This study aimed to investigate the level of understanding of Euclidian geometry, in terms of theoretical knowledge as well as its problem-solving application, in pre-service mathematics education (PME) students at the University of Pretoria. In order to do so, a one group pre-test/ post-test procedure was conducted around an intensive geometry module, and a representational group of students was interviewed before and after the module to discuss their high school experiences of learning geometry and to analyse their attitudes towards the subject. The van Hiele Theory of Levels of Thought in Geometry was used as the theoretical framework for this study. The PME students in this study, prior to their completion of the geometry module, lacked the content knowledge, skills and insight in Euclidian geometry that is expected at matric level (Level 3). The pre-test results revealed that half the group could only be classified as being on Level 0. By the time the post-test was written, 60% of the group had moved onto Level 1 as their maximum competence level. This implies that these students were all brought to greater insight by the teaching they received during the geometry module. However, the overall improvement in the group as revealed in the post-test results, consisted of an upward movement of only one level. Therefore, the geometry module offered did not bring about sufficient improvement for these students to be able to teach geometry adequately (Level 3 is required). The students who were interviewed for this study uniformly expressed their dislike or fear of Euclidian geometry in general, but described the positive change in their attitude during the course of the module because of the way it was presented. Training of students for a career as mathematics educators which includes an in-depth van Hiele-based geometry module would facilitate the acquisition of insight and relational understanding. / Dissertation (MEd)--University of Pretoria, 2008. / Curriculum Studies / MEd / unrestricted

The purpose of this study was to investigate (a) the role of a yearlong geometry course on high school geometry students' logical thinking and proof construction abilities, (b) the linkage between students' logical thinking and proof construction abilities, and (c) the impact of dynamic geometry software on students' performance. In addition, this study also ventured to determine if the type of geometry course had any impact on students' logical thinking and proof construction achievement. The sample for the study consisted of 1,325 high school geometry students enrolled in regular, honors, and mastery courses in four high schools from the school district affiliated with the Local Education Agency (LEA) during the academic year 2004-2005. Geometer's Sketchpadä (GSP) was assumed to represent the dynamic geometry software. Responses of students on two pre-tests and two post-tests, each with one on logical thinking and one on proof, were analyzed to address the research questions. Results of the analyses indicated no significant effect of the yearlong geometry course on the performance of students on proof tests but a fairly significant effect on the tests of logical thinking. Use of GSP was found to have some impact on honors and mastery students' performance on proof post-tests. Honors students showed a higher logical thinking level than their regular and mastery counterparts in both GSP and non-GSP groups. There was a significant positive correlation between students' performance on the tests of logical thinking and proof.

L'apprentissage des preuves mathématiques déductives est fondamental dans l'enseignement des mathématiques. Pourtant, la dernière enquête TIMSS (Trends in International Mathematics and Science Study) menée par l'IEA ("International Association for the Evaluation of Educational Achievement") en mars 2015, le niveau général des étudiants en mathématiques est en baisse et les étudiants éprouvent de plus en plus de difficultés pour comprendre et écrire les preuves mathématiques déductives.Pour aborder ce problème, plusieurs travaux en didactique des mathématiques utilisent l’apprentissage collaboratif en classe.L'apprentissage collaboratif consiste à regrouper des étudiants pour travailler ensemble dans le but d'atteindre un objectif commun. Il repose sur le débat et l'argumentation. Les étudiants s'engagent dans des discussions pour exprimer leurs points de vue sous forme d'arguments et de