Dissertations / Theses on the topic 'Mathematics syllabus'

The topic of this half-thesis was conceived after the author had noted with interest, that an alarmingly high number of pupils in the Mdantsane Central District of the Ciskei were registering for Higher Grade (HG) mathematics for their final (Standard 10) year. What mattered most was that the failure rate was abnormally high. As a teacher of the subject for 14 years as at the end of 1988, the author had seen very weak pupils passing through his hands who had nevertheless insisted on registering for the examination in HG mathematics. A look through the Human Sciences Research Council (HSRC) Report of 1971 which recommended the syllabus differentiation, revealed that the intention of the relevant HSRC Committee was to offer a subject at a level in accordance with a pupil's interest, aptitude and ability. In a survey undertaken by the author, samples of teachers, pupils and parents indeed confirmed the existence of a very high number of HG candidates, as suspected. The survey also revealed an aversion to the standard grade (SG) course by some pupils, teachers and even principals. A desire for university education was shown to be in the minds of most pupil respondents. Well sourced registration statistics clearly showed that the Department of Education and Training (DET) in the Republic of South Africa and the Department of Education in the Ciskei had by far the greatest proportion of pupils on the HG in mathematics in comparison with the other race groups.Standard 10 (Std 10) results for two years taken separately confirmed the author's suspicion that aptitude and/or ability were not taken into consideration by most pupils, in the choice of grade. The results were not good at all. In most cases the uninformed pupil was acting alone in taking the important decision, perhaps with only peer group pressure to contend with. Three independent opinions suggested that there was probably a general feeling that Black education was already inferior, even without the syllabus differentiation hence the reluctance to register for the SG option. Furthermore, the investigation indicated that there was no adequate provision for technical education for Blacks in either the RSA or Ciskei which would provide the alternative to a university career for a pupiL Amongst other things, a mild form of pupil screening and a well structured careers guidance programme are suggested in the concluding chapters

This study was motivated by the latest revision of the mathematics syllabuses of the Cape Education Department. The most important changes to content in the Senior Secondary Mathematics Syllabus (July 1984) are the introduction of calculus and linear programming, the substitution of a section on analytical geometry for vector algebra and the recall of the remainder and factor theorems. The way in which these changes were introduced left the task of integrating them into the teaching process in the hands of individual teachers. This is a task of extreme importance. If one's classroom practice is to simply plough one's way through the syllabus, one loses many opportunities to make the study of mathematics meaningful and worthwhile. Accepting the view of the spiral nature of the curriculum where one returns to concepts and procedures at increasing levels of sophistication, one needs to identify the position of topics in this spiral and to trace their conceptual foundations. Analytical geometry is in particular need of this treatment. Similarly there are many opportunities for preparing for the introduction of calculus. If the teaching of calculus is left until the last moments of the Standard 10 year without proper groundwork, the pupil will be left with little time to develop an understanding of the concepts involved. It is the advent of calculators which presents the greatest challenge to mathematics education. We ignore this challenge to the detriment of our teaching. Taken seriously calculators have the potential to exert a radical influence on the content of curricula and examinations. They bring into question the time we spend on teaching arithmetic algorithms and the priority given to algebraic manipulation. Numercial methods gain new prominence. Calculators can even breathe new life into the existing curriculum. Their computing power can be harnessed not only to carry out specific calculations but also to introduce new topics and for concept reinforcement. The purpose of this study has been to bring about a proper integration of the new sections into the existing syllabus and to give some instances of how the calculator can become an integral part of the teaching/learning process

