This study focused on the evaluative listening practices of four teachers who participated in an algebra professional development involving lesson study. This instrumental case study operationalizes the enactment of teacher listening followed by teacher questions and responses to define listening-to-question. Also, questioning-to-listen is operationalized as the enactment of purposefully posing questions to posture oneself to listen to students' mathematical thinking. Because of the tacit aspect of teacher listening and the visibility of teacher questioning, interrelating listening and questioning affords teachers an accessible point of entry into developing listening practices. In response to participants wondering as to when evaluative listening is appropriate in the mathematics classroom, this study discusses six instances of teaching excerpts along a continuum of listening orientations from directive to observational to responsive. The results indicate positive aspects of evaluative listening towards an observational and responsive listening stance. Results of the study also confirm a reliance on low-order gathering information questions as the predominant type of teacher question posed in mathematics teaching. This study reveals the necessity of contextualizing teacher questions to inform appropriate uses of evaluative listening. Future professional development should consider emphasizing positive aspects of evaluative listening in mathematics teaching.

Orientador: Marcelo Reicher Soares / Resumo: O objetivo desse trabalho é apresentar, de forma rigorosa, como a matemática aborda o conceito de infinito e propor uma sequência de atividades para que o professor possa explorar esse tema com seus alunos de forma inovadora e estimulante. Muito do que é compreendido acerca do infinito se deve às ideias desenvolvidas por Georg Cantor, que estabeleceu a teoria dos números cardinais transfinitos, gerando uma série de resultados surpreendentes, que serão apresentados ao longo dessa dissertação. Cantor descobriu que existem diversos tipos de infinito e definiu critérios para classificá-los e compará-los. Para compreender esta teoria, é fundamental recordar os conceitos básicos da teoria de conjuntos e funções. Além disso, serão apresentados formalmente os números naturais através dos axiomas de Peano, bem como suas operações e propriedades. A partir deste, será construído o conjunto dos números inteiros, racionais e reais. Dessa forma, será possível definir formalmente a noção de conjunto finito e infinito, bem como a noção de conjuntos enumeráveis, e não-enumeráveis, e estabelecer critérios para comparar a cardinalidade de tais conjuntos. O trabalho é finalizado com a apresentação de uma proposta didática voltada para os alunos de ensino médio, sustentado no relato de duas experiências de sua aplicação. O tema é abordado utilizando atividades diferenciadas e fundamentadas no cotidiano, visando com isto contribuir para que os alunos apresentem um maior interesse e uma participaçã... (Resumo completo, clicar acesso eletrônico abaixo) / Abstract: The aim of this work is to present in a rigorous way how mathematics approaches the concept of the in nite and to propose a sequence of activities so that the teacher can explore this theme with his students in an innovative and stimulating way. Much of what is understood about infinite is due to the ideas developed by Georg Cantor who established the theory of transfinite cardinal numbers generating a series of surprising results that will be presented throughout this dissertation. Cantor found that there are several types of infinite and defined criteria for classifying and comparing them. To understand this theory it is essential to remember the basic concepts of set and function theory. In addition natural numbers will be formally presented through Peano axioms as well as their operations and properties. From the natural numbers the sets of integers, rationals and reals will be constructed. Then it will be possible to formally de ne the notions of finite and infinite sets as well as the notions of countable and uncountable sets and establish criteria for comparing the cardinality of such sets. The work is concluded with the presentation of a didactic proposal aimed at high school students supported by the report of two experiences of its application. The theme is presented through difierent activities, based on daily life, aiming to contribute to the students to show more interest and participate more actively in the classes. / Mestre

