This thesis explores two aspects of mathematical reasoning: affect and gender. I started by looking at the reasoning of upper secondary students when solving tasks. This work revealed that when not guided by an interviewer, algorithmic reasoning, based on memorising algorithms which may or may not be appropriate for the task, was predominant in the students reasoning. Given this lack of mathematical grounding in students reasoning I looked in a second study at what grounds they had for different strategy choices and conclusions. This qualitative study suggested that beliefs about safety, expectation and motivation were important in the central decisions made during task solving.  But are reasoning and beliefs gendered? The third study explored upper secondary school teachers conceptions about gender and students mathematical reasoning. In this study I found that upper secondary school teachers attributed gender symbols including insecurity, use of standard methods and imitative reasoning to girls and symbols such as multiple strategies especially on the calculator, guessing and chance-taking were assigned to boys. In the fourth and final study I found that students, both male and female, shared their teachers view of rather traditional feminities and masculinities. Remarkably however, this result did not repeat itself when students were asked to reflect on their own behaviour: there were some discrepancies between the traits the students ascribed as gender different and the traits they ascribed to themselves. Taken together the thesis suggests that, contrary to conceptions, girls and boys share many of the same core beliefs about mathematics, but much work is still needed if we should create learning environments that provide better opportunities for students to develop beliefs that guide them towards well-grounded mathematical reasoning.

This dissertation consists of four articles and a summary. The main focus of the studies is students' explorations in upper secondary school mathematics. In the first study the central research question was to find out if the students could learn something difficult by using the graphing calculator. The students were working with questions connected to factorisation of quadratic polynomials, and the factor theorem. The results indicate that the students got a better understanding for the factor theorem, and for the connection between graphical and algebraical representations. The second study focused on a the last part of an investigation, the verification of an idea or a conjecture. Students were given three conjectures and asked to decide if they were true or false, and also to explain why the conjectures were true or false. In this study I found that the students wanted to use rather abstract mathematics in order to verify the conjectures. Since the results from the second study disagreed with other research in similar situations, I wanted to see what Swedish teachers had to say of the students' ways to verify the conjectures. The third study is an interview study where some teachers were asked what expectations they had on students who were supposed to verify the three conjectures from the second study. The teachers were also confronted with examples from my second study, and asked to comment on how the students performed. The results indicate that teachers tend to underestimate students' mathematical reasoning. A central focus to all my three studies is explorations in mathematics. My fourth study, a revised version of a pilot study performed 1998, concerns exactly that: how students in upper secondary school explore a mathematical concept. The results indicate that the students are able to perform explorations in mathematics, and that the graphing calculator has a potential as a pedagogical aid, it can be a support for the students' mathematical reasoning.

The overall problem that formed the basis for this thesis is that students get limited opportunity to develop their mathematical reasoning ability while, at the same time, there are dynamic mathematics software available which can be used to foster this ability. The aim of this thesis is to contribute to knowledge in this area by focusing on task design in a dynamic software environment and by studying the reasoning that emerges when students work on tasks in such an environment. To analyze students’ mathematical reasoning, a new analytical tool was developed in the form of an expanded version of Toulmin’s model. Results from one of the studies in this thesis show that exploratory tasks in a dynamic software environment can promote mathematical reasoning in which claims are formulated, examined and refined in a cyclic process. However, this reasoning often displayed a lack of the more conceptual, analytic and explanatory reasoning normally associated with mathematics. This result was partly confirmed by another of the studies. Hence, one key question in the thesis has been how to design tasks that promote conceptual and explanatory reasoning. Two articles in the thesis deal with task design. One of them suggests a model for task design with a focus on exploration, explanation, and generalization. This model aims, first, to promote semantic proof production and then, after the proof has been constructed, to encourage further generalizations. The other article dealing with task design concerns the design of prediction tasks to foster student reasoning about exponential functions. The research process pinpointed key didactical variables that proved crucial in designing these tasks. / Baksidestext Det övergripande problem som legat till grund för denna avhandling är att elever får begränsad möjlighet att utveckla sin resonemangsförmåga samtidigt som det finns dynamiska matematikprogram som kan utnyttjas för att stimulera denna förmåga. Syftet med avhandlingen är att bidra till den samlade kunskapen inom detta problemområde, dels genom att fokusera på design av uppgifter i en lärandemiljö med dynamiska matematikprogram och dels genom att studera och karakterisera de resonemang som utvecklas när elever jobbar med olika uppgifter i denna miljö. För att analysera elevernas resonemang utvecklades ett nytt analysverktyg i form av en utökad version av Toulmins modell. Resultat från en av studierna i avhandlingen visar att dynamiska matematikprogram i kombination med utforskande uppgifter kan stimulera till matematiska resonemang där hypoteser formuleras, undersöks och förfinas i en cyklisk process. Samtidigt visar samma studie att de resonemang som utvecklas i stor utsträckning saknar matematiskt grundade förklaringar. Detta resultat bekräftas till viss del av ytterligare en studie.  Frågan hur uppgifter bör designas för att främja matematiskt grundade resonemang har därför varit central i avhandlingen. Två av artiklarna behandlar uppgiftsdesign, men utifrån olika utgångspunkter.

