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This thesis presents the first Swedish empirical evidence on how teachers employ a dynamic mathematical software when teaching mathematics in upper secondary school. The study examines: a) How teachers formulate mathematical problems? b) How they use the experience the students have gained? and c) What use they make of the software’s potential? These questions are examined through classroom observations followed up by discussions with the teachers. The study comprises three teachers and shows that they have very different mathematical experiences and teaching skills. A questionnaire was sent to the teachers prior to the classroom visits to collect relevant background information; e.g., the teachers were asked to describe their teacher training, their view of mathematics and of how a dynamic software could contribute to their teaching. The results show that the teachers’ ability to pose thought-provoking openended problems is the most important factor as it significantly influences what the students learn. The way a mathematical problem is formulated could, in conjunction with a dynamic software, actually limit the students’ achievement. However, this study confirms that it could also provide an opportunity for students to discover new mathematical relations, draw conclusions, generalise and formulate hypotheses. This could in turn lead to an in formally proving a mathematical relation. A conclusion of the study is that to be successful, teachers need a good mathematical background with a firm knowledge base and an understanding of the software’s potential, but they also need the skill to formulate open-ended problems that will enable their students to work successfully with a dynamic mathematical software.

The purpose of this study was to investigate plane geometry problem solving strategies of pre-service elementary mathematics teachers in technology and paper-and-pencil environments after receiving an instruction with GeoGebra. Qualitative research strategies were used to investigate teacher candidates&lsquo / solution strategies. The data was collected and analyzed by means of a multiple case study design. The study was carried out with 7 pre-service elementary mathematics teachers. The main data sources were classroom observations and interviews. After receiving a three-week instructional period, the participants experienced data collection sessions during a week. The data was analyzed by using records of the interviews, answers to the instrument, and transcribing and examining observation records. Results revealed that the participants developed three solution strategies: algebraic, geometric and harmonic. They used mostly algebraic solutions in paper-and-pencil environment and v geometric ones in technology environment. It means that different environments contribute separately pre-service teachers&lsquo / mathematical problem solving abilities. Different from traditional environments, technology contributed students&lsquo / mathematical understanding by means of dynamic features. In addition, pre-service teachers saved time, developed alternative strategies, constructed the figures precisely, visualized them easily, and measured accurately and quickly. The participants faced some technical difficulties in using the software at the beginning of the study but they overcome most of them at the end of instructional period. The results of this study has useful implications for mathematics teachers to use technology during their problem solving activities as educational community encourages to use technology in teaching and learning of mathematics.

The aim of this study was to investigate the effects of using a dynamic geometry environment together with open-ended explorations on sixth grade students&rsquo / performance in polygons and congruency and similarity of polygons. Two groups of sixth grade students were selected for this study: (1) An experimental group composed of 66 students whom 34 were boys and 32 were girls / and (2) a control group composed of 68 students whom 35 were boys and 33 were girls. While the students in the control group received instruction via traditional methods, the students in the experimental group studied the same topics by open-ended explorations in a dynamic geometry environment. Geometry Test (GT) and Computer Attitude Scale (CAS) were used as data collection instruments. All students had taken the GT as pre-test, post-test, and delayed post test. However, CAS was administered only to the experimental group at the end of the instruction. Furthermore, some qualitative data were collected through video-taped classroom observations and interviews with selected students. Pre-test scores showed no statistical difference between control and experimental group students in terms of their performances in polygons and congruency and similarity of polygons before the study. On the other hand, the results of the post and delayed-post tests which are analyzed by independent t test showed that experimental group achieved significantly better than the control group students. In addition, a statistically significant correlation between CAS and GT was observed. Those results were also supported by the qualitative data. In conclusion, the results indicated that dynamic geometry environment together with open-ended explorations significantly improved students&rsquo / performances in polygons and congruency and similarity of polygons.

The aims of this study were to investigate the effects of two different methods of dynamic geometry based computer instruction on seventh grade students&rsquo / attitudes towards geometry, attitudes toward mathematic and technology and spatial abilities compared to traditional textbook based instruction and to get the students&rsquo / views related to the effects of computer based instruction on their learning. The sample consisted of 57 seventh grade students from a private elementary school in Kayseri. The study was conducted in the 2006-2007 academic year, lasting 14 lesson hours (two weeks). The data were collected through spatial ability test, mathematics and technology attitude scale, geometry attitude scale, and interviews. The quantitative analyses were carried out by using multivariate covariance analyses. The results revealed that two different methods of dynamic geometry based computer instruction didn&rsquo / t have a significangt effect on students&rsquo / spatial abilities compared to traditional textbook based instruction. The results also indicated that two different methods of dynamic geometry based instruction had a significant effect on students&rsquo / attitudes toward geometry, mathematics and technology compared to traditional textbook based instruction. The results of the interviews indicated that computers created a dynamic learning environment which supported students&rsquo / development and computers also helped students to explore mathematic in a far more meaningful way.

