As a result of two years working with the pre-service primary teachers in a College in Fiji I became aware of the difficulty many of the students were having understanding the primary school mathematics they would be required to teach. During that time I had attempted to help them overcome the difficulties by using different teaching approaches and activities but was far from satisfied with my efforts. Hence I decided to make a concerted effort to help the students by planning, implementing and partially evaluating a mathematics education unit, known as the Teaching Program for the first semester of their course. This work formed the basis of my study. For the Teaching Program I chose a constructivist teaching approach with number sense as the underlying theme. To examine the aspects of the Program I used my observations and those of the students especially ones reported in their mathematics journals. To evaluate the effectiveness of the Teaching Program I collected and analysed quantitative data from traditional testing of the class of forty students as well as data from case studies of six of the pre-service teachers in the class. To determine what features of the Teaching Program were linked to positive changes my main source of data was the case studies, especially entries from their journal writings. The findings suggested that a significant development of the cognitive aspects of the students’ number sense did occur during the time of the Teaching Program but not as much as was hoped for. As a result of the analysis of the data I came to a greater realisation of the importance of the non-cognitive aspects of number sense and the necessity for a greater consideration of them in the development of a Program. I also realise now that a major development that did occur was in my understanding of the knowledge and learning of mathematics. My ideas of a teaching paradigm of social constructivism had not guided me sufficiently to incorporate activities and procedures to develop the non-cognitive aspects. I suggest that a paradigm which extends the theory of social constructivism to give greater consideration of these aspects of learning in general, and hence numeracy and number sense in particular, was needed. As a result of this study, my introduction to the theory of enactivism appears to be giving me some direction in this search at this stage.

This is a report of an investigation of children’s beliefs about the nature of mathematics, the nature of learning and helping factors for learning mathematics. The study aimed to investigate whether beliefs held by eight learners of eight to nine years of age could be articulated and portrayed. It aimed also to develop procedures to facilitate this process, to portray children’s beliefs from their responses to the research procedures, to provide insights into possible complexities and subtleties of young learners’ beliefs, to reflect upon the significance for the mathematics classroom of the insights gained, and to reflect upon the value of the procedures developed for the study. The research took the form of individual case studies of four girls and four boys of eight to nine years of age from two schools in suburban Melbourne. Four children were teacher-perceived low achievers in mathematics and four were teacher-perceived high achievers in mathematics. The children were each interviewed on ten occasions over a five-month period using thirty semi-structured, creative interviewing procedures that were developed or adapted for the study that included drawing, writing, discussing scenarios presented through photographs, video snippets and other children’s drawings, ordering of descriptors, and responding to questionnaires presented verbally. The interview data consisted of transcripts and artefacts. Some class administered tasks, lesson observations and interviews with the mathematics teachers provided background information. Analysis of interview responses was undertaken through a criss-cross examination in which themes were drawn from each child’s data. Responses were not judged for correctness or for a match to any predetermined categories and the researcher sought to take a stance of neutrality to the phenomena under study. The research suggests that teachers and others involved in the education of young learners of mathematics should know that: it is possible to gain insights into children’s beliefs about maths (the term used most commonly by the children), learning, and helping factors for learning maths; to gain insights into young children’s beliefs, it is important to have dialogue with the children to avoid making assumptions about their interpretations or meanings; the creative interviewing procedures developed for the present research are helpful as they can stimulate reflection and prompt conversation; young children’s beliefs can be complex, subtle, broad and deep; young children’s beliefs are individually constructed and differ from child to child; children may not see mathematics concepts in the same ways as their teachers and other adults; beliefs are sufficiently diverse and significant to affect the way children see the mathematics learning situation; although the beliefs of children of eight to nine years of age may, on the surface, appear simplistic and naïve, they are not necessarily so. Young learners are able to reflect on their own and others’ experiences and often construct complex beliefs. There is a lot happening in the minds of these children. The research suggests also that it is important that educators do not to make assumptions about: what children see as maths (or mathematics); what children see as learning; and what children see as helping factors for learning maths. A key factor facilitating children’s reflection and expression was the range of visual, verbal, and text-based creative interviewing strategies developed for the present study. The individual procedures provided suitable prompts to allow young children to articulate or represent their beliefs. The semi-structured procedures, through which ideas were explored on multiple occasions, followed by theme-based, criss-cross analysis of interview transcripts and artefacts, resulted in rich and trustworthy portrayals of beliefs, increasing the validity of the findings. The research provides the education community with insights into young children’s beliefs that are unlikely to emerge within the day to day activity of the classroom and, through the availability of the research procedures, facilitates further gaining of insights into beliefs either by classroom teachers or other researchers.

