This licentiate thesis examines mathematical activities in a preschool. More precisely, the aim is to create knowledge of how mathematical activities emerge and are constructed in children's interactions with each other and with their teachers. The empirical material consists of video recordings and field notes collected through participant observations during a six-month period in two preschool units for children 3-5 years old. Theoretically the study is based on an ethnomethodological (EM) and conversation analytic (CA) perspective. Video recordings were analyzed using conversation analytic methods, involving a close and detailed analysis of sources in situated mathematical activities. Through the use of an EM and CA perspective this study contributes with new theoretical and methodological approaches to research on mathematical activities in preschools. In the close analysis of children's actions in interaction, an active child with ideas, interests, and commitment emerges, a child who uses a variety of communicative resources when participating in mathematical activities. Whether it is the children or the teachers who initiate the activity the children are actively involved in the construction of the mathematical content. Geometric shapes and concepts as well as different aspects of children's number sense are a couple of the mathematical topics covered in the study. In the activities the childrens display knowledge of math verbally as well as with their bodies, something that is analyzed by using the concept of epistemic stance. The preschool teachers sometimes used occasions when children display specific knowledge as an educational resource for other children's learning. The study also shows that children as well as their teachers follow each other's initiatives in the activities. This means that children change and enlarge the mathematical content within the activities and that the teachers follow the children's initiative. Through this reciprocity the mathematical content of the activity is maintained.

This licentiate thesis deals with some influencing factors to develop mathematicalabilities among mathematical gifted adolescents. Krutetskii’s structureof the mathematical abilities and Mönks’ triadic model of giftedness isused as a theoretical framework.The thesis consists of two articles with different aims. The first aim is toinvestigate to what extent the students had participated in various mathematicalactivities during their years in school and what impact the students attachto these activities. The second aim was to examine some aspects of the importanceof motivation for the mathematically gifted adolescents.To answer the research questions data was collected with a questionnaireand an interview study of a total of 27 finalists in a national mathematicalcompetition for students in Swedish upper secondary schools.Generally the students were positive about the activities they had participatedin. Specifically acceleration in the subject and mathematical competitionsstand out as particularly significant activities according to the students.The study shows the significance of mathematical activities providing aframework to relate to, which will make the progression more visible for thestudents. Such activities could be mathematical competition problem solvingor acceleration in the subject.The results of the study indicates that intrinsic motivation together withextrinsic motivation with integrated or identified regulation are the most importanttypes of motivation. All students in the study had both intrinsic motivationand some type of extrinsic motivation. / Denna licentiatuppsats handlar om påverkansfaktorer som bidrar till att utvecklamatematiska förmågor hos matematiskt begåvade ungdomar. Somövergripande teoretiskt ramverk för studien används Krutetskiis struktur av dematematiska förmågorna samt Mönks begåvningsmodell.Uppsatsen består av två artiklar med olika syften. Den första artikeln syftartill att undersöka i vilken utsträckning studiens ungdomar har deltagit i olikamatematiska aktiviteter under sina år i skolan och vilken betydelse de tillmäterdessa aktiviteter. Den andra artikelns syfte är att undersöka några aspekter avmotivationens betydelse hos de matematiskt begåvade ungdomarna.För att besvara frågeställningarna samlades data in med en enkät- och intervjustudiemed totalt 27 finalister i Skolornas matematiktävling.Generellt uttalade sig eleverna positivt om de verksamheter som de hadedeltagit i under skoltiden. Speciellt framkom acceleration i ämnet och matematiktävlingarsom särskilt betydelsefulla. Studien indikerar betydelsen av attde matematiska verksamheterna ger en ram att relatera till, vilket gör utvecklingenmer synlig för eleverna. Sådana aktiviteter kan vara problemlösninginom tävlingsmatematik eller acceleration i ämnet.Resultaten av den andra studien visar att inre motivation tillsammans medyttre motivation med integrerad eller identifierad kontroll är de viktigaste formernaav motivation hos studiens deltagare. I studien framkommer också attingen av deltagarna endast hade inre motivation för ämnet. Tvärtom hadesamtliga deltagare både inre motivation och autonom yttre motivation.

