Problem-posing is recognized as an important component in the nature of mathematical thinking (Kilpatrick, 1987). More recently, there is an increased emphasis on giving students opportunities with problem posing in mathematics classroom (English& Grove, 1998). These research has shown that instructional activities as having students generate problems as a means of improving ability of problem solving and their attitude toward mathematics (Winograd, 1991). In this study, teaching Taxicab Geometry which is a non-Euclidean geometry is aimed to mathematics teacher candidates by means of computer game-Simcity- using real life problems posing. This studies’ participants are forty mathematics teacher candidates taking geometry course. Because of using Simcity computer game, this game is based on Taxicab Geometry. Firstly, students had been given Taxicab geometry theory for two weeks and then seperated six each of groups. Each of groups is wanted to posing problem and solving from real life problems at Taxicab geometry. In addition to, students applied to problem solving at Simcity computer game. Studens were model into Simcity game. They founded ideal city, healty village, university campus, holiday village, etc. interesting of each others.

Podemos dizer que as Catedrais Góticas, verdadeiras bíblias de pedra, são signos medievais que podem ser lidos já como o resultado da intercomunicação entre matemática-ciência-arte, uma vez que tais edificações surgiram de projeções arquitetônicas, da utilização de uma dada geometria assim como da execução de determinados conjuntos escultóricos. Podemos ainda ressaltar que essa intercomunicação se intensifica durante todo o Renascimento, exemplo máximo da união entre esses três campos do conhecimento. No Renascimento, a geometria dominante e a Euclidiana; os artistas enfrentavam as questões espaciais a partir de um ponto de vista fixo. A historia se transforma quando alguns matemáticos por volta de 1800 começam a pensar na possibilidade de outra geometria que não a de Euclides. Surge, então, um tipo de geometria que ficaria conhecida como geometria não-Euclidiana, uma geometria para ser utilizada em espaços curvos. As implicações dessa nova Geometria foram tão abrangentes que influiu na elaboração da Teoria da Relatividade, de Einstein. Um novo tipo de intercomunicação entre matemática-ciência-arte, que ajudou a resolver questões ligadas a quadrimensionalidade. Enfim, trata-se de uma intercomunicação que influenciou na produção de artistas como Picasso, Duchamp e Dali. / We can say that the Gothic Cathedral, veritable Bibles of stone, are medieval sign that can be read as a result of the intercommunication among mathematicsscience- art, since that one buildings appear from an architectonic projection, from the utilization of a given geometry just as from the execution of a group of sculpture. We can salient that this intercommunication intensifies during Renaissance, example maximum of the union among those three fields of knowledge. Into the Renaissance, the geometry dominant is the Euclidean, the artists faced the special questions from one fixed viewpoint. The story becomes different when some mathematicians around 1800 begin thinking on the possibility of another geometry that doesn\'t that of Euclids. Appears, then, a kind of geometry that would be known as non-Euclidean Geometry: a geometry to be used in curved space. The implications of that new Geometry was so in-depth that influenced the elaboration of Einsteins Relativity Theory. Therefore a new kind of intercommunication among mathematics-science-art, which it helped to resolve questions linked together to the fourth dimension. An intercommunication that influenced the production of artists like Picasso, Duchamp and Dali.

Problem-posing is recognized as an important component in the nature of mathematical thinking (Kilpatrick, 1987). More recently, there is an increased emphasis on giving students opportunities with problem posing in mathematics classroom (English& Grove, 1998). These research has shown that instructional activities as having students generate problems as a means of improving ability of problem solving and their attitude toward mathematics (Winograd, 1991). In this study, teaching Taxicab Geometry which is a non-Euclidean geometry is aimed to mathematics teacher candidates by means of computer game-Simcity- using real life problems posing. This studies’ participants are forty mathematics teacher candidates taking geometry course. Because of using Simcity computer game, this game is based on Taxicab Geometry. Firstly, students had been given Taxicab geometry theory for two weeks and then seperated six each of groups. Each of groups is wanted to posing problem and solving from real life problems at Taxicab geometry. In addition to, students applied to problem solving at Simcity computer game. Studens were model into Simcity game. They founded ideal city, healty village, university campus, holiday village, etc. interesting of each others.