The fundamental question that this study set out to investigate was: Can the advantages of hypermedia be extended to curriculum materials that are for the sole use of teachers? To consider this question, three areas needed to be investigated: hypermedia (the medium); teachers (the target) and curriculum documents (the content). Hypermedia has a long history dating back to Bush (1986) who in 1945 imagined his Memex system as building information trails between ideas. However, it was not until the mid 1980s that technology caught up with the theory and hypermedia came of age. The evaluation of hypermedia documents is still in its infancy and design standards are still being formlulated. Social acceptability and usability will be of major concern in the evaluation process of hypermedia. Therefore this study needed to investigate whether this medium of presentation is socially acceptable to teachers? Advances in Information Technology (IT), both in hardware and software in the last few years have brought the potential of hypermedia to the personal computer (PC). Information, be it text, sound, graphics or video, or a mixture of these, can now be presented on the same screen and the movement between screens can be seamless. The movement between screens is no longer limited to sequential movement as it is when the information is presented in a hard copy form, but can be randomly accessed. This access allows the user to move about the information as they would move about within their own minds, that is, by association. Already commercial hypermedia products are being produced for the education and "leisure" markets. Teachers' work loads are increasing, as they take on more curriculum responsibilities, while at the same time, information is expanding at a rapid rate. The challenge today is to encourage teachers to use new information technology to overcome these problems. However, since their inception into schools fifteen years ago, computers have not delivered the results that had been expected of them. Can the access to hypermedia curriculum documents help teachers to lessen their work load and encourage them to use IT? Firstly, it is important to consider whether curriculum materials for teacher use are suitable for hypermedia presentation. The literature indicated that textual materials that are not meant to be read sequentially like a novel, arc suitable to be presented in hypermedia form. At present, curriculum materials for teachers contain the content in hard copy form but the presentation is lacking in quality. This hard copy material is expensive, hard to correct and slow to update. Hypermedia offers the potential to overcome these limitations and to provide easy access to much more information. This new medium could allow teachers for the first ti.me to truly integrate their teaching programme by enabling them to access multiple curriculum documents. The methodology used in this study was based on two types of descriptive research, survey and correlation methods. The target population for this study was all K-7 teachers using the Western Australia Mathematics syllabus within Western Australia. The instrument was a mailed survey questionnaire that consisted of five parts. The first part consisted of collecting personal data such as age and gender. The second part was the Computer Attitude Scale (CAS), designed by Loyd and Gressard (1984), and was used to measure attitudes towards learning and using computers. The third part consisted of questions that asked teachers for their views and impressions on the social acceptability and utility of the present hard copy. The fourth part consisted of questions on computer experience and use, both in and outside the classroom. The final part consisted questions on the likely acceptance and usefulness of a hypermedia copy of the syllabus. This study found that the likely medium-based anxiety for this type of application is low for the teachers sampled, with 70 percent indicating that they were likely to accept this type of application. The findings indicated that the acceptance rate increased as the teachers' positive attitude towards computers increased. Teachers that rated themselves competent at using a computer were also more likely to accept this type of application. Time spent using a computer at school showed that teachers who frequently use them at least several times a week were more likely to accept this type of application. The study also found that the majority of teachers sampled considered the ability to link the syllabus to other teaching material was very useful. Many of the problems identified by the teachers sampled concerning the usability of the present hard copy could be overcome using a hypermedia version.

Since the textbook has a prominent role in Swedish mathematics education it is of importance that textbooks has a focus on everyday life application that matches the requirements of the syllabuses and provide examples of practical usage of mathematics. The purpose of this qualitative case study of four randomly chosen textbooks in mathematics for the ninth grade, published during the 60s, 70s, 80s and 90s, from which a stratified random sample of word problems has been drawn, is to study the prevalence of word problems related to everyday life as well as to see how much of the everyday life related instructions in the syllabuses these textbooks implement. The result shows that the proportion of exercises that is made up from word problems has increased steadily from the 1960s to the 1990s in the textbooks that were analysed and at the same time has the proportion of exercises related to everyday life among the word problems decreased. The overarching impression regarding instructions related to everyday life in the syllabuses is that rather few of them have been implemented via word problems.
Då läroboken har en dominerande roll i den svenska matematikundervisningen är det av vikt att dessa böcker har ett fokus på vardagslivskopplingen som motsvarar budskapet i kursplanerna samt ger eleverna exempel på matematikens vardagliga tillämpningsområden. Syftet med denna kvalitativa fallstudie av fyra slumpmässigt valda läroböcker i matematik för årskurs 9, utgivna under 60-, 70-, 80- och 90-talet, ur vilka ett stratifierat slumpmässigt urval av textuppgifter dragits, är att studera förekomsten av vardagsrelaterade textuppgifter samt hur läroböckerna uppfyller anvisningarna i kursplanerna rörande den vardagslivsrelaterade kopplingen. Resultatet visar att andelen uppgifter som utgörs av textuppgifter har ökat stadigt från 60- till 90-talet i de studerade läroböckerna samtidigt som de vardagsrelaterade uppgifternas andel av textuppgifterna har minskat. Beträffande kursplanernas anvisningar är det övergripande intrycket att tämligen lite har implementerats via textuppgifterna.