<p>Syftet med föreliggande studie är att, genom intervjuer med tolv elever ur år fem, undersöka deras uppfattningar om matematik. För att studera detta har vi använt oss av en kvalitativ metod med inspiration av den fenomenografiska ansatsen. Utifrån studiens syfte och frågeställningar har tre temaområden framtagits inom vilka elevernas uppfattningar analyserats och därefter kategoriserats. Ur resultatet framkommer en relativt snäv syn på matematiken där läroboken är dominerande i elevernas uppfattningar. Boken fungerar som utgångspunkt både i definitionen av matematikämnet och vid beskrivningen av hur ämnet upplevs. Detta kan ha sin grund i, vilket framgår i elevernas beskrivning av matematiklektionerna, att undervisningen till stor del kännetecknas av räknande i läroboken. Ett stort antal av eleverna uppfattar undervisning i matematik som mer eller mindre lustfyllt beroende på hur långt de kommit i läroboken. Att ligga långt fram och att räkna fort framstår som betydande för hur duktiga eleverna upplever sig vara. Flertalet elever uttrycker dessutom en önskan om variation i undervisningen med förslag såsom miljöombyte, nyttjande av konkret material eller möjlighet till arbete närmare det egna intresseområdet. Samtliga elever beskriver dock matematikämnet som viktigt där de flesta argumenten är av funktionell karaktär. Den praktiska användbarheten av matematiken framkommer i liten utsträckning, då främst i anknytning till handel och pengahantering.</p> / <p>The aim of the present study is, through interviews with 12 students in year five, to investigate their views on mathematics. In order to study this, we have used a qualitative approach inspired by the phenomenographic methodology. Based on the study’s aims, three areas have been identified, in which the pupils’ views were analyzed and then categorized. The results showed a somewhat limited view of mathematics, where the course textbook heavily influenced pupils’ perceptions of the subject. The study shows that the textbook largely forms the basis both for how the subject is defined and perceived. This could be based on teaching being largely based on arithmetic exercises in the textbook itself, as is shown in students’ description of mathematics lessons. A large number of pupils correlate their enjoyment of mathematics with how far they have come in the courseware.  <em>To be a long way into the book </em>and <em>to be able to calculate quickly </em>are both key indicators of how mathematically able students consider themselves to be. In addition a majority of pupils express a desire for more variation in teaching methods, with proposals such as a change of scenery, the use of real world examples, or the possibility to concentrate on an area in which they were particularly interested. However, all pupils consider the subject of mathematics to be important, where the most common arguments are of a functional nature. The practical usefulness of mathematics was considered to be limited, mainly confined to shopping and managing money.</p>

<p>Syftet med uppsatsen är att undersöka förekomsten av värden och värderingar i matematikläroböcker för gymnasieskolans kurs Matematik A. En modell som tidigare använts vid analys av värden i matematikläroböcker i Singapore och Australien ligger till grund för granskningen. Modellen utvecklas och tillämpas för analys av sammanlagt 472 övningsuppgifter i tre kapitel i tre utvalda matematikläroböcker. De utvalda kapitlen behandlar området algebra. Modellen visade sig vara tillämpbar i en svensk utbildningskontext. Resultatet av innehållsanalysen visar på förekomst av värden och värderingar. Av resultatet framkommer även att vissa värden ges mer utrymme än andra och att en diskrepans finns mellan det erhållna resultatet och de värden som lyfts fram i kursplanen för Matematik A. Avslutningsvis förs en diskussion kring det erhållna resultatet i förhållande till en demokratisk matematikutbildning.</p>