[pt] O objetivo deste trabalho é argumentar em favor de uma nova interpretação para o papel dos diagramas nas demonstrações da geometria euclideana. À luz de trabalhos recentes acerca do tema, pretende-se promover, em particular, uma nova avaliação daquele que é considerado o primeiro sistema dedutivo rigoroso na história da matemática: a geometria de Euclides, sistematizada nos seus Elementos. Com efeito, a utilização dos diagramas como partes essenciais das demonstrações neste sistema fez com que, na modernidade, tal sistema fosse considerado um exemplo de sistema informal, no qual as demonstrações são meros esboços do que seriam verdadeiras demonstrações. Estas, de acordo com a concepção de demonstração que se tornou comum na modernidade, devem ser compostas exclusivamente de fórmulas, as quais podem ser derivadas umas das outras apenas com base em regras lógicas ou princípios explícitos de antemão. Uma vez que tal concepção tornou-se dominante, por conta de diversos fatores nem sempre interligados, os diagramas que faziam parte das demonstrações euclideanas passaram a ser vistos como uma das principais causas de uma alegada falta de rigor por parte das mesmas. Para devolver às demonstrações matemáticas o rigor que lhes é necessário, autores como Hilbert e Pasch propuseram reconstruções formais da obra de Euclides, nas quais as demonstrações prescindem totalmente dos diagramas. No presente trabalho pretende-se reconstruir a seqüência de eventos que levou ao declínio das representações diagramáticas em geometria, bem como mostrar que é possível uma interpretação da obra de Euclides que leve em conta a participação dos diagramas nas demonstrações, sem que com isso as demonstrações sejam deficientes em termos de rigor. Serão rebatidas as críticas dos que defendem a concepção de demonstração acima mencionada, e, assim, será requerida uma revisão de tal postura - visando tanto a adoção de uma concepção mais abrangente de demonstração, quanto uma interpretação da geometria euclideana que faça mais justiça ao seu sucesso. / [en] The main concern of this work is to argue for new interpretations regarding the role of the diagrams in Euclidean geometry. Taking into account recent works on the subject, the goal here is to present alternative ways to evaluate the system which is considered the first rigorous deductive system in the history of mathematics: Euclid`s Elements. In fact, the use of diagrams as parts of its demonstrations has been considered as a flaw of that formal system. According to the standard conception of demonstration in modern times, a demonstration must be a chain of formulae, each of them being either a principle (accepted without demonstration) or a formula that follows from some principle by logical inference. As this conception became influent, the diagrams in Euclidean geometry turned out to be seen as one of the main reasons for an alleged lack of rigor of its demonstrations. In face of this, authors like Pasch and Hilbert worked on a formalization of Euclidean geometry in modern fashion, i.e., suppressing the diagrams from its demonstrations. The present work aims at a reconstruction of the main events which led to the decline of diagrammatic representations in geometry. It will be shown that an alternative view is possible. This view takes into account the importance of diagrams for the demonstrations without denying their deductive rigor. It will be argued against the conception of demonstration mentioned above, and for a revision of such conception in order to achieve a broader and fairer conception of Euclidean geometry.

In this thesis we study projective perspective on planar euclidean geometry. First we take an euclidean construction and transform it into the projective language. Then we discover and show principles of this transformation. We show equivalence between complex points I, J and some euclidean structures. Moreover we study conics, triangles, polygons and circles. We build this thesis on examples. 1

Much research has been conducted in the past 25 years related to the teaching and learning of proof in Euclidean geometry. However, very little research has been done focused on preservice secondary school mathematics teachers’ notions of proof in Euclidean geometry. Thus, this qualitative study was exploratory in nature, consisting of four case studies focused on identifying preservice secondary school mathematics teachers’ current notions of proof in Euclidean geometry, a starting point for improving the teaching and learning of proof in Euclidean geometry. The unit of analysis (i.e., participant) in each case study was a preservice mathematics teacher. The case studies were parallel as each participant was presented with the same Euclidean geometry content in independent interview sessions. The content consisted of six Euclidean geometry statements and a Euclidean geometry problem appropriate for a secondary school Euclidean geometry course. For five of the six Euclidean geometry statements, three justifications for each statement were presented for discussion. For the sixth Euclidean geometry statement and the Euclidean geometry problem, participants constructed justifications for discussion. A case record for each case study was constructed from an analysis of data generated from interview sessions, including anecdotal notes from the playback of the recorded interviews, the review of the interview transcripts, document analyses of both previous geometry course documents and any documents generated by participants via assigned Euclidean geometry tasks, and participant emails. After the four case records were completed, a cross-case analysis was conducted to identify themes that traverse the individual cases. From the analyses, participants’ current notions of proof in Euclidean geometry were somewhat diverse, yet suggested that an integration of justifications consisting of empirical and deductive evidence for Euclidean geometry statements could improve both the teaching and learning of Euclidean geometry.