Text in Afrikaans, abstract in Afrikaans and English / Hierdie studie is daarop gemik om te bepaal tot welke mate wiskundeleerlinge op skool en onderwysstudente in wiskunde, onderrig in logika ontvang as agtergrond vir strenge bewysvoering. Die formele aspek van wiskunde op hoerskool en tersiere vlak is besonder belangrik. Leerlinge en studente kom onvermydelik met hipotetiese argumente in aanraking. Hulle leer ook om die kontrapositief te gebruik in bewysvoering. Hulle maak onder andere gebruik van bewyse uit die ongerymde. Verder word nodige en voldoende voorwaardes met stellings en hulle omgekeerdes in verband gebring. Dit is dus duidelik dat 'n studie van logika reeds op hoerskool nodig is om aanvaarbare wiskunde te beoefen. Om seker te maak dat aanvaarbare wiskunde beoefen word, is dit nodig om te let op die gebrek aan beheer in die ontwikkeling van 'n taal, waar woorde meer as een betekenis het. 'n Kunsmatige taal moet gebruik word om interpretasies van uitdrukkings eenduidig te maak. In so 'n kunsmatige taal word die moontlikheid van foutiewe redenering uitgeskakel. Die eersteordepredikaatlogika, is so 'n taal, wat ryk genoeg is om die wiskunde te akkommodeer. Binne die konteks van hierdie kunsmatige taal, kan wiskundige toeriee geformaliseer word. Verskillende bewystegnieke uit die eersteordepredikaatlogika word geidentifiseer, gekategoriseer en op 'n redelik eenvoudige wyse verduidelik. Uit 'n ontleding van die wiskundesillabusse van die Departement van Onderwys, en 'n onderwysersopleidingsinstansie, volg dit dat leerlinge en studente hierdie bewystegnieke moet gebruik. Volgens hierdie sillabusse moet die leerlinge en studente vertroud wees met logiese argumente. Uit die gevolgtrekkings waartoe gekom word, blyk dit dat die leerlinge en studente se agtergrond in logika geheel en al gebrekkig en ontoereikend is. Dit het tot gevolg dat hulle nie 'n volledige begrip oor bewysvoering het nie, en 'n gebrekkige insig ontwikkel oor wat wiskunde presies behels. Die aanbevelings om hierdie ernstige leemtes in die onderrig van wiskunde aan te spreek, asook verdere navorsingsprojekte word in die laaste hoofstuk verwoord. / The aim of this study is to determine to which extent pupils taking Mathematics at school level and student teachers of Mathematics receive instruction in logic as a grounding for rigorous proof. The formal aspect of Mathematics at secondary school and tertiary levels is extremely important. It is inevitable that pupils and students become involved with hypothetical arguments. They also learn to use the contrapositive in proof. They use, among others, proofs by contradiction. Futhermore, necessary and sufficient conditions are related to theorems and their converses. It is therefore apparent that the study of logic is necessary already at secondary school level in order to practice Mathematics satisfactorily. To ensure that acceptable Mathematics is practised, it is necessary to take cognizance of the lack of control over language development, where words can have more than one meaning. For this reason an artificial language must be used so that interpretations can have one meaning. Faulty interpretations are ruled out in such an artificial language. A language which is rich enough to accommodate Mathematics is the first-order predicate logic. Mathematical theories can be formalised within the context of this artificial language. Different techniques of proof from the first-order logic are identified, categorized and explained in fairly simple terms. An analysis of Mathematics syllabuses of the Department of Education and an institution for teacher training has indicated that pupils should use these techniques of proof. According to these syllabuses pupils should be familiar with logical arguments. The conclusion which is reached, gives evidence that pupils' and students' background in logic is completely lacking and inadequate. As a result they cannot cope adequately with argumentation and this causes a poor perception of what Mathematics exactly entails. Recommendations to bridge these serious problems in the instruction of Mathematics, as well as further research projects are discussed in the final chapter. / Curriculum and Institutional Studies / D. Phil. (Wiskundeonderwys)