O uso da geometria no dia a dia das pessoas tem importÃncia significativa por ser um assunto que utiliza desenhos, formas e teoremas como elementos de estudos para comprovar sua atuaÃÃo nos mais diversos campos da sociedade tais como engenharias, siderÃrgicas, arquiteturas, topografias, etc. Neste contexto, podemos afirmar que construÃÃes geomÃtricas propiciam a descoberta de valiosas ideias que auxiliam à compreensÃo das propriedades geomÃtricas. As avaliaÃÃes em larga escala apresentadas nos indicadores da educaÃÃo pÃblica no Estado do Cearà retratam claramente as dificuldades de aprendizagem por parte dos alunos quando relacionados aos conceitos geomÃtricos especificamente o teorema do Ãngulo externo tanto na teoria (conceito algÃbrico) como na prÃtica (conceito geomÃtrico). A partir desta analise, propomos realizar uma investigaÃÃo atravÃs da presente pesquisa que conseguisse identificar possÃveis entraves existentes no ensino de geometria para que pudesse obter avanÃos que visam melhorar no ensino relacionado ao Teorema do Ãngulo Externo e suas ConsequÃncias usando os ambientes de aprendizagens Velho Papel e Caneta (VPC) e o Ambiente virtual de Aprendizagem (AVA) com a operacionalidade do software educativo de geometria dinÃmica GeoGebra. O trabalho teve a participaÃÃo de um grupo de 12(doze) professores de matemÃtica em formaÃÃo continuada de um Curso de EspecializaÃÃo no Ensino de MatemÃtica da Universidade Vale do Acaraà (UVA) na cidade de Cascavel-Ce. O uso operacional e pedagÃgico do software de geometria dinÃmica GeoGebra foi aplicado em aulas expositivas com questionÃrios de problemas envolvendo o teorema do Ãngulo externo que busca avaliar o desempenho dos estudantes participantes da pesquisa em relaÃÃo suas prÃticas de sala de aula com o ensino de geometria. Adotamos abordagens qualitativa, exploratÃria e pesquisa-aÃÃo para caracterizar a pesquisa e buscamos tomar como base os pressupostos teÃricos e reflexivos segundo as concepÃÃes de Valente, Michele Artigue, Pais e Fiorentini e Lorenzato. A pesquisa revelou avanÃos no processo de aprendizagem dos estudantes participantes que se mostraram entusiasmados com os conhecimentos que construÃram e que os possibilitou estabelecerem um relacionamento colaborativo entre os grupos envolvidos (estudantes e professor-pesquisador) / The use of geometry in everyday life people have significant importance because it is a subject that uses designs, shapes and theorems as studies of evidence to make its activities in various fields of society such as engineering, steel, architecture, topography, etc. In this context, we can say that geometric constructions provide the discovery of valuable ideas that help the understanding of geometric properties. The large-scale assessments presented in public education indicators in the State of Ceara clearly portray the difficulties of learning by students when related to geometric concepts specifically the exterior angle theorem in theory (algebraic concept) and in practice (geometric concept). From this analysis, we propose to conduct an investigation through this research that could identify possible barriers in existing geometry teaching so he could obtain advances to improve the teaching related to the External Angle Theorem and its Consequences using the old learning environments and Paper pen (VPC) and the virtual Learning Environment (VLE) with the operation of educational software of dynamic geometry GeoGebra. The work was attended by a group of twelve (12) mathematics teachers in continuing education of a Specialization Course in Teaching of Mathematics at the University Vale do Acaraà (UVA) in the city of Cascavel-Ce. The operational and pedagogical use of dynamic geometry software GeoGebra was applied in lectures with questionnaires problems involving the exterior angle theorem that seeks to assess the performance of students participating in the survey regarding their classroom practices with the teaching of geometry. We adopted a qualitative, exploratory and action research approaches to characterize the research and seek to build on the theoretical and reflexive assumptions according to Valente conceptions, Michele Artigue, Parents and Fiorentini and Lorenzato. The survey showed progress in the learning process of participating students that were excited by the knowledge that built and that allowed establish a collaborative relationship between the groups involved (students and teacher-researcher).