This dissertation deals with aspects of axiomatization, intuition and visualization in the history of mathematics. Particular focus is put on the end of the 19th century, before David Hilbert's (1862–1943) work on the axiomatization of Euclidean geometry. The thesis consists of three papers. In the first paper the Swedish mathematician Torsten Brodén (1857–1931) and his work on the foundations of Euclidean geometry from 1890 and 1912, is studied. A thorough analysis of his foundational work is made as well as an investigation into his general view on science and mathematics. Furthermore, his thoughts on geometry and its nature and what consequences his view has for how he proceeds in developing the axiomatic system, is studied. In the second paper different aspects of visualizations in mathematics are investigated. In particular, it is argued that the meaning of a visualization is not revealed by the visualization and that a visualization can be problematic to a person if this person, due to a limited knowledge or limited experience, has a simplified view of what the picture represents. A historical study considers the discussion on the role of intuition in mathematics which followed in the wake of Karl Weierstrass' (1815–1897) construction of a nowhere differentiable function in 1872. In the third paper certain aspects of the thinking of the two scientists Felix Klein (1849–1925) and Heinrich Hertz (1857–1894) are studied. It is investigated how Klein and Hertz related to the idea of naïve images and visual thinking shortly before the development of modern axiomatics. Klein in several of his writings emphasized his belief that intuition plays an important part in mathematics. Hertz argued that we form images in our mind when we experience the world, but these images may contain elements that do not exist in nature.