The purpose of this study is to examine discursive change regarding mathematics and language in the Swedish primary school curriculums. The study analyzes pedagogical discourses represented within the curriculums and how they represent different pedagogical theories in relation to discursive and social practice. The analysis shows a change towards a more language oriented curriculum discourse in mathematics which still carries an intertextual and discursive heritage from the prior curriculums. In relation to the early curriculums of the Swedish primary schools the former explicit differentiation of students becomes implicit as the language skills become of increasingly importance.

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This research analyzes the constitution's movements of the University in Europe and in Brazil; the creation of Mathematics’ first course in Brazil; the constitution of the contents which make up the Linear Algebra discipline syllabus and, finally, its current discipline syllabus in the presential courses in Mathematics Degree of the Brazilian Federal Universities. The purpose of the mentioned analysis is to answer the following research question: how did occur the contents' constitution that compose the syllabus of Linear Algebra in Mathematics degrees of Brazilian Federal Universities? In order to reach this objective, the bibliographical and documentary study were used as methodology, especially classical, history and yearbooks, legislation and the syllabus. The theoretical reference is based on Historical-Cultural Theory assumptions, which is scientific, but also political (MOURA, 2017; VYGOTSKI, 1995; CEDRO, 2008; OLIVEIRA, 2002; ZANELLA, 2007), through the study of logical- historical movement – defended by Kopnin, 1978; Sousa, Panossian, Cedro, 2014; Sousa, 2004; Fazenda, 2000 – of Algebra Linear discipline constitution, according to Pires, 2006; Dorier, 2000, Silva, 2003; Moore, 1995. It was necessary to understand the reasons of its inclusion in the university context and its contribution to the professional formation of Mathematics teacher, according to Bastos (1980; 2006), Boyer (1974), Cavalari (2012), Celestino (2000), Garding (1981), Kopnin (1978), Caraça (1959), Ríbnikov (1987), Tubino (1984), Kleiner (2007), among others. It was analyzed 55 Federal Brazilian Universities that offer presential course of Degree in Mathematics. As categories of analysis were stablished the content of syllabus, the workload and the basic bibliographic reference of compulsory discipline of Linear Algebra of these institutions. As more relevant conclusions, it was observed that Linear Algebra began to be developed and taught in Universities to solve practical problems of diverse intellectuals of engineering and exact sciences, and that the transfer of its indexes in an unrelated way may contribute to the difficulties in teaching and learning process present in this discipline.
Matemática das Universidades Federais Brasileiras. O objetivo dessa análise é o de responder à seguinte questão de pesquisa: como se deu a constituição dos conteúdos que compõem a ementa da disciplina de Álgebra Linear nos cursos de Licenciatura em Matemática das Universidades Federais Brasileiras? Para atingir o objetivo, utilizou-se como metodologia o estudo bibliográfico e documental, mormente de obras clássicas, livros de história, anuários, legislação e ementas. O referencial teórico é pautado nos pressupostos da Teoria Histórico-Cultural, que se constitui científica, mas também política (MOURA, 2017; VYGOTSKI, 1995; CEDRO, 2008; OLIVEIRA, 2002; ZANELLA, 2007), mediante o estudo do movimento lógico-histórico – apregoado por Kopnin, 1978; Sousa, Panossian, Cedro, 2014; Sousa, 2004; Fazenda, 2000 – de constituição da disciplina de Álgebra Linear (PIRES, 2006; DORIER, 2000, SILVA, 2003; MOORE, 1995). Fez-se necessário compreender o que motivou a sua inserção no contexto universitário e qual a sua contribuição para a formação do professor de Matemática, com base em Bastos (1980; 2006), Boyer (1974), Cavalari (2012), Celestino (2000), Garding (1981), Kopnin (1978), Caraça (1959), Ríbnikov (1987), Tubino (1984), Kleiner (2007), entre outros. Foram analisadas 55 Universidades Federais Brasileiras que oferecem o curso presencial de Licenciatura em Matemática. Estabeleceram-se como categorias de análise os conteúdos das ementas, a carga horária e a referência bibliográfica básica da disciplina obrigatória de Álgebra Linear oferecida nestas instituições. Como principais conclusões, observou-se que a disciplina de Álgebra Linear começou a ser desenvolvida e ensinada nas Universidades para resolver problemas práticos de diversos intelectuais das engenharias e ciências correlatas, e que o repasse de seus conteúdos de forma desvinculada pode contribuir para as dificuldades do processo de ensino e aprendizagem presentes nessa disciplina.