This thesis explores two aspects of mathematical reasoning: affect and gender. I started by looking at the reasoning of upper secondary students when solving tasks. This work revealed that when not guided by an interviewer, algorithmic reasoning, based on memorising algorithms which may or may not be appropriate for the task, was predominant in the students reasoning. Given this lack of mathematical grounding in students reasoning I looked in a second study at what grounds they had for different strategy choices and conclusions. This qualitative study suggested that beliefs about safety, expectation and motivation were important in the central decisions made during task solving.  But are reasoning and beliefs gendered? The third study explored upper secondary school teachers conceptions about gender and students mathematical reasoning. In this study I found that upper secondary school teachers attributed gender symbols including insecurity, use of standard methods and imitative reasoning to girls and symbols such as multiple strategies especially on the calculator, guessing and chance-taking were assigned to boys. In the fourth and final study I found that students, both male and female, shared their teachers view of rather traditional feminities and masculinities. Remarkably however, this result did not repeat itself when students were asked to reflect on their own behaviour: there were some discrepancies between the traits the students ascribed as gender different and the traits they ascribed to themselves. Taken together the thesis suggests that, contrary to conceptions, girls and boys share many of the same core beliefs about mathematics, but much work is still needed if we should create learning environments that provide better opportunities for students to develop beliefs that guide them towards well-grounded mathematical reasoning.

No description available.