Orientador: Ernandes Rocha de Oliveira / Resumo: Este trabalho aborda o ensino de Matemática através da metodologia de Resolução de Problemas e, mediante essa metodologia, busca investigar qual o tratamento dado à Resolução de Problemas na avaliação externa Prova Brasil, assim como a forma como essa metodologia é trabalhada na Apostila do Aluno das escolas de Ensino Fundamental de Novo Horizonte - SP. Para dar suporte ao estudo do fenômeno de interesse, o trabalho é norteado pelas premissas: I) Como a Resolução de Problemas é abordada na Apostila do Aluno no Ensino Fundamental de Novo Horizonte - SP? II) Como é abordado a Resolução de Problemas na Prova Brasil, caso seja abordada? III) Caso haja Resolução de Problemas na Apostila do Aluno, eles desenvolvem as habilidades que norteiam a Prova Brasil? Nesse sentido, a pesquisa tem caráter exploratório e descritivo, com apresentação de análises documentais, em que foram analisadas: a Apostila de Matemática do Aluno do Ensino Fundamental; a Apostila de Matemática do Professor do Município de Novo Horizonte – SP e a Matriz de Referência que norteia a avaliação Prova Brasil. Inicialmente, temos uma revisão teórica, por meio de levantamento bibliográfico do conceito de resolução de problemas e sua importância para o ensino, tendo por base autores como Onuchic, Allevato e Van de Walle. Levando em consideração as concepções dos pesquisadores citados, é discutida a noção de aprendizagem associada à essa metodologia e apresentada de forma detalhada a Prova Brasil, para entendermos com... (Resumo completo, clicar acesso eletrônico abaixo) / Abstract: This work addresses the teaching of Mathematics through the Problem Solving methodology and, through this methodology, seeks to investigate the treatment given to the Problem Solving in the external evaluation Prova Brasil, as well as the way this methodology is worked in the Student Handout of elementary schools in Novo Horizonte - SP. To support the study of the phenomenon of interest, the work is guided by the premises: I) How is the Problem Resolution approached in the Student Handout in the Elementary School of Novo Horizonte - SP? II) How is the Problem Resolution approached in the Brazil Test, if it is addressed? III) If there is problem solving in the Student Handout, do they develop the skills that guide the Brazil Test? Therefore, the research has an exploratory and descriptive character, with the presentation of documentary analyzes, in which the Mathematics Handout of the Elementary School Student, the Mathematics Handout of the Teacher of the Municipality of Novo Horizonte -SP, as well as the Reference Matrix that guides the Prova Brasil evaluation. Initially, we have a theoretical review, through a bibliographic survey of the concept of problem solving and its importance for teaching, based on authors such as Onuchic, Allevato and Van de Walle. Taking into account the conceptions of the aforementioned researchers, the notion of learning associated with this methodology is discussed, and the Prova Brasil is conceptualized to understand how data related to teachin... (Complete abstract click electronic access below) / Mestre

This diploma thesis deals with the relations between mathematics and geography in teaching at grammar school at implemented curriculum level. In theoretical part it deals with conceptions of teaching which support connection of educational content of individual subjects as well as general possibilities for realization interdisciplinary relations of mathematics or geography. There are also concrete interdisciplinary relations between mathematics and geography presented and there are identified main reasons why the developing of connection berween the mentioned subjects is important. Problems of the realization interdisciplinary relations at teaching conducts a survey by halfstructured interview with mathematics teachers. Their answers provide the view of realized curriculum and they uncover opportunities and limitations the tachers of mathematics at level of interdisciplinary connection deals with in practice. There are mainly teachers with mathematics and geography qualification, who can give their opinion on relations of these two subjects. There were prepared six lessons in this diploma thesis focused on active use of interdisciplinary relations of mathematics and geography. Then the autor taught it at grammar school and it made practical review of possibilities and restrictions of such teaching....

