TITLE: Didactical situations in mathematics at lower secondary level. The classification of quadrilaterals based on selected characteristics. AUTHOR: Bc. Kateřina Vladyková DEPARTMENT: Department of Mathematics and Mathematical Education SUPERVISOR: Prof. RNDr. Jarmila Novotná, CSc. ABSTRACT: This thesis deals with the possible use of the Theory of didactical situations in mathematics at the lower secondary level of Czech school. It is specifically focused on the classification of quadrilaterals based on selected characteristics. The thesis consists of two parts, theoretical and practical. The theoretical part deals mainly with the introduction of Theory of didactical situations, defining and explaining their basic concepts. The thesis also briefly introduces currently existing curricula in the Czech Republic, in which quadrilaterals were the author's main focus. Furthermore, the topic is analyzed from educational materials (textbooks) used in Czech primary schools. In the experimental part of the thesis the author presents a detailed script of educational unit. It is a practical demonstration of a particular theory of didactic situations in teaching mathematics at the Czech elementary school, which is developed and implemented according to the principles of the theory of didactic situations. The thesis...

Title: Construction of elementary pupils' geometric schemas via motivating learning environments Author: Jaroslava Kloboučková Department: Department of mathematics and mathematics education Supervisor: doc. RNDr. Darina Jirotková, Ph.D. Abstract: The aim of the dissertation is to discuss teaching geometry as integral part of mathematics education at the primary school level. The thesis also documents a longitudinal teaching study which was initiated in 2010 and which gives us a base for discussion of some fundamental questions regarding the process of learning geometry for pupils in their early school years. The main objective here is to attempt to answer the following four didactic questions: In which way do pupils learn about geometrical objects? How do they share their geometrical knowledge, experience and discoveries with one another? How much (at what level) are they able to understand mathematical concepts that the official curricular documents (the Czech Framework for Education Program) place in later years of schooling? What phenomena are they able to grasp and describe using their mother tongue? The theoretical framework focuses on the learning process and the typology of mathematical problems in geometry. Four specific engaging environments (Cube Buildings, Origami, Wooden Sticks, and Tiles) and...

Made available in DSpace on 2016-04-27T17:13:00Z (GMT). No. of bitstreams: 1 Alvesmar Ferreira da Silva Filho.pdf: 4341581 bytes, checksum: 7d920ecf354e0137a2bbca7e034ddc90 (MD5) Previous issue date: 2007-05-24 / Made available in DSpace on 2016-08-25T17:25:37Z (GMT). No. of bitstreams: 2 Alvesmar Ferreira da Silva Filho.pdf.jpg: 1943 bytes, checksum: cc73c4c239a4c332d642ba1e7c7a9fb2 (MD5) Alvesmar Ferreira da Silva Filho.pdf: 4341581 bytes, checksum: 7d920ecf354e0137a2bbca7e034ddc90 (MD5) Previous issue date: 2007-05-24 / Secretaria da Educação do Estado de São Paulo / The aim of this study was to involve a group of mathematics teachers from the same school in the development of a teaching sequence related to quadrilaterals and their properties. Inspired by the aims of the project AProvaMe (Argumentation and proof in school mathematics), a collaborative group was organised in the school where the teacher-researcher works, with the intention of creating a space within which teachers could engage in discussions about the theme of the research: the teaching and learning of proof. Before initiating the reunions of the group, a first version of the teaching sequence was conceived to serve as a focus for the discussions. The design of the activities of the sequence was informed by the work of Parsysz about the teaching of geometry in general and, more specifically, aspects related to justifying and proving were based in the perspectives about types of proof and their functions of Balacheff and De Villiers. With the sequence in hand, two teachers were invited to compose, together with the teacher-researcher, a collaborative group within which to work through, discuss and, where necessary, modify the proposed activities. On the basis of the initial survey of the conceptions of the teachers about prova and its teaching, it was identified that the area of the curriculum is not one woth which they are well familiar or comfortable and, in particular, both the teachers indicated that they expect their students only to understand arguments of a pragmatic and not a conceptual nature.. Given this fact, perhaps the most significant contribution of their participation in the collaborative group meetings was the oportunity to reflect upon possible teaching strategies that might help smooth the passage from pragmatic to conceptual proofs / Esse trabalho tem como objetivo envolver um grupo de professores da Matemática de uma mesma escola no desenvolvimento de uma seqüência didática sobre quadriláteros e suas propriedades. Inspirado nos objetivos do projeto AprovaME (Argumentação e Prova na Matemática Escola), resolvemos organizar um grupo colaborativo na escola onde trabalhamos, com a intenção de criar um espaço que possibilitasse a discussão sobre o tema dessa pesquisa: o ensino e aprendizagem de prova. Antes de iniciar as reuniões do grupo, concebemos uma primeira versão de uma seqüência didática para servir como base para as discussões. A elaboração das atividades da seqüência foi baseada nos trabalhos de Parsysz (2000) sobre o ensino de Geometria em geral e, mais especialmente, as partes referentes à justificativa e prova foram baseadas nas perspectivas sobre tipos de provas e suas funções de Balacheff (1988) e De Villiers (2001). Em seguida, convidamos duas professoras da Matemática para formarem, juntamente com o pesquisador/professor, um grupo colaborativo no qual realizamos, discutimos e, na medida necessária, modificamos as atividades propostas, inicialmente, na seqüência. A partir de levantamento inicial das concepções das professoras sobre provas e seu ensino, identificamos que provas não representam um tópico com qual as professoras sentiam-se muito seguras e, em particular, ambas indicaram que esperavam que seus alunos apenas entendessem provas de natureza pragmática. Assim, a contribuição mais significativa da participação delas nos encontros do grupo colaborativo foi a possibilidade de refletir sobre estratégias de ensino que pudessem facilitar a passagem de provas pragmáticas a provas conceituais