contre-arguments dans le but de résoudre un problème posé.L’argumentation utilisée dans ces approches est basée sur des discussions informelles qui permettent aux étudiants d'exprimer publiquement leurs déclarations et de les justifier pour construire des preuves déductives. Ces travaux ont montré que l’argumentation est une méthode efficace pour l’apprentissage des preuves mathématiques : (i) elle améliore la pensée critique et les compétences métacognitives telles que l'auto-surveillance et l'auto-évaluation (ii) augmente la motivation des étudiants par les interactions sociales et (iii) favorise l'apprentissage entre les étudiants. Du point de vuedes enseignants, certaines difficultés surgissent avec ces approches pour l'évaluation des preuves déductives. En particulier, l'évaluation des résultats, qui comprend non seulement la preuve finale mais aussi les étapes intermédiaires, les discussions, les conflits qui peuvent exister entre les étudiants durant le débat. En effet, cette évaluation introduit une charge de travail importante pour les enseignants.Dans cette thèse, nous proposons un système pour la construction et l'évaluation des preuves mathématiques déductives. Ce système a un double objectif : (i) permettre aux étudiants de construire des preuves mathématiques déductives à partir un débat argumentatif structuré (ii) aider les enseignants à évaluer ces preuves et toutes les étapes intermédiaires afin d'identifier les erreurs et les lacunes et de fournir un retour constructif aux étudiants.Le système offre aux étudiants un cadre structuré pour débattre durant la construction de la preuve en utilisant les cadres d'argumentation proposés en intelligente artificielle. Ces cadres d’argumentation sont utilisés aussi dans l’analyse du débat qui servira pour représenter le résultat sous différentes formes afin de faciliter l’évaluation aux enseignants. Dans un second temps, nous avons implanté et validé le système par une étude expérimentale pour évaluer son acceptabilité dans la construction collaborative des preuves déductives par les étudiants et dans l’évaluation de ces preuves par les enseignants. / Learning deductive proofs is fundamental for mathematics education. Yet, many students have difficulties to understand and write deductive mathematical proofs which has severe consequences for problem solving as highlighted by several studies. According to the recent study of TIMSS (Trends in International Mathematics and Science Study), the level of students in mathematics is falling. students have difficulties to understand mathematics and more precisely to build and structure mathematical proofs.To tackle this problem, several approaches in mathematical didactics have used a social approach in classrooms where students are engaged in a debate and use argumentation in order to build proofs.The term "argumentation" in this context refers to the use of informal discussions in classrooms to allow students to publicly express claims and justify them to build proofs for a given problem. The underlying hypotheses are that argumentation: (i) enhances critical thinking and meta-cognitive skills such as self monitoring and self assessment; (ii) increases student's motivation by social interactions; and (iii) allows learning among students. From instructors' point of view, some difficulties arise with these approaches for assessment. In fact, the evaluation of outcomes -- that includes not only the final proof but also all intermediary steps and aborted attempts -- introduces an important work overhead.In this thesis, we propose a system for constructing and evaluating deductive mathematical proofs. The system has a twofold objective: (i) allow students to build deductive mathematical proofs using structured argumentative debate; (ii) help the instructors to evaluate these proofs and assess all intermediary steps in order to identify misconceptions and provide a constructive feedback to students. The system provides students with a structured framework to debate during construction of proofs using the proposed argumentation frameworks in artificial intelligence. These argumentation frameworks are also used in the analysis of the debate which will be used to represent the result in different forms in order to facilitate the evaluation to the instructors. The system has been implemented and evaluated experimentally by students in the construction of deductive proofs and instructors in the evaluation of these proofs.