Studiens syfte är att beskriva hur lärare i grundskolan F-3 arbetar med de fem förmågorna som grundar sig i kursplanen för matematik i sin matematikundervisning för att stödja elevers lärande. Studien har genomförts med en kvalitativ metod i form av enskilda semistrukturerade intervjuer med sex behöriga och verksamma lärare i grundskolan F-3. Insamlade data har transkriberats och sorterats i fem kategorier som är kopplade till forskningsfrågorna. Därefter analyserades data utifrån studiens teoretiska perspektiv, som är det didaktiska kontraktet och beliefs. Det framkommer i resultatet att lärare har skilda uppfattningar och visar osäkerhet över de fem förmågorna i kursplanen för matematik. Läromedlen dominerar lärares planering och undervisning i matematik. Vidare framkom att bedömningen är övervägande summativ utifrån färdiga diagnoser och lärarna tar inte alltid hänsyn till alla fem förmågor. Studiens slutsats är att lärarna har goda kunskaper och uppfattningar om vad matematik är och hur elever lär sig matematik, men trots det har lärarna svårt att omsätta det i matematikundervisningen för att stödja alla elevers lärande. / The aim of this study is to describe how teachers in primary class up to grade 3 works with the five competencies that the curricula in mathematics are based on, to support pupils' learning. The study was conducted with individual qualitative semi-structured interviews with six primary school teachers. The collected data has been transcribed and sorted into five categories linked to the research questions. The data were then analyzed from the theoretical perspective of the study, which is the didactic contract and beliefs. The result shows that teachers have different views and show uncertainty about the five competencies in the mathematics curricula. Textbooks dominate teachers' planning and teaching in mathematics. Furthermore, it was found that the assessment is mostly summative, based on completed diagnoses where the teachers don’t always consider all five competencies. The conclusion of this study is that teachers have good knowledge and perceptions of what mathematics is and how students learn mathematics, but despite this, teachers find it difficult to convert it into mathematics education to support all pupils' learning.

Syftet med studien är att undersöka huruvida elevers attityder till matematik korrelerar med deras prestationer i ämnet, vilka faktorer som påverkar attityden, och vad lärare kan göra för att förbättra elevers attityder till matematik. Studien fokuserar på elever som befinner sig i grundskolans senare åldrar och gymnasieskolan. Vi vill besvara frågeställningarna “Vad säger tidigare forskning om hur elevers attityd till matematik påverkar deras prestationer i ämnet?”, “Vilka påverkansfaktorer lyfter tidigare forskning kring elevers attityd till matematik?” och “Vad kan urskiljas i tidigare forskning angående hur lärare kan påverka elevers attityd till matematik?”. För att få svar på dessa frågor har vi utfört en innehållsanalys av 12 artiklar som publicerats i vetenskapliga tidskrifter. Vårt resultat visar att det finns en korrelation mellan attityder till matematik och prestationer i ämnet, och vi har utifrån de vetenskapliga artiklarna lyft fram påverkansfaktorer som kategoriseras i: föräldrars attityd till matematik, personliga bakgrundskaraktäristiker, arbetssätt och lärarens inverkan. / The purpose of this essay is to investigate whether students' attitudes towards mathematics correlate with their performance in the subject, what factors influence attitude, and what teachers can do to improve students' attitudes towards mathematics. The study focuses on students between the ages of 12-18. We want to answer the questions "What does previous research say about how students' attitudes towards mathematics affect their performance in the subject?", "What influencing factors on students' attitudes towards mathematics are highlighted in previous research?" and "What can be discerned in previous research regarding how teachers can influence students' attitudes towards mathematics?”. To answer these questions we have performed a content analysis of 12 articles published in scientific journals. Our results show that there is a correlation between attitudes towards mathematics and performance in the subject, and we have found and categorized influencing factors into: parents' attitudes towards mathematics, personal background characteristics, working methods and the teacher's influence.

