A thesis submitted to the Faculty of Science, University of the Witwatersrand in fulfillment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 3 April 2017. / The effect of suction and blowing at the base on the horizontal spreading under gravity of a two-dimensional thin fluid film and an axisymmetric liquid drop is in- vestigated. The velocity vn which describes the suction/injection of fluid at the base is not specified initially. The height of the thin film satisfies a nonlinear diffusion equation with vn as a source term. The Lie group method for the solution of partial differential equations is used to reduce the partial differential equations to ordinary differential equations and to construct group invariant solutions. For a group invari- ant solution to exist, vn must satisfy a first order linear partial differential equation. The two-dimensional spreading of a thin fluid film is first investigated. Two models for vn which give analytical solutions are analysed. In the first model vn is propor- tional to the height of the thin film at that point. The constant of proportionality is β (−∞ < β < ∞). The half-width always increases to infinity as time increases even for suction at the base. The range of β for the thin fluid film approximation to be valid is determined. For all values of suction and a small range of blowing the maximum height of the film tends to zero as time t → ∞. There is a value of β corresponding to blowing for which the maximum height remains constant with the blowing balancing the effect of gravity. For stronger blowing the maximum height tends to infinity algebraically, there is a value of β for which the maximum height tends to infinity exponentially and for stronger blowing, still in the range for which the thin film approximation is valid, the maximum height tends to infinity in a finite time. For blowing the location of a stagnation point on the centre line is determined by solving a cubic equation approximately by a singular perturbation method and then exactly using a trigonometric solution. A dividing streamline passes through the stagnation point which separates the flow into two regions, an upper region consisting of fluid descending due to gravity and a lower region consisting of fluid rising due to blowing. For sufficiently strong blowing the lower region fills the whole of the film. In the second model vn is proportional to the spatial gradient of the height with constant of proportionality β∗ (−∞ < β∗ < ∞). The maximum height always decreases to zero as time increases even for blowing. The range of β∗ for the thin fluid film approximation to be valid is determined. The half-width tends to infinity algebraically for all blowing and a small range of weak suction. There is a value of β∗ corresponding to suction for which the half-width remains constant with the suction balancing the spreading due to gravity. For stronger suction the half-width tends to zero as t → ∞. For even stronger suction there is a value of β∗ for which the half-width tends to zero exponentially and a range of β∗ for which it tends to zero in a finite time but these values lie outside the range for which the thin fluid film approximation is valid. For blowing there is a stagnation point on the centre line at the base. Two dividing streamlines passes through the stagnation point which separate fluid descending due to gravity from fluid rising due to blowing. An approximate analytical solution is derived for the two dividing streamlines. A similar analysis is performed for the axisymmetric spreading of a liquid drop and the results are compared with the two-dimensional spreading of a thin fluid film. Since the two models for vn are still quite general it can be expected that general results found will apply to other models. These include the existence of a divid- ing streamline separating descending and rising fluid for blowing, the existence of a strength of blowing which balances the effect of gravity so the maximum height remains constant and the existence of a strength of suction which balances spreading due to gravity so that the half-width/radius remains constant. / MT 2017

Este estudo se debruça sobre a questão da formação de professores de matem ática para a incorporação de recursos tecnológicos em sala de aula. No cenário da Educação a Distância (EAD), esta pesquisa se propõe a analisar a seguinte questão: na modalidade EAD, como organizar uma proposta de formação que vise a capacitação de professores de matemática para o uso do potencial dos registros dinâmicos de representa- ção semiótica que se tem no software GeoGebra? O processo de apropriação de softwares de Matemática Dinâmica (MD) por professores é analisado à luz da Abordagem Instrumental. O conhecimento matemático estabelecido culturalmente possui representação peculiar, que in uencia o processo de aprendizagem matemática, e é analisado através dos Registros de Representação Semióica Na articulação destes aportes teóricos, identi ca-se o registro dinâmico presente nos softwares de MD, e analisa-se a apropriação do recurso pelos professores através da exploração do potencial presente nesse registro. A partir da metodologia da Engenharia Didática, é apresentado o planejamento de Arquitetura Pedagógica para uma disciplina de curso especialização, cursada a distância por professores de matemática em formação continuada. O texto apresenta o relato da implementação da disciplina e, analisando as produções dos professores-alunos, identi ca-se o desenvolvimento de esquemas de utiliza- ção do software GeoGebra, indicativos do início de apropriação desse recurso por parte dos professores de matemática participantes do estudo. / This study focuses on the question of mathematics teacher training for the incorporation of technological resources in the classroom. In the Distance Education scenario, this research aims to analyze the question: in the distance education modality, how to organize a training proposal aimed at the training of mathematics teachers to use the potential of the dynamic register of semiotic representation at the GeoGebra software? The process of appropriation of Dynamic Mathematics software for teachers is analyzed under the perspective of Instrumental Approach. The mathematical knowledge culturally established has peculiar representation which in uence the process of learning mathematics, and is analyzed through Registers of Semiotic Representation At the junction of these theoretical contributions, identi es the dynamic record in Dynamic Mathematics software, and analyzes the appropriation of the resource by teachers through the exploitation of this potential in this register. Using the methodology of the Didactic Engineering, it presents the Pedagogical Architecture planning, for a especialization course's discipline given for math teachers in continuing education. The text reports the record of achievement of the course and analyzing the productions of student teachers, identi es the development of GeoGebra software utilization schemes, indicating the start of the appropriation of this appeal by math teachers in the study.

