The aim of the study was to investigate what influence the use of computers using MS Excel and RJS Graph software has on grade 11 Eritrean students' understanding of functions in the learning of mathematics. An empirical investigation using quantitative and qualitative research methods was carried out. A pre-test (task 1) and a post-test (task 2), a questionnaire and an interview schedule were used to collect data. Two randomly selected sample groups (i.e. experimental and control groups) of students were involved in the study. The experimental group learned the concepts of functions, particularly quadratic functions using computers. The control group learned the same concepts through the traditional paper-pencil method. The results indicated that the use of computers has a positive impact on students' understanding of functions as reflected in their achievement, problem-solving skills, motivation, attitude and the classroom environment. / Educational Studies / M. Ed. (Math Education)

The study was an investigation into the teaching decisions that mathematics teachers make over time. I view a mathematics classroom as a didactical system of teacher, learners and content within an educational institution, where content is the material that brings teachers and learners together. Within such a system I view the teacher's role as making content available to learners. Prior research has often investigated the teacher's role by comparing teaching practices nationally or internationally, but these comparisons have tended to use the lesson as the unit of analysis. I propose that how teachers make content available can change over the course of a series of lessons and so my study used the lesson series as the unit of analysis. I purposefully designed the study so that it involved four cases, which allowed me to explore the role of the teacher and the topic in how content was made available. To investigate how teachers made content available to learners in each case, I developed an analytical approach from which I could study the modes of teacher interaction that featured across the lesson series, the forms of mathematical content made available and the sequencing of these forms. Attending to forms of content - rather than content itself - allowed for comparison of teaching of different topics. This original analytical approach represents a contribution to both mathematics education and mixed methods research. Within this small sample of cases, quantifiable differences were identified in how content was made available between classwork and seatwork, from lesson to lesson and between cases. Between-case differences in the nature of teaching 'between-the-desks' during seatwork were also identified. These differences illuminated teaching decisions to which teachers and classroom researchers may not routinely attend. The findings therefore contribute - and identify additional lines of enquiry that might contribute further - to a more extensive understanding of teaching practices.

This is a report on research conducted in Lebowa (Northern Province) Primary Mathematics Project schools. In view of the high failure rate of matric students, the researcher believes that it is necessary that ways should be devised to improve mathematics understanding from the first level of schooling. A research study was made of constructivist and investigative teaching and learning methods as employed by teachers in a number of primary schools in the area. The researcher believes that investigative and constructivist teaching approaches produce better results than the traditional approach. He further believes that children learn better in a co-operative non-threatening classroom environment. A pilot study was made with two experimental schools and two control schools. The schools were selected from both urban and rural areas. The experimental schools are operating under the PMP and the control schools are not. The experimental schools are also supported by expert teachers called key teachers. Many of these key teachers have attended courses at Leeds University, while others have been trained locally in the theory and practice of constructivist and investigative teaching and learning. Chapter 3 illustrates clearly what is done in the Project schools. The methodology employed in the research included qUestionnaire responses from 174 teachers. Written tests by four schools (350 pupils), and interviews with 55 people, including directors of education, inspectors, principals, teachers and parents. Observations in classes were also done. Another questionnaire was given to 484 pupils. The tests were mitten on two occasions: an initial test was written during November 1993, the second year of the PMP project, while a second test was written the following year. Data collected was analysed and positive results obtained. The results from the tests indicated that pupils in experimental schools where constructivist and investigative approaches are used, perform better than those from schools where the traditional approach is still used. They indicated that children in PMP schools develop a better understanding of mathematics. This would seem to indicate that the constructivist and investigative approach to teaching produces better results than the traditional approaches. The reader will find graphs indicating the results and their analysis in Chapters 4 and 5. It is recommended by the researcher that: - Constructivist and investigative teaching and learning methods be introduced to all schools. - The services of key teachers be supported by the Department - The Department should equip all schools with the necessary materials for proper teaching and learning, or provide materials for schools to make their own teaching aids. - Teachers be involved with materials production where they are given guidance on how teaching aids can be made. - The curriculum for primary school mathematics be revised and changed, especially in view of the fact that at present it does not cater for local needs. It was planned by whites and it caters mainly for those with an European cultural background.