Text in Afrikaans, abstract in Afrikaans and English / Hierdie studie is daarop gemik om te bepaal tot welke mate wiskundeleerlinge op skool en onderwysstudente in wiskunde, onderrig in logika ontvang as agtergrond vir strenge bewysvoering. Die formele aspek van wiskunde op hoerskool en tersiere vlak is besonder belangrik. Leerlinge en studente kom onvermydelik met hipotetiese argumente in aanraking. Hulle leer ook om die kontrapositief te gebruik in bewysvoering. Hulle maak onder andere gebruik van bewyse uit die ongerymde. Verder word nodige en voldoende voorwaardes met stellings en hulle omgekeerdes in verband gebring. Dit is dus duidelik dat 'n studie van logika reeds op hoerskool nodig is om aanvaarbare wiskunde te beoefen. Om seker te maak dat aanvaarbare wiskunde beoefen word, is dit nodig om te let op die gebrek aan beheer in die ontwikkeling van 'n taal, waar woorde meer as een betekenis het. 'n Kunsmatige taal moet gebruik word om interpretasies van uitdrukkings eenduidig te maak. In so 'n kunsmatige taal word die moontlikheid van foutiewe redenering uitgeskakel. Die eersteordepredikaatlogika, is so 'n taal, wat ryk genoeg is om die wiskunde te akkommodeer. Binne die konteks van hierdie kunsmatige taal, kan wiskundige toeriee geformaliseer word. Verskillende bewystegnieke uit die eersteordepredikaatlogika word geidentifiseer, gekategoriseer en op 'n redelik eenvoudige wyse verduidelik. Uit 'n ontleding van die wiskundesillabusse van die Departement van Onderwys, en 'n onderwysersopleidingsinstansie, volg dit dat leerlinge en studente hierdie bewystegnieke moet gebruik. Volgens hierdie sillabusse moet die leerlinge en studente vertroud wees met logiese argumente. Uit die gevolgtrekkings waartoe gekom word, blyk dit dat die leerlinge en studente se agtergrond in logika geheel en al gebrekkig en ontoereikend is. Dit het tot gevolg dat hulle nie 'n volledige begrip oor bewysvoering het nie, en 'n gebrekkige insig ontwikkel oor wat wiskunde presies behels. Die aanbevelings om hierdie ernstige leemtes in die onderrig van wiskunde aan te spreek, asook verdere navorsingsprojekte word in die laaste hoofstuk verwoord. / The aim of this study is to determine to which extent pupils taking Mathematics at school level and student teachers of Mathematics receive instruction in logic as a grounding for rigorous proof. The formal aspect of Mathematics at secondary school and tertiary levels is extremely important. It is inevitable that pupils and students become involved with hypothetical arguments. They also learn to use the contrapositive in proof. They use, among others, proofs by contradiction. Futhermore, necessary and sufficient conditions are related to theorems and their converses. It is therefore apparent that the study of logic is necessary already at secondary school level in order to practice Mathematics satisfactorily. To ensure that acceptable Mathematics is practised, it is necessary to take cognizance of the lack of control over language development, where words can have more than one meaning. For this reason an artificial language must be used so that interpretations can have one meaning. Faulty interpretations are ruled out in such an artificial language. A language which is rich enough to accommodate Mathematics is the first-order predicate logic. Mathematical theories can be formalised within the context of this artificial language. Different techniques of proof from the first-order logic are identified, categorized and explained in fairly simple terms. An analysis of Mathematics syllabuses of the Department of Education and an institution for teacher training has indicated that pupils should use these techniques of proof. According to these syllabuses pupils should be familiar with logical arguments. The conclusion which is reached, gives evidence that pupils' and students' background in logic is completely lacking and inadequate. As a result they cannot cope adequately with argumentation and this causes a poor perception of what Mathematics exactly entails. Recommendations to bridge these serious problems in the instruction of Mathematics, as well as further research projects are discussed in the final chapter. / Curriculum and Institutional Studies / D. Phil. (Wiskundeonderwys)