nÃo hà / A formaÃÃo inicial no Curso de Pedagogia da Faculdade de EducaÃÃo (FACED) da Universidade Federal do Cearà (UFC) envolve a compreensÃo por parte do aluno (futuro-professor) nÃo somente dos conteÃdos a serem trabalhados com seus alunos, mas tambÃm sobre como utilizar as prÃticas pedagÃgicas que melhor facilitem à transposiÃÃo didÃtica desses conhecimentos. Diante de um ensino trabalhado por vezes repleto de regras a serem memorizadas e sem qualquer significaÃÃo para o aluno, em particular dos conteÃdos matemÃticos, objetivamos analisar as estratÃgias matemÃticas apresentadas pelos alunos do curso de Pedagogia, visando a classificaÃÃo de problemas matemÃticos no que diz respeito aos raciocÃnios: (i) concreto; (ii) grÃfico (iii) aritmÃtico; e, (iv) algÃbrico, com vistas à construÃÃo e nÃo apenas à memorizaÃÃo de fatos e fÃrmulas, levantando questÃes relativas à formaÃÃo do professor de matemÃtica. Esta pesquisa de natureza qualitativa se deu, em parte, com: (a) observaÃÃo das aulas de matemÃtica; e, (b) realizaÃÃo de um conjunto de problemas matemÃticos, essas atividades foram desempenhadas durante a disciplina de ensino de matemÃtica na turma do Curso de Pedagogia no semestre de 2016.1. Anteriormente, tambÃm buscamos em livros, teses e periÃdicos, pesquisas sobre ensinar e aprender matemÃtica nos anos iniciais do ensino fundamental. Coletadas as informaÃÃes, iniciamos as anÃlises sobre as estratÃgias utilizadas pelos alunos na resoluÃÃo dos problemas categorizamos as respostas a partir da classificaÃÃo feita por Johannot (1947) quanto ao raciocÃnio matemÃtico. Os resultados indicam para novos e melhores espaÃos de reflexÃo tanto na formaÃÃo inicial de professores como na atuaÃÃo direta com os alunos da EducaÃÃo BÃsica, em especial nos anos iniciais do ensino fundamental. Mostramos com esta pesquisa a relevÃncia no que se refere a compreensÃo de como os alunos de Pedagogia constroem suas estratÃgias de resoluÃÃo de problemas e as consequÃncias disso para o ensino dos conteÃdos matemÃticos. Embora que o objeto de estudo desta pesquisa seja o entendimento das estratÃgias apresentadas pelos alunos sobre o raciocÃnio matemÃtico, para este entendimento tivemos como forte apoio as observaÃÃes Ãs aÃÃes mediadas pela professora, em particular, na construÃÃo de novos saberes por meio da SequÃncia Fedathi. / The initial training in the Pedagogy Course of the Faculty of Education (FACED) of the Federal University of Cearà (UFC) involves an understanding by the student (future-teacher) not only of contents to be worked with his students, but also pedagogical practices that better facilitate the didactic transposition of knowledge. Faced with a teaching that is sometimes filled with rules to be memorized and with no meaning for the student, in particular the mathematical contents, we aim to analyze how mathematical strategies presented by the students of the Pedagogy course, aiming at a classification of mathematical problems not referring to reasoning : (i) concrete; (ii) graph (iii) arithmetic; and (iv) algebraic, with a view to the construction and not only the memorization of facts and formulas, raising questions related to the formation of the mathematics teacher. This research of a qualitative nature occurred in part with: (a) observation of mathematics classes; and (b) accomplishment of a set of mathematical problems, activities with the unemployed during a course of mathematics teaching in the course of the Course of Pedagogy semester of 2016.1. Previously, we also searched in books, theses and periodicals, research on teaching and learning math in the early years of elementary school. Collected as information, we began as analyzes on how strategies for students in solving the problems categorized as answers from the series made by Johannot (1947) regarding mathematical reasoning. The results indicate new and better spaces for reflection both in initial teacher training and in updating with students of Basic Education, especially in the initial years of elementary education. We show with this research the relevance in terms of an understanding of how Pedagogy students build their problem solving strategies and as consequences for the teaching of mathematical contents. What is the object of study of this research is the understanding of the strategies presented by the students on the mathematical reasoning, for the understanding as the strong support as observations to the actions mediated by teacher, in particular, in the construction of new knowledge through the Fedathi Sequence.

Det är en etablerad åsikt att en stor del av matematikundervisningen idag har sin utgångspunkt i läroböcker, och därför är det viktigt att granska läroböckernas innehåll eftersom det inte längre existerar någon statlig kontroll av läromedel i Sverige. Studiens syfte ämnar granska ett läromedel i årskurs tre för att undersöka på vilket sätt och i vilken utsträckning läroboken låter eleverna arbeta med problemlösning. Undersökningen har utgått från en kvalitativ textanalys och en kvantitativ innehållsanalys för att granska, klassificera och kategorisera innehållet i utvalda och specifika delar i Mera Favorit matematik 3A och sedan granskat samma avsnitt för att fastställa om problemuppgifterna kräver kreativa matematiska resonemang för att lösas. Resultatet visar att det existerar tydliga skillnader mellan vad läroboken anser är problemlösning, och vad aktuell studies analys menar är problemlösning, med grund i kreativa matematiska resonemang. / It is an established opinion that a large part of mathematics education today has its main point in textbooks, so it’s important to review the content of the textbooks because there is no longer any state approval of teaching materials in Sweden. The purpose of the study is to examine a mathematical textbook in grade three to investigate the way and extent to which the textbook allows students to work with problem solving. The survey has been based on a qualitative text analysis and a quantitative content analysis to review, classify and categorize the content of selected and specific parts of “Mera Favorit Matematik 3A” and then review the same section again to determine if the problem assignments requires creative mathematical reasoning to be solved. The result shows that there are clear differences between what the textbook considers is problem solving, and what current studies analysis means are problem solving, based on creative mathematical reasoning.

