Made available in DSpace on 2016-04-27T16:58:33Z (GMT). No. of bitstreams: 1 Mirtes Fatima Pasini.pdf: 22771511 bytes, checksum: 984ad26489e7839b0fb9fc255399a645 (MD5) Previous issue date: 2007-10-16 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This work is inserted the research project Argumentation and Proof in School Mathematics (AProvaME), which aims to study the teaching and learning of mathematical proofs during compulsory schooling. The main research question of this contribution to the project relates to how proof is treated in particular geometry topics in one collection of mathematics textbooks for secondary school students. More specifically, the study aims to identify how the passage from empiricism to deduction is contemplated in the textbook activities as well as to document the interventions and strategies necessary on the part of the mathematics teacher in order to manage this transition. The types of proofs in the classification of Balacheff (1988) and the functions of proof identified by de Villiers (2001) serve as the principle theoretical tools for these analyses. Following a survey of the activities related to proof and proving in topics related to the theorem of Pythagoras and properties of straight lines and triangles, teaching sequences based on these activities were developed with students from the 8th Grade of a secondary school within the public school system of the municipal of Jacupiranga in the State of São Paulo. The main findings of the study indicate that the teacher has at his or her disposal material that permit a broad approach to proof and proving, although the passage from exercises involving reliance on empirical manipulations for validation to the construction of proofs based on mathematical properties is not very explicitly addressed, with the result that intense teacher intervention is necessary at this point. A particular difficulty faced by the teacher is knowing how to intervene without assuming responsibility for the resolution of the task in question. Finally, a dynamic geometry activity is presented, as an attempt to provide a learning situation which might enable students to engage more spontaneously in the transition from evidence-based arguments to valid mathematical proofs / Nosso trabalho está inserido no Projeto Argumentação e Prova na Matemática Escolar (AProvaME), que tem como objetivo estudar o ensino e aprendizagem de provas matemáticas na Educação Básica. A questão principal da pesquisa consiste em analisar o tratamento deste tema em determinados conteúdos geométricos de uma coleção de livros didáticos do Ensino Fundamental. Mais especificamente, o estudo busca identificar como a passagem do empirismo à dedução é contemplada nas atividades dos livros e quais as intervenções e estratégias necessárias por parte do professor para gerenciar essa passagem. Os tipos de prova na classificação de Balacheff (1988) e as funções de prova identificadas por De Villiers (2001) foram as principais ferramentas teóricas utilizadas para estas análises. Após um levantamento das atividades relacionadas à prova nos conteúdos Teorema de Pitágoras, Retas Paralelas e as propriedades dos Triângulos, seqüências baseadas nessas atividades foram desenvolvidas com alunos de 8.ª Série do Ensino Fundamental de uma escola pública no Município de Jacupiranga, do Estado da São Paulo. Concluímos que o professor tem à sua disposição material consistente para trabalhar com seus alunos, embora exista o problema na passagem brusca de exercícios empíricos em diversos níveis de verificação para as demonstrações formais, sendo necessária intervenção do professor por meio de revisões pertinentes, proporcionando ao aluno esclarecimentos para desenvolver uma atividade. A principal dificuldade para o professor foi interferir sem assumir a responsabilidade de resolver a situação em questão. Por fim, apresenta-se uma atividade no ambiente de geometria dinâmica, visando proporcionar uma transição mais espontânea entre argumentos baseados em evidência e argumentos baseados em propriedades matemáticas

Submitted by Silvana Teresinha Dornelles Studzinski (sstudzinski) on 2017-04-19T13:42:18Z No. of bitstreams: 1 Débora de Lima Velho Junges_.pdf: 2121896 bytes, checksum: 8dd2c6b16aeb8c2f81a64929adaf8938 (MD5) / Made available in DSpace on 2017-04-19T13:42:18Z (GMT). No. of bitstreams: 1 Débora de Lima Velho Junges_.pdf: 2121896 bytes, checksum: 8dd2c6b16aeb8c2f81a64929adaf8938 (MD5) Previous issue date: 2017-02-17 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / A Tese tem como objetivo analisar como a escola e, em particular, a matemática escolar, operavam como parte dos processos de subjetivação de escolares descendentes de imigrantes alemães no Rio Grande do Sul, no período da Campanha de Nacionalização. De modo mais específico, identifica, nas enunciações dos entrevistados, rituais escolares que operavam como tática de manifestação da verdade de que “os alemães são superiores” e analisa esses rituais. Discute, também, os rituais da matemática escolar nos quais a tática da manifestação da verdade operava, analisando os jogos de linguagem matemáticos que estavam presentes nesses rituais, como eram ensinados e que conhecimentos matemáticos eram transmitidos. As ferramentas teóricas do estudo estão vinculadas às teorizações de Michel Foucault e de Ludwig Wittgenstein. O material de pesquisa consiste em