<p>Detta arbete är en textanalys av hur några svenska läroböcker i matematik introducerar algebra speglat i van Hieles teorier om tankenivåer vid inlärning. Van Hieles teorier poängterar språket som kunskapsbärare i matematik vilket går som en röd tråd genom analysen. Generellt börjar läroböckerna på van Hieles tankenivå 3. Enligt van Hieles teorier borde undervisningen i algebra börja på nivå 1, vilket då blir lärarens uppgift att göra utan stöd av matematikboken. Förslag på arbetssätt för nivå 1 och 2 ingår.</p> / <p>This paper is a text analysis of how some Swedish textbooks in mathematics introduce algebra filtered by van Hiele theories of levels of thinking. The van Hiele theories emphasize that the language constitutes the knowledge objects in mathematics, wich is used all through the analysis. Generally the textbooks start at the third van Hiele’s level of thinking. According to the van Hiele theories the teaching of algebra should start at level 1. This then becomes the teachers’ task to do without the support of the textbook in mathematics. Ideas on teaching level 1 and 2 are included.</p>

Detta arbete är en textanalys av hur några svenska läroböcker i matematik introducerar algebra speglat i van Hieles teorier om tankenivåer vid inlärning. Van Hieles teorier poängterar språket som kunskapsbärare i matematik vilket går som en röd tråd genom analysen. Generellt börjar läroböckerna på van Hieles tankenivå 3. Enligt van Hieles teorier borde undervisningen i algebra börja på nivå 1, vilket då blir lärarens uppgift att göra utan stöd av matematikboken. Förslag på arbetssätt för nivå 1 och 2 ingår. / This paper is a text analysis of how some Swedish textbooks in mathematics introduce algebra filtered by van Hiele theories of levels of thinking. The van Hiele theories emphasize that the language constitutes the knowledge objects in mathematics, wich is used all through the analysis. Generally the textbooks start at the third van Hiele’s level of thinking. According to the van Hiele theories the teaching of algebra should start at level 1. This then becomes the teachers’ task to do without the support of the textbook in mathematics. Ideas on teaching level 1 and 2 are included.

The purpose of this thesis is the analysis of the 2nd round of the mathematical olympiad, including the statistics of the success rate of the students in particular problems, analysis of their procedures and error analysis. This thesis should serve as a survey of the problems of 56th year of MO and as a study text for the mathematical olympiad participants.

Skolmatematiken och matematikdidaktiken har under de senaste årtiondena genomgått en förändring från ett historiskt fokus på ren räkning och utantillkunskaper mot alltmer processorientering. Det pågår en aktiv debatt om hur framgångsrik dagens matematikundervisning egentligen är då de svenska elevernas resultat i internationella jämförelser så som PISA är inte lysande. Historiskt har den svenska matematikundervisningen hämtat influenser från amerikansk matematikdidaktikutveckling. I detta examenarbete görs jämförelse av förespråkad matematikdidaktik vid två olika lärarkurser i Sverige och en lärarkurs på Stanford, USA. Syftet är att undersöka likheter och skillnader i synen på ”god” matematikundervisning på dessa kurser. Som huvudsaklig analysmetod valdes en diskursanalys. De tre olika lärarkurserna ses som tre diskurser. Fyra frågor ställts till respektive diskurs: Vad lyfts fram om matematikdidaktik? Hur talas det omdetta? Vad utesluts eller tonas ner? Vad framställs som god matematik-undervisning? Huvudinriktningen mot en processorienterad matematik är tydlig i alla tre diskurserna. Samtidigt så nämns i diskurserna att ”kunna vissa saker utantill är också viktigt” så det är inte helt entydigt men ändå en tydlig riktning. Alla diskurser tar också upp uppgifternas betydelse för lärandet. Val av uppgifter är en viktig del av matematikdidaktiken. Några skillnader som framkommer är att den amerikanska diskursen lyfter fram betydelsen av mjuka faktorer som attityd, självförtroende, motivation, tilltro, uppmuntran betydligt mer än de två svenska. Sammanfattningsvis visar min analys av de tre diskurserna att den amerikanska diskursen tydligare lyfter fram värderingar och undervisar lärarstudenterna i vad som är god matematikundervisning. God matematikundervisning innefattar många mjuka aspekter som motivation, självförtroende och jämlikhet. Budskapet i de två svenska diskurserna är sakligare och med mer bredd – god matematikundervisning omfattar ett spektrum av förmågor, kunskaper, ämnesområden. Lärarstudenten får ett ”smörgåsbord” och får sedan, på gott och ont, plocka ihop sin egen tallrik av hur matematikundervisningen ska bedrivas. / In recent decades, school mathematics and mathematics education have undergone a change from a historical focus on pure arithmetic and facts knowledge towards an increasingly process orientation. There is an active debate about how successful today's mathematics education really is and the Swedish students' results in international comparisons such as PISA are not brilliant. Historically, Swedish mathematics teaching has taken influences from American mathematics didactic trends. In this thesis, a comparison is made of advocated mathematic education at two different teacher courses in Sweden and a teacher course at Stanford, USA. The purpose is to investigate similarities and differences in the view of “good mathematics education” in these courses. A discourse analysis was chosen as the main analysis method. The three different teacher courses are seen as three discourses. Four questions are asked for each discourse: What mathematics didactics is highlighted? How is this talked about? What is excluded or toned down? What is presented as good mathematics teaching? In all three discourses a clear focus on a process-oriented mathematics is seen. At the same time, it is mentioned in the discourses that "knowing certain things by heart is also important" so it is not completely unambiguous but still a clear direction. All discourses also address the importance of the math problems. Choice of problems and exercises is an important part of mathematics didactics. One difference that emerge is that the American discourse highlights the importance of soft factors such as attitude, self-confidence, motivation, confidence, encouragement significantly more than the other two. In summary, my analysis of the three discourses shows that the American discourse more clearly highlights values and educates student teachers what is good mathematics teaching. Good mathematics education includes many soft aspects such as motivation, self-confidence and equality. The message in the two Swedish discourses is more objective and with more breadth - good mathematics education encompasses a spectrum of abilities, knowledge, subject areas. The teacher student gets a "smorgasbord" and then has to fill his own plate with theories and methods how the mathematics teaching should be conducted.