Orientador: Prof. Dr. Márcio Fabiano da Silva / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Mestrado Profissional em Matemática em Rede Nacional - PROFMAT, 2014. / Baseados no artigo de M. Josefsson, zemos nossos estudos dos quadriláteros convexos, mais precisamente dos trapézios, e buscamos por fundamentações que os caracterizassem. Estudamos seus ângulos, seus lados, as medidas de suas áreas e estabelecemos relações entre seus elementos, lançando mão de diversos teoremas, como o Postulado de Pasch, mas nossa principal fundamentação está no Postulado das Paralelas. / Based on the paper of M. Josefsson we have elaborated our studies on the convex quadrilaterals, more precisely the trapezoids, and searched for fundamentations which would characterize them. We have studied their angles, their sides, their areas and stabilished relations to their elements, making use of several theorems as the Pasch's Postulate, however, our main fundamentation is on the Parallel Postulate.

Made available in DSpace on 2016-04-27T16:58:57Z (GMT). No. of bitstreams: 1 MARCIA CRISTINA DOS SANTOS AMORIM.pdf: 3371817 bytes, checksum: cbce7dedf280961f108d7f7e5606f8b7 (MD5) Previous issue date: 2009-10-20 / The presente work has a objective a sequence of activities which, with the help of dynamic geometry provided by the Cabri Geometry software, might empower high school students with new ways of thinking and establishing links between information and properties within a meaningful approach to mathematical reasoning. The sequence of activities is linked to the properties of a quadrilateral, which are of an empirical and exploratory nature so as to encourage a deductive approach in students. Our hypothesis is that these activities help students understand quadrilateral concepts and properties, and with the aid of the software tools, enable them to simulate and manipulate objects. Thus, these activities make for a meaningful and effective way of learning and dealing with Mathematics. It is hoped that with this sequence of activities students probe and discuss their conjectures, and put forth mathematically-grounded arguments and justifications to bear them out. The methodology adopted for the elaboration of activities is based on the principles of didactic engineering, which furnished analytical tools for the study of each activity devised. The results were examined according to Balacheff's (1988) classification of proof types. The conclusion drawn is that, thanks to all involved experimentation, manipulation and investigation, dynamic geometry has laid on a meaningful learning environment. As to reasoning and proof, it appears that students find it difficult to break free from specific cases when sustaining their arguments. Developing teaching-learning skills so as to improve construction of mathematical proof is of paramount importance / O presente trabalho tem como objetivo apresentar uma seqüência de atividades que possibilitem a alunos do Ensino Médio novas formas de pensar, relacionar informações e propriedades em uma abordagem significativa para justificativas matemáticas, com o auxílio da geometria dinâmica proporcionada pelo software Cabri-Géomètre. A sequência de atividades está relacionada com as propriedades dos quadriláteros e tem um caráter empírico e exploratório com a preocupação de fomentar no aluno a necessidade da demonstração dedutiva. Temos como hipótese que o desenvolvimento de atividades contribui para auxiliar o aluno na compreensão dos conceitos e propriedades dos quadriláteros, assim, com o uso das ferramentas do software será possível simular e manipular objetos oportunizando uma maneira eficiente e significativa de aprender e fazer Matemática. Com esta seqüência de atividades, esperamos que os alunos investiguem, discutam suas conjecturas e produzam argumentos ou justificativas matemáticas que as validem ou não. A metodologia utilizada para a elaboração das seqüências se baseou em noções da engenharia didática, que forneceu subsídios como fonte de observação para realizarmos uma análise de cada atividade aplicada. Os resultados foram examinados segundo a classificação dos tipos de provas de Balacheff (1988).Concluímos que a geometria dinâmica proporcionou um ambiente de aprendizagem significativo, com base na experimentação, manipulação e investigação. Quanto à argumentação e prova, percebemos que o aluno não consegue desprender-se dos casos particulares para concretizar a argumentação. Após este trabalho refletimos que desenvolver habilidades para elevar o nível de conhecimento quanto à construção de provas em Matemática é elemento essencial no processo de ensino e aprendizagem

