows that when games of the type that has been studied here are used the students show the ability to reason in a nuanced way and engineering skills are also present.

A presente pesquisa se situa no contexto das investigações que buscam contribuir para o ensino de matemática nas séries iniciais da escolaridade. As investigações nesta área sugerem que as habilidades metacognitivas do indivíduo devam se tornar o foco da instrução em sala de aula. A literatura sobre educação matemática destaca as atividades de resolução de problemas como especialmente significativas para a investigação dos processos metacognitivos do aluno. Além disto, o tema problemas aritméticos verbais com história tem gerado numerosos artigos e livros que analisam as diversas categorias de problemas existentes, entre eles os problemas de adição/subtração e de multiplicação/divisão. Assim, o presente trabalho se propõe a investigar a eficácia de procedimento de desenvolvimento de habilidades metacognitivas em matemática, utilizando-se de problemas aritméticos verbais com história em um ambiente lúdico de aprendizagem. A amostra foi composta com 100 alunos de três turmas de segunda série do Ensino Fundamental. Todos os alunos foram avaliados por meio da Prova de Problemas Aritméticos Verbais com História (de adição, subtração, multiplicação e divisão) e o Subteste de Aritmética do Teste de Desempenho Escolar TDE. A partir dos resultados obtidos nestas duas avaliações, cada classe foi dividida em duas metades, a primeira, com resultados superiores à mediana, compôs o grupo de controle superior, e a segunda, com resultados inferiores à mediana, foi novamente subdividida, sendo que, um quarto compôs o grupo de controle inferior e o outro quarto, o grupo de intervenção. Este grupo recebeu o treinamento em habilidades metacognitivas em matemática em um ambiente lúdico de aprendizagem, ao longo do segundo semestre letivo, num total de 11 sessões, enquanto os outros dois grupos de controle participaram de atividades placebo. No final de cada semestre letivo, todos os alunos foram novamente avaliados, como no seu início. A análise estatística dos resultados obtidos no TDE e na Prova de Problemas Aritméticos revelou diferença significativa nas duas avaliações apenas para os alunos do Grupo de Intervenção. Para os dois Grupos de Controle, a diferença foi significativa somente no TDE. Assim, foi possível concluir que o treinamento realizado com o Grupo de Intervenção foi eficaz no sentido de promover uma melhoria nas habilidades metacognitivas em matemática. / This research situates within the context of investigations that seek to contribute to the teaching of mathematics in the early grades of schooling. Investigations in this area suggest that the metacognitive skills of the individual should become the focus of instruction in the classroom. The literature on mathematics education highlights the activities of problem solving as particularly significant for the investigation of the metacognitive processes of the student. Moreover, the theme of \"verbal arithmetic problems with history\" has generated numerous articles and books about the different categories of problems, including the problems of addition / subtraction and multiplication / division. The present study aims to investigate the effectiveness of the procedure of developing metacognitive skills in mathematics, using the \"verbal arithmetic problems with the story\" in a playful learning environment. The sample is composed of 100 students from three classes of second grade of elementary school. All students were assessed using the Test of Verbal Arithmetic Problems with History (addition, subtraction, multiplication and division) and the arithmetic subtest of the Test of Educational Achievement - TDE. From the results obtained in these two evaluations, each class was divided into two halves, the first are better than the median, composed the Control Higher Group, and second, with results below the median was again divided, with one quarter composed the Control Lower Group and the other fourth the Intervention Group. This group received training in metacognitive skills in mathematics in a playful learning environment, during the second semester, a total of eleven sessions, while the other two control groups participated in activities placebo. At the end of each semester all students were re-evaluated, as in the beginning. Statistical analysis of results obtained in the TDE and Problem Arithmetic Test revealed significant differences in the two ratings for the students in the intervention group. For the two control groups, the difference was significant only in the TDE. Thus, we concluded that the training carried out with the group intervention was effective in promoting an improvement in metacognitive skills in mathematics.