<p>I denna uppsats studeras förekomsten av värden i geometrikapitel i läromedel för gymnasieskolans Matematik A. En metod baserad på innehållsanalys som använts i en tidigare studie om förekomst av värden i algebrakapitel har vidareutvecklats. Sammanlagt 955 uppgifter i geometrikapitlen i fem olika läroböcker har analyserats. Resultatet visar på en skillnad mellan intentionerna i gymnasieskolans styrdokument och det utrymme olika värden ges. Även skillnader mellan det utrymme studerade läromedel ger åt olika värden har påvisats.</p>

Syftet med uppsatsen är att undersöka förekomsten av värden och värderingar i matematikläroböcker för gymnasieskolans kurs Matematik A. En modell som tidigare använts vid analys av värden i matematikläroböcker i Singapore och Australien ligger till grund för granskningen. Modellen utvecklas och tillämpas för analys av sammanlagt 472 övningsuppgifter i tre kapitel i tre utvalda matematikläroböcker. De utvalda kapitlen behandlar området algebra. Modellen visade sig vara tillämpbar i en svensk utbildningskontext. Resultatet av innehållsanalysen visar på förekomst av värden och värderingar. Av resultatet framkommer även att vissa värden ges mer utrymme än andra och att en diskrepans finns mellan det erhållna resultatet och de värden som lyfts fram i kursplanen för Matematik A. Avslutningsvis förs en diskussion kring det erhållna resultatet i förhållande till en demokratisk matematikutbildning.

Syftet med föreliggande studie är att, genom intervjuer med tolv elever ur år fem, undersöka deras uppfattningar om matematik. För att studera detta har vi använt oss av en kvalitativ metod med inspiration av den fenomenografiska ansatsen. Utifrån studiens syfte och frågeställningar har tre temaområden framtagits inom vilka elevernas uppfattningar analyserats och därefter kategoriserats. Ur resultatet framkommer en relativt snäv syn på matematiken där läroboken är dominerande i elevernas uppfattningar. Boken fungerar som utgångspunkt både i definitionen av matematikämnet och vid beskrivningen av hur ämnet upplevs. Detta kan ha sin grund i, vilket framgår i elevernas beskrivning av matematiklektionerna, att undervisningen till stor del kännetecknas av räknande i läroboken. Ett stort antal av eleverna uppfattar undervisning i matematik som mer eller mindre lustfyllt beroende på hur långt de kommit i läroboken. Att ligga långt fram och att räkna fort framstår som betydande för hur duktiga eleverna upplever sig vara. Flertalet elever uttrycker dessutom en önskan om variation i undervisningen med förslag såsom miljöombyte, nyttjande av konkret material eller möjlighet till arbete närmare det egna intresseområdet. Samtliga elever beskriver dock matematikämnet som viktigt där de flesta argumenten är av funktionell karaktär. Den praktiska användbarheten av matematiken framkommer i liten utsträckning, då främst i anknytning till handel och pengahantering. / The aim of the present study is, through interviews with 12 students in year five, to investigate their views on mathematics. In order to study this, we have used a qualitative approach inspired by the phenomenographic methodology. Based on the study’s aims, three areas have been identified, in which the pupils’ views were analyzed and then categorized. The results showed a somewhat limited view of mathematics, where the course textbook heavily influenced pupils’ perceptions of the subject. The study shows that the textbook largely forms the basis both for how the subject is defined and perceived. This could be based on teaching being largely based on arithmetic exercises in the textbook itself, as is shown in students’ description of mathematics lessons. A large number of pupils correlate their enjoyment of mathematics with how far they have come in the courseware.  To be a long way into the book and to be able to calculate quickly are both key indicators of how mathematically able students consider themselves to be. In addition a majority of pupils express a desire for more variation in teaching methods, with proposals such as a change of scenery, the use of real world examples, or the possibility to concentrate on an area in which they were particularly interested. However, all pupils consider the subject of mathematics to be important, where the most common arguments are of a functional nature. The practical usefulness of mathematics was considered to be limited, mainly confined to shopping and managing money.