Matematiken är en stor del av vårt vardagsliv. Matematikundervisningen är idag till stor del läroboksbaserad och abstrakt vilket inte bidrar till långsiktiga användbara kunskaper. Då lärare vittnar om att elevernas matematiska förståelse är låg, är syftet med denna kunskapsöversikt att undersöka hur matematikläroboken kan kompletteras för att eleverna ska utveckla matematisk förståelse. Matematisk förståelse i skolan är nära förknippat med meningsskapande. Meningsskapande handlar om att se matematikens användbarhet, som är ett värdefullt redskap för individen och samhällsmedborgaren för att lösa vardagliga problem.      Detta är en kunskapsöversikt som består av åtta vetenskapliga artiklar. Artiklarna är inhämtade från en forskningsdatabas. Artiklarna är tematiserade och analyserade för att kunna svara på vår frågeställning. Vårt huvudsakliga fokus är lågstadiet.   Slutsatsen av denna kunskapsöversikt är att läroböcker ska ses som en resurs som behöver kompletteras för att uppnå matematisk förståelse. Undervisning behöver bli vardagsanknuten, erfarenhetsbaserad och ämnesöverskridande, där läraren intar en ansvarsfull ledarroll istället för att agera förmedlare av ett matematiskt innehåll. Arbetsmetoden konkret-illustrerande-abstrakt, som är en översättning från engelskans concrete – pictorial – abstract (CPA), utgår ifrån det konkreta och arbetar mot det abstrakta. Genom att undervisningen utgår ifrån CPA   och laborativt material gynnas elevernas matematiska förståelse. Det väsentliga blir att synliggöra sambandet mellan den konkreta vardagen och den abstrakta läroboksmatematiken. / Mathematics is a large part of our everyday life. Mathematics teaching today, is largely influenced by mathematics textbooks and are abstract, and does not contribute to long-term useful knowledge. Teachers testify that students mathematical understanding is low. Because of this, the aim of this research is to examine how the mathematic teaching has to change to make the students achieve this. Mathematical understanding is closely associated with meaning making. Meaning making is about seeing the usefulness of mathematics, as a useful tool for the individual and the citizen of society that can be used to solve everyday problems.      This is a knowledge overview. The result data consist of eight scientific articles that were collected from a research database. The articles were themed and reviewed to be able to answer our question. Our main focus for the conclusion is primary school.   The result shows that mathematic textbooks have to be seen as a resource that needs to be supplemented to achieve mathematical understanding. The teaching has to become everyday connected, experience-based and interdisciplinary, where the teacher is taking a responsible leading role, instead of being an intermediary of the mathematical content. The working method concrete - pictorial - abstract (CPA), starts with the concrete and work towards the abstract. Through teaching with CPA and manipulatives, befriends the student´s mathematical understanding. The key point is to make the connection between the concrete everyday life, and the abstract textbook.

Denna studie fokuserar på problemet där flesta elever tappar intresse, motivation och lust att lära sig matematik eftersom de upplever att det är ett svårt ämne. Syftet med detta arbete å andra sidan är att undersöka om hur matematikundervisning som har vardagliga samband påverkar elevernas lärande. Studien vill också belysa hur lärare arbetar så att undervisningen i matematik kan bidra till att elever utvecklar intresse för matematik och förtroende för deras förmåga att använda matematik i olika sammanhang. Resultaten är baserade på en analys av elevers och lärares enkäter. Det finns tjugotvå (22) elever och fyra (4) legitimerade matematiklärare som deltog i denna studie. Eleverna går i årskurs sex (6) och lärarna undervisar olika klasser från årskurs ett (1) till sex (6). Enkäten behandlade olika frågor bland annat; elevernas intresse, motivation och lärande när lektioner har vardagliga samband.När det gäller motivation, 19 elever svarade att de är mer motiverade att lösa matematiska uppgifter när de använder eller arbetar med material eller något som hjälper dem att lösa uppgifterna. Med samma siffror svarade å andra sidan att de tappar lusten eller motivation när lektionerna inte är kopplade till saker som de vet eller gillar. När det gäller inlärning/ undervisning gick både elever och lärare i samma håll. Alla 22 eleverna svarade att de lär sig bättre när läraren använder vardagliga samband för att lösa eller förklara matematiska problem eller begrepp medan 3 lärare svarade att dem anser att deras elever lär sig matematik bäst när lektionerna är kopplade till saker de vet eller gillar. Resultaten av denna studie visar emellertid att elevernas intresse, motivation och prestation i matematik ökar när lektioner är kopplade till vardagliga samband.Nyckelord: intresse, matematikundervisning, motivation, lärande, vardagsanknytning. / This study focuses on the problem where most pupils lose interest, motivation and the desire to learn mathematics because they perceive it to be a difficult subject. The purpose of this thesis on the other hand is to investigate how mathematics teaching that has everyday life connections affects pupils' learning. The study also wants to shed light on how teachers work so that the teaching of mathematics will contribute to pupils developing interest in mathematics and confidence in their ability to use mathematics in different contexts. The results are based on an analysis of pupils’ and teachers’ questionnaires. There are twenty (22) pupils and four (4) licensed mathematics teachers that took part in this study. The pupils are in grade six (6) and the teachers teach different grades from grade one (1) to six (6). The questionnaires addressed different issues among them; pupils' interest, motivation and learning using everyday connections. When it comes to motivation, 19 pupils answered that they are more motivated to solve math tasks when they use or work with materials or something that helps them solve the tasks. The same number on the other hand answered that they lose the desire or motivation when the lessons are not connected to things they know or like. On learning/teaching, both pupils and teachers went into the same direction. All the 22 pupils answered that they learn better when the teacher uses everyday connections to solve or explain mathematical problems or concepts while 3 out of 4 teachers answered that their pupils learn mathematics best when the lessons are connected to things they know or like. The results of this study however show that pupils' interest, motivation and performance in mathematics increases when lessons are linked to everyday connections.Keywords: everyday connection, interest, mathematics teaching, motivation, understanding.

