According to recent research a focus in teaching mathematics to children is the development of problem solving abilities. Problem solving means the process of applying mathematical knowledge and skills to unfamiliar situations. A case study was done using a problem solving approach to the learning and teaching of mathematics with a sample of teachers registered at the Umlazi College For Further Education. These teachers were familiar with the traditional approach of teaching mathematics through drill and practice methods. The new syllabus that is to be implemented emphasises a problem solving approach to the teaching of mathematics. This study set out to implement a problem solving approach with primary school mathematics teachers so that they would be someway prepared for the innovations of the new syllabus. Workshops were conducted using an action research approach with discourse and practice leading to reconstruction with improvements. Early theorists like Piaget and Bruner offered ways of understanding children's learning, to help the teacher develop his teaching. Dienes introduced an element of play and Dewey spoke of the importance of experience. Dienes and Dewey show the first positive signs of recognising the importance of social interaction in the learning situation. Social interaction lays emphasis on language and discussion in the mathematics classroom. A social constructivist model of teaching and learning was used for the research. This research includes a study of the established ideas on developing a problem solving approach to mathematics teaching. These ideas were incorporated into the workshops that the group of teachers attended. During the workshops teachers were gradually exposed to the essence of problem solving techniques through much group discussion and doing practical exercises, which they could then implement in their classes. The teachers reported back at each subsequent workshop. A non-participant observer evaluated the development at the workshops. The workshops' success was evident from the change in the teachers' attitudes and behaviour as well as their feedback of what transpired in the classroom. They reported on the change in their roles as information suppliers to facilitators where the thinking process was focused on, rather than the importance of a correct answer. In the workshops the teachers themselves moved from passive listeners to active participants. It would appear from this preliminary investigation that through using a problem solving approach in workshops, inservice teachers can benefit constructively from this approach and will attempt to use it in their own teaching.

This half-thesis gives an overview of the influence of the microcomputer on the way in which mathematics is done, taught and learnt. The nature of mathematics and the nature of the tutor, tool and tutee modes of microcomputer usage are discussed as background. A case is made for the use of action research methods and a classroom-based curriculum development model to facilitate innovation and the integration of the microcomputer into the mathematics classroom. A curriculum development cycle of situation analysis. planning, trial and evaluation is advocated. This approach is used to develop a microcomputer-based course aimed at enhancing the reasoning skills of standard 7 pupils. Pupils, working in groups of three, interact with the PROLOG system to build up databases of facts and rules. The microcomputer is used in tutee mode. In "teaching" this tutee, pupils discover the need for formal language and logical reasoning. Active learning is promoted by pupils' interaction with the PROLOG system and by discussions within groups. In this environment the teacher becomes a consultant and constructive critic rather than a lecturer. Findings suggest that the microcomputer plays an important role in terms of pupil motivation and that the microcomputer-based course enables pupils to experience formal language usage and logical reasoning·as relevant activities. Pupil databases provide evidence of the pupils' ability to make appropriate use of rules and to distinguish between and-conditions and or-conditions. The objective of making pupils think was largely achieved. It is recommended that the course be incorporated in the standard 1 or standard 8 curriculum to complement or replace parts of the Euclidean geometry sections as a vehicle for developing logical reasoning skills. Suggestions for the further use of the microcomputer as an investigative tool in mathematics classes and for further microcomputer-inspired courses are also made. The provision of appropriate training to enable teachers to make effective and innovative use of the microcomputer in mathematics lessons is advocated.

The proliferation in the number of schools offering junior secondary education in Namibia since independence in 1990 has led to an increase in the number of learners in the classroom and has created a wide range of mathematical proficiency among learners entering senior secondary education in grade 11. This broad range of basic mathematical ability among these learners, together with increasing classroom numbers has caused problems for the senior secondary mathematics teachers (Batchelor, 2004). The study shows that diagnostic testing can prove to be useful in assessing learners’ mathematical preparedness by identifying learners’ areas of weakness, which have hindered their mathematics learning and performance. Taking the results of a diagnostic test into consideration could help teachers cater for their learners who need remediation classes as early as possible before extending the mathematics curriculum. Setting and using diagnostic testing requires careful consideration; there are many pitfalls that are highlighted in this research. These include question coverage and general analysis of category totals.