Title: The Collection of Exercises About Qaudrangles Author: Bc. Pavel Nohál Department: Department of Mathematics Education Supervisor: doc. RNDr. Jarmila Robová, CSc., Department of Mathematics Education Abstract: The master thesis is focused on exercises regarding qaudrangles. This gives a chance to pupils of primary schools and lower secondary schools to repeat and practice math curriculum with a view of further study, but also develop pupils' common sense and logical thinking. The quadrangles are being encountered by us every day and therefore the exercises in this area are largely connected with the practical life. As not many current Collections on the market are dealing exclusively with quadrangles, author would like to fill this vacancy by this thesis. Keywords: Quadrilaterals and their division Parallelograms and trapezoids Circumference and area Tops, sides and interior angles

This paper examines the mathematical knowledge for teaching (MKT) in Indonesia, specifically in school geometry content. A translated and adapted version of the MKT measures developed by the Learning Mathematics for Teaching (LMT) project was administered to 210 Indonesian primary and junior high teachers. Psychometric analyses revealed that items related to classification of quadrilaterals were difficult for these teachers. Further interactions with teachers in a professional development setting confirmed that teachers held a set of exclusive definitions of quadrilaterals.

This paper examines the mathematical knowledge for teaching (MKT) in Indonesia, specifically in school geometry content. A translated and adapted version of the MKT measures developed by the Learning Mathematics for Teaching (LMT) project was administered to 210 Indonesian primary and junior high teachers. Psychometric analyses revealed that items related to classification of quadrilaterals were difficult for these teachers. Further interactions with teachers in a professional development setting confirmed that teachers held a set of exclusive definitions of quadrilaterals.