narrativas de sete pessoas que estudaram em escolas da imigração alemã no Rio Grande do Sul, no período da Campanha de Nacionalização. Os principais resultados da investigação apontam que o Deutschtum operava na vida dos imigrantes alemães e seus descendentes, subjetivando-os de modo a se perceberem como colonos na qual a descendência alemã era priorizada. Foram identificados três rituais escolares que operaram em favor do discurso de manutenção do Deutschtum e como uma tática de manifestação da verdade de que os alemães eram “indivíduos superiores”. Sobre a matemática escolar, constatou-se que para os familiares dos participantes da pesquisa era importante que seus filhos dominassem as regras e os jogos de linguagem da matemática escolar, para, com isso “honrar” sua descendência alemã e preservar o Deutschtum. Também foi possível identificar dois rituais da matemática escolar que operavam como forma de reforçar a manifestação da verdade de que os “alemães eram superiores”. O primeiro ritual tratou da realização de exercícios nas aulas de matemática e se observou que as listas de exercícios eram extensas e apresentavam questões que os professores esperavam/exigiam que os alunos aplicassem as mesmas regras gramaticais e os mesmos jogos de linguagem ensinados na explicação e nos exemplos apresentados, os quais eram marcados pelo formalismo e pela abstração da matemática escolar. Apropriar-se dos jogos de linguagem da matemática escolar era valorizado tanto pelos professores, quanto pelos familiares, conduzindo os escolares a considerar que saber a matemática escolar era condição necessária para que fossem identificados como “bons alemães”. O segundo ritual da matemática escolar se centrou na prática de realização de contas consideradas “difíceis”. Aqueles alunos que as realizavam corretamente eram posicionados como inteligentes e exemplos a serem seguidos; eram reconhecidos como “alemães de verdade”, uma vez que consideravam a matemática como uma disciplina de difícil aprendizagem. Esses resultados oferecem elementos que permitem inferir que, nas formas de vida da imigração alemã no Rio Grande do Sul, no período da Campanha de Nacionalização, era assumida como uma verdade que os descendentes alemães eram “indivíduos superiores”, sendo a matemática escolar utilizada para reforçar tal manifestação. / The thesis aims to analyze how the school and, in particular, the school mathematics, operated as part of the process of subjectivation of students who was descendants of German immigrants in the estate of Rio Grande do Sul (far South of Brazil) during the happening of what was called “Campaign of Nationalization”. Specifically, identifies, in the talk of respondents, school rituals which operated as a tactic of manifestation of the truth that "the Germans are superior" and analyzes these rituals. Discusses also the rituals of school mathematics in which the tactics of the manifestation of truth operated by analyzing the mathematical language games that were present in these rituals, the way they were taught and that math skills were transmitted. The theoretical tools of the study are linked to the theorizing of Michel Foucault and Ludwig Wittgenstein. The research material consists of narrations of seven people who have studied in schools of German immigration in Rio Grande do Sul, in the period of the Campaign of Nationalization. The main results of the investigation indicate that the Deutschtum operated in the lives of German immigrants and their descendants, making them to perceive themselves above all as German settlers. Were identified three school rituals that operated in favour of the maintenance of Deutschtum and as a tactic for manifestation of the truth that the Germans were "superior individuals". About the mathematics, it was found that for the families of the participants of the survey it was important that his children dominate the rules and language games of the school mathematics to "honor" their German ancestry and preserve the Deutschtum. It was also possible to identify two rituals of school mathematics which operated as a way of strengthening the manifestation of the truth that the "Germans were superior." The first ritual was about making exercise in mathematic class and was noted that the lists of exercises were extensive and proposed questions that teachers expected/required that students apply the same grammatical rules and the same language games taught in the explanation and the examples presented, which were marked by the formalism and abstraction of school mathematics. Take ownership of the school mathematics language games was valued by both the teachers and the family, leading schoolchildren to consider that knowing the school mathematics was a necessary condition to be identified as "good Germans". The second rite of school mathematics focused on practice of realization of accounts considered "difficult". Those students that answered correctly, were placed as smart and examples to be followed; they were recognized as "real Germans", once mathematics were considered as a discipline of hard learning. These results provide elements that allow to infer that, in the form of life of German immigration in Rio Grande do Sul, during the period of the Campaign of Nationalization, was assumed to be a fact that the Germans were "superior individuals", being the school mathematics used to reinforce such manifestation.