At the time when this study began, there were nine boards offering Advanced Level Mathematics syllabuses. Some of the boards offered three,and sometimes four versions of A-level Mathematics. The study looks at these various forms from a number of different standpoints. The first of these is a consideration of the 'readability' of the question papers themselves, using the Cloze Procedure. The data is analysed by a three-way, fully crossed, analysis of variance. The work then moves on to consider the structure of the various papers. It then proceeds to analyse questions from the papers under various headings. The method is a substantial modification of a method used by the GCE examination boards in cross-moderation studies. A questionnaire was developed to explore the opinions of sixth form teachers, regarding the various versions of A-level Mathematics. The opinions of university staff were also sought. The question as to whethersome A-level courses are better preparation for university mathematics courses is addressed. Results of students at A-level and in the first year university mathematics examinations are compared. The students who participated in the 'readability' exercise were also interviewed, after looking at further questions from a selection of A-level Mathematics papers. During the course of this study, a number of boards have started to offer modular, or unit based, courses. This significant development is considered towards the end of the study. Two schools and a sixth form college were visited, each one using a different modular A-level syllabus. An account of the observations is given. The study closes with a discussion of the findings from the various themes and makes suggestions for possible improvements.

PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO
CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO
Este trabalho visa a estudar a influência da Reforma Benjamin Constant no currículo de matemática do Colégio Pedro II. Benjamin Constant foi professor e Ministro da Instrução, dos Correios e Telégrafos, em 1890-1. Ele quis reformar o ensino brasileiro, desde a escola primária até os cursos superiores. Essa Reforma teve caráter enciclopédico, inchando de conteúdos os programas das disciplinas, sobretudo os de matemática. O Colégio Pedro II foi criado em 1837-8 para servir de modelo ao ensino secundário brasileiro, que se caracterizou por dirigir-se à elite socioeconômica; na prática, o Colégio não correspondeu bem à expectativa inicial. A Reforma alterou os programas de matemática do Colégio; dentre outras modificações, introduziu neles, a partir de 1895, noções de cálculo diferencial e integral. Para analisar a influência da Reforma nos mencionados programas, o trabalho compara os currículos vigentes de 1880 a 1890 aos posteriores (1890- 1900). As bases teórico-metodológicas da pesquisa são Ivor Goodson e Jean- François Sirinelli. O trabalho visa, ainda, a caracterizar Benjamin Constant como intelectual e a examinar se e como esse seu status teria influído na Reforma. A pesquisa usa numerosos documentos: atos normativos (como os decretos que instituíram reformas educacionais no Império e no início da República), documentos curriculares (programas de matemática do Colégio Pedro II) e documentos produzidos pelo próprio B. Constant.
This paper intends to study the influence of the Benjamin Constant Reform in the Mathematics curriculum of the Pedro II School. Benjamin Constant was a Mathematics teacher and the Head of the Instruction, Mail and Telegraph Ministry (1890-1). He intended to reform all levels of Brazilian education and decreed the Reform named after him. The Reform had an encyclopedic nature, inflating the content of the programs for each subject, especially the one for Mathematics. The Pedro II School was founded in 1837-8 to serve as a model for Brazilian secondary education, which was characterized as targeted to the social/economical elite; however, the School does not seem to have fulfilled the initial expectations. The Reform altered the Mathematics syllabus of the Pedro II School; among other changes, it added, from 1895 on, notions of differential and integral calculus. To analyze the influence of the Reform in the aforementioned syllabi, this paper compares the Mathematics curriculum in use in 1880-1890 to the ones used later on (1890-1900). The theoretical and methodological bases for this research were Ivor Goodson and Jean-François Sirinelli. The paper also aims to establish Benjamin Constant as an intellectual, and to examine if and how this status would have affected the Reform. The research uses several documents: normative acts (such as the decrees that established educational reforms in Brazil, during the Empire and in the early Republic), curriculum documents (Mathematics syllabi of the Pedro II School in the 1880-1900 period) and documents produced by B. Constant himself.

