This thesis concerns the teaching of mathematics at university level, with a particular focus on the teaching of the function concept. The main aim of the thesis is describing and analysing the teaching practices of university mathematics teachers regarding the function concept, and how this concept is constituted through these practices. To this end, video recordings of lectures by seven mathematics teachers at three Swedish universities were analysed using a discursive perspective, Sfard’s commognitive framework. The observed teaching was traditional in form, with teachers using “chalk talk” – simultaneously talking and writing on the board. The results show that the teaching practices of the teachers belong to two distinct but intertwined discourses – a mathematical discourse, and a discourse of mathematics teaching. Classifications of important aspects of these discourses are presented, and it is found that the teachers’ discursive practices, while sharing overall form, still display considerable differences. Other results include an analysis of the levels of objectification displayed by the teachers in their discursive constitution of the function concept. The study contributes to a small but growing body of empirical research on university mathematics teaching practice.

Selvan Naidoo, Student no: 0215998E. MSc Education, Faculty of Science, 2006. / This study used concept mapping to explore South African Grade 11 learners’ understanding of the function concept. Learners’ understanding of the function concept was investigated by examining the relationships learners made between the function concept and other mathematical concepts. The study falls within a social constructivist framework and is underpinned by the key educational notion of understanding. The research method employed was a case study. Data for the study was collected through a concept mapping task, a task on functions and individual learner interviews. In the analysis four key issues are identified and discussed. They are concerned with (a) learners who make most connections; (b) issues related to learners’ omission and addition of concepts; (c) learners’ use of examples in concept mapping and (d) the nature of connections learners made. The study concludes that concept mapping is an effective tool to explore learners’ understanding of the function concept. The report concludes with recommendations for classroom practice, teacher education and further research, particularly given the context of school mathematics practice in the South African curriculum where concept mapping (i.e. use of metacogs) has recently been incorporated as an assessment tool.

This research developed a framework describing students’ developing understanding of function. The research started with the problem: How might typical learning paths of secondary school students’ developing understanding of function be described and assessed? The following principles and research questions guided the development of the framework. Principle 1. The framework should be research-based. Principle 2. The framework should include key aspects of the function concept. Principle 3. The framework should be in a form that would enable teachers to assess and monitor students’ developing understanding of this concept. Principle 4. The framework should reflect students’ big ideas or growth points which describe students’ key cognitive strategies, knowledge and skills in working with function tasks. Principle 5. The framework should reflect typical learning trajectories or a general trend of the growth points in students’ developing understanding of function. The following questions guided the development of the framework of growth points: 1. What are the growth points in students’ developing understanding of function? 2. What information on students’ understanding of function is revealed in the course of developing the framework of growth points that would be potentially useful for teachers? The framework considered four key domains of the function concept: Graphs, Equations, Linking Representations and Equivalent Functions. Students’ understanding of function in each of these domains was described in terms of growth points. Growth points are descriptions of students’ “big ideas”. The description of each growth point highlights students’ developing conceptual understanding rather than merely procedural understanding of a mathematical concept. For example, growth points in students’ understanding of function under Equations were: 1) interpretations based on individual points; 2) interpretations based on holistic analysis of relationships; 3) interpretations based on local properties; and, 4) manipulations and transformations of functions (in equation form) as objects. he growth points in each domain are more or less ordered according to the likelihood that these “big ideas” would emerge. o identify and describe these growth points, Year 8, 9 and 10 students in Australia and the Philippines were given tasks involving function that would highlight thinking in terms of the process-object conception and the property-oriented conception of function. Students’ performance on these tasks and their strategies served as bases for the identification and description of the growth points. he research approach was interpretive and exploratory during the initial stages of analysis. The research then moved to a quantitative approach to identify typical patterns across the growth points, before returning to an interpretive phase in refining the growth points in the light of these data. The main data were collected from students in the Philippines largely through two written tests. Interviews with a sample of students also provided insights into students’ strategies and interpretations of tasks. he research outputs, the research-based framework and the assessment tasks, have the potential to provide teachers with a structure through which they can assess and develop students’ growth in the understanding of function, and their own understanding of the function concept.