The Namibian curriculum favours knowledge with understanding and application of knowledge and skills, because facts that are learned with understanding are easier to remember. This made learners to be able to represent mathematical situations in different ways and for different purposes. It also motivates teachers to contextualise the content to make mathematics teaching and learning more interesting and enjoyable to teachers and learners. However, some mathematics teachers are challenged by being required to teach multigrade classes, due to a massive response to improved access to education.Multigrade teaching is seen as a difficult practice especially when teachers are not well prepared to teach combined grades. This case study focused on the teaching of mathematics at Upper Primary Phase. An Interpretive paradigm was used to understand the teachers’ experiences and their teaching ethods. Hence the study aimed to investigate how teachers deliver their mathematics lessons in multigrade classrooms, considering the good practices and challenges that may occur. The findings revealed that teachers mostly use their monograde pedagogical knowledge,resulting in using quasi-monograde with a common timetable approach when teaching mathematics in multigrade settings. Although learners have access to the common mathematics syllabus, the quality of teaching the subject in multigrade classrooms seems to be affected due to the lack of teacher training in multigrade teaching. Also, equity and democracy need to be considered more than it currently is in terms of resource distribution to multigrade schools. Good practice of multigrade teaching existed in building on lower grade competencies, introducing lessons with common activities, as well as the concentration on lower grades which make learners independent. However, insufficient time, lack of knowledge in multigrade and curriculum knowledge in particular seem to challenge teachers and learners. Therefore, multigrade teachers need to be part of curriculum development for them to master the syllabus. These teachers will assist in developing multigrade resources for integration, contextualisation and more enrichment activities for high achievers.

Orientador: Miriam Godoy Penteado / Banca: Andreia Osti / Banca: Luciano Feliciano de Lima / Resumo: A presente pesquisa busca compreender "O que dizem os idosos que participam de atividades oferecidas na universidade, sobre a sua relação com a Matemática?". As pessoas da Terceira Idade foram questionados sobre seu envolvimento com a Matemática, no passado, no presente e no futuro. É uma pesquisa de caráter qualitativo que se insere no campo da inclusão social, com o objetivo de identificar as representações sociais (RS) de idosos que participam de atividades oferecidas na universidade, sobre a Matemática. Com base na teoria das Representações Sociais, de Moscovici (2005) foram realizadas entrevistas semiestruturadas com pessoas da Terceira Idade que frequentavam atividades oferecidas na Unesp, Câmpus de Rio Claro, envolvendo temas como escola, cotidiano, trabalho, qualidade de vida e futuro. A análise dessas entrevistas permitiu a identificação das seguintes representações sociais: A Matemática está em tudo; A Matemática ajuda na Qualidade de Vida; É bom para o idoso conhecer Matemática; A relação com a Matemática melhora com o passar do tempo; e A Matemática é para poucos. Os resultados demonstram que as RS dos idosos estão baseadas em informações que são advindas do senso comum. Palavras que estão presentes no mundo acadêmico da Matemática e da Educação Matemática, o universo reificado, são incorporadas no discurso das pessoas sem conhecer o contexto em que foram criadas. A presença da Matemática em suas vidas foi apresentada na realização de tarefas do dia a dia, evidenc... (Resumo completo, clicar acesso eletrônico abaixo) / Abstract: The present research aims to understand "What do the elderly people, who participate in activities offered at university, say about their relationship with Mathematics?" These elderly people were questioned about their involvement with Mathematics in the past, present and future. It is a qualitative research which is inserted in the social inclusion field with the objective to identify as social representations (SR) of elderly people who participate in activities offered at the University about Mathematics. Based on the Social Representations theory from Moscovici (2005), semi-structured interviews were conducted with Third Age people whom attended activities offered at Unesp, Rio Claro Campus, involving themes such as school, daily life, work, quality of life, and future. These interviews analysis allowed identification of the following representations: Mathematics is in everything; Mathematics helps in life quality; It is good for the elderly to understand Mathematics; The relationship with Mathematics improves over time; and Mathematics is for a few. The results demonstrate that SR of the elderly people are based on information that comes from traditional wisdom. Words that are present at the academic world of Mathematics and Mathematics Education, the reified universe, are incorporated into the discourse of people without knowing the context in which they were created. The presence of Mathematics in their lives was presented in the accomplishment of the day to day tasks, ... (Complete abstract click electronic access below) / Mestre

Orientador: Irineu Bicudo / Banca: Carlos Roberto de Moraes / Banca: Gustavo Barbosa / Banca: Henrique Lazari / Banca: Renata Geromel Meneghetti / Resumo: Essa narrativa tenta reconstituir a história de vida de Benedito Castrucci. Trouxe aspectos da vida pessoal, a origem dos seus antecedentes, algumas circunstâncias da infância, juventude e do convívio familiar com esposa e filhos. A sua formação acadêmica na educação básica e superior foi retratada. Revelou as diversas escolas que frequentou, indicando que recebeu elevado padrão de ensino, conhecimento específico e cultural de vanguarda. Identificou algumas características do processo educativo, da relação com professores e do seu rendimento escolar. A trajetória profissional como docente por diversas instituições exemplificam sua dedicação por muitos anos ao ensino e aprendizagem da matemática, na formação de jovens, professores de matemática e também engenheiros. Como pesquisador e educador, contribuiu diretamente para o desenvolvimento das atividades de pesquisa em matemática e na estruturação e consolidação da Educação Matemática brasileira. Sua obra destinada ao estudo e ensino da geometria aponta que ele foi um dos maiores autores e investigadores desse tema no Brasil / Abstract: This narrative tries to reconstitute the life history of Benedito Castrucci. Brought aspects of personal life, the source of his antecedents, some circumstances of childhood, youth and family living with his wife and children. His academic background in basic and higher education was portrayed. Revealed the various schools he attended, indicating that received a high standard of education, specific knowledge and cultural. Identified some characteristics of the educational process, the relationship with teachers and school performance. The professional career as a teacher by various institutions exemplify his dedication for many years to teaching and learning of mathematics, on the training of young people, teachers of mathematics and engineers also. As a researcher and educator, contributed directly to the development of research activities in mathematics and in structuring and consolidation of Brazilian Mathematics Education. His work for the study and teaching of geometry points out that he was one of the greatest authors and researchers of this theme in Brazil / Doutor