M.Ed. (Subject Didactics) / Research shows that "the aims of secondary school's teaching of mathematics are often not realized with many pupils leaving the school with passive knowledge of mathematics" (H.S.R.C. 1981:8). This means that knowledge of mathematical facts are reproduced on demand, instead of active mathematical knowledge " which is congruent with the aims of teaching secondary mathematics" (Crooks, 1988 : 6/7). Active knowledge of mathematics implies and characterised by the understanding of concepts, principles that underlie facts and ideas and principles and concepts that are connected to each other" (Entwistle & Entwistle, 1992 : 2). Active knowledge also enables pupils to act intellectually independently. One reason for the previously mentioned predicament is that "teaching often encourage passive knowledge because the teaching practice of mathematics teachers are often not in accordance with their educational aims" (Gravett, 1994 :6). Thus, a discrepancy exists between teacher's intentions of teaching mathematics and their conduct during teaching. It can be argued also that teachers teach mathematics in the classroom but that the pupils not always effectively learn. It is from the perception above that a constructivistic view of learning as a conceptual change underlies the idea that teaching "as the creation of a classroom context conducive to learning" (Strike & Posner, 1985:117). Biggs (1993 : 74) thus argues that "if knowledge is constructed, rather than recorded as received, it does not make sense to think of teaching as imparting knowledge, but rather as creating learning environments that enhance the process of mathematical knowledge construction". Russell (1969: 14) mentions that "mathematics is a subject in which we never know what we are talking about, nor whether what we are saying is true". The views, amongst others Oosthuizen, Swart and Gildenhuys (1992:2) see mathematics as "an essential language of a creative but deductive process which has its origins in the problems of the physical world", In the light of this, the origin of mathematics in the real world, it can be argued that from a "constructivistic perspective, mathematical learning is an active process by which pupils construct their own mathematical knowledge in the light of their existing knowledge and through interaction with the world around them" (Gravett, 1994 : 6/7). "Construction, not absorption or unfocused discovery, enables learning" (Leder, 1993 : 13). Mathematics is not something discovered by mankind, mathematics is a creation of mankind and is transmitted and changed from one generation to the next.

M.Ed. / The importance of a multicultural curriculum for communities with a multiracial, ethnic and diverse constititution cannot be overemphasised in a modern approach to education. In the South African context their exists an urgent need for a mathematics curriculum which is able to accommodate the specific cultural background of every individual. The reality of cultural diversity in South Africa emphasises the importance of the implementation of a multicultural approach to the teaching of Mathematics. Mathematics is a social process which constitutes a fundamental part of education. It remains a dynamic and living cultural product, while remaining part and parcel of the social construction of a community. The recognition of this reality creates a viable foundation for a multicultural approach to the teaching of Mathematics. The Mathematics curriculum can be implemented to emphasise a person's own culture and to provide information regarding a community, as well as that which is relevant to its multicultural character. Pupils represent diverse cultural-, class- and linguistic backgrounds. Other aspects which have to be considered in the creation of a multicultural curriculum are the different approaches, points of view and thinking patterns of pupils. In addition to this, there remains a difference between the levels of education of parents as well as the premium they place on literacy. The degree in which multicultural education will realise, however, depends mainly on the teacher's attitude and classroom skills. The primary aim of this study is to conduct a research into the different ways in which a teacher can forge a constructive link with children from diverse ethnic communities, via his/her own perceptions, educational aims and strategies, usage of language in the classroom, as well as classroom skills and techniques. Themes and practical examples which also exhibits multicultural characteristics, are included and can be implemented by the teacher in his/her own classroom techniques. Teachers have a professional responsibility to remove all elements of prejudice from the classroom, as well as to acknowledge and respect the diversity of cultures.

In many cases topics for research are presented to a student in capsulated, clearly defined terms, either as the result of his own experience or as a request by some institution. In other cases the topic takes shape but gradually, very often as the result of a student slowly becoming aware of a field of research through repeated observation of related factors. In some cases the aim of research is to determine whether there is a relationship between various factors; or disprove such in others the main aim may be to prove relationship in unequivocal terms. A large body of research is, however, concerned mainly with the statement of a problem or the finding of facts. The work presented in the following pages can be regarded as falling in the last-mentioned category. A vague suspicion was gradually strengthened by observation and experience until it finally crystallised to form the basis of the research. Facts and figures were gathered and analysed and some conclusions drawn, conclusions that gave rise to more questions and problems than fall within the scope of this work. It was, in fact, found that this research raised more questions than were answered by it and served mainly to underline the magnitude of the problem rather than to offer a solution.