It is no unknown fact that South African learners underachieve in mathematics. Due to the fact that the quality of mathematics teaching is one possible factor that has an influence on learners' mathematics achievement, there are valid reasons questioning the conceptual mathematical knowledge of mathematics teachers. In order to facilitate conceptual understanding teachers themselves must possess profound mathematical knowledge. Apart from the influence of a teacher's knowledge for teaching, teachers' attitudes and beliefs play a meaningful role in the way they teach mathematics. The deficient nature of prospective and practising teachers' knowledge of school mathematics, as well as their attitudes and beliefs towards mathematics has serious implications for the training of prospective mathematics teachers. Literature reveals that a technologically enhanced environment can improve the conceptual learning of prospective mathematics teachers. The purpose of this study was to determine the influence of a dynamic technologically enhanced environment on the attitudes and beliefs, as well as the conceptualisation of prospective mathematics teachers regarding the function concept. In this study, prospective teachers were exposed to Geometer's Sketchpad®, a dynamic software programme providing a powerful learning context that promotes the investigation of algebraic relationships. In order to answer the research question, an explanatory mixed method design was used. In the quantitative part of the study, the Study Orientation Questionnaire in Mathematics and a function test were administered to prospective teachers. In the latter, conceptualisation of the function concept was measured in terms of four competence components, namely interpretation, modelling, translation and reification. In the qualitative part of the investigation semi-structured and task-based interviews were held with a group of prospective teachers. Analysis of the results revealed that the dynamic technologically enhanced environment did not contribute to an improvement of the prospective teachers' attitudes and beliefs. In fact, these decreased visibly. Regarding their conceptualisation, only the reification component showed a practically significant improvement. It therefore appears as if prospective teachers are not being prepared to benefit from the dynamic technologically enhanced environment. A model is proposed for the effective use of such a learning environment. The model involves that diagnostic assessment be made of prospective teachers' basic knowledge of the function concept, their study habits, their attitudes and beliefs with respect to mathematics, as well as their mathematics anxiety. The second component comprises recommendations made to prospective teachers as a result of the diagnostic assessment, as well as continuous support being offered as an integrated part of the mathematics module. Support is offered with respect to cognitive and meta-cognitive skills, affective factors and the creation of an advantageous technologically enhanced learning environment. Despite the restricted value of generalisation of the findings from this study, I still recommend the expansion, refining and implementation of the model so that prospective mathematics teachers can effectively benefit from a technologically enhanced environment. Key words for indexing: mathematics education, mathematics teacher education, teacher knowledge, prospective mathematics teachers, function concept, conceptual learning, tertiary education. / Thesis (Ph.D. (Education))--North-West University, Potchefstroom Campus, 2009.

