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.

Our paper presents some preliminary observation from a collaborative exploratory study linking mathematics, science and reading within a technology enhanced problem-based learning scenario conducted at one French Canadian Elementary and Middle School. Presented in a form of dialogue between teacher and researcher, our findings give some meaningful insight in how an innovative mathematics teaching can be developed and implemented using a real-world problem solving. Instead of a traditional presentation of material about lighting up homes, participating mathematics, science and French teachers were working collaboratively with the ICT integration mentor and two university professors helping students investigate a problem from various perspectives using a variety of cognitive and metacognitive strategies, discussing and sharing the finding with peers and presenting them to a larger audience using media tools. Our preliminary results may prompt further investigation of how innovation in teaching and learning can help students become better critical thinkers and scientifically empowered citizens.

No description available.

Submitted by Izabel Franco (izabel-franco@ufscar.br) on 2016-09-08T17:16:42Z No. of bitstreams: 1 DissKAD.pdf: 6658012 bytes, checksum: d55e92d194481c3b4ed161eb948cbb72 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-09-12T17:22:57Z (GMT) No. of bitstreams: 1 DissKAD.pdf: 6658012 bytes, checksum: d55e92d194481c3b4ed161eb948cbb72 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-09-12T17:23:43Z (GMT) No. of bitstreams: 1 DissKAD.pdf: 6658012 bytes, checksum: d55e92d194481c3b4ed161eb948cbb72 (MD5) / Made available in DSpace on 2016-09-12T17:23:51Z (GMT). No. of bitstreams: 1 DissKAD.pdf: 6658012 bytes, checksum: d55e92d194481c3b4ed161eb948cbb72 (MD5) Previous issue date: 2015-02-27 / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / This present research, of qualitative nature, was guided by the question: what characterizes and what are the functions of school demonstrations in different level textbooks for teaching mathematics? In order to answer this question, three high school and three Junior high school textbook collections –which are assessed and approved by the National Textbook Program (PNLD, in Portuguese), the National Curricular Parameters (PCN, in Portuguese) and the PNLD guides for the same presented school levels –were considered as documents. Inspired by the Depth Hermeneutics methodological referential, these demonstrations were considered symbolical. Firstly, the way how research in academic mathematics and Mathematical Education discuss the demonstrations was pointed out. Afterwards, the demonstrations, in different historical periods, were exposed –such as in the literary work “The Elements” by Euclid and in textbooks which were published during reforms in the teaching of mathematics in Brazil. Through a referential of the sociology of the science, the symbolic value of these school demonstrations was discussed. The development of this study has pointed to a discussion about the naturalization processes of the uniqueness and verity of logical values. In addition, the demonstration was presented as a belief linked to symbols of stringency, precision, mathematical scientificity, proof, subdual, respectability and authority. Upon performing content analysis on the selected books, four categories of school demonstrations were set: (1) through experiments and particular cases; (2) through deductive reasoning with an exploratory character; (3) through formal elements of classical reasoning; (4) through particular cases, generalization and explanation. The analysis has shown changes regarding: the methodology for the development of a demonstration; the type of language applied; the way a procedure was introduced and concluded; the use of pictures and inductive, intuitive and visual procedures. In order to complement interpretations and understand the different demonstration fashions expressed in these categories, official documents were used, what made it possible to identify that the school demonstrations in the textbooks fulfill their role in preparing students for the understanding and future development of a formal demonstration, besides being aligned with the goal of developing deductive reasoning and approximation in forming a professional mathematician. Upon that, it can be understood that the formal demonstration is appreciated and motivated in curricular propositions which, despite guiding the use of different demonstration forms adapted to each teaching degree, seek to build a unique idea of demonstration. / A presente pesquisa, de natureza qualitativa, se orientou pela questão: o que caracteriza e quais funções cumprem as demonstrações escolares em livros didáticos dos diferentes níveis do ensino de matemática? A fim de responder essa questão tomamos como documentos três coleções de livros didáticos dos anos finais do ensino fundamental e três do ensino médio, avaliadas e aprovadas pelo Programa Nacional do Livro Didático (PNLD); os Parâmetros Curriculares Nacionais (PCN); e os guias do PNLD para os mesmos níveis de ensino citados. Inspiramo-nos no referencial metodológico da Hermenêutica de Profundidade (HP) e concebemos as demonstrações como formas simbólicas. Primeiramente, destacamos como as pesquisas da área da matemática acadêmica e da Educação Matemática discutem a demonstração e, em seguida, expomos as demonstrações em momentos históricos, como na obra “Os Elementos” de Euclides e em livros didáticos publicados durante reformas do ensino de matemática no Brasil. Por meio de um referencial da sociologia da ciência discutimos o valor simbólico das demonstrações escolares. Os encaminhamentos desse estudo nos apontaram para uma discussão acerca dos processos de naturalização da unicidade, verdade e de valores da lógica. Além disso, a demonstração se apresentou como uma crença atrelada a símbolos de rigor, de precisão, de cientificidade da matemática, comprovação, pujança, respeitabilidade e de autoridade. Da análise de conteúdo realizada nos livros didáticos selecionados foram organizadas quatro categorias para as demonstrações escolares: (1) via experimentos e casos particulares; (2) lógico-dedutiva com caráter de exploração; (3) formal com elementos da lógica clássica; (4) mediante casos particulares, generalização e explicação. A análise nos indicou mudanças quanto: à metodologia para o desenvolvimento de uma demonstração; ao tipo de linguagem empregada; à maneira de se introduzir e concluir um procedimento; ao uso de figuras e de procedimentos indutivos, intuitivos e visuais. Para complementar as interpretações e compreender as diferentes formas de demonstrar expressas nas categorias recorremos aos documentos oficiais, que nos permitiram identificar que as demonstrações escolares nos livros didáticos cumprem o papel de preparação dos estudantes para a compreensão e desenvolvimento futuro de uma demonstração formal, além de estarem atreladas ao objetivo de desenvolvimento do raciocínio lógico e a aproximação ao fazer matemático profissional. Com isso entendemos que a demonstração formal é valorizada e incentivada em propostas curriculares que, apesar de orientarem o uso de diferentes formas de se demonstrar adequadas a cada nível de ensino, almejam a construção de uma ideia única de demonstração.