No description available.

Algebra är ett grundläggande nyckelområde inom matematiken och tillsammans med matematisk resonemangsförmåga essentiellt inom skolan för elevers utveckling av matematisk färdighet. Studien analyserar gymnasieelevers kollektiva algebraiska resonemang vid algebraisk uppgiftslösning med avseende på resonemangsargumentens förankring i matematiska egenskaper, samt det algebraiska tänkande som indikeras av dessa. Resultatet visar att eleverna över lag för resonemang utifrån matematisk grund, med undantag av slutsatsernas utvärdering, och att även om uppgiftens matematiska innehåll är relevant för stimulerande av matematiskt grundat resonemang kan argumentationen indikera olika aspekter på algebraiskt tänkande beroende på elevernas lösningskonstruktion. Studien särskiljer sig gentemot tidigare forskning vilken företrädelsevis har inriktat sig på karaktärisering av individuellt resonemang i olika typer och vid studier av kollektivt resonemang har man fokuserat på yngre elever. Det studien också bidrar med är en beskrivning av hur elever kan skapa resonemang tillsammans, och identifierar en ny typ av argument, konventionella argument, kopplade till frågor om notation. Diskussionen berör bland annat vikten av förståelse för den matematik på vilken elever grundar sitt resonemang, som en aspekt att anpassa undervisning efter.

Studien grundas i att studera hur den aktuella digitaliseringen kan komma och påverka den traditionella geometriundervisningen. Syftet med denna systematiska litteraturstudie är att sammanställa befintlig forskning där eleverna använder digitala verktyg inom geometriundervisning samt att svara på frågan: Hur kan digitala verktyg användas på olika sätt för att skapa möjligheter till geometriska resonemang i matematikundervisning? Hughes (2000) ramverk som identifierar syftet med digitala verktyget ligger till grund för studiens analys. Vidare tar studien en teoretisk utgångspunkt i Lithners (2008) teori om olika matematiska resonemang som eleven använder sig av vid arbete med matematikuppgifter. Resultatet visar att beroende på om det digitala verktyget har ett ersättande, förstärkande eller transformerande syfte kommer det möjliggöra för eleverna att tillämpa olika geometriska resonemang. Detta ställer höga krav på lärares digitala kompetens då det är användningen av det digitala verktyget som påverkar elevernas kunskapsutveckling. Med denna studie kan verksamma lärare skapa en vidare förståelse för hur ett digitalt verktyg kan användas på flera olika sätt. / Studien bygger på att undersöka hur dagens digitalisering kan komma och påverka traditionell geometriundervisning. Syftet med denna systematiska litteraturstudie är att sammanställa befintlig forskning där elever använder digitala verktyg i geometriundervisning och att besvara frågan: Hur kan digitala verktyg användas på olika sätt för att skapa möjligheter för geometriska resonemang i matematikundervisningen? Hughes (2000) ramverk, som identifierar syftet med digitala verktyg, ligger till grund för studiens analys. Vidare tar studien en teoretisk utgångspunkt i Lithners (2008) teori om olika matematiska resonemangsstrategier som elever använder sig av när de arbetar med matematiska uppgifter. Resultaten visar att beroende på om det digitala verktyget har ett ersättande, förstärkande eller transformerande syfte, gör det möjligt för eleverna att tillämpa olika geometriska resonemang. Detta ställer höga krav på lärares digitala kompetens eftersom det är användningen av det digitala verktyget som påverkar elevernas kunskapsutveckling. Med denna studie kan verksamma lärare få en bredare förståelse för hur ett digitalt verktyg kan användas på olika sätt.