In recent years, schools have become increasingly multilingual, which must considered in mathematics teaching as the proportion of multilingual pupils is constantly increasing. Teachers ideas about teaching in multilingual mathematics classrooms, towards multilingual students and multilingualism are thus crucial to the students’ school achievements and identity information. Therefore, in-service teachers in compulsory school need to acknowledge multilingual pupils and language and cultural resources. Mathematics teaching needs to be adapted to a linguistic and cultural diversity in the classroom, where the students' different languages are used, and are seen as good resources.The method for collecting data relevant for this study: qualitative studies, which includes interviews and observations with active teachers in compulsory school. Relevant scientific articles were found in various databases, secondary sources, and also consultation with our supervisors and other course mates. However, the basis for this study is based on the empirical material that emerged in the qualitative interviews and observation, linked to relevant research.The result in this study are designed to provide knowledge about which aspects, teachers consider to be important, and knowledge development for multilingual students in mathematics education.

Syftet med denna kvalitativa studie är att fördjupa kunskaper om grundskollärares användning av olika representationsformer och lärarnas syn på hur de olika representationsformerna kan hjälpa elever i årskurs 1–3 att utveckla sin problemlösningsförmåga. Representationsformer såsom bilder, konkret material, ord och symboler kan användas av eleverna för att förstå problemet, komma fram till och visa sina lösningar. I studien har vi genomfört semistrukturerade intervjuer med åtta lärare i årskurserna 1–3. Intervjusvaren har tolkats utifrån ett sociokulturellt perspektiv där bland annat mediering och den proximala utvecklingszonen ingår. Studiens resultat visar på ett samband mellanatt elevernas val och användning av olika representationsformer och kan påverkas av lärarens egen användning av representationsformer. Lärarna anger att eleverna med hjälp av representationsformer kan synliggöra sina tankar och på så vis förstå ett problem och dess lösning. Resultatet visar att lärarna använder sig utav flera olika representationsformer men för det mesta bara en eller två åt gången istället för att regelbundet kombinera dem. / The purpose of this qualitative study is to investigate and deepen the understanding of primary school teachers' view on their use of different representations and in what way the different representations can aid students in grades 1-3 in developing their problem-solving ability. Representations such as pictures, concrete objects, words and symbols can be used by the students understand the problem, reach and show their solutions. In this study we have conducted semi-structured interviews with eight grade 1-3 teachers. The interview responses have been interpreted with a sociocultural perspective where, among other things, mediation and the zone of proximal development are included. The result of the study shows that the students' choice and use of different representations can be affected by the teacher's own use of representations. The result of the study showed a correlation between the students' choice and use of different representations and the teacher's own use of representations. The teachers state that the students with the aid of representations can visualise their thoughts and with that understand a problem and its solution. The results show that the teachers use several representations but for the most part only one or two at the time instead of regularly combining them.