Orientador: Antonio Miguel / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Educação / Made available in DSpace on 2018-08-10T22:23:44Z (GMT). No. of bitstreams: 1 Gomes_MarcosLuis_M.pdf: 3633774 bytes, checksum: dd656fb177d316fbb33fdb88eee39a65 (MD5) Previous issue date: 2008 / Resumo: Este trabalho tem como objeto de estudo as práticas culturais de mobilização da história da matemática realizadas por autores de livros didáticos de matemática que escreveram livros para o Ensino Médio. Estas práticas são aqui concebidas como formas simbólicas e, assim, o estudo dessas formas de mobilização da história foi realizado com base em uma análise de cunho hermenêutico. Realizamos o cruzamento entre três tipos de fontes documentais: coleções de livros didáticos constantes no PNLEM 2005; entrevistas realizadas com os autores destas coleções e os pareceres constantes no catálogo do PNLEM relativos a essas coleções. Esta análise nos remeteu a empreender uma interpretação personalizada dos padrões semióticos pelos quais teriam se pautado alguns autores de livros didáticos de matemática, no sentido de procurarem estabelecer um diálogo com a história da matemática a fim de fazerem-na participar de seus textos didáticos destinados à educação matemática escolar / Abstract: This work has as study objects the cultural practices of mobilization of history of mathematics accomplished by authors of mathematics text books that wrote books for the high school. These practices are conceived here as symbolic forms and, like this, the study of these mobilization forms of the history was accomplished with base in an analysis of hermeneutic stamp. For this analysis, we accomplished the crossing among three types of documental sources: collections of text books present in PNLEM 2005; interviews accomplished with the authors of these collections and your analysis present in the PNLEM catalog about these collections. Through this crossing, we noticed that the cultural practices of mobilization of the history of mathematics happen more frequently at the present time, and one of the decisive factors for this was the interference of the Federal Government in relation to the purchases of text books for the high school. The didactic collections analyzed did not become necessarily homogeneous because of the criteria of the official evaluation, but the coming of the mobilization of the history constitutes a positive factor for the students' continuous formation and teachers of the high school / Mestrado / Educação Matematica / Mestre em Educação