[pt] Este trabalho tem por objetivo mostrar como o conceito de proporcionalidade foi sendo abordado em livros didáticos de matemática adotados no Brasil do século XX. Considerar o conceito de proporcionalidade sob a perspectiva do seu desenvolvimento histórico e sob o ponto de vista da fenomenologia didática contribuiu para o entendimento dos textos escolares, pois mostrou o quanto esse conceito está situado na confluência de muitos outros e, com isso, trouxe esclarecimentos sobre sua importância para o ensino. Foram analisados livros didáticos que tiveram repercussão no ensino fundamental. Com base nas análises realizadas foi possível identificar mudanças no tratamento do conceito de proporcionalidade, considerando-se os aspectos teórico, algorítmico e didático das abordagens, e o quadro atual se particulariza em relação ao que se encontrou em abordagens anteriores. / [en] The purpose of this work is to show how the proportionality concept has been approached in mathematics textbooks adopted in Brazil during the 20th Century. To consider the proportionality concept under the perspective of its historical development and under the point of view of the didactical phenomenology contributed to the understanding of the textbooks, because it showed that this concept is placed in the confluence of many others and, thus also brought explanations on its importance for the teaching of mathematics. We analized elementary school textbooks which were important and widely used. Based on these analyses it was possible to identify changes in the treatment of the proportionality concept, considering the theoretic, algorithmic and didactic aspects of the approaches, and the current presentations of the proportionality concept is compared to former approaches.

<p>Syftet med avhandlingen är att beskriva och analysera argument för matematik i grundskolan och att förstå varför och hur de officiella argumenten förändras, från de argument som återfinns i styrdokument till de argument som förs fram av undervisande matematiklärare. En utgångspunkt är att skolmatematikens villkor och verklighet kan beskrivas genom analys av officiella argument och av lärarstudenters och lärares personliga argument för matematik i grundskolan. Specifika forskningsfrågor i anslutning till syftet är:</p><p>- Vilka argument för lärarstudenten fram inför yrkesdebuten?</p><p>- Vilka argument för läraren fram under sina första år i yrket?</p><p>- Vilka beskrivningar av skolmatematikens villkor ger lärarna?</p><p>En longitudinell studie har genomförts där en grupp lärarstudenter följts genom utbildningen och under de första åren i yrket. Resultatet visar att lärarstudenter under utbildningen utvecklar en syn på matematik och matematikundervisning som stämmer väl med läroplanen och kursplanen i matematik enligt Lpo 94. De nyblivna lärarna med undervisning i matematik och NO-ämnen upplever i början av yrkeskarriären skilda villkor på olika skolor. Gemensamt för de lärare som undervisar i senare delen av grundskolan är upplevelser av krav på att ”hinna med kursen” inför det nationella provet i årskurs 9. Lärarnas mål med matematikundervisningen i grundskolan blir därför att förbereda eleverna för det nationella provet. En faktor som påverkar är kravet på att elever skall ha betyget godkänd för att vara behöriga till gymnasieskolans nationella program. De nyblivna lärarna upplever en konflikt mellan olika officiella argument för matematik i grundskolan. Faktorer som påverkar lärarnas och matematikämnets villkor och verklighet i grundskolan är bl.a. skolornas organisation i arbetslag och lärarnas kombination av undervisningsämnen.</p> / <p>The aim of this thesis is to describe and analyse arguments for mathematics in compulsory school and to understand why and how the official arguments change, from the arguments written in the national curriculum and course syllabus for mathematics to the arguments presented by mathematics teachers. The point of departure is that the conditions and the reality for school mathematics can be understood through an analysis of official arguments and of personal arguments given by teacher students and teachers. A longitudinal investigation has been carried out; a group of teacher students has been followed during their teacher education and the first three years after their professional debut. The result shows that during their education the teacher students develop a view on mathematics and mathematics education harmonizing with the goals of mathematics in the national syllabus. The