The purpose of this study has been to compare the present with the future syllabus of mathematics. With a text analytical approach has questions about how the structure is different, how the look at knowledge has changed and which learning theory that characterizes the two syllabus been answered. As a theoretical framework I adopt three different learning theories, behaviourism, cognitivism and the socio-cultural perspective. The pervious research I have looked at is the concept of knowledge an the underlying theories to previous curriculum. During the increase results in the greatest difference seen in the structure. The next syllabus structure is based on one level as opposed to before when it was built on two levels of objectives, which makes the upcoming syllabus clearer and more concrete than the current syllabus in mathematics.

This study was conducted to analyse aspects of mental computation within primary school mathematics curricula and to formulate recommendations to inform future revisions to the Number strand of mathematics syllabuses for primary schools. The analyses were undertaken from past, contemporary, and futures perspectives. Although this study had syllabus development in Queensland as a prime focus, its findings and recommendations have an international applicability. Little has been documented in relation to the nature and role of mental computation in mathematics curricula in Australia (McIntosh, Bana, & Farrell, 1995,p. 2), despite an international resurgence of interest by mathematics educators. This resurgence has arisen from a recognition that computing mentally remains a viable computational alternative in a technological age, and that the development of mental procedures contributes to the formation of powerful mathematical thinking strategies (R. E. Reys, 1992, p. 63). The emphasis needs to be placed upon the mental processes involved, and it is this which distinguishes mental computation from mental arithmetic, as defined in this study. Traditionally, the latter has been concerned with speed and accuracy rather than with the mental strategies used to arrive at the correct answers. In Australia, the place of mental computation in mathematics curricula is only beginning to be seriously considered. Little attention has been given to teaching, as opposed to testing, mental computation. Additionally, such attention has predominantly been confined to those calculations needed to be performed mentally to enable the efficient use of the conventional written algorithms. Teachers are inclined to associate mental computation with isolated facts, most commonly the basic ones, rather than with the interrelationships between numbers and the methods used to calculate. To enhance the use of mental computation and to achieve an improvement in performance levels, children need to be encouraged to value all methods of computation, and to place a priority on mental procedures. This requires that teachers be encouraged to change the way in which they view mental computation. An outcome of this study is to provide the background and recommendations for this to occur. The mathematics education literature of relevance to mental computation was analysed, and its nature and function, together with the approaches to teaching, under each of the Queensland mathematics syllabuses from 1860 to 1997 were documented. Three distinct time-periods were analysed: 1860-1965, 1966-1987, and post-1987. The first of these was characterised by syllabuses which included specific references to calculating mentally. To provide insights into the current status of mental computation in Queensland primary schools, a survey of a representative sample of teachers and administrators was undertaken. The statements in the postal, self-completion opinionnaire were based on data from the literature review. This study, therefore, has significance for Queensland educational history, curriculum development, and pedagogy. The review of mental computation research indicated that the development of flexible mental strategies is influenced by the order in which mental and written techniques are introduced. Therefore, the traditional written-mental sequence needs to be reevaluated. As a contribution to this reevaluation, this study presents a mental-written sequence for introducing each of the four operations. However, findings from the survey of Queensland school personnel revealed that a majority disagreed with the proposition that an emphasis on written algorithms should be delayed to allow increased attention on mental computation. Hence, for this sequence to be successfully introduced, much professional debate and experimentation needs to occur to demonstrate its efficacy to teachers. Of significance to the development of efficient mental techniques is the way in which mental computation is taught. R. E. Reys, B. J. Reys, Nohda, and Emori (1995, p. 305) have suggested that there are two broad approaches to teaching mental computation,,Ya behaviourist approach and a constructivist approach. The former views mental computation as a basic skill and is considered an essential prerequisite to written computation, with proficiency gained through direct teaching. In contrast, the constructivist approach contends that mental computation is a process of higher-order thinking in which the act of generating and applying mental strategies is significant for an individual's mathematical development. Nonetheless, this study has concluded that there may be a place for the direct teaching of selected mental strategies. To support syllabus development, a sequence of mental strategies appropriate for focussed teaching for each of the four operations has been delineated. The implications for teachers with respect to these recommendations are discussed. Their implementation has the potential to severely threaten many teachersf sense of efficacy. To support the changed approach to developing competence with mental computation, aspects requiring further theoretical and empirical investigation are also outlined.