It is no unknown fact that South African learners underachieve in mathematics. Due to the fact that the quality of mathematics teaching is one possible factor that has an influence on learners' mathematics achievement, there are valid reasons questioning the conceptual mathematical knowledge of mathematics teachers. In order to facilitate conceptual understanding teachers themselves must possess profound mathematical knowledge. Apart from the influence of a teacher's knowledge for teaching, teachers' attitudes and beliefs play a meaningful role in the way they teach mathematics. The deficient nature of prospective and practising teachers' knowledge of school mathematics, as well as their attitudes and beliefs towards mathematics has serious implications for the training of prospective mathematics teachers. Literature reveals that a technologically enhanced environment can improve the conceptual learning of prospective mathematics teachers. The purpose of this study was to determine the influence of a dynamic technologically enhanced environment on the attitudes and beliefs, as well as the conceptualisation of prospective mathematics teachers regarding the function concept. In this study, prospective teachers were exposed to Geometer's Sketchpad®, a dynamic software programme providing a powerful learning context that promotes the investigation of algebraic relationships. In order to answer the research question, an explanatory mixed method design was used. In the quantitative part of the study, the Study Orientation Questionnaire in Mathematics and a function test were administered to prospective teachers. In the latter, conceptualisation of the function concept was measured in terms of four competence components, namely interpretation, modelling, translation and reification. In the qualitative part of the investigation semi-structured and task-based interviews were held with a group of prospective teachers. Analysis of the results revealed that the dynamic technologically enhanced environment did not contribute to an improvement of the prospective teachers' attitudes and beliefs. In fact, these decreased visibly. Regarding their conceptualisation, only the reification component showed a practically significant improvement. It therefore appears as if prospective teachers are not being prepared to benefit from the dynamic technologically enhanced environment. A model is proposed for the effective use of such a learning environment. The model involves that diagnostic assessment be made of prospective teachers' basic knowledge of the function concept, their study habits, their attitudes and beliefs with respect to mathematics, as well as their mathematics anxiety. The second component comprises recommendations made to prospective teachers as a result of the diagnostic assessment, as well as continuous support being offered as an integrated part of the mathematics module. Support is offered with respect to cognitive and meta-cognitive skills, affective factors and the creation of an advantageous technologically enhanced learning environment. Despite the restricted value of generalisation of the findings from this study, I still recommend the expansion, refining and implementation of the model so that prospective mathematics teachers can effectively benefit from a technologically enhanced environment. Key words for indexing: mathematics education, mathematics teacher education, teacher knowledge, prospective mathematics teachers, function concept, conceptual learning, tertiary education. / Thesis (Ph.D. (Education))--North-West University, Potchefstroom Campus, 2009.

The study presents a pragmatic evaluation of the influence of inductive teaching on grade 11 learners in two high schools in Tshwane West District in the Gauteng province in a form of pseudo experiment complemented with a qualitative investigation. The study focussed on the influence of inductive teaching on the nature of conceptualisation of and the learning achievement with regard to functions in Grade 11. A model adopted by O'Callaghan that identifies and applies the four competencies of modelling a function, interpreting a function, translating and reifying a function proved to be relevant for the investigation and hence was adapted for the study. The methodology used included data collection through pretest-posttest control group experimental design complemented with unstructured interviews. The verification of the reliability of research instruments and data analysis was done with the assistance of the Northwest University (Potchefstroom Campus) Statistical Consultation Services and through identification of common perceptions and experiences of participants. The results of the study did indicate positive influence of inductive teaching on the nature and quality of conceptual learning of the function concept. / Thesis (M.Ed.)--North-West University, Potchefstroom Campus, 2009.

The study presents a pragmatic evaluation of the influence of inductive teaching on grade 11 learners in two high schools in Tshwane West District in the Gauteng province in a form of pseudo experiment complemented with a qualitative investigation. The study focussed on the influence of inductive teaching on the nature of conceptualisation of and the learning achievement with regard to functions in Grade 11. A model adopted by O'Callaghan that identifies and applies the four competencies of modelling a function, interpreting a function, translating and reifying a function proved to be relevant for the investigation and hence was adapted for the study. The methodology used included data collection through pretest-posttest control group experimental design complemented with unstructured interviews. The verification of the reliability of research instruments and data analysis was done with the assistance of the Northwest University (Potchefstroom Campus) Statistical Consultation Services and through identification of common perceptions and experiences of participants. The results of the study did indicate positive influence of inductive teaching on the nature and quality of conceptual learning of the function concept. / Thesis (M.Ed.)--North-West University, Potchefstroom Campus, 2009.

Orientador: Ariane Luzia dos Santos / Resumo: Este texto tem como objetivo, reconhecendo a devida importância do tema Funções, apresentar uma proposta de atividades apoiadas pelo software Geogebra como forma de potencializar a aprendizagem da matemática, onde enfatizamos o conceito de função, as particularidades de suas representações e as propriedades, nos utilizando da transição entre elas como sugestão para sanar as dificuldades encontradas pelos alunos na compreensão desse tema. Também incluímos uma noção dos conceitos de limite e continuidade entendendo que a abordagem de alguns tópicos do cálculo no Ensino Médio seria de grande valia para auxiliá-los nessa tarefa, ressaltando que utilizamos a tecnologia em conjunto com os atores tradicionais (oralidade e escrita) através de aulas expositivas dialogadas. / Abstract: This paper aims, recognizing the importance of the theme Functions, to present a proposal of activities supported by Geogebra software as a way to enhance the learning of mathematics, where we emphasize the concept of function, the particularities of its representations and the properties, using of the transition between them as a suggestion to remedy the di culties encountered by students in understanding this theme. We also included a notion of the concepts of limit and continuity, understanding that the approach to some topics of calculus in High School would be of great value to assist them in this task, emphasizing that we use technology with traditional actors (oral and written) through dialogic expositive classes. / Mestre