Denna studie utgår från en tes om framgångsrik undervisning som baseras på Hatties m.fl. (2017) och Grevholms (2012) forskning. Uppsatsen är en jämförande studie som utgår från tre undervisningsmodeller i matematik, traditionell undervisningsmodell, montessorimodellen och singaporemodellen. Fokus är på matematikundervisning i årskurs 1-3, svensk skola. Studien omfattar dels en mindre litteraturstudie, dels en intervjustudie med tre lärare som arbetar enligt de tre modellerna. Jag utgår från ett sociokulturellt perspektiv på lärande, undervisning och kunskap. Analyserna baseras på en riktad kvalitativ innehållsanalys. Resultatet visar på för och nackdelar med de tre undervisningsmodellerna och hur väl de uppfyller kraven enligt tesen för en framgångsrik undervisning. I resultatet tydliggörs genom lärarnas utsagor problematiken med de olika undervisningsmodellerna. / The proposal in this study define successful teaching from Hattie et al. (2017) and Grevholm (2012) research results. This is a comparative study with three teaching models in mathematics, Traditional model, Montessori model and Singapore model. The focus is on mathematics teaching in primary school, in Sweden.  The study embrace a minor literature study and an interview study with three teachers, who work according the three models in their teaching. I use a socio-cultural perspective on learning, teaching and knowledge. A directed qualitative content analyze is used to analyze both the literature and the interviews.  The results show the pros and cons of the three teaching models, according to the requirements of the proposed ideas of successful teaching. The results of the teachers´ statements clarify and problematize the differences between the three educational models in practice.

Implementering av digitala verktyg har haft inverkan på hur lärare väljer att undervisa i matematikklassrummet de senaste åren. Därför som blivande matematiklärare blir det intressant att undersöka hur lärare ser på hur digitala verktyg påverkar den egna undervisningen och elevernas lärande samt undersöka lärares val av undervisningsmetoder i geometriämnet när användningen av digitala hjälpmedel står i fokus. För att samla in data genomfördes intervjuer av sju matematiklärare från tre olika skolar. De teoretiska ramverken som använts för analys i vår studie är SAMR-modellen (SAMR är en akronym som står för fyra engelska begrepp: Substitution, Augmentation, Modification och Redefinition) som beskriver hur digitala verktyg används i olika steg i undervisning och Drijvers orkestreringstyper för att identifiera olika undervisningsmetoder och beskriva deras kännetecken. Studien visade att alla intervjuade lärare använder olika digitala verktyg i sin geometriundervisning samt att de överlag är positiva till att använda verktygen. Lärarna beskriver att verktygen understödjer undervisningen utifrån de behov som uppstår i praktiken, till exempel behov av individanpassade uppgifter, olika redovisningsformer medmera. Resultatet av analysen visade att flera av de intervjuade lärare når steg tre i SAMR-modellen, vilket innebär att synen på elevernas lärande och undervisningen har börjat förändrats. Studien visade också att lärarna varierar och utnyttjar olika orkestreringstyper för att organisera sin undervisning för specifika lärandemål. / The aim of this thesis is to execute digital tools in mathematics teaching. The study focuses its attention on how teachers choose the type of the digital tools and carry out mathematics teaching, especially in the subject of geometry. As a future teacher, it will be interesting to investigate the teachers’ views concerning the effect of digitizing on their teaching and students’ learning, as well as the teacher's choice of teaching methods and the design of the teaching within that method, will depend on the lessons of geometry where the use of digital technology is the focus in the classroom. The empirical materials were collected through interviews with seven teachers from three different schools. The data were analyzed with a focus on the ways by which teachers employ digital tools to support and work with mathematical content. The theoretical frameworks in our study are the SAMR model (SAMR is an acronym that stands for four English concepts Substitution, Augmentation, Modification, and Redefinition) that describes how digital tools are used at different stages in teaching and Drijver's orchestration type to identify different teaching methods and describe their characteristics. The results based on the interviews show that most teachers meet requirements for reaching step three in the SAMR model. The study also shows an overview of different teaching methods in mathematics teaching where digital technology is used. The results also show that the teachers utilize several types of orchestration in the work with the subject of geometry in their teaching. In total, this study shows that teachers use up to five different types of orchestration. Generally, all teachers have positive opinions concerning the employment of digital technology in mathematics education, and the utilities of digitization that they spotlight include assisting and supporting students' mathematical knowledge as well as developing their understanding of geometric concepts.