Made available in DSpace on 2016-04-27T16:58:19Z (GMT). No. of bitstreams: 1 marcia.pdf: 567454 bytes, checksum: 549773993b0ea2b0efbe76289c983b75 (MD5) Previous issue date: 2002-11-19 / Conselho Nacional de Desenvolvimento Científico e Tecnológico / This work is about Mathematics teacher education. It ains to offer a contribution in this area, in terms of understanding the processes by which new mathematical content is adquired and existing knowledge extended. We hope that it helps teachers to develop suitable strategies for working with Geometry in the classroom. This study is founded in BROUSSEAU s theory of situations and in DUVAL s studies about the use of several registers of semiotic representation. We prepared a workshop composed of activities whose focus was quadrilaterals, which was attended by a group of teachers from elementary and secondary school levels. During the workshop, we discussed the theoretical references that had served as a base for selectiing the activities and for the manners in which these were presented. The research question is: how can we work with teachers in ways which result in the acquisition of geometrical knwledge and, at the some time, provide them with inherent didactic knowledge related the geometrical content? Our main hypothesis is that the proposed activities will contribute to the content acquisition and the discussion of the theoretical references to the didactic knowledge improvement. Analyses of the teachers behavior and discussion, during 33 workshop hours, reveals that the activities provoked considerations about definitions, assumpitions, properties of quadrilaterals, theorems and proofs, as well as helping teachers to discover the difficulties they have in using different registers of representation in Geometry. The discussion of the theoretical references made the teachers understand that in their classroom they have usually omitted the action, formulation and validity stages, discussed by BROUSSEAU, presenting Geometry in an institutionalized way / Esse trabalho trata da formação de professores de matemática. Nosso objetivo é oferecer uma contribuição nessa área, tanto no que se refere à aquisição de conteúdos, quanto no aprimoramento de conhecimentos que auxiliem os professores na elaboração de estratégias adequadas para o trabalho com geometria em sala de aula. Fundamentados na Teoria das situações de BROUSSEAU e nos estudos de DUVAL sobre a utilização de diversos registros de representação semiótica, elaboramos uma oficina composta por atividades envolvendo os quadriláteros e a desenvolvemos com um grupo de professores do ensino fundamental e médio. Durante o desenvolvimento da oficina, discutimos com os participantes o referencial teórico que embasou a seleção das atividades e a maneira utilizada para apresentar o conteúdo. A questão investigada é: como trabalhar com formação de professores de forma a contribuir com a aquisição de conteúdos em geometria, proporcionando ao professor conhecimentos didáticos inerentes a esses conteúdos? Nossas principais hipóteses supõem que o desenvolvimento das atividades contribuirá com a aquisição de conteúdos, e a discussão do referencial teórico, com aprimoramento de conhecimentos didáticos inerentes à geometria. A análise das discussões e comportamento dos professores durante as trinta e três horas de oficina, revelaram-nos que as atividades provocaram reflexões sobre definições, conjeturas, propriedades dos quadriláteros, teoremas e demonstrações, bem como ajudou os professores a descobrirem a dificuldade que têm em utilizar diferentes registros de representação em geometria. A discussão do referencial teórico fez com que os professores notassem que, geralmente, têm omitido em suas aulas, as fases de ação, formulação e validação discutidas por BROUSSEAU, apresentando a geometria de forma já institucionalizada