Climate change will increase the risk of floods from storms and rising sea levels in many countries with coasts, seas and rivers. In Sweden, climate change is going to affect the intensity and frequency of precipitation which, together with rising sea levels, will result in more floods in the future. In the last few years amphibious buildings have emerged as an alternative solution to building protective infrastructure such as walls and levees around communities.  There are several different strategies for how communities can be protected against floods. Amphibious architecture is a strategy were buildings are working with nature instead of trying to force water away. Amphibious buildings have the same amenities and appearance as traditional building constructions and help residents to stop living in fear of potential flood and start living in harmony with water. The building technique is applicable globally and minimizes material damage, financial losses and mental suffering in the event of a flood. The purpose of this master thesis is to contribute to increased knowledge of amphibious buildings as a building technique by summarizing experience and knowledge from other countries and identifying opportunities and limitations for building that type of constructions in Sweden. A literature study was conducted in order to find published knowledge on the subject and to form a basis for an interview study regarding opportunities and limitations to build amphibious buildings in Sweden.  Results from the literature and interview studies indicate that one of the main opportunities of the building technique is the possibility to use more land for buildings and that amphibious buildings are adapted for future climate changes. By using an alternative strategy against floods where buildings work with the water less impacts in nature and ecosystems occur. One of the limitations of amphibious construction is the limited research on the subject. Amphibious buildings also requires low vertical water flow rates and waves. Knowledge of amphibious buildings as a building technique is low among many professionals in the construction sector and many countries’ laws are not adapted to this type of construction which makes it more difficult to build amphibious houses.  Furthermore, results indicate that amphibious buildings could be an effective solution for building in flood areas. Proposals for future studies include how amphibious buildings work in a Nordic climate. Another proposal is to investigate what is needed for amphibious buildings to become an accepted alternative solutions in Swedish flood areas. / När klimaten förändras ökar risken för översvämningar från stormar, skyfall och stigande havsnivåer i många länder. Klimatförändringar kommer påverka Sverige på flera olika sätt där en konsekvens är ökad nederbörd och stigande havsnivåer vilket kommer resultera i ökat antal översvämningar i framtiden. På senare tid har alternativa lösningar lyfts fram istället för bygga skyddande infrastruktur som exempelvis skyddsmurar runt samhällen. Det finns flera olika strategier hur samhällen kan skyddas mot översvämningar där amfibisk arkitektur är en strategi som innebär natur inte motarbetas utan byggnaden samarbetar med vattenmassorna. Byggnaderna har samma faciliteter och utseende som traditionella konstruktioner men vid en översvämning flyter huset på vattnet. Byggtekniken hjälper individer som lever inom riskzon för översvämningar att sluta leva i rädsla och börja leva i harmoni med vattnet. Byggmetoden är applicerbar globalt och har potential att minimerar materiella skador, ekonomiska förluster och psykiskt lidande för inblandade vid en eventuell översvämning. Syfte med studien är att bidra till ökad förståelse om amfibiska hus som byggnadsmetod, vilka möjligheter och begränsningar som finns för att använda sig av tekniken i Sverige, sammanställa kunskap och erfarenhet från andra länder samt vilka faktorer som påverkar utformningen. Målet är att identifiera kunskaper och erfarenheter om byggtekniken och sammanfatta dessa. En litteraturstudie genomfördes i syfte att identifiera och sammanställa redan känd kunskap inom ämnet och ta fram underlag till en intervjustudie. Intervjustudien fokuserade på möjligheter och begränsningar att bygga amfibiska hus i Sverige. Intervjustudien behandlade tre perspektiv; myndighetsperspektiv, finans- och försäkringsperspektiv samt sakfrågeperspektiv.  Resultatet från litteratur- och intervjustudien indikerar att en av de främsta möjligheterna  med byggtekniken är att mer markyta kan utnyttjas och att byggtekniken är anpassad för framtida klimatförändringar. Genom att använda sig av en alternativ strategi mot översvämningar som arbetar med vattenmassorna krävs mindre ingrepp i naturen och ekosystemet. Byggtekniken är applicerbar i många olika länder och kan anpassas efter individens ekonomiska förutsättningar.  En begränsande faktor är bristen på dokumenterad kunskap som kan försvåra användning av byggtekniken. Byggmetoden passar inte alla lokaliseringar utan det är platsen förutsättningar som styr huruvida amfibiska hus kan användas. Kunskapen om amfibiska hus är låg hos många aktörer inom byggsektorn och många länders lagstiftningar är inte anpassad för denna typ av byggteknik vilket försvårar byggandet av sådana objekt.  Förslag på framtida studier är att se över hur byggtekniken kan användas för större byggnader och hur byggtekniken påverkas av nordiskt klimatet. Ett annat förslag är vilka utredningar som krävs för att svenska myndigheter ska ta ställning till byggtekniken och godkänna anpassad bebyggelse på orter med översvämningsrisk.