novice teachers have mathematics and sciences as their teaching subjects and they experience quite different conditions when they start to work as teachers. Common for those teaching in school years 7–9 is the experience of pressure to “cover the course” before the pupils shall take the national test in school year 9. Preparing the pupils for the national test becomes the most important goal for the novice teachers. A factor influencing the mathematics teacher is the qualification requirement in mathematics from compulsory school to go into the national programs in the upper secondary school. The novice teachers experience a conflict between different goals in the national curriculum and course syllabus for mathematics. Factors that have an influence on mathematics as a school subject are the organization of teachers at the local schools and the teachers’ combination of teaching subjects.</p>

Syftet med avhandlingen är att beskriva och analysera argument för matematik i grundskolan och att förstå varför och hur de officiella argumenten förändras, från de argument som återfinns i styrdokument till de argument som förs fram av undervisande matematiklärare. En utgångspunkt är att skolmatematikens villkor och verklighet kan beskrivas genom analys av officiella argument och av lärarstudenters och lärares personliga argument för matematik i grundskolan. Specifika forskningsfrågor i anslutning till syftet är: - Vilka argument för lärarstudenten fram inför yrkesdebuten? - Vilka argument för läraren fram under sina första år i yrket? - Vilka beskrivningar av skolmatematikens villkor ger lärarna? En longitudinell studie har genomförts där en grupp lärarstudenter följts genom utbildningen och under de första åren i yrket. Resultatet visar att lärarstudenter under utbildningen utvecklar en syn på matematik och matematikundervisning som stämmer väl med läroplanen och kursplanen i matematik enligt Lpo 94. De nyblivna lärarna med undervisning i matematik och NO-ämnen upplever i början av yrkeskarriären skilda villkor på olika skolor. Gemensamt för de lärare som undervisar i senare delen av grundskolan är upplevelser av krav på att ”hinna med kursen” inför det nationella provet i årskurs 9. Lärarnas mål med matematikundervisningen i grundskolan blir därför att förbereda eleverna för det nationella provet. En faktor som påverkar är kravet på att elever skall ha betyget godkänd för att vara behöriga till gymnasieskolans nationella program. De nyblivna lärarna upplever en konflikt mellan olika officiella argument för matematik i grundskolan. Faktorer som påverkar lärarnas och matematikämnets villkor och verklighet i grundskolan är bl.a. skolornas organisation i arbetslag och lärarnas kombination av undervisningsämnen. / The aim of this thesis is to describe and analyse arguments for mathematics in compulsory school and to understand why and how the official arguments change, from the arguments written in the national curriculum and course syllabus for mathematics to the arguments presented by mathematics teachers. The point of departure is that the conditions and the reality for school mathematics can be understood through an analysis of official arguments and of personal arguments given by teacher students and teachers. A longitudinal investigation has been carried out; a group of teacher students has been followed during their teacher education and the first three years after their professional debut. The result shows that during their education the teacher students develop a view on mathematics and mathematics education harmonizing with the goals of mathematics in the national syllabus. The novice teachers have mathematics and sciences as their teaching subjects and they experience quite different conditions when they start to work as teachers. Common for those teaching in school years 7–9 is the experience of pressure to “cover the course” before the pupils shall take the national test in school year 9. Preparing the pupils for the national test becomes the most important goal for the novice teachers. A factor influencing the mathematics teacher is the qualification requirement in mathematics from compulsory school to go into the national programs in the upper secondary school. The novice teachers experience a conflict between different goals in the national curriculum and course syllabus for mathematics. Factors that have an influence on mathematics as a school subject are the organization of teachers at the local schools and the teachers’ combination of teaching subjects.

