Det finns en problematik i övergången för svenska studenter mellan gymnasiet och högskolan. En faktor i den problematiken är att bevis och bevisföring hanteras på olika sätt på de olika utbildningsnivåerna. Dessutom är forskning kring hur läroböcker hanterar bevis och bevisföring begränsad. I denna studie undersöks två vanliga läroböcker på gymnasiet med avseende på bevis och bevisföring. Utifrån de i ämnesplanen befintliga ämnesområdena granskas teoriavsnitten i läroböckerna med fokus på bevis och bevisföring. Dessutom undersöks bevisföringsuppgifter med utgångspunkt i G. Stylianides (2009) ramverk. Resultaten visar att läroböckerna är inkonsekventa i sitt hanterande av bevis och att bevis ofta osynliggörs i teoriavsnitten. Ett annat resultat är att den matematiska strukturen är svår att följa. Läroböckerna innehåller 6,7 % respektive 13,7 % bevisföringsuppgifter och vi har funnit 19 olika typer av bevisföringsuppgifter varav två typer är väldigt dominerande. Vi argumenterar att didaktiskt värdefulla syften kan uppnås genom att synliggöra bevis bättre i läroböcker. / There exists a problem in Swedish students’ transition between high school and college. One factor of this problem stems from the fact that proofs and proving are handled in different ways at these different levels of education. In addition, research on how textbooks deal with proofs and proving is limited. This study examines proofs and proving in two common math textbooks intended for upper secondary high school students in Sweden. Based on the contents of the curriculum, the theoretical sections in the textbooks are examined with an added focus on proofs and proving. Also, the textbooks’ tasks are examined with the help of a modified framework based on G. Stylianides (2009) framework. The results show that the textbooks are inconsistent in their handling of proofs and that proofs are often made invisible in the theoretical sections. Another result is that the mathematical structure is difficult to follow. The textbooks contain 6.7% and 13.7% proving tasks respectively and we have found 19 different types of proving tasks among which two types are very dominant. We argue that didactically valuable objectives can be achieved by making proofs more visible in textbooks.

<p>This thesis investigates grade seven learners perceived experiences in learning algebra.Things that learners do and say during algebra lessons and about algebra were investigated. The study was done at one of the previously disadvantaged schools in Cape Town, South Africa.The data were collected through observations, a questionnaire and interviews. Observations were made from the day the topic was started in two grade seven classes. Two different teachers taught the two classes. Focus group interviews were conducted, two group of learners, ten learners from each of the two classes were interviewed. Learners devised a number of strategies for solving problems related to sums and differences. The principal learning difficulties experienced by learners in algebra related to the transition from arithmetic conventions to those of algebra, the meaning of literal symbols and the recoginition of structures. It became obvious then that developing algebraic thinking is not necessarily dependent upon algebraic notation and that the presence of algebraic notation says little about the level of problem solving.</p>

This study aimed to develop a teaching and learning model, based on principles derived from the fields of constructivist theory, schema theory, critical literacy theory, and design theory, to inform the development of hypermedia-mediated learning environments that facilitate the construction of mathematical knowledge by secondary school students in Thailand. In this study, the participants were a group of three secondary school students from the Demonstration school attached to the Faculty of Education at Khon Kaen University (Thailand). In order to ascertain how mathematical learning could be facilitated by the process of designing a web page that could be used to introduce other students to analytic geometry, all three participants were asked to work collaboratively to design an analytic geometry web page. The process of designing the web page was informed by a theoretical model derived from an analysis and synthesis from the research literature on constructivist theory, schema theory, critical literacy theory, and design theory. Findings from the study indicated that the creation of a web page facilitated and enhanced the Thai students' learning about analytic geometry. The major outcomes from the study are a revised theoretical framework to inform the integration of the design of mathematical web pages into Thai mathematics classrooms and a conceptual map framework to assess qualitative and quantitative changes to students' repertoires of knowledge about analytic geometry that emerge during the process of designing a webpage.

The diploma thesis deals with ways of teaching combinatorics at a secondary school. Specifically, I analyzed selected mathematics textbooks for secondary schools in terms of introducing concepts and operations of combinatorics and in terms of types of tasks used. I carried out interviews with six secondary school mathematics teachers and observations of their lessons in order to describe their method of teaching combinatorics. Using results of tests written by these teachers' pupils, I examined whether and how their solving strategies and errors might be influenced by their teachers' approach to teaching combinatorics. Finally, I compared my results with the existing results of mathematics education research on pupils' combinatorial reasoning. The work is divided into four chapters; the first three are theoretical (curricular documents for selected schools, analysis of textbooks on combinatorics in terms of the implementation of combinatorial concepts and operations, selected research about pupils' solving strategies and errors for combinatorial problems, methods of checking the correctness of their solutions. and the impact of ways of teaching combinatorics on pupils' performance). Chapter 4 focuses on my own research which consists of interviews with teachers, observations of lessons on combinatorics, the...

This investigation of twentieth century trends in mathematics education includes the survey of existing literature and questionnaires conducted with retired and active Texas teachers. Historical events, trends in curriculum, instruction, learning theories, and contradictions of twenty-year periods are delineated. Questionnaire responses are tabulated along the same periods and vignettes of typical classrooms are drawn from the data. Results of the survey show the impact of societal forces on mathematics curricula, a continued downward expansion of content into lower grades and expanding knowledge of learning processes. A unified mathematics curriculum, classroom-related learning theory research, and further development of team-teaching are postulated as future trends. Recommendations include further examination of trends through isolation of other variables such as region and ethnicity.

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

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.

Although the Internet is widely used in many areas, its use in school mathematics is at best in its infancy. Studies show that Mathematics teachers have fewer uses for the Internet than teachers of other disciplines. Hence, this research adopted a mixed method approach to investigate what mathematics materials are on the Internet, how teachers are teaching mathematics with the Web and mathematic students' perceptions and engagement with the Internet. This research reviewed the World Wide Web for mathematics materials and found three major groupings of online resources namely interactive resources, non-interactive resources, and communications possibilities. A typology of Web objects was constructed and a database based on a Task-Web object approach was proposed for teacher use. A broad survey was used to elicit information about Internet usage among mathematic teachers. A total of 103 mathematics teachers responded and 15 were interviewed to gain further insight into their usage. Observations of Internet use were also conducted in the classrooms of 4 teachers. The results show that most teachers would like to use the Internet more in their teaching of mathematics but many do not know where and how to do so in an effective way. Statistics, Business Mathematics and Number operations appear to be the more popular topics. Using statistics data from the Web seem to be the Web feature that is most common followed by using the Internet as a resource centre for word problems. Web communications are seldom used. Common constraints teachers face include lack of time, difficulty in planning, lack of knowledge of good Web sites that map to curricula, slow download times, and limited booking times. Students perceive doing activities on the Internet as better than from the textbook because of the amount and variety of information, the better explanations and the change in mode of presentation. Students who have a low comfort level with mathematics wish their teachers would use the Internet. The power of interactive activities on the Internet to engage and motivate these students is due to a variety of reasons such as the element of game play, a change from the routine, its ability to present different conceptual visuals, the independent self paced learning, and quick feedback that came with the use of the Internet. The Internet also enabled students to access difficult to find information and saved them time. The findings also suggest that teachers' persistence in using the Internet could bring about a routine that helps students settle down to the task and stay on task. Teachers' choice and discernment of Web-based activities that are engaging and motivating are paramount to the success of this learning tool. Four Web-based strategies for teaching mathematics were documented and a model of underlying knowledge for teacher practice with the Web was suggested.

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)