This study was conducted to analyse aspects of mental computation within primary school mathematics curricula and to formulate recommendations to inform future revisions to the Number strand of mathematics syllabuses for primary schools. The analyses were undertaken from past, contemporary, and futures perspectives. Although this study had syllabus development in Queensland as a prime focus, its findings and recommendations have an international applicability. Little has been documented in relation to the nature and role of mental computation in mathematics curricula in Australia (McIntosh, Bana, & Farrell, 1995,p. 2), despite an international resurgence of interest by mathematics educators. This resurgence has arisen from a recognition that computing mentally remains a viable computational alternative in a technological age, and that the development of mental procedures contributes to the formation of powerful mathematical thinking strategies (R. E. Reys, 1992, p. 63). The emphasis needs to be placed upon the mental processes involved, and it is this which distinguishes mental computation from mental arithmetic, as defined in this study. Traditionally, the latter has been concerned with speed and accuracy rather than with the mental strategies used to arrive at the correct answers. In Australia, the place of mental computation in mathematics curricula is only beginning to be seriously considered. Little attention has been given to teaching, as opposed to testing, mental computation. Additionally, such attention has predominantly been confined to those calculations needed to be performed mentally to enable the efficient use of the conventional written algorithms. Teachers are inclined to associate mental computation with isolated facts, most commonly the basic ones, rather than with the interrelationships between numbers and the methods used to calculate. To enhance the use of mental computation and to achieve an improvement in performance levels, children need to be encouraged to value all methods of computation, and to place a priority on mental procedures. This requires that teachers be encouraged to change the way in which they view mental computation. An outcome of this study is to provide the background and recommendations for this to occur. The mathematics education literature of relevance to mental computation was analysed, and its nature and function, together with the approaches to teaching, under each of the Queensland mathematics syllabuses from 1860 to 1997 were documented. Three distinct time-periods were analysed: 1860-1965, 1966-1987, and post-1987. The first of these was characterised by syllabuses which included specific references to calculating mentally. To provide insights into the current status of mental computation in Queensland primary schools, a survey of a representative sample of teachers and administrators was undertaken. The statements in the postal, self-completion opinionnaire were based on data from the literature review. This study, therefore, has significance for Queensland educational history, curriculum development, and pedagogy. The review of mental computation research indicated that the development of flexible mental strategies is influenced by the order in which mental and written techniques are introduced. Therefore, the traditional written-mental sequence needs to be reevaluated. As a contribution to this reevaluation, this study presents a mental-written sequence for introducing each of the four operations. However, findings from the survey of Queensland school personnel revealed that a majority disagreed with the proposition that an emphasis on written algorithms should be delayed to allow increased attention on mental computation. Hence, for this sequence to be successfully introduced, much professional debate and experimentation needs to occur to demonstrate its efficacy to teachers. Of significance to the development of efficient mental techniques is the way in which mental computation is taught. R. E. Reys, B. J. Reys, Nohda, and Emori (1995, p. 305) have suggested that there are two broad approaches to teaching mental computation,,Ya behaviourist approach and a constructivist approach. The former views mental computation as a basic skill and is considered an essential prerequisite to written computation, with proficiency gained through direct teaching. In contrast, the constructivist approach contends that mental computation is a process of higher-order thinking in which the act of generating and applying mental strategies is significant for an individual's mathematical development. Nonetheless, this study has concluded that there may be a place for the direct teaching of selected mental strategies. To support syllabus development, a sequence of mental strategies appropriate for focussed teaching for each of the four operations has been delineated. The implications for teachers with respect to these recommendations are discussed. Their implementation has the potential to severely threaten many teachers�f sense of efficacy. To support the changed approach to developing competence with mental computation, aspects requiring further theoretical and empirical investigation are also outlined.