<p>Students’ use of superficial reasoning seems to be a main reason for learning difficulties in mathematics. It is therefore important to investigate the reasons for this use and the components that may affect students’ mathematical reasoning development. Assessments have been claimed to be a component that significantly may influence students’ learning.</p><p>The purpose of the study in Paper 1 was to investigate the kind of mathematical reasoning that is required to successfully solve tasks in the written tests students encounter in their learning environment. This study showed that a majority of the tasks in teacher-made assessment could be solved successfully by using only imitative reasoning. The national tests however required creative mathematically founded reasoning to a much higher extent.</p><p>The question about what kind of reasoning the students really use, regardless of what theoretically has been claimed to be required on these tests, still remains. This question is investigated in Paper 2.</p><p>Here is also the relation between the theoretically established reasoning requirements, i.e. the kind of reasoning the students have to use in order to successfully solve included tasks, and the reasoning actually used by students studied. The results showed that the students to large extent did apply the same reasoning as were required, which means that the framework and analysis procedure can be valuable tools when developing tests. It also strengthens many of the results throughout this thesis. A consequence of this concordance is that as in the case with national tests with high demands regarding reasoning also resulted in a higher use of such reasoning, i.e. creative mathematically founded reasoning. Paper 2 can thus be seen to have validated the used framework and the analysis procedure for establishing these requirements.</p><p>Paper 3 investigates the reasons for why the teacher-made tests emphasises low-quality reasoning found in paper I. In short the study showed that the high degree of tasks solvable by imitative reasoning in teacher-made tests seems explainable by amalgamating the following</p><p>factors: (i) Limited awareness of differences in reasoning requirements, (ii) low expectations of students abilities and (iii) the desire to get students passing the tests, which was believed easier when excluding creative reasoning from the tests.</p><p>Information about these reasons is decisive for the possibilities of changing this emphasis. Results from this study can also be used heuristically to explain some of the results found in paper 4, concerning those teachers that did not seem to be influenced by the national tests.</p><p>There are many suggestions in the literature that high-stake tests affect practice in the classroom. Therefore, the national tests may influence teachers in their development of classroom tests. Findings from paper I suggests that this proposed impact seem to have had a limited effect, at least regarding the kind of reasoning required to solve included tasks. What about other competencies described in the policy documents?</p><p>Paper 4 investigates if the Swedish national tests have had such an impact on teacher-made classroom assessment. Results showed that impact in terms of similar distribution of tested competences is very limited. The study however showed the existence of impact from the national tests on teachers test development and how this impact may operate.</p>

Arbetets syfte är att undersöka hur högstadieelever för matematiskt resonemang. De frågeställningar som studien inriktas på är vilka lösningsstrategier elever väljer då de resonerar matematiskt såväl som vad  det finns för skillnader och likheter mellan de yngre elevernas lösningar och de äldre elevernas lösningar. Undersökningen genomfördes i två klasser, den ena i årskurs 8 och den andra i årskurs 9, på en grundskola. Eleverna fick lösa uppgifter, vilka uppmanade dem att föra matematiskt resonemang, individuellt. Resultatet av studien visar att majoriteten av undersökta elever har valt att resonera deduktivt. Jämförelsen av elevers lösningar i två årskurser visar att årskurs 9 elevers resonemangsföring präglas av större förtrogenhet med den algebraiska demonstrationen. Resultatet visar även att elever med högre kunskaper om algebra oftare visar benägenheter till att vidaregeneralisera de givna påståendena. / The purpose of this study is to examine secondary school students’ strategies of reasoning. The study inquires into which strategies students choose when reasoning mathematically as well as differences and similarities between the younger students’ solutions and the older students’ solutions. The study was conducted in two classes, in years 8 and 9 respectively, at a secondary school. The students were asked to solve tasks, which encouraged them to reason mathematically, on individual basis. The study revealed that the majority of students had chosen to reason deductively. The comparison of students’ presented answers in two years showed that the ninth-graders’ solutions are characterized of greater skill when it comes to algebraic demonstrations. The results of the study also reveal that students with stronger algebraic abilities attempt more often to generalize the given mathematical statements further.