Arbetets syfte är att undersöka hur högstadieelever för matematiskt resonemang. De frågeställningar som studien inriktas på är vilka lösningsstrategier elever väljer då de resonerar matematiskt såväl som vad  det finns för skillnader och likheter mellan de yngre elevernas lösningar och de äldre elevernas lösningar. Undersökningen genomfördes i två klasser, den ena i årskurs 8 och den andra i årskurs 9, på en grundskola. Eleverna fick lösa uppgifter, vilka uppmanade dem att föra matematiskt resonemang, individuellt. Resultatet av studien visar att majoriteten av undersökta elever har valt att resonera deduktivt. Jämförelsen av elevers lösningar i två årskurser visar att årskurs 9 elevers resonemangsföring präglas av större förtrogenhet med den algebraiska demonstrationen. Resultatet visar även att elever med högre kunskaper om algebra oftare visar benägenheter till att vidaregeneralisera de givna påståendena. / The purpose of this study is to examine secondary school students’ strategies of reasoning. The study inquires into which strategies students choose when reasoning mathematically as well as differences and similarities between the younger students’ solutions and the older students’ solutions. The study was conducted in two classes, in years 8 and 9 respectively, at a secondary school. The students were asked to solve tasks, which encouraged them to reason mathematically, on individual basis. The study revealed that the majority of students had chosen to reason deductively. The comparison of students’ presented answers in two years showed that the ninth-graders’ solutions are characterized of greater skill when it comes to algebraic demonstrations. The results of the study also reveal that students with stronger algebraic abilities attempt more often to generalize the given mathematical statements further.

The aim of this dissertation is to describe and analyze how discourse as a theoretical and didactical concept can help in advancing knowledge about the teaching of mathematics in school. The dissertation has been written within a socio-cultural perspective where active participation and support from artefacts and mediation are viewed as important contributions to the development of understanding. Discourse analysis was used as a theoretical point of departure to grasp language use, knowledge construction and mathematical content in the teaching practises. The collection of empirical data was made up of video and audio tape recordings of the interaction of teachers and pupils in mathematics classrooms when they deal with problem-solving tasks, as well as discussions between student teachers as they engage in planning a teaching situation in mathematics. Discourse analysis was used as a tool to shed light upon how pupils learn and develop understanding of mathematics. The results of my studies demonstrate that discussions very often are located in either a mathematical or in an every-day discourse. Furthermore, the results demonstrate how change between every-day and mathematical language often takes place unknowingly. Also the results underline that a specific and precise dialogue can contribute towards teachers’ and pupils’ conscious participation in the learning process. Translated into common vocabulary such as speak, think, write, listen and read teachers and pupils would be able to interact over concepts, signs, words, symbols, situations and phenomena in every-day discourse and its mathematical counterpart. When teachers and pupils become aware of discursive boundary crossing in mathematics an understanding of mathematical phenomena can start to develop. Teachers and pupils can construct a meta-language leading to new knowledge and new learning in mathematics. / Syftet med avhandlingen är att beskriva och analysera hur diskurs som teoretiskt- didaktiskt begrepp kan bidra till att utveckla kunskap i och om matematik i skolan. Avhandlingen skrivs utifrån ett sociokulturellt perspektiv där aktivt deltagande med hjälp av artefakter och mediering är viktiga bidrag till förståelsen. För att få syn på språket, kunskapen och matematiken i matematikundervisningen används diskursanalys som teoretisk utgångspunkt och metod. Data insamlingen består av video- och ljudbandsinspelningar av lärares och elevers samtal i ett matematikklassrum då de arbetar med problemlösning och lärarstudenters samtal när de planerar en undervisningssituation i matematik. Diskusanalys har använts som ett redskap för att upptäcka hur elever lär och utvecklar sin förståelse av matematik. Resultaten visar att i mina studier befinner sig samtalet ofta antingen i en matematisk eller i en vardaglig diskurs. Växlingen mellan vardagligt och matematiskt språk sker ofta omedvetet. I avhandlingen understryks att ett specifikt och precist samtal i matematik underlättar ömsesidig förståelse och kan bidra till att lärare och elever blir delaktiga i lärandet. Med hjälp av orden tala, tänka, skriva, lyssna och läsa skulle lärare och elever kunna interagera kring begrepp, tecken, ord, symboler, situationer och företeelser i den vardagliga och den matematiska diskursen och förståelse skulle lättare kunna äga rum i matematik. Genom att lärare och elever blir medvetna om hur de passerar diskursiva gränser både i matematiken och mellan matematik och vardag kan förståelse klargöras. Lärare och elever skulle kunna utveckla ett metaspråk som leder till ny kunskap och nytt lärande inom matematiken i skolan.