Vi har under året fått larmrapporter om att Sveriges matematikundervisning är undermålig. Enligt Skolverkets statistik får vi också detta bekräftat. Undersökningen avser att ta reda på hur lärare uttrycker sig kring undervisning, elevens självbild och kunskapsbedömning när det gäller att minimera antalet elever som lämnar grundskolan utan betyg i matematik. I denna kvalitativa studie har vi valt att intervjua sex matematiklärare i grundskolans senare år. Med dessa intervjuer vill vi synliggöra tänkbara orsaker till att svenska elever tappar mark när det gäller matematikkunskaper. Vi har valt att fokusera på hur undervisning och kunskapsbedömning ser ut på skolor med högt respektive lågt antal elever som ej uppnått målen för matematik i grundskolan. Vi har dessutom valt att lägga fokus på hur elevens självbild påverkar förmågan att tillägna sig matematikkunskaper. I vårt resultat visar det sig att våra sex respondenter inte skiljer sig anmärkningsvärt i hur de undervisar och kunskapsbedömer sina elever utifrån högt eller lågt antal elever som ej uppnått målen för grundskolan i matematik. Vad vi däremot kan se är att matematiklärarna är kritiska till hur undervisningen fungerar i de tidigare åldrarna. Respondenterna ger också en tydlig bild av att vår samhällsstruktur har förändrats vilket ger dem elever med skiftande social och kulturell bakgrund. Vi har behandlat dessa iakttagelser i vår diskussionsdel där vi med stöd från olika litteratur och styrdokument ger tänkbara orsaker om hur vi kan åtgärda resultatet utifrån frågor som var ansvaret ligger men också påvisa olika faktorers samspel i den komplexa verklighet vår skola befinner sig.

We have during the year received alarming reports concerning the inferior state of the Swedish mathematics teaching. According to the statistics from the Department of Education we also get this confirmed. The survey intends to find out how teachers express themselves about teaching, the pupil's self-image, and judging of knowledge when it comes to minimizing the number of pupil's who leave elementary school without any grades in mathematics. In this qualitative study we have chosen to interview six teachers of mathematics from the later years of the elementary school. With these interviews we would like to make visible possible causes why Swedish pupils are falling behind when it comes to knowledge in mathematics. We have chosen to focus on how teaching and judging of knowledge look like at schools with a high respectively low number of pupils who haven't reached the goals set up for mathematics in elementary school. We have also chosen to focus on how the self-image of the pupil affects the ability to acquire knowledge in mathematics. Our results show that our six respondents do not in a considerable way differ in the way they teach and judge the knowledge from a point of view where a high or low number of pupils that have not reached the goals set for the elementary school is concerned. What we can see on the other hand is that the teachers of mathematics are critical about the way the teaching works in the earlier years. The respondents also provide a clear picture that the structure of our society has changed which gives them pupils of shifting social and cultural background. We have treated these observations in our discussion section where we with support from different literature and steering documents present thinkable causes about how we can take measures from questions concerning where the responsibility lies but also show the interplay between different factors in the complex reality where our school is.

There are numerous problems associated with the teaching of Euclidean geometry at secondary schools today. Students do not see the necessity of proving results which have been obtained intuitively. They do not comprehend that the validity of a deduction is independent of the 'truth' of the initial assumptions. They do not realise that they cannot reason from diagrams, because these may be misleading or inaccurate. Most importantly, they do not understand that Euclidean geometry is a particular interpretation of physical space and that there are alternative, equally valid interpretations. A possible means of addressing the above problems is tbe introduction of nonEuclidean geometry at school level. It is imperative to identify those students who have the pre-requisite knowledge and skills. A number of interesting teaching strategies, such as debates, discussions, investigations, and oral and written presentations, can be used to introduce and develop the content matter.
Mathematics Education
M. Sc. (Mathematics)