This volume contains the papers presented at the International Conference on “Models in Developing Mathematics Education” held from September 11-17, 2009 at The University of Applied Sciences, Dresden, Germany. The Conference was organized jointly by The University of Applied Sciences and The Mathematics Education into the 21st Century Project - a non-commercial international educational project founded in 1986. The Mathematics Education into the 21st Century Project is dedicated to the improvement of mathematics education world-wide through the publication and dissemination of innovative ideas. Many prominent mathematics educators have supported and contributed to the project, including the late Hans Freudental, Andrejs Dunkels and Hilary Shuard, as well as Bruce Meserve and Marilyn Suydam, Alan Osborne and Margaret Kasten, Mogens Niss, Tibor Nemetz, Ubi D’Ambrosio, Brian Wilson, Tatsuro Miwa, Henry Pollack, Werner Blum, Roberto Baldino, Waclaw Zawadowski, and many others throughout the world. Information on our project and its future work can be found on Our Project Home Page http://math.unipa.it/~grim/21project.htm It has been our pleasure to edit all of the papers for these Proceedings. Not all papers are about research in mathematics education, a number of them report on innovative experiences in the classroom and on new technology. We believe that “mathematics education” is fundamentally a “practicum” and in order to be “successful” all new materials, new ideas and new research must be tested and implemented in the classroom, the real “chalk face” of our discipline, and of our profession as mathematics educators. These Proceedings begin with a Plenary Paper and then the contributions of the Principal Authors in alphabetical name order. We sincerely thank all of the contributors for their time and creative effort. It is clear from the variety and quality of the papers that the conference has attracted many innovative mathematics educators from around the world. These Proceedings will therefore be useful in reviewing past work and looking ahead to the future.

This volume contains the papers presented at the International Conference on “Turning Dreams into Reality: Transformations and Paradigm Shifts in Mathematics Education” held from September 11-17, 2011 at Rhodes University, Grahamstown, South Africa. The Conference was organized jointly by Rhodes University and The Mathematics Education into the 21st Century Project - an international educational project founded in 1986. Our Project is dedicated to the improvement of mathematics education world-wide through the publication and dissemination of innovative ideas. Many prominent mathematics educators have supported and contributed to the project, including the late Hans Freudental, Andrejs Dunkels and Hilary Shuard, as well as Bruce Meserve and Marilyn Suydam, Alan Osborne and Margaret Kasten, Mogens Niss, Tibor Nemetz, Ubi D’Ambrosio, Brian Wilson, Tatsuro Miwa, Henry Pollack, Werner Blum, Roberto Baldino, Waclaw Zawadowski, and many others throughout the world. Information on our project and its future work can be found on our Project home page http://math.unipa.it/~grim/21project.htm . In this year, 2011, we celebrate the 25th anniversary of the founding of our Project, when Manmohan Singh Arora suggested the idea to Fayez Mina and myself around a swimming pool in Bahrain (of all places!) That first meeting was, however, typical of the multi-cultured and global character of our Project and it’s subsequent conferences throughout the world. These Proceedings begin with the Plenary Papers and then the other contributions in alphabetical name order of the principal authors. We sincerely thank all of the contributors for their time and creative effort. It is clear from the variety and quality of the papers that the conference has attracted many innovative mathematics educators from around the world.