Students’ use of superficial reasoning seems to be a main reason for learning difficulties in mathematics. It is therefore important to investigate the reasons for this use and the components that may affect students’ mathematical reasoning development. Assessments have been claimed to be a component that significantly may influence students’ learning. The purpose of the study in Paper 1 was to investigate the kind of mathematical reasoning that is required to successfully solve tasks in the written tests students encounter in their learning environment. This study showed that a majority of the tasks in teacher-made assessment could be solved successfully by using only imitative reasoning. The national tests however required creative mathematically founded reasoning to a much higher extent. The question about what kind of reasoning the students really use, regardless of what theoretically has been claimed to be required on these tests, still remains. This question is investigated in Paper 2. Here is also the relation between the theoretically established reasoning requirements, i.e. the kind of reasoning the students have to use in order to successfully solve included tasks, and the reasoning actually used by students studied. The results showed that the students to large extent did apply the same reasoning as were required, which means that the framework and analysis procedure can be valuable tools when developing tests. It also strengthens many of the results throughout this thesis. A consequence of this concordance is that as in the case with national tests with high demands regarding reasoning also resulted in a higher use of such reasoning, i.e. creative mathematically founded reasoning. Paper 2 can thus be seen to have validated the used framework and the analysis procedure for establishing these requirements. Paper 3 investigates the reasons for why the teacher-made tests emphasises low-quality reasoning found in paper I. In short the study showed that the high degree of tasks solvable by imitative reasoning in teacher-made tests seems explainable by amalgamating the following factors: (i) Limited awareness of differences in reasoning requirements, (ii) low expectations of students abilities and (iii) the desire to get students passing the tests, which was believed easier when excluding creative reasoning from the tests. Information about these reasons is decisive for the possibilities of changing this emphasis. Results from this study can also be used heuristically to explain some of the results found in paper 4, concerning those teachers that did not seem to be influenced by the national tests. There are many suggestions in the literature that high-stake tests affect practice in the classroom. Therefore, the national tests may influence teachers in their development of classroom tests. Findings from paper I suggests that this proposed impact seem to have had a limited effect, at least regarding the kind of reasoning required to solve included tasks. What about other competencies described in the policy documents? Paper 4 investigates if the Swedish national tests have had such an impact on teacher-made classroom assessment. Results showed that impact in terms of similar distribution of tested competences is very limited. The study however showed the existence of impact from the national tests on teachers test development and how this impact may operate.

This paper is about problem solving. One common result of previous research is that pupils’ learning is greater in classrooms where they are engaged in problem solving tasks that are cognitively demanding. Therefore the purpose of this study is to analyse the didactic choices two teachers make when they are using mathematical problem solving in their classrooms. One teacher in Sweden who teaches pupils’ mathematics in grade 4 and one teacher in Malta that teaches mathematics in Year 6 takes part in the study. The pupils in both Sweden and in Malta are 10 years of age. Three questions directed this study: How cognitively challenging are problem solving tasks in the Swedish and Maltese mathematics classroom? How does a teacher in Sweden and a teacher in Malta define and exemplify the concept of problem solving? What similarities and differences are there between the Swedish and Maltese mathematical classrooms regarding two teachers’ didactic choice of mathematical problem solving, and the teachers working methods around these tasks? The study was based on observations and interviews. The purpose of the interviews was to get answers to how the teachers interpret the concept of problem solving and how they relate to it. The aim of the observations was to see how the teachers involved incorporate problem solving during the maths lessons and how the tasks carried out are cognitive. The results show that in the Swedish and the Maltese mathematical classroom, the teachers give reference to and work with cognitive demanding tasks that require a connection to conceptual and procedural knowledge. The differences between the two observed classrooms are significant. The teacher in Sweden provides space for discussion and creative reasoning and pupils often solve mathematical problems individually. The teacher in Malta provides students with mathematical skills of how pupils can solve mathematical problems using different models. The pupils in Malta are often working on problems together, with the teacher in a dialog.