Orientação: Maria de Lurdes de Pina Manique Ferreira Braga de Figueiredo Pereira
O presente estudo tem, como objetivo, analisar e comparar criticamente os últimos programas de Matemática, confrontando o programa homologado em 1991 (DGEBS, 1991) com o de 2007 (Ponte et al., 2007), tendo em conta o processo de ensino-aprendizagem, a diversificação de estratégias, atividades e instrumentos, a autonomia da aprendizagem, bem como a auto e a heteroavaliação. Durante dois anos letivos, analisamos o impacto da implementação do programa de 2007 (Ponte te al., 2007) junto de alunos e professores. O processo de investigação é de natureza qualitativa, pois envolve a obtenção de dados descritivos, provenientes do contacto do investigador com a situação em estudo. Na recolha de dados privilegiamos não só as tarefas matemáticas, mas também recorremos a outros instrumentos – relatório matemático, teste em duas fases, diário de bordo, entrevistas a professores e ao relatório final de avaliação do PM II/PMEB. A análise dos dados revelou que, de uma forma geral, perante diferentes abordagens, os alunos tiveram uma aprendizagem mais eficaz, mostrando-se mais autónomos e competentes. Tal foi comprovado em diversos momentos de auto e heteroavaliação, dado que os alunos se mostraram capazes de refletir individualmente, em pares ou em grupo, numa perspetiva construtivista de avaliação reguladora.
The aim of the present study is to critically analyze and compare the latest programs of Mathematics, confronting the program approved in 1991 (DGEBS, 1991) to with the one from 2007 (Ponte et al., 2007), taking into account the process of teaching and learning, the variety of strategies, activities and tools and learning autonomy, as well as self and hetero assessment. For two academic years we have analyzed the impact of the implementation of the 2007 program (Ponte et al., 2007) with students and teachers. The research process is of qualitative nature because it involves obtaining descriptive data from the contact of the researcher with the situation under study. In collecting data we focused not only on the mathematical tasks, but we also resorted to other instruments – mathematical report, test in two phases, the logbook, teacher interviews and the final report of the PM II/PMEB. The data analysis revealed that, in general, faced with different approaches, students had more effective learning, being at the same time more autonomous and competent. This was confirmed on several moments of self and hetero assessment, since the students were able to reflect individually, in pairs or in groups, in a constructivist perspective for regulatory assessment.

The teaching of probability in schools provides a good opportunity for examining how a new topic is integrated into a school curriculum. Furthermore, because probabilistic thinking is quite different from the deterministic thinking traditionally found in mathematics classrooms, such an examination is particularly able to highlight significant forces operating within educational practice. After six chapters which describe relevant aspects of the philosophical, cultural, and intellectual environment within which probability has been taught, a 'Broad-Spectrum Ecological Model' is developed to examine the forces which operate on a school system. The Model sees school systems and their various participants as operating according to general ecological principles, where and interprets actions as responses to situations in ways which minimise energy expenditure and maximise chances of survival. The Model posits three principal forces-Physical, Social and Intellectual-as providing an adequate structure. The value of the Model as an interpretative framework is then assessed by examining three separate aspects of the teaching of probability. The first is a general survey of the history of the teaching of the topic from 1959 to 1994, paying particular attention to South Australia, but making some comparisons with other countries and other states of Australia. The second examines in detail attempts which have been made throughout the world to assess the understanding of probabilistic ideas. The third addresses the influence on classroom practice of research into the teaching and learning of probabilistic ideas. In all three situations the Model is shown to be a helpful way of interpreting the data, but to need some refinements. This involves the uniting of the Social and Physical forces, the division of the Intellectual force into Mathematics and Mathematics Education forces, and the addition of Pedagogical and Charismatic forces. A diagrammatic form of the Model is constructed which provides a way of indicating the relative strengths of these forces. The initial form is used throughout the thesis for interpreting the events described. The revised form is then defined and assessed, particularly against alternative explanations of the events described, and also used for drawing some comparisons with medical education. The Model appears to be effective in highlighting uneven forces and in predicting outcomes which are likely to arise from such asymmetries, and this potential predictive power is assessed for one small case study. All Models have limitations, but this one seems to explain far more than the other models used for mathematics curriculum development in Australia which have tended to see our practice as an imitation of that in other countries.
Thesis (Ph.D.)--Graduate School of Education and Department of Pure Mathematics, 2001.