This study aimed to investigate the effects of using dynamic geometry software while teaching by guided discovery compared to paper-and-pencil based guided discovery and traditional teaching method on sixth grade students&rsquo / van Hiele geometric thinking levels and geometry achievement. The study was conducted in one of the private schools in Ankara and lasted six weeks. The sample of the study consisted 47 sixth grade students in the school. The present study was designed as pretest-posttest control group quasi-experimental study.In order to gather data, Geometry Achievement Test (GAT) and Van Hiele Geometric Thinking Level Test (VHL) were used. At the end of the research, the data were analyzed by means of analysis of covariance. The results of the study indicated that there was a significant effect of methods of teaching on means of the collective dependent variables of the sixth grade students&rsquo / scores on the POSTVHL after controlling their PREVHL scores, and there was a significant effect of methods of teaching on means of the collective dependent variables of the sixth grade students&rsquo / scores on the POSTGAT after controlling their PREGAT scores.

Denna undersökning syftar till att belysa vad som motiverar gymnasieelever att använda GeoGebra i matematiken. Vidare undersöktes även eventuella skillnader i motivation, både typ och grad, hos gymnasieelever som använder GeoGebra ofta respektive sällan i matematiken. Undersökningen grundar sig i motivationsteori och specifikt förväntan-värde teorin och utfördes med kvantitativa- och kvalitativa metoder. Datainsamlingen skedde med en enkät vilken besvarades av 72 gymnasieelever från fyra skolor i Norrbottens län, Stockholms län och Västra Götalands län. Resultatet från undersökningen visar att gymnasieelever främst motiveras av nytto- och kostnadsvärdet som GeoGebra medför. Med andra ord, att GeoGebra ger dem fördelar i matematiken genom att göra det enklare och mer tidseffektiv att hantera matematikproblem med hjälp av GeoGebra. Vidare så visade även resultatet att gymnasieelever som använder GeoGebra ofta motiverades mer av nyttan som verktyget medförde, än gymnasieelever som använder GeoGebra sällan motiverades av det. / The focus of this research was to enlighten what motivates high school students to make use of GeoGebra in mathematics studies. The research also aimed at finding out if there were any differences in motivation, both type and degree, between high school students who used GeoGebra frequently or rarely in mathematics studies. The study was based on motivational theory and more specific expectancy-value theory and used quantitative- and qualitative methods. The data collection was through a survey in which 72 high school students from four schools in Norrbottens county, Stockholms county and Västra Götalands county participated. The result from this study showed that the main reason why high school students are motivated to use GeoGebra in their mathematics studies is because of the utility and cost value the instrument entails. In other means, high school students use the programme because it's easier and more time efficient to solve mathematical problems with it. The study also showed that high school students who use GeoGebra more frequently in mathematics studies are more motivated by the utility value that the instrument brings, than high school students who only use GeoGebra sometimes were.

Text in Afrikaans / Real empowerment of school learners requires preparing them for the age of technology. This empowerment can be achieved by developing their higher-order thinking skills. This is clearly the intention of the proposed South African FET National Curriculum Statements Grades 10 to 12 (Schools). This research shows that one method of developing higher-order thinking skills is to adopt an integrated curriculum approach. The research is based on the assumption that an integrated curriculum approach will produce learners with a more integrated knowledge structure which will help them to solve problems requiring higher-order thinking skills. These assumptions are realistic because the empirical results of several comparative research studies show that an integrated curriculum helps to improve learners' ability to use higher-order thinking skills in solving nonroutine problems. The curriculum mentions four kinds of integration, namely integration across different subject areas, integration of mathematics with the real world, integration of algebraic and geometric concepts, and integration into and the use of dynamic geometry software in the learning and teaching of geometry. This research shows that from a psychological, pedagogical, mathematical and historical perspective, the theme conic sections can be used as an integrating factor in the new proposed FET mathematics curriculum. Conics are a powerful tool for making the new proposed curriculum more integrated. Conics can be used as an integrating factor in the FET band by means of mathematical exploration, visualisation, relating learners' experiences of various parts of mathematics to one another, relating mathematics to the rest of the learners' experiences and also applying conics to solve real-life problems. / Mathematical Sciences / D.Phil. (Wiskundeonderwys)

The widespread use of Dynamic Geometry Software (DGS) is raising many interesting questions and discussions as to the necessity, usefulness and meaning of proof in school mathematics. With these questions in mind, a didactical sequence on the topic “Conics” was developed in a teacher education course tailored for pre-service secondary math methods course. The idea of the didactical sequence is to introduce “Conics” using a concrete manipulative approach (paper folding) then an explorative DGS-based construction activity embedding the need for a proof. For that purpose, the DGS software serves as an intermediary tool, used to bridge the gap between the physical model and the formal symbolic system of proof. The paper will present an analysis of participants’ geometric thinking strategies, featuring proof as an embedded process in geometric construction situations.