Το περίφημο Ευκλείδειο Αίτημα (5ο αίτημα), όπως διατυπώνεται στα Στοιχεία του Ευκλείδη, απασχόλησε τον μαθηματικό κόσμο για περίπου 2000 χρόνια. Ξεκινώντας λοιπόν από το βιβλίο που αποτέλεσε ορόσημο για τη μαθηματική σκέψη, αναφερόμαστε σε ορισμένες αδυναμίες (κυρίως στο βαθμό αυστηρότητας) που έχουν επισημάνει σε αυτό οι κριτικοί και στεκόμαστε στο εξής γεγονός: Ο Ευκλείδης δεν έδωσε αποδείξεις για ορισμένες ιδέες και δηλώσεις του. Επειδή όμως αυτές οι δηλώσεις ήταν απαραίτητες για τις περαιτέρω μελέτες του τις έθεσε ως αληθινές. Η ιδέα ότι ορισμένες προτάσεις, μέσα στο πλαίσιο μιας θεωρίας, θα πρέπει να λαμβάνονται ως αληθινές χωρίς απόδειξη, είναι πολύ αρχαιότερη του Ευκλείδη. Ήδη ο Αριστοτέλης είχε εκθέσει στα «Αναλυτικά» του, μια θεωρητική επεξεργασία αυτής της αναγκαιότητας. Ο Ευκλείδης ακολουθεί την παγιωμένη αυτή τακτική προτάσσοντας τα πέντε αιτήματά του στο πρώτο βιβλίο των Στοιχείων του. Πολλές προσπάθειες απόδειξης του 5ου αιτήματος έγιναν από σεβαστό αριθμό μαθηματικών. Όμως η εμφάνιση απόδειξης στο πρόβλημα δεν φαινόταν να «επιθυμεί» να έρθει στο φως. Έτσι, και ενώ είχε περάσει ένα αρκετά μεγάλο χρονικό διάστημα, τελικά μέσα από την άρνηση του ίδιου του 5ου αιτήματος ήρθαν στο προσκήνιο οι Μη Ευκλείδειες Γεωμετρίες. Η άρνηση του 5ου αιτήματος οδήγησε στην άποψη πως είναι δυνατή η ύπαρξη μίας Γεωμετρίας ανεξάρτητης από το 5ο αίτημα θέτοντας έτσι τη βάση για την ανάπτυξη μίας νέας λογικά συνεπούς θεωρίας, η οποία έμελε να εκφράζει πιο πιστά αυτό που πράγματι συμβαίνει γενικά στη φύση και όχι σε μια ειδική περιοχή της . Σε πρώτο στάδιο, για να παρουσιάσουμε μία πλήρη ιστορική αναδρομή, χρησιμοποιούμε ως "σημείο εκκίνησης" τα χρόνια που προηγήθηκαν της συγγραφής των Στοιχείων. Μέσω αυτής της αναδρομής στόχος μας είναι να αναδειχθούν τόσο η φύση, όσο και ο σημαντικός ρόλος του Ευκλείδειου αιτήματος στη μαθηματική εξέλιξη. Στην καταγραφή αυτή, είναι δυνατό να συναντήσει κανείς πληροφορίες για το κλίμα που ευνόησε τη συγγραφή των Στοιχείων, ιδιαίτερα χαρακτηριστικά του συγγραφέα τους, αλλά και του ίδιου του έργου, μέσα από μία γενική θεώρηση που στόχο έχει πάντα την βαθύτερη κατανόηση του 5ου αιτήματος. Στη συνέχεια και έχοντας εξετάσει εν συντομία τα ιδιαίτερα αλλά και τα βασικά χαρακτηριστικά των Στοιχείων και του συγγραφέα τους μεταβαίνουμε στο βασικό θέμα της εργασίας. Πρόκειται, αρχικά, για την έκθεση των πέντε αιτημάτων, ενώ ακολουθεί η εκτενής παρουσίαση του 5ου αιτήματος. Βασικό αντικείμενο μελέτης μας σε αυτό το στάδιο είναι οι διαφορετικές διατυπώσεις που χρησιμοποιήθηκαν για να καταγραφεί το ίδιο ακριβώς θέμα, καθώς επίσης και οι ποικίλες προσπάθειες απόδειξής του. Παρουσιάζουμε ορισμένες από τις βασικότερες αποδείξεις του 5ου αιτήματος, τα δυνατά σημεία τους αλλά και τις αδυναμίες/ σφάλματα που επισημάνθηκαν από τους μελετητές. Το δέκατο ένατο αιώνα, οι μαθηματικοί άλλαξαν τακτική και επιχείρησαν να δείξουν ότι το 5ο αίτημα έπεται από τα άλλα τέσσερα: για να το κάνουν αυτό, πήραν τα τέσσερα αξιώματα και την άρνηση του 5ου και προσπάθησαν να εντοπίσουν τυχόν αντιφάσεις. Μόνο που αντί για αντιφάσεις, ανακάλυψαν μια καινούρια, διαφορετική, εσωτερικά συνεπή γεωμετρία. Το βασικότερο βήμα προς την ανακάλυψη των μη Ευκλείδειων γεωμετριών έγινε με την άρνηση του 5ου αιτήματος. Η καινούρια ιδέα που ήρθε στο προσκήνιο πρότεινε ουσιαστικά την αντικατάσταση του 5ου αιτήματος από την άρνησή του. Επομένως, εάν επιχειρούσαμε να καταγράψουμε το περιεχόμενο της εργασίας συνοπτικά θα καταλήγαμε στα εξής: Πρόκειται για μία ιστορική αναδρομή που έχει βασικό της θέμα, αρχικά την παρουσίαση του Ευκλείδειου αιτήματος, έπειτα τις προσπάθειες απόδειξής του και τέλος την ανακάλυψη των Μη Ευκλείδειων Γεωμετριών μέσω της άρνησής του. / --