M.Ed. (Curriculum Studies) / Mathematics is felt to be one of the most important subjects in the school curriculum by educators, parents and society at large. As Bishop (1988: 1) puts it: "Anyone who cants to get on today, needs to study mathematics,. and preferably computing too." Diab (1987) also argues along these lines and says that mathematics has been a key subject in the school curriculum and it is still a basic ingredient in the educational make-up of a person who wishes to find his place in today's increasingly technological world. Mathematics promotes the development of the mental, social, emotional and occupational life of a person (Grove & Hauptfleisch, 1979: 228). ~ardner, et al. (1973: 18) outline the reasons why a person should learn mathematics. They argue that, inter alia, mathematics should be learned and taught at schools for the following reasons: ii) Living. Mathematics for living, refers to those aspects of mathematics which an individual must know in order to function adequately as a member of society. At primary level this clearly includes such topics as number, time, money, length and weight. More and more information is presented in statistical form and this trend will continue. An educated person must be able to evaluate and interpret such data effectively if he is to playful and useful part in society. Most people will, at some time or other, be involved with such complex activities as house purchase and insuarance...

It is common for students to make mistakes while solving mathematical problems. Some of these mistakes might be caused by the false ideas, or misconceptions, that students developed during their learning or from their practice. Calculus courses at the undergraduate level are mandatory for several majors. The introductory course of calculus—Calculus I—requires fundamental skills. Such skills can prepare a student for higher-level calculus courses, additional higher-division mathematics courses, and/or related disciplines that require comprehensive understanding of calculus concepts. Nevertheless, conceptual misunderstandings of undergraduate students exist universally in learning calculus. Understanding the nature of and reasons for how and why students developed their conceptual misunderstandings—misconceptions—can assist a calculus educator in implementing effective strategies to help students recognize or correct their misconceptions. For this purpose, the current study was designed to examine students’ misconceptions in order to explore the nature of and reasons for how and why they developed their misconceptions through their thought process. The study instrument—Calculus Problem-Solving Tasks (CPSTs)—was originally created for understanding the issues that students had in learning calculus concepts; it features a set of 17 open-ended, non-routine calculus problem-solving tasks that check students’ conceptual understanding. The content focus of these tasks was pertinent to the issues undergraduate students encounter in learning the function concept and the concepts of limit, tangent, and differentiation that scholars have subsequently addressed. Semi-structured interviews with 13 mathematics college faculty were conducted to verify content validity of CPSTs and to identify misconceptions a student might exhibit when solving these tasks. The interview results were analyzed using a standard qualitative coding methodology. The instrument was finalized and developed based on faculty’s perspectives about misconceptions for each problem presented in the CPSTs. The researcher used a qualitative methodology to design the research and a purposive sampling technique to select participants for the study. The qualitative means were helpful in collecting three sets of data: one from the semi-structured college faculty interviews; one from students’ explanations to their solutions; and the other one from semi-structured student interviews. In addition, the researcher administered two surveys (Faculty Demographic Survey for college faculty participants and Student Demographic Survey for student participants) to learn about participants’ background information and used that as evidence of the qualitative data’s reliability. The semantic analysis techniques allowed the researcher to analyze descriptions of faculty’s and students’ explanations for their solutions. Bar graphs and frequency distribution tables were presented to identify students who incorrectly solved each problem in the CPSTs. Seventeen undergraduate students from one northeastern university who had taken the first course of calculus at the undergraduate level solved the CPSTs. Students’ solutions were labeled according to three categories: CA (correct answer), ICA (incorrect answer), and NA (no answer); the researcher organized these categories using bar graphs and frequency distribution tables. The explanations students provided in their solutions were analyzed to isolate misconceptions from mistakes; then the analysis results were used to develop student interview questions and to justify selection of students for interviews. All participants exhibited some misconceptions and substantial mistakes other than misconceptions in their solutions and were invited to be interviewed. Five out of the 17 participants who majored in mathematics participated in individual semi-structured interviews. The analysis of the interview data served to confirm their misconceptions and identify their thought process in problem solving. Coding analysis was used to develop theories associated with the results from both college faculty and student interviews as well as the explanations students gave in solving problems. The coding was done in three stages: the first, or initial coding, identified the mistakes; the second, or focused coding, separated misconceptions from mistakes; and the third elucidated students’ thought processes to trace their cognitive obstacles in problem solving. Regarding analysis of student interviews, common patterns from students’ cognitive conflicts in problem solving were derived semantically from their thought process to explain how and why students developed the misconceptions that underlay their mistakes. The nature of how students solved problems and the reasons for their misconceptions were self-directed and controlled by their memories of concept images and algorithmic procedures. Students seemed to lack conceptual understanding of the calculus concepts discussed in the current study in that they solved conceptual problems as they would solve procedural problems by relying on fallacious memorization and familiarity. Meanwhile, students have not mastered the basic capacity to generalize and abstract; a majority of them failed to translate the semantics and transliterate mathematical notations within the problem context and were unable to synthesize the information appropriately to solve problems.