The purpose of this study was to reveal and define the problems middle school mathematics teachers experienced in applying transformational geometry and the solutions they proposed to overcome these problems. A total of six elementary mathematics teachers (grades 5-8) in Ankara participated in the study. The data were collected by means of one-to-one interviews with the participants. The findings indicated that the participants&rsquo / problems divided into three parts. These problems were problems arising from teachers, problems arising from students and problems arising from resources. The participants expressed challenges in teaching due to lack of materials, textbooks, and visualization ability of teachers, classroom size, and time. According to the findings, rotation was the most problematic issue. The participants claimed insufficient technological materials were the reason of this problem. Participants did not feel confidence enough to implement transformational geometry especially in rotation since they lacked adequate training and support. The participants claimed that the Ministry&rsquo / s support should be increased, concrete and technological materials should be sufficient in number, and the duration of transformational geometry lesson should be increased.

Views of teaching: implications for the didactic training of mathematics teachers. School mathematics teaching is an essential learning area in South African schools. Owing to persistent traditional positivist-based views and approaches, it still suffers from a variety of teaching-learning problems. Various national attempts have already been made to develop an effective teaching-learning program for school mathematics. Prominent researchers reveal that the failure of teaching-learning programmes often have to be attributed to the lack of an underlying grounded didactic theory. Therefore this study focused on the development of a grounded teaching-theoretical framework for school mathematics teaching. A further problem regarding school mathematics is that its teaching and learning traditionally are viewed from a narrow school subject disciplinary perspective. Therefore this study departed from a general didactic-theoretical perspective, creating the opportunity to approach and solve problems from a wider angle. A constructivist-based post-positivist view of effective teaching was developed, before entering the field of school mathematics. In this way an integrated ontologicalcontextual view of teaching was developed in terms of six identified ontological essential features, and their contextual coherence, namely: intention, teacher, leamer, interaction, content and context. Contrary to traditional positivist views, no causal relationship between teaching and learning was imposed, and teaching was not qualified in terms of learning products. Instead, teaching was characterised and qualified on ontological grounds, departing from the phenomenon itself. In this way the limitations of positivist process-product views of teaching could be identified, explained and overcome. Alternatively, a dynamic integrated view of teaching as a human act, directed at the facilitation of relevant and meaningful learning, was grounded and developed. Based on this general ontological-contextually based view, a specific ontologicalcontextual view of effective school mathematics teaching was grounded and developed. To this end a variety of prominent contemporary views of and approaches to school mathematics, and its teaching and learning, needed to be analysed in a critical way. According to this analysis school mathematics, and its teaching and learning should be viewed and approached from a constructivist-based dynamic change-and-grow perspective as human acts. In addition, it could have been proved that the perspective concerned can facilitate the treatment and solving of the currently experienced teaching-learning problems. This requires the reconsideration, from a similar perspective, of the current school mathematics curriculum, as well as the preservice didactic training of mathematics teachers. Specific implications of the developed ontological-contextual view of effective school mathematics teaching were identified, and practically tested in the corresponding preservice didactic training situation in the North West Province. Based on this an integrated model for the training concerned was formulated. It was found that the current training largely contributed to the continuation of traditional views of and approaches to school mathematics teaching, and its essential features. From the developed integrated ontological-contextual perspective definitive proposals regarding the transformation of school mathematics teaching and the corresponding didactic training were made and motivated. Further areas for investigation and development, resulting from this study, were identified, as well. This study aimed at investigating, and revealing for further exploration, the specific and broadening interaction between the general teaching and subject didactical fields and research, particularly in the two contexts of effective school mathematics teaching and the corresponding preservice didactical training. A particular attempt was made to accomplish this in a grounded and integrated way, to the benefit of both fields. / Thesis (PhD)--PU for CHE, 1998.