For the problem of producing pedagogical step-by-step solutions to mathematical problems in education, standard methods and algorithms used in construction of computer algebra systems are often not suitable. A method of using rules to manipulate mathematical expressions in small steps is suggested and implemented. The problem of creating a step-by-step solution by choosing which rule to apply and when to do it is redefined as a graph search problem and variations of the A* algorithm are used to solve it. It is all put together into one prototype solver that was evaluated in a study. The study was a questionnaire distributed among high school students. The results showed that while the solutions were not as good as human-made ones, they were competent. Further improvements of the method are suggested that would probably lead to better solutions. / För problemet att producera pedagogiska steg-för-steg-lösningar till matematiska problem inom utbildning, är vanliga metoder och algoritmer som används i konstruktion av datoralgebrasystem ofta inte lämpliga. En metod som använder regler för att manipulera matematiska uttryck i små steg föreslås och implementeras. Problemet att välja vilka regler som ska appliceras och när de ska göra det för att skapa en steg-för-steg-lösning omdefineras som ett grafsökningsproblem och varianter av algoritmen A* används för att lösa det. Allt sätts ihop till en prototyp av en lösare vilken utvärderas i en studie. Studien var ett frågeformulär som delades ut till gymnasiestudenter. Resultaten visade att även fast lösningar skapade av programmet inte var lika bra som lösningar skapade av människor, så var de anständiga. Fortsatta föbättringar av metoden föreslås, vilka troligtvis skulle leda till bättre lösningar.

Syftet med denna studie är att undersöka lärarnas upplevelse av elevers missuppfattningar vid inlärning av bråk och sannolikhet samt om det är möjligt att minska missuppfattandet med hjälp av kategorisering av dessa. Genom att fråga matematiklärare om de upplever att eleverna de undervisar ofta har missuppfattningar och om samma missuppfattningar är återkommande, kan vi få en bild av vilka delar av de matematiska begreppen som kan uppfattas svåra av eleverna. De missuppfattningar som tenderar att återkomma kan komma att behöva mer fokus på förklaring. Studien genomfördes genom en surveyundersökning i enkätform som publicerades i grupper som samlar matematiklärare på sociala medier, samt skickades till matematiklärare via mail. Det resulterade i 41 enkätsvar som analyserades genom beskrivande statistik i kombination med en induktiv innehållsanalys. Studien visar att orsaken till att missuppfattningar kopplade till matematiska begrepp kan bero på ett för stort fokus på procedurinriktad undervisning i de tidigare skolåren. Detta upplever lärarna medför att eleverna inte har tillräcklig begreppsförståelse när de börjar på gymnasiet. Det vanligaste åtgärdsförslaget är kopplat till undervisningsstrategier med mer sociokulturella inslag i undervisningen. De allra flesta av studiens deltagare upplever att begreppsförståelse är viktigt och utgör en förutsättning för att klara av både problemlösning och mer avancerad matematik. För att skapa förståelse för matematiska begrepp är det nyttigt för lärare att känna till vanliga missuppfattningar. Kategorisering av missuppfattningar kan därför vara till nytta för lärarna i undervisningen, som ett stöd i lektionsplanering och som ett pedagogiskt verktyg för att utveckla elevernas matematiska kunskaper. / The purpose of this study is to investigate the teachers' experience of students' misconceptions when learning fractions and probability, and whether it is possible to reduce misconceptions by categorizing them. By asking mathematics teachers if they feel that the students they teach often have misconceptions and if the same misconceptions are repeated, we can get a picture of which parts of the mathematical concepts may be perceived as difficult by the students. The misconceptions that tend to recur may need more focus on explanation. The study was carried out through a survey in questionnaire form that was published in groups that bring together mathematics teachers on social media and was also sent to mathematics teachers via email. This resulted in 41 survey responses that were analysed through descriptive statistics in combination with an inductive content analysis. The study shows that the reason for misconceptions connected to mathematical concepts may be due to too much focus on procedure-oriented teaching in the earlier school years. The teachers feel that this means that the students do not have sufficient conceptual understanding when they start high school. The most common proposed measure is linked to teaching strategies with more socio-cultural elements in the teaching. The vast majority of the study's participants feel that conceptual understanding is important and constitutes an essentiality for being able to cope with both problem solving and mathematics at more advanced levels. In order to create an understanding of mathematical concepts, it is useful for teachers to know about common misconceptions. Categorization of misconceptions can therefore be useful for teachers in teaching, as a support in lesson planning and as a pedagogical tool to develop students' mathematical knowledge.