A presente pesquisa se situa no contexto das investigações que buscam contribuir para o ensino de matemática nas séries iniciais da escolaridade. As investigações nesta área sugerem que as habilidades metacognitivas do indivíduo devam se tornar o foco da instrução em sala de aula. A literatura sobre educação matemática destaca as atividades de resolução de problemas como especialmente significativas para a investigação dos processos metacognitivos do aluno. Além disto, o tema problemas aritméticos verbais com história tem gerado numerosos artigos e livros que analisam as diversas categorias de problemas existentes, entre eles os problemas de adição/subtração e de multiplicação/divisão. Assim, o presente trabalho se propõe a investigar a eficácia de procedimento de desenvolvimento de habilidades metacognitivas em matemática, utilizando-se de problemas aritméticos verbais com história em um ambiente lúdico de aprendizagem. A amostra foi composta com 100 alunos de três turmas de segunda série do Ensino Fundamental. Todos os alunos foram avaliados por meio da Prova de Problemas Aritméticos Verbais com História (de adição, subtração, multiplicação e divisão) e o Subteste de Aritmética do Teste de Desempenho Escolar TDE. A partir dos resultados obtidos nestas duas avaliações, cada classe foi dividida em duas metades, a primeira, com resultados superiores à mediana, compôs o grupo de controle superior, e a segunda, com resultados inferiores à mediana, foi novamente subdividida, sendo que, um quarto compôs o grupo de controle inferior e o outro quarto, o grupo de intervenção. Este grupo recebeu o treinamento em habilidades metacognitivas em matemática em um ambiente lúdico de aprendizagem, ao longo do segundo semestre letivo, num total de 11 sessões, enquanto os outros dois grupos de controle participaram de atividades placebo. No final de cada semestre letivo, todos os alunos foram novamente avaliados, como no seu início. A análise estatística dos resultados obtidos no TDE e na Prova de Problemas Aritméticos revelou diferença significativa nas duas avaliações apenas para os alunos do Grupo de Intervenção. Para os dois Grupos de Controle, a diferença foi significativa somente no TDE. Assim, foi possível concluir que o treinamento realizado com o Grupo de Intervenção foi eficaz no sentido de promover uma melhoria nas habilidades metacognitivas em matemática. / This research situates within the context of investigations that seek to contribute to the teaching of mathematics in the early grades of schooling. Investigations in this area suggest that the metacognitive skills of the individual should become the focus of instruction in the classroom. The literature on mathematics education highlights the activities of problem solving as particularly significant for the investigation of the metacognitive processes of the student. Moreover, the theme of \"verbal arithmetic problems with history\" has generated numerous articles and books about the different categories of problems, including the problems of addition / subtraction and multiplication / division. The present study aims to investigate the effectiveness of the procedure of developing metacognitive skills in mathematics, using the \"verbal arithmetic problems with the story\" in a playful learning environment. The sample is composed of 100 students from three classes of second grade of elementary school. All students were assessed using the Test of Verbal Arithmetic Problems with History (addition, subtraction, multiplication and division) and the arithmetic subtest of the Test of Educational Achievement - TDE. From the results obtained in these two evaluations, each class was divided into two halves, the first are better than the median, composed the Control Higher Group, and second, with results below the median was again divided, with one quarter composed the Control Lower Group and the other fourth the Intervention Group. This group received training in metacognitive skills in mathematics in a playful learning environment, during the second semester, a total of eleven sessions, while the other two control groups participated in activities placebo. At the end of each semester all students were re-evaluated, as in the beginning. Statistical analysis of results obtained in the TDE and Problem Arithmetic Test revealed significant differences in the two ratings for the students in the intervention group. For the two control groups, the difference was significant only in the TDE. Thus, we concluded that the training carried out with the group intervention was effective in promoting an improvement in metacognitive skills in mathematics.

Students only learn what they get the opportunity to learn. This means, for example, that students do not develop their reasoning- and problem solving competence unless teaching especially focuses on developing these competencies. Despite the fact that it has for the last 20 years been pointed out the need for a reform-oriented mathematics education, research still shows that in Sweden, as well as internationally, an over-emphasis are placed on rote learning and procedures, at the cost of promoting conceptual understanding. Mathematical understanding can be separated into procedural and conceptual understanding, where conceptual understanding can be connected to a reform oriented mathematics education. By developing a reasoning competence conceptual understanding can also be developed. This thesis, which deals with students’ opportunities to learn to