Views of teaching: implications for the didactic training of mathematics teachers. School mathematics teaching is an essential learning area in South African schools. Owing to persistent traditional positivist-based views and approaches, it still suffers from a variety of teaching-learning problems. Various national attempts have already been made to develop an effective teaching-learning program for school mathematics. Prominent researchers reveal that the failure of teaching-learning programmes often have to be attributed to the lack of an underlying grounded didactic theory. Therefore this study focused on the development of a grounded teaching-theoretical framework for school mathematics teaching. A further problem regarding school mathematics is that its teaching and learning traditionally are viewed from a narrow school subject disciplinary perspective. Therefore this study departed from a general didactic-theoretical perspective, creating the opportunity to approach and solve problems from a wider angle. A constructivist-based post-positivist view of effective teaching was developed, before entering the field of school mathematics. In this way an integrated ontologicalcontextual view of teaching was developed in terms of six identified ontological essential features, and their contextual coherence, namely: intention, teacher, leamer, interaction, content and context. Contrary to traditional positivist views, no causal relationship between teaching and learning was imposed, and teaching was not qualified in terms of learning products. Instead, teaching was characterised and qualified on ontological grounds, departing from the phenomenon itself. In this way the limitations of positivist process-product views of teaching could be identified, explained and overcome. Alternatively, a dynamic integrated view of teaching as a human act, directed at the facilitation of relevant and meaningful learning, was grounded and developed. Based on this general ontological-contextually based view, a specific ontologicalcontextual view of effective school mathematics teaching was grounded and developed. To this end a variety of prominent contemporary views of and approaches to school mathematics, and its teaching and learning, needed to be analysed in a critical way. According to this analysis school mathematics, and its teaching and learning should be viewed and approached from a constructivist-based dynamic change-and-grow perspective as human acts. In addition, it could have been proved that the perspective concerned can facilitate the treatment and solving of the currently experienced teaching-learning problems. This requires the reconsideration, from a similar perspective, of the current school mathematics curriculum, as well as the preservice didactic training of mathematics teachers. Specific implications of the developed ontological-contextual view of effective school mathematics teaching were identified, and practically tested in the corresponding preservice didactic training situation in the North West Province. Based on this an integrated model for the training concerned was formulated. It was found that the current training largely contributed to the continuation of traditional views of and approaches to school mathematics teaching, and its essential features. From the developed integrated ontological-contextual perspective definitive proposals regarding the transformation of school mathematics teaching and the corresponding didactic training were made and motivated. Further areas for investigation and development, resulting from this study, were identified, as well. This study aimed at investigating, and revealing for further exploration, the specific and broadening interaction between the general teaching and subject didactical fields and research, particularly in the two contexts of effective school mathematics teaching and the corresponding preservice didactical training. A particular attempt was made to accomplish this in a grounded and integrated way, to the benefit of both fields. / Thesis (PhD)--PU for CHE, 1998.