The function concept is one of the most important concepts in the learning of mathematics (Dubinsky & Harel, 1992), yet it is considered by many researchers to be one of the least understood and most difficult concepts to master in the learning of high school mathematics (Eisenberg, 1992, Sfard, 1992). To this end, problems concerning its teaching and learning are often confronted (Mann, 2000) and few teachers know how learners come to understand functions (Yoon, 2007). As a result, most teachers teach functions using the conventional approach which starts by stating definitions followed by examples and then a few applications. The nature of this approach has not encouraged teachers to engage learners and their ways of reasoning in knowledge construction and adequately addressing their difficulties. The purpose of this study was to use design research to improve the teaching and learning of functions at grade 11 level. This was achieved by adapting design cycles of Wademan’s (2005) Generic Design Research model in which each cycle comprised different iterative APOS (Action, Process, Object, Schema) analysis, design, development and implementation of hypothetical learning trajectories (HLTs). I started by interrogating twelve grade 11 learners of a particular rural high school on the June 2011 mathematics paper 1 examination they had written to determine the APOS theory conception level each learner was operating at, and their difficulties. Learners’ difficulties from initial interviews and literature were grouped under the function definition and representation. I then designed instruction based on HLTs embedded with Realistic Mathematics Education (RME) activities and two separate tasks on the definition and representation as a form of intervention to help learners move up from their initial conception levels to the next and to overcome their difficulties. After each design cycle I interviewed learners based on the task for a particular concept and learners’ responses were analysed using APOS theory and used to design further instruction to help learners approximate the schema level of understanding concepts related to functions. The major findings of this study were that the use of learners’ conceptions and RME activities in designing instruction helped learners to progress smoothly through APOS theory conception levels though they did not fully reach the intended schema level. In addition, design research cycles and their HLTs implemented in a constructivist environment enabled learners to collectively derive working definitions of the function concept and to improve their conceptual understanding of the process of switching from a graph to an equation. Another contribution of this study has been a deeper understanding of the extent to which design research can be used to improve learners’ understanding of functions and an addition of some insights to the teaching and learning of functions. / Thesis (PhD)--University of Pretoria, 2013. / gm2014 / Science, Mathematics and Technology Education / unrestricted

In the school mathematics curriculum functions play a pivotal role in accessing and mastering algebra and the whole of mathematics. The study investigated the extent to which pupils with little experience in algebra would develop the function concept and was motivated by the need to bring the current Zimbabwean mathematics curriculum in line with reform ideas that introduce functions early in the secondary school curriculum. An instrument developed from literature review was used to assess the extent to which the Form1/Grade 8 pupils developed the concept. The teaching experiment covered a total of 26 lessons, a period of about eight weeks spread over two terms starting in the second term of the Zimbabwean school calendar. The problem-centred teaching approach based on the socio-constructivist view of learning formed the background to facilitate pupils' individual and social construction of knowledge. Data was collected from the pupils' written work, audio taped discussions and interviews with selected pupils. The extent to which each pupil of the seven pupils developed the aspects of function, change, relationship, rule, representation and strategies, was assessed. The stages of development and thinking levels of functional reasoning at the beginning of the experiment, then during the learning phase and finally at the end of the experiment, were compared. The results showed that functions can be introduced at Form 1 and pupils progressed in the understanding of most of the aspects of a function. / Educational Studies / M. Ed. (Mathematics Education)