Το πρόβλημα Fermat-Torricelli για n μη συγγραμμικά σημεία με βαρύτητες στον R^3 (b.FT) διατυπώνεται ως εξής: Δοθέντος n μη συγγραμμικών σημείων στον R^3 να βρεθεί ένα σημείο το οποίο ελαχιστοποιεί το άθροισμα των αποστάσεων με θετικές βαρύτητες του σημείου αυτού από τα n δοσμένα σημεία. Το αντίστροφο πρόβλημα Fermat-Torricelli για n μη συγγραμμικά και μη συνεπίπεδα σημεία με βαρύτητες στον R^3 (αντ.FT) διατυπώνεται ως εξής: Δοθέντος ενός σημείου που ανήκει στο εσωτερικό ενός κλειστού πολυέδρου που σχηματίζεται από n δοσμένα μη συγγραμμικά και μη συνεπίπεδα σημεία στον R^3, υπάρχει μοναδικά προσδιορίσιμο σύνολο τιμών για τις βαρύτητες που αντιστοιχούν σε κάθε ένα από τα n δοσμένα σημεία, ώστε το σημείο αυτό να επιλύει για τις τιμές αυτές των βαρυτήτων το πρόβλημα b.FT στον R^3; Στην παρούσα διατριβή, αποδεικνύουμε μία γενίκευση της ισογώνιας ιδιότητας του σημείου b.FT για ένα γεωδαισιακό τρίγωνο σε ένα Κ-επίπεδο (Σφαίρα, Υπερβολικό επίπεδο, Ευκλείδειο επίπεδο). Στη συνέχεια, δίνουμε μία αναγκαία συνθήκη για να είναι το σημείο b.FT εσωτερικό σημείο ενός τετραέδρου και ενός πενταέδρου (πυραμίδες) στον R^3. Η δεύτερη ομάδα αποτελεσμάτων της διατριβής περιλαμβάνει τη θετική απάντηση στο αντ.FT πρόβλημα για τρία μη γεωδαισιακά σημεία στο Κ-επίπεδο και στο αντ.FT πρόβλημα για τέσσερα μη συγγραμμικά και μη συνεπίπεδα σημεία στον R^3. Η αρνητική απάντηση στο αντ.FT για τέσσερα μη συγγραμμικά σημεία στον R^2 θα μας οδηγήσει σε σχέσεις εξάρτησης των βαρυτήτων που ονομάζουμε εξισώσεις της δυναμικής πλαστικότητας των τετραπλεύρων. Ομοίως, δίνοντας αρνητική απάντηση στο αντ.FT πρόβλημα για πέντε μη συνεπίπεδα σημεία στον R^3, παίρνουμε τις εξισώσεις δυναμικής πλαστικότητας , διατυπώνουμε και αποδεικνύουμε την αρχή της πλαστικότητας των κλειστών εξαέδρων στον R^3, που αναφέρει ότι: Έστω ότι πέντε προδιαγεγραμμένα ευθύγραμμα τμήματα συναντώνται στο σημείο b.FT, των οποίων τα άκρα σχηματίζουν ένα κλειστό εξάεδρο. Επιλέγουμε ένα σημείο σε κάθε ημιευθεία που ορίζει το προδιαγεγραμμένο ευθύγραμμο τμήμα, τέτοιο ώστε το τέταρτο σημείο να βρίσκεται πάνω από το επίπεδο που σχηματίζεται από την πρώτη και δεύτερη προδιαγεγραμμένη ημιευθεία και το τρίτο και πέμπτο σημείο να βρίσκονται κάτω από το επίπεδο που σχηματίζεται από την πρώτη και δεύτερη προδιαγεγραμμένη ημιευθεία. Τότε η μείωση της τιμής της βαρύτητας που αντιστοιχεί στην πρώτη, τρίτη και τέταρτη προδιαγεγραμμένη ημιευθεία προκαλεί αύξηση στις βαρύτητες που αντιστοιχούν στη δεύτερη και πέμπτη προδιαγεγραμμένη ημιευθεία.Τέλος, ένα σημαντικό αποτέλεσμα της διατριβής αφορά την επίλυση του γενικευμένου προβλήματος του Gauss για κυρτά τετράπλευρα στο Κ-επίπεδο, θέτοντας δύο σημεία στο εσωτερικό του κυρτού τετραπλεύρου με ίσες βαρύτητες, τα οποία στη συνέχεια αποδεικνύουμε ότι είναι δύο σημεία b.FT με συγκεκριμμένες βαρύτητες, αποτέλεσμα το οποίο γενικεύει το πρόβλημα b.FT για τετράπλευρα στο Κ-επίπεδo. / The weighted Fermat-Torricelli for n non-collinear points in R^3 states the following: Given n non-collinear points in R^3 find a point (b.FT point) which minimizes the sum of the distances multiplied by a positive number which corresponds to a given point (weight). The inverse Fermat-Torricelli problem for n non-collinear points with weights in R^3 (inv.FT) states the following: Given a point that belongs to the interior of a closed polyhedron which is formed between n given non-collinear points in R^3, does there exist a unique set of weights which corresponds to each one of the n points such that this point solves the weighted Fermat-Torricelli problem for this particular set of weights? In the present thesis, we prove a generalization of the isogonal property of the b.FT point for a geodesic triangle on the K-plane (Sphere, Hyperbolic plane, Euclidean plane). We proceed by giving a sufficient condition to locate the b.FT point at the interior of tetrahedra and pentahedra (pyramids) in R^3. The second group of results contains a positive answer on the inv.FT problem for three points that do not belong to a geodesic arc on the K-plane and on the inv.FT problem for four non collinear points and non coplanar in R^3. The negative answer with respect to the inv.FT problem for four non-collinear points in R^2 lead us to the relations of the dependence between the weights that we call the equations of dynamic plasticity for quadrilaterals. Similarly, by giving a negative answer with respect to the inv.FT problem for five points which do not belong in the same plane in R^3, we derive the equations of dynamic plasticity of closed hexahedra and we prove a plasticity principle of closed hexahedra in R^3, which states that: Considering five prescribed rays which meet at the weighted Fermat-Torricelli point, such that their endpoints form a closed hexahedron, a decrease on the weights that correspond to the first, third and fourth ray, causes an increase to the weights that correspond to the second and fifth ray, where the fourth endpoint is upper from the plane which is formed from the first ray and second ray and the third and fifth endpoint is under the plane which is formed from the first ray and second ray. Finally, a significant result of this thesis deals with the solution of the generalized Gauss problem for convex quadrilaterals on the K-plane in which by setting two points at the interior of the convex quadrilateral with equal weights we prove that these points are weighted Fermat-Torricelli points with specific weights, that generalizes the b.FT problem for quadrilaterals on the K-plane.