reason mathematically, includes three studies (with data from Swedish upper secondary school, year ten and mathematics textbooks from twelve countries). These opportunities have been studied based on a textbook analysis and by studying students&amp;#039; work with textbook tasks during normal classroom work. Students’ opportunities to learn to reason mathematically have also been studied by examining the relationship between students&amp;#039; reasoning and their beliefs. An analytical framework (Lithner, 2008) has been used to categorise and analyse reasoning used in solving tasks and required to solve tasks.Results support previous research in that teaching and mathematics textbooks are not necessarily in harmony with reform-oriented mathematics teaching. And that students indicated beliefs of insecurity, personal- and subject expectations as well as intrinsic- and extrinsic motivation connects to not using mathematical reasoning when solving non-routine tasks. Most commonly students used other strategies than mathematical reasoning when solving textbook tasks. One common way to solve tasks was to be guided, in particular by another student. The results also showed that the students primarily worked with the simpler tasks in the textbook. These simpler tasks required mathematical reasoning more rarely than the more difficult tasks. The results also showed a negative relationship between a belief of insecurity and the use of mathematical reasoning. Furthermore, the results show that the distributions of tasks that require mathematical reasoning are relatively similar in the examined textbooks across five continents.Based on the results it is argued for a teaching based on sociomathematical norms that leads to an inquiry based teaching and textbooks that are more in harmony with a reform-oriented mathematics education. Elever kan bara lära sig de det de får möjlighet att lära sig. Detta innebär till exempel att elever inte utvecklar sin resonemangs- och problemlösningsförmåga i någon större utsträckning om inte deras undervisning fokuserar på just dessa förmågor. Forskning, nationellt och internationellt visar att det finns en överbetoning på utantillinlärning och på procedurer. Detta verkar ske på bekostnad av en konceptuell förståelse, trots att det under 20 års tid pekats på behovet av en reforminriktad matematikundervisning. Matematisk förståelse kan delas in i procedurell- och konceptuell förståelse där en konceptuell förståelse kan kopplas till en reforminriktad matematikundervisning. Genom att utveckla förmågan att resonera matematiskt utvecklas också den konceptuella förståelsen. Denna avhandling, som inbegriper tre studier (med empiri från gymnasiet år ett och matematikläroböcker från tolv länder) behandlar elevers möjlighet att lära sig att resonera matematiskt. Dessa möjligheter har studerats utifrån att undersöka vilka möjligheter läroboken ger att lära sig matematiska resonemang, dels via en läroboksanalys och dels genom att studera elevers arbete med läroboksuppgifter i klassrumsmiljö. Elevers möjligheter att lära sig att resonera matematiskt har också studerats genom att undersöka relationen mellan elevers matematiska resonemang och deras uppfattningar om matematik. Ett analytiskt ramverk (Lithner, 2008) har används för att kategorisera och analysera resonemang som använts för att lösa uppgifter och som behövs för att lösa en uppgift.Resultaten från studierna har givit stöd åt tidigare forskning vad gäller att undervisning och läroböckerna inte nödvändigtvis harmonierar med en reforminriktad matematikundervisning. Och att elever har uppfattningar om matematik som bygger på osäkerhet, förväntan på ämnet och sin egen förmåga samt motivation och att dessa uppfattningar delvis kan kopplas till att eleverna inte använder matematiska resonemang för att försöka lösa icke-rutinuppgifter. Det vanligaste sättet att lösa läroboksuppgifter var att välja andra strategier än att använda sig av matematiska resonemang. Ett vanligt sätt att lösa uppgifter var att låta sig guidas, av främst en annan elev. Eleverna arbetade framförallt med de enklare uppgifterna i läroböckerna. Bland dessa enklare uppgifter var det mer sällsynt med uppgifter som krävde matematiska resonemang för att lösas relativt de svårare uppgifterna. Resultaten visade även att det fanns en negativ relation mellan en uppfattning av osäkerhet hos elever och ett användande av matematiska resonemang. Resultaten visade vidare att fördelningen av uppgifter som krävde matematiska resonemang var relativt lika i alla undersökta läroböcker från fem världsdelar.Utifrån resultaten argumenteras för en förändrad undervisning mot en undersökande undervisning och läroböcker som är mer i harmoni med en reforminriktad matematikundervisning. / Elever kan bara lära sig de det de får möjlighet att lära sig. Detta innebär till exempel att elever inte utvecklar sin resonemangs- och problemlösningsförmåga i någon större utsträckning om inte deras undervisning fokuserar på just dessa förmågor. Forskning, nationellt och internationellt visar att det finns en överbetoning på utantillinlärning och på procedurer. Detta verkar ske på bekostnad av en konceptuell förståelse, trots att det under 20 års tid pekats på behovet av en reforminriktad matematikundervisning. Matematisk förståelse kan delas in i procedurell- och konceptuell förståelse där en konceptuell förståelse kan kopplas till en reforminriktad matematikundervisning. Genom att utveckla förmågan att resonera matematiskt utvecklas också