Bibliography: pages 82-84. / This dissertation is concerned with aspects of the role of the textbook in school mathematics. An attempt is made to uncover control strategies used by the teacher in textbook use in the classroom, and those implicit in a mathematics textbook. It is argued that these forms of regulation place constraints on the transformative role sometimes attributed to textbooks. The following research question is addressed: how does the teacher recruit the textbook in the classroom, how is he/she 'recruited' by it and how are both recruited by school mathematics? A case study methodology is described, involving a video-recording of a fifty minute mathematics lesson and a follow-up interview with the teacher. Transcriptions are used and a fine-grained analysis of data is attempted. A literature survey examines other research in the areas of content selection, content control and content expression. Content selection refers to choices and omissions, content control refers to sequencing, pacing and authority in the pedagogic relationship, and content expression includes verbal and textual modes of expressing content. Theoretical ideas are drawn from Bernstein (1976, 1991, 1993) and Dowling (1993). Although these works are methodologically different, they both describe aspects of regulation and control. Of particular interest are Bernstein's notions of classification and framing, and Dowling's ideas on discourse and procedure. The hypothesis is put forward here that there is a dialectical relationship involving the positioning of teacher and textbook. The teacher recruits the textbook to regulate pupils and knowledge, but s/he is at the same time constrained by strategies implicit in the textbook. In other words the teacher both positions and is positioned by the textbook. Both in tum are positioned by school mathematics. The data analysis examines the 'how', 'what' and 'who' of control. It considers the regulation of speech, silence, working and listening, as well as the sequencing, pacing, selecting, presenting and authorising of content. It argues that the teacher both recruits and is 'recruited' by the textbook, and that although the framing is strong and the teacher has a high degree of control in the pedagogic relationship, the classification is also strong and the teacher lacks control over what she can teach and the relationship between contents. The research concludes by suggesting that the transformative role sometimes attributed to the textbook is problematic. The strategies of regulation and control operating in the classroom, implicit in the textbook and in school mathematics, limit the possibilities of how textbooks can be used by the teacher and constrain transformation to a significant degree.

Although proof has long been viewed as a cornerstone of mathematical activity, incorporating the mathematical practice of proving into classrooms is not a simple matter. In this dissertation I aim to advance research on proof by addressing this issue. In particular, I explore the role proof might play in promoting the learning of mathematics in the classroom. I do this in a series of three articles (organized as three chapters), which are preceded by an introductory chapter. The introductory chapter situates the remaining chapters in the context of mathematics education research. In the second chapter I explore what the literature on proof tells us about what role proof might play in the promotion of learning in the mathematics classroom. In this chapter I also compare the ways in which proof is purported to promote learning in the mathematics classroom with the roles it is purported to play in the field of research mathematics. In the third chapter I look at empirical data to explore ways engaging in proof and proving might create opportunities for student learning. In particular, my analysis led me to focus on how identifying and reflecting on the key idea of a proof can create opportunities for learning mathematics. The final chapter is an article for a practitioner journal and discusses implications for practice based on the two preceding articles.