The purpose of this study was to investigate the effects of teaching linear equations with Dynamic Mathematics Software (GeoGebra) on seventh grade students&rsquo / achievement compared to the regular instruction. Randomized posttest-only control group design was utilized in the study. 60 seventh grade students (32 girls and 28 boys) of a public school in Yenimahalle district in Ankara participated in the study. The study was conducted in 2011-2012 fall semester, lasting 9 class hours in three weeks. The data was collected by three Mathematics Achievement Tests: Cartesian coordinate system achievement test (MAT1), linear relation achievement test (MAT2) and graph of linear equation achievement test (MAT3). The quantitative analysis was conducted by using analysis of covariance (ANCOVA). The results revealed that teaching Cartesian coordinate system and linear relation by using Dynamic Mathematics Software had no significant effect on seventh grade students&rsquo / achievement compared to the regular instruction. On the other hand, the results also indicated that teaching graph of linear equations by using Dynamic Mathematics Software had a significant effect on seventh grade students&rsquo / achievement positively.

The overall problem that formed the basis for this thesis is that students get limited opportunity to develop their mathematical reasoning ability while, at the same time, there are dynamic mathematics software available which can be used to foster this ability. The aim of this thesis is to contribute to knowledge in this area by focusing on task design in a dynamic software environment and by studying the reasoning that emerges when students work on tasks in such an environment. To analyze students’ mathematical reasoning, a new analytical tool was developed in the form of an expanded version of Toulmin’s model. Results from one of the studies in this thesis show that exploratory tasks in a dynamic software environment can promote mathematical reasoning in which claims are formulated, examined and refined in a cyclic process. However, this reasoning often displayed a lack of the more conceptual, analytic and explanatory reasoning normally associated with mathematics. This result was partly confirmed by another of the studies. Hence, one key question in the thesis has been how to design tasks that promote conceptual and explanatory reasoning. Two articles in the thesis deal with task design. One of them suggests a model for task design with a focus on exploration, explanation, and generalization. This model aims, first, to promote semantic proof production and then, after the proof has been constructed, to encourage further generalizations. The other article dealing with task design concerns the design of prediction tasks to foster student reasoning about exponential functions. The research process pinpointed key didactical variables that proved crucial in designing these tasks. / Baksidestext Det övergripande problem som legat till grund för denna avhandling är att elever får begränsad möjlighet att utveckla sin resonemangsförmåga samtidigt som det finns dynamiska matematikprogram som kan utnyttjas för att stimulera denna förmåga. Syftet med avhandlingen är att bidra till den samlade kunskapen inom detta problemområde, dels genom att fokusera på design av uppgifter i en lärandemiljö med dynamiska matematikprogram och dels genom att studera och karakterisera de resonemang som utvecklas när elever jobbar med olika uppgifter i denna miljö. För att analysera elevernas resonemang utvecklades ett nytt analysverktyg i form av en utökad version av Toulmins modell. Resultat från en av studierna i avhandlingen visar att dynamiska matematikprogram i kombination med utforskande uppgifter kan stimulera till matematiska resonemang där hypoteser formuleras, undersöks och förfinas i en cyklisk process. Samtidigt visar samma studie att de resonemang som utvecklas i stor utsträckning saknar matematiskt grundade förklaringar. Detta resultat bekräftas till viss del av ytterligare en studie.  Frågan hur uppgifter bör designas för att främja matematiskt grundade resonemang har därför varit central i avhandlingen. Två av artiklarna behandlar uppgiftsdesign, men utifrån olika utgångspunkter.