Η παρούσα διδακτορική διατριβή εστιάζεται στο πρόβλημα της “μετάβασης” από την δευτεροβάθμια εκπαίδευση στην τριτοβάθμια προτείνοντας την σύνδεση των Στοιχειωδών με τα Ανώτερα Μαθηματικά. Η σύνδεση αυτή επιχειρείται να γίνει μέσω κατάλληλα επιλεγμένων θεμάτων και οι μη-Ευκλείδειες Γεωμετρίες αποτελούν ένα ελκυστικό αντικείμενο για έρευνα στο παραπάνω πλαίσιο. Η διατριβή αυτή, αφορά τις μη Ευκλείδειες Γεωμετρίες και τα μοντέλα τους και ιδιαίτερα την Ελλειπτική Γεωμετρία: καθώς αυτή μοντελοποιείται πάνω στη σφαίρα, θα μπορούσε να αποτελέσει τη “γέφυρα” για το πέρασμα από την Ευκλείδεια στις μη-Ευκλείδειες Γεωμετρίες. Γίνεται μια σύντομη ιστορική αναδρομή από την ανακάλυψη των μη Ευκλείδειων Γεωμετριών μέχρι την αξιωματική θεμελίωση του Hilbert. Περισσότερη έμφαση δίνεται στα μοντέλα των μη Ευκλειδείων και προτείνεται μια κατηγοριοποίηση αυτών για παιδαγωγικούς σκοπούς καθώς και ένα μοντέλο της Ελλειπτικής Γεωμετρίας για πιθανή διδακτική χρήση. Επίσης αναλύεται ένα νέο θεωρητικό πλαίσιο έρευνας για την διδακτική της γεωμετρίας και στην συνέχεια θα περιγραφεί η ερευνητική μεθοδολογία και το κυρίως διδακτικό πείραμα. Στο διδακτικό πείραμα, στο οποίο βασίζεται η διατριβή αυτή, συμμετείχαν φοιτητές του Μαθηματικού Τμήματος του Πανεπιστημίου Πατρών οι οποίοι ασχολήθηκαν, μεταξύ άλλων, με ζητήματα όπως: η συντομότερη διαδρομή μεταξύ δύο σημείων πάνω στην επιφάνεια της σφαίρας και η κατασκευή μοντέλου της Ελλειπτικής Γεωμετρίας. / In this thesis, we investigate university students' interaction in an attempt to construct a model of elliptic geometry. In order to study the several ways in which geometrical meaning is produced through context and practices, we introduce three different types of use of geometrical concepts, namely as (1) elements of representation of spatial experience, (2) objects of traditional school practice, and (3) constituents of an abstract mathematical theory. The analysis of students' dialog according to this framework reveals how students develop their ability to communicate mathematically and negotiate their meanings. Students with a different use of geometrical concepts were able to interact and understand their peers.

Orientador: Clotilzio Moreira dos Santos / Banca: Tatiana Miguel Rodrigues / Banca: Tatiana Bertoldi Carlos / Resumo: Na Grécia Antiga, os sábios buscaram a resolução de problemas que se baseavam na construção geométrica utilizando exclusivamente dois instrumentos: a régua não graduada e o compasso. Alguns desses problemas se tornaram clássicos por exigirem, dentro do desenvolvimento da Matemática, grandes esforços para se chegar a uma solução. São eles: a duplicação de um cubo, determinando o lado de um cubo, cujo volume é o dobro do volume de um outro cubo dado, a trisseção de um ângulo, que é dividir um ângulo em três partes iguais ou três ângulos de medidas exatamente iguais e a quadratura de um círculo, que consiste em construir um quadrado com área igual à de um círculo dado. Neste trabalho apresentaremos algumas construções geométricas com régua não graduada e compasso, algumas soluções encontradas que não estavam de acordo com as regras estabelecidas e desenvolveremos a fundamentação algébrica que demonstra a insolubilidade dos três problemas clássicos citados / Abstract: In ancient Greece, the sages sought to solve problems that were based on geometric construction using only two instruments: non-graduated ruler and compass. Some of these problems have become classics because they require within the development of Mathematics, great efforts to reach a solution. They are: the duplication of the cube, the side of a cube whose volume is twice the volume of a given cube; the trisseção of an angle, which is to divide an angle into three equal parts or three measures angles exactly equal and the squaring of a circle, which consists of constructing a square with the same area as a given circle. In this work we present some geometric constructions with non-graded ruler and compass, some solutions that were not in accordance with the rules laid down and develop the algebraic reasoning which demonstrates the insolubility of the three classic problems cited / Mestre