Matematiskt begåvade elever är i behov av lika mycket stöd, variation och utmaningar som alla andra elever. Därför är det viktigt att dessa elever inte kommer i skymundan och får ”klara sig själva”. Syftet med denna studie är att synliggöra matematiskt begåvade elevers uppfattning om hur deras matematikundervisning ser ut idag, hur de själva lär sig bäst, samt hur de anser att matematikundervisningen ska utformas för att de ska ha möjlighet att utvecklas efter sin fulla förmåga. Med stöd i teori och tidigare forskning valdes sex matematiskt begåvade elever från år 4 ut för att medverka i denna undersökning. Metoden som valdes var att intervjua dessa sex elever för att få en djupare förståelse för hur de själva uppfattar matematikundervisningen idag, deras egna lärstilar, samt hur de vill arbeta i matematiken för att ha möjlighet utvecklas till sin fulla förmåga. Resultatet från intervjuerna visar att stor del av undervisningen idag genomsyras av ensamarbete och där dessa elever inte får det stöd och den hjälp de behöver. Det visade sig också att dessa elever var väldigt olika i sina lärstilar, precis som alla individer, och därför behöver en varierad undervisning för att alla lärstilar ska inkluderas. Det visade sig att alla eleverna i denna studie söker utmaning, mer arbete i grupp och stöd från lärare för att få möjlighet att utvecklas till sin fulla förmåga. / The mathematic gifted pupil needs support, variation and challenge as much as other children. Therefore it is important that these children do not end up in the background and that the teacher does not think that ”they can handle it on their own”. The aims of this study are to explore the mathematic gifted pupils´ beliefs about mathematics teaching, their learning styles and how they want to work to have the opportunity to develop their full mathematical capacity.  With support from theory and earlier research we chose six mathematical gifted pupils in the fourth grade to participate in the study. As the research method we chose to interview these pupils in order to get a deeper understanding of how they believe that their own learning in mathematic looks like, how they think they learn the best and how they want to work to have the opportunity to develop their full mathematical capacity. The result and conclusion of the interviews show that according to the pupils, a majority of the learning situations includes the pupils’ individual work and by that these children do not get the help and support that they need. The results from the interviewees also show that these pupils have different ways of learning, like everybody else, and therefore the learning situations have to be varying.  Finally, the result shows that the pupils want to have challenges, work together in groups and more support from teachers to be able to develop their full mathematical capacity.

This essay explores the issue of organizing mathematics education by means of syllogism. Two aspects turn out to be particularly significant. One is the syllogistic analysis while the other is the cunning of reason. Thus the exploration is directed towards gathering evidence of their existence and showing by examples their usefulness within mathematics education. The syllogistic analysis and the cunning of reason shed also new light on Chevallard's theory of didactic transposition. According to the latter, each piece of mathematical knowledge used inside school is a didactic transposition of some other knowledge produced outside school, but the theory itself does not indicate any way of transposing, and this empty space can be filled with the former. A weak prototype of syllogism considered here is Freudenthal's change of perspective. Some of the major difficulties in mathematics learning are connected with the inability of performing change of perspective. Consequently, to ease the difficulties becomes a significant issue in mathematics teaching. The syllogistic analysis and the cunning of reason developed in this essay are the contributions to the said issue.

This thesis which is about working with mathematics teaching has no effect/impact on performance and understanding of pupils. The study is intended to gain insight on working in mathematics education which has an impact on performance and understanding of students who are ethnically Swedish and foreign ethnic. In order to compare the ethnic Swedish and foreign students of ethnic understanding, I judged from their performance on the tests taken in year three. The qualitative approach permeated the work, both interviews and observations which were included in the survey. The qualitative study was chosen to get an understanding of how mathematics teaching is conducted in two different areas. The results of the investigation show that working in mathematics education is relevant to performance and understanding of students who are ethnically Swedish and foreign ethnic. In addition, it led to ethnically Swedish students having higher results than students who are ethnically non-resident in the comparison schools.