Este estudo se debruça sobre a questão da formação de professores de matem ática para a incorporação de recursos tecnológicos em sala de aula. No cenário da Educação a Distância (EAD), esta pesquisa se propõe a analisar a seguinte questão: na modalidade EAD, como organizar uma proposta de formação que vise a capacitação de professores de matemática para o uso do potencial dos registros dinâmicos de representa- ção semiótica que se tem no software GeoGebra? O processo de apropriação de softwares de Matemática Dinâmica (MD) por professores é analisado à luz da Abordagem Instrumental. O conhecimento matemático estabelecido culturalmente possui representação peculiar, que in uencia o processo de aprendizagem matemática, e é analisado através dos Registros de Representação Semióica Na articulação destes aportes teóricos, identi ca-se o registro dinâmico presente nos softwares de MD, e analisa-se a apropriação do recurso pelos professores através da exploração do potencial presente nesse registro. A partir da metodologia da Engenharia Didática, é apresentado o planejamento de Arquitetura Pedagógica para uma disciplina de curso especialização, cursada a distância por professores de matemática em formação continuada. O texto apresenta o relato da implementação da disciplina e, analisando as produções dos professores-alunos, identi ca-se o desenvolvimento de esquemas de utiliza- ção do software GeoGebra, indicativos do início de apropriação desse recurso por parte dos professores de matemática participantes do estudo. / This study focuses on the question of mathematics teacher training for the incorporation of technological resources in the classroom. In the Distance Education scenario, this research aims to analyze the question: in the distance education modality, how to organize a training proposal aimed at the training of mathematics teachers to use the potential of the dynamic register of semiotic representation at the GeoGebra software? The process of appropriation of Dynamic Mathematics software for teachers is analyzed under the perspective of Instrumental Approach. The mathematical knowledge culturally established has peculiar representation which in uence the process of learning mathematics, and is analyzed through Registers of Semiotic Representation At the junction of these theoretical contributions, identi es the dynamic record in Dynamic Mathematics software, and analyzes the appropriation of the resource by teachers through the exploitation of this potential in this register. Using the methodology of the Didactic Engineering, it presents the Pedagogical Architecture planning, for a especialization course's discipline given for math teachers in continuing education. The text reports the record of achievement of the course and analyzing the productions of student teachers, identi es the development of GeoGebra software utilization schemes, indicating the start of the appropriation of this appeal by math teachers in the study.

Este estudo se debruça sobre a questão da formação de professores de matem ática para a incorporação de recursos tecnológicos em sala de aula. No cenário da Educação a Distância (EAD), esta pesquisa se propõe a analisar a seguinte questão: na modalidade EAD, como organizar uma proposta de formação que vise a capacitação de professores de matemática para o uso do potencial dos registros dinâmicos de representa- ção semiótica que se tem no software GeoGebra? O processo de apropriação de softwares de Matemática Dinâmica (MD) por professores é analisado à luz da Abordagem Instrumental. O conhecimento matemático estabelecido culturalmente possui representação peculiar, que in uencia o processo de aprendizagem matemática, e é analisado através dos Registros de Representação Semióica Na articulação destes aportes teóricos, identi ca-se o registro dinâmico presente nos softwares de MD, e analisa-se a apropriação do recurso pelos professores através da exploração do potencial presente nesse registro. A partir da metodologia da Engenharia Didática, é apresentado o planejamento de Arquitetura Pedagógica para uma disciplina de curso especialização, cursada a distância por professores de matemática em formação continuada. O texto apresenta o relato da implementação da disciplina e, analisando as produções dos professores-alunos, identi ca-se o desenvolvimento de esquemas de utiliza- ção do software GeoGebra, indicativos do início de apropriação desse recurso por parte dos professores de matemática participantes do estudo. / This study focuses on the question of mathematics teacher training for the incorporation of technological resources in the classroom. In the Distance Education scenario, this research aims to analyze the question: in the distance education modality, how to organize a training proposal aimed at the training of mathematics teachers to use the potential of the dynamic register of semiotic representation at the GeoGebra software? The process of appropriation of Dynamic Mathematics software for teachers is analyzed under the perspective of Instrumental Approach. The mathematical knowledge culturally established has peculiar representation which in uence the process of learning mathematics, and is analyzed through Registers of Semiotic Representation At the junction of these theoretical contributions, identi es the dynamic record in Dynamic Mathematics software, and analyzes the appropriation of the resource by teachers through the exploitation of this potential in this register. Using the methodology of the Didactic Engineering, it presents the Pedagogical Architecture planning, for a especialization course's discipline given for math teachers in continuing education. The text reports the record of achievement of the course and analyzing the productions of student teachers, identi es the development of GeoGebra software utilization schemes, indicating the start of the appropriation of this appeal by math teachers in the study.

This thesis concerns the challenge of integrating dynamic software environments into the teaching of mathematics. It investigates particular aspects of the design of tasks which employ this type of computer-based system, with a focus on improvement, both of the tasks themselves and of the design process through which they are developed and refined. The thesis reports two research projects: a small initial one preceding a larger main project. The initial case study, involving two graduate students in mathematics, develops a task design model for geometrical locus problems. The main study constitutes the first iteration of a design-based study, conducted in collaboration with four upper-secondary school teachers and their classes. It seeks to identify task design characteristics that foster students’ mathematical reasoning and proficient use of software tools, and examines teachers’ organisation of ‘follow-up’ lessons. The findings concern three particular aspects: features of tasks and task environment relevant to developing a specific plan of action for a lesson; orchestration of a particular task environment to support the instrumental genesis of specific dynamic software tools; how to follow up students’ work on computer-based tasks in a whole-class discussion.