den konceptuella förståelsen. Denna avhandling, som inbegriper tre studier (med empiri från gymnasiet år ett och matematikläroböcker från tolv länder) behandlar elevers möjlighet att lära sig att resonera matematiskt. Dessa möjligheter har studerats utifrån att undersöka vilka möjligheter läroboken ger att lära sig matematiska resonemang, dels via en läroboksanalys och dels genom att studera elevers arbete med läroboksuppgifter i klassrumsmiljö. Elevers möjligheter att lära sig att resonera matematiskt har också studerats genom att undersöka relationen mellan elevers matematiska resonemang och deras uppfattningar om matematik. Ett analytiskt ramverk (Lithner, 2008) har används för att kategorisera och analysera resonemang som använts för att lösa uppgifter och som behövs för att lösa en uppgift. Resultaten från studierna har givit stöd åt tidigare forskning vad gäller att undervisning och läroböckerna inte nödvändigtvis harmonierar med en reforminriktad matematikundervisning. Och att elever har uppfattningar om matematik som bygger på osäkerhet, förväntan på ämnet och sin egen förmåga samt motivation och att dessa uppfattningar delvis kan kopplas till att eleverna inte använder matematiska resonemang för att försöka lösa icke-rutinuppgifter. Det vanligaste sättet att lösa läroboksuppgifter var att välja andra strategier än att använda sig av matematiska resonemang. Ett vanligt sätt att lösa uppgifter var att låta sig guidas, av främst en annan elev. Eleverna arbetade framförallt med de enklare uppgifterna i läroböckerna. Bland dessa enklare uppgifter var det mer sällsynt med uppgifter som krävde matematiska resonemang för att lösas relativt de svårare uppgifterna. Resultaten visade även att det fanns en negativ relation mellan en uppfattning av osäkerhet hos elever och ett användande av matematiska resonemang. Resultaten visade vidare att fördelningen av uppgifter som krävde matematiska resonemang var relativt lika i alla undersökta läroböcker från fem världsdelar. Utifrån resultaten argumenteras för en förändrad undervisning mot en undersökande undervisning och läroböcker som är mer i harmoni med en reforminriktad matematikundervisning.

Läroplanen, Lgr 11, vilar på en sociokulturell syn på lärande där lärande genom interaktion och kommunikation är en naturlig del. Kommunikationens men också de matematiska resonemangens betydelse för elevers matematiklärande råder samstämmighet kring inom forskningen. En viktig del i att skapa givande samtal är lärares frågor. Syftet med studien är att undersöka vilka slags frågor matematiklärare använder sig av vid problemlösningslektioner och i vilken uträckning frågorna leder till så kallade utforskande samtal. Undersökningen genomfördes genom strukturerade observationer via ett egenkonstruerat analysverktyg utifrån teorier om olika slags frågor och utforskande samtal. Studien visar att de vanligast förekommande frågorna är de som gäller fakta eller procedur och att de utforskande samtalen är sparsamt förekommande. Vidare synliggörs att det finns stora skillnader lärare emellan gällande vilka frågor som ställs och vad de ger upphov till samt att förekomsten av både frågor som uppmuntrar till resonemang och utforskande interaktion är vanligare vid grupparbete än i helklassinteraktion. Dominansen av fakta och procedurfrågor visar att traditionella frågemönster består, även om en förändring eventuellt kan skönjas. Studien har gett en inblick i förekommande frågor vid matematikundervisning samt huruvida de ger upphov till utforskande samtal. Därmed har kunskap skapats om elevernas möjlighet att kommunicera och resonera matematiskt med utgångspunkt i lärares frågor. Studien har även bidragit till att lärares frågor och hantering av elevsvar uppmärksammas, något som kan leda till att djupare matematiska diskussioner blir mer vanligt förekommande / The Swedish curriculum is grounded in a sociocultural view of learning where learning through interaction and communication comes naturally. Research unanimously emphasizes the role of communication and mathematical reasoning for students’ learning in mathematics. Teachers’ questions are important when creating fruitful discussions. The purpose of this study is therefore to examine what kind of questions mathematics teachers pose in problem-solving lessons and the extent to which the questions lead to so-called exploratory talks. The study was conducted through structured observations via a self-constructed analysis tool which was based on theories regarding questions and exploratory talks. The study shows that the most common questions are about facts or procedures and that exploratory talks sparsely occur. Furthermore, there are great differences between teachers regarding what kind of questions are asked and what kind of communication these generate. Questions that encourage reasoning and exploratory talks are more common in group work than in whole-class interaction. The domination of factual and procedure questions shows that traditional question-patterns persist, although a change may be discernible. The study has provided an insight into the kind of questions mathematics teachers pose in their teaching and whether they generate exploratory talks. Thus, knowledge has been created about students’ possibilities to communicate and reason mathematically based on teachers’ questions. In addition, this study has drawn attention to teachers’ questions and handling of students’ answers, which can lead to deeper mathematical discussions becoming more common.