I det svenska skolsystemet sker olika stadieövergångar och övergången från högstadiet till gymnasiet är en sådan. I ämnet matematik visar tidigare forskning att en skillnad mellan dessa stadier är ett ökat studietempo och en förskjutning mot en mer formell matematik i det senare stadiet. Syftet med denna studie har varit att undersöka vilka innehållsmässiga likheter och skillnader det finns mellan dessa stadier inom området linjära funktioner. Detta har gjorts via en innehållsanalys av några läroböcker som används för respektive stadie då dessa kan ses som den potentiellt realiserade läroplanen. Vidare har studien baserats på Anna Sfards teori om operationell respektive strukturell begreppsuppfattning som ett sätt att få syn på och kontrastera det innehåll som behandlas i de olika läroböckerna. Resultatet visar att det finns likheter och skillnader mellan de olika stadierna, utifrån hur detta uttrycks via läroböckerna, och gymnasiet tenderar att fokusera mer på det strukturella i funktionsbegreppet medan högstadiet i högre grad betonar det operationella. Vidare finns det skillnader mellan de olika läroböckerna inom samma stadie där studien visar att beroende på kombination av läromedel för högstadiet respektive gymnasiet kan det bli olika grad av repetition på gymnasiet. Vissa kombinationer kan ge en större överlappning mellan innehållet i stadierna medan andra kombinationer riskerar att istället skapa ett glapp mellan stadierna. / In the Swedish school system, different stage transitions take place and the transition from lower secondary school to upper secondary school is one such. In the subject of mathematics, previous research shows that a difference in these stages is an increased pace of study and a shift towards more formal mathematics. The purpose of this study has been to investigate what content-related similarities and differences there are between junior high school and high school mathematics in the area of ​​linear functions. This has been done via a content analysis of some textbooks that are used for the different stages, as these can be seen as the potentially implemented curriculum. Furthermore, the study has been based on Anna Sfard's theory of operational and structural concepts as a way to gain insight into and contrast the content covered in the various textbooks. The result shows that there are similarities and differences in the different stages, from how this is expressed via the textbooks, and the upper secondary school tends to focus more on the structural concept of function, while the lower secondary emphasizes the operational aspects to a greater degree. Furthermore, there are differences between the different textbooks within the same stage, where this study shows that depending on the combination of textbooks for lower- and upper secondary school, there may be different degrees of repetition in the latter. Some combinations can provide a greater overlap between the content of the stages, while other combinations risk instead creating a gap between the stages.

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)