This study investigates the types of mathematics teaching aids available at both public and private secondary schools in Windhoek. The study characterises their usage and source as well as teachers’ perceptions towards the use of such teaching resources in the Mathematics classroom. The study is grounded in an interpretive paradigm and employed a mixed methods approach to generate both quantitative and qualitative data in two sequential phases. Phase 1 of the research process, which involved 75 Mathematics teachers, took the form of an audit of the availability and use of teaching aids at 25 secondary schools in Windhoek. A case study methodology was adopted in Phase 2 which focused on five purposively selected schools that displayed different characteristics in terms of the availability of teaching resources. The study found that the majority of teachers at secondary schools in Windhoek have a positive attitude towards the importance and role of teaching aids in Mathematics, seeing them as promoters of hands-on engagement, visual reasoning, active participation and motivation amongst learners. However, in some instances schools are underresourced with respect to certain types of teaching aids, specifically graph boards, geoboards, geometric models and computers. A need for appropriate in-school support on the use of teaching aids was also identified

The influence of modern technology on education is becoming more noticeable and has implications for the curriculum and the teaching methods of mathematics. The microcomputer can be used effectively as a powerful teaching and learning aid within the mathematics classroom. This study considers the role of the computer as an aid to teachers and pupils in the teaching and learning of mathematics and shows it as having great potential. At present relatively few schools in South Africa are using the computer as a teaching aid in the mathematics classroom. The researcher proposes that some reasons are a shortage of suitable programs, the fact that few teachers have adequate skills in educative uses of the computer and most teachers are not skilled in programming techniques. Based on this assessment of the problem, spreadsheet programs related to some mathematics lessons, together with teaching notes and pupils' material have been developed. Besides describing the potential of the computer in mathematics, this study outlines the development of three spreadsheet packages and suitable teaching methodologies used for each package, and assesses an action research investigation undertaken by the researcher, teachers and pupils when using these spreadsheet packages in six classes. The findings of the investigation are most encouraging. The overall conclusion is that computer spreadsheet packages can assist the teacher in making the learning of mathematics more effective, more interesting and more enjoyable.

There is a variety of possible ways in which computers can be used to enhance mathematics education. This thesis attempts to identify, analyse and classify these possibilities, particularly at the secondary school level. It describes and exemplifies applications ranging from drill-and-practice through games and simulations to problem solving by computer programming. Software evaluation procedures are considered in some depth. Illuminative evaluations of various items of software and there classroom use are reported. The underlying methodology is small-scale action research. Insights gained during the process of investigating each class of software lead to the eventual formulation of a scheme for classifying mathematics education software by means of 'multidimensional attributes'. It is contended that this scheme will help mathematics teachers to make well informed and sound professional judgements regarding the evaluation and use of computer programs for teaching/learning purposes. Also, it is hoped that this scheme and the thesis as a whole will contribute towards the establishment of well founded standards and procedures for software development in the field of mathematics education. Several implications of the computer for the mathematics curriculum in South Africa are suggested. A note of caution is sounded regarding possible detrimental effects of the computer and several questions requiring further research are posed. A recommendation arising from the thesis is that in-service training courses concerning computer applications in mathematics education should be run for secondary school teachers.

The investigator's experience of teaching mathematics at a college of education since 1983 has reinforced his conviction that trainee students come to college with significant gaps, weaknesses and faults in their (mathematical) conceptual structures, probably as a result of shortcomings in the mathematics teaching to which they have been exposed. The theme of this investigation is thus a natural choice that appeared to be of immediate relevance to secondary school mathematics teaching. The analysis of the issue leads to a unified perspective: the problem is placed in a theoretical framework where Bruner [spiral curriculum], Ausubel [structured learning] and Skemp [relational understanding] are brought together. How the curriculum, textbooks and examination influence school mathematics teaching is examined in some depth and the consequences investigated. Two specific topics, viz. the generalised Pythagorean relation and absolute value are investigated in relation to published work, curriculum and textbooks, and each (topic) is presented as a unifying theme in secondary mathematics to standard 9 pupils. The classroom exercise is assessed to test the hypothesis that structured, integrated presentation around a spiral curriculum promotes "relational understanding". Analysis of results supports the hypothesis.