DOCSLIB.ORG

Projective Geometry in the Primary School Curriculum : Children's Spatial-Perceptual Abilities

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Projective Geometry in the Primary School Curriculum : Children's Spatial-Perceptual Abilities This electronic thesis or dissertation has been downloaded from the King’s Research Portal at https://kclpure.kcl.ac.uk/portal/ Projective geometry in the primary school curriculum : children's spatial-perceptual abilities. Salisbury, Andrew J The copyright of this thesis rests with the author and no quotation from it or information derived from it may be published without proper acknowledgement. END USER LICENCE AGREEMENT Unless another licence is stated on the immediately following page this work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International licence. https://creativecommons.org/licenses/by-nc-nd/4.0/ You are free to copy, distribute and transmit the work Under the following conditions: Attribution: You must attribute the work in the manner specified by the author (but not in any way that suggests that they endorse you or your use of the work). Non Commercial: You may not use this work for commercial purposes. No Derivative Works - You may not alter, transform, or build upon this work. Any of these conditions can be waived if you receive permission from the author. Your fair dealings and other rights are in no way affected by the above. Take down policy If you believe that this document breaches copyright please contact [email protected] providing details, and we will remove access to the work immediately and investigate your claim. Download date: 09. Oct. 2021 1 PROJECTIVE GEOMETRY IN THE PRIMARY SCHOOL CURRICULUM : CHILDREN'S SPATIAL-PERCEPTUAL ABILITIES ANDREW 3. SALISBURY Thesis submitted in fulfilment of the requirements for the Ph.D. degree of the University of London Centre for Science and Mathematics Education Chelsea College, University of London, 1983. 2 ASTRACT Some fundamental questions are posed concerning the teaching and learning of geometry in primary schools. How can geometry be made more interesting, vital and relevant; and how can it be related to other subject areas? How may critical learning ex- periences be identified which can provide a conceptual framework and an intellectual clarity within a primary school curriculum? Specific aspects of projective geometry are formulated to resolve the problem of disparate emphases in topological and euclidean approaches, thereby linking them both logically and psychologically; and enabling the learning of key concepts to be integrated with other subjects such as geography and art. Implications for teaching are considered, especially those inherent in the historical development of geometry, the relation- ship between projective geometry and perspective, the influence of the secondary school curriculum on primary, geometry, and the development of children's perceptions through projective experiences. The study investigates ways in which children interpret and use line drawings. In addition, it is shown that children have strong preferences f or projective rather than euclidean. transformations. A description is given of a projective geometry curriculum together with an account of its application in a primary school where it was evaluated in a variety of ways. Modes of geometrical thinking are suggested, the significances of these for teachers and learners are considered, and possible avenues for further research are discussed. 3 ACKNOWLEDG'1ENT This study developed from ideas generated during discussions at Chelsea College and during investigations at George White Middle School, Norwich. My thanks are due to Professor David Johnson who supervised this study always providing criticism, guidance, and encouragement; to the pupils, staff, and head teacher of George White School for their considerable help and consideration; and to Norman Wordsworth1 John Nicholls, and Roy Baker for their adv"ice. I am also grateful for the support and forebearance of my family. 1.4. TABLE OF CONTENTS page number TITLE PAGE I ABSTRACT 2 ACKNCWLGEMENT 3 TABLE OF CONTENTS 14. LIST OF TABLES 10 LIST OF FIGURES 11 CHAPTL ONE 12 Introduction An examination of the Problems in Primary School Geometry 12 The Main Theme of thi Study 12 A g efinition of Projective Geometry 13 A Confusion in Present Day Primary Geometry 13 Increasing Geometric Relevance across the Curriculum 14• A Projective Geometry Curriculum 11.4. Outline of the Thesis 15 CHAPTER TWO 17 Geometry in Education Geometry : Its Nature and Historical Development 17 Geometry as useful 17 Geometry as an abstraction 17 Geometry as a logical structure 17 Geometry as a study of relationships 18 Geometry as exhibited by behavioural changes Geometry as a process 19 5 TABLE OF COTE7TS COINI page number A great name in the history of geometry : Euclid 19 Euc1ids postulates : The fifth postulate 20 Other early geometrical developments 22 Renaissance art 22 Non euclidean geometry 22 Desargues and others 27 Projections and cross ratios 28 Differences between projective geometry and perspective 30 Topology and Hubert 31 Primary and Secondary School Geometry Changing Approaches 31 Difficulties in analysis of school geometry 31 Euclidean geometry in the secondary school 32 Comparison with classics as a discipline 33 A period of transition : emphasis on the linear map 3L. Primary geometry 3o Projective geometry in school : texts and suggestions 39 A Review of Research in Learning Geometry with an emphasis on Projective Ideas 1 Piaget, Inhelder, and Szeminska 14.1 General spatial ability Pictorial space. spontanethus and copying exercises 14.5 Perspective ability 14.7 Views supporting Piaget's research findings Views questioning Piaget's research findings '9 Perception 52 6 TABLE OF CO'TEI'TS CO?TINtJ cage number Projective Geometry : Its Relationships with other Selected Curriculum Areas in the Primary School 56 Science 56 Geography 59 Art 62 Curricuium Development in Projective Geometry Theoretical Considerations, Pratica3. Problems, and Suggested Approaches 63 Theoretical implications in the relationships between children's topology and projective geometry 6k Practical problems : the role of the teaher of geometry in the primary school 70 Suggested approaches 72 CHAPT TIEE 7k Children's Projective Geometric Abilities An Exploratory Phase of the Research 7k Straightness as a concept 7k Interpretations of line drawings 77 Research into Children's Proective Preferences 99 Outline of the research background 99 0 The research design I R esu3. ts 114 A further experiment 115 7 TABLE OF CONTENTS C0NTINU page number CH.APT FOUR 120 The develo'oment of a Protective Geometry Curriculum in the Primary School Setting up the Research 120 Introduction 120 The Initial pilot tests 122 The Main Research : Aspects of Suzmnative Evaluation 123 Changes in the research design 123 Summative evaluation I 2 Pretest marking scheme 1 1 Pretest and post-test results lLf4 The Main Research : Classification of Children's Responses I Lf 8 The main programme i+8 A topological theme li+9 A projective theme 151 An affine theme 155 A theme in visual illusions 158 A theme on co-ordination of views 160 Testing Materials developed late in the Research loo A new pilot class 166 An analysis of children's responses 167 Research Results and Interpretations 170 Operating principles 170 Post-test discussion 171 General implications 172 8 TABLE OF COE1'TS CONTINU page number CKAPT FIVE 17k The Promotion of Geometrical Thinking in Primary Schools An Examination of Problems in Primary School Geometry 17k Preamble 17k The unique and distinctive nature of geometry 17k The function of geometry in the school curriculum 175 The links between geometry and. other subject areas 175 Difficulties associated with children's learning of geometrical concepts in relation to the sometimes contradictory findings of research 175 Children's geometric interpretations 1 their projective preferences and abilities 176 Research brought about by the development of a new approach, that is, through a projective geometry curriculum 177 Problems in the Promotion of Geometrical Thinking 177 Modes of thought 177 Perception of the environment. 179 Viewing and. visualization I ao Affine representation I 80 Projective co-ordination i8i Straightness and polygons 182 A general projective perception 182 The Significance for Teachers and Learners 182 Possible Avenues or Further esearch i EJ4 9 TABLE OF CONTES CONT IN page number BLIOGRAPHY 190 APPENDICES 1: A Projective Geometry Curriculum for Children in Primary School 212 2: Getting Geometry into Perspective : A bird's eye view 336 3: Geometrical, Spatial Experiences for 5 to 12 year olds 338 k: Children's Interpretations of Line Drawings 3L 5: Children's Projective Preferences 3Lf 5 6: Swiunative Evaluation : Pretest and Post-test Raw Data 3L4.7 7: Some Strategy games using Desargues' Theorem 353 10 LIST OF TABLES :able number Table Caption Page number 1 Straightness scores for (a) Rectagular border and (b) Curved border experiments 76 2 Results of preference choices 114 3 Results of further preference choices 118 4 Pretest and post-test instructions 13? 5 Pretest and post-test marking scheme 1+2 6 Items analysed in pretest, post-test comparison 1+ 7 Pretest, post-test t statistics for matched pairs for the experimental and control group 147 8 Results of 144 children on Co-ordination of Views activity Where are you? i6'+ 9 Scores for Where is the middle? activity for 29 children aged nine 168 10 Scores for What will it look like from other places? for 27 children aged nine 169 11 LIST OF FIGURES Figure number Figure caption page number 1 Lobatchevesky' s postulate 23 2 A classification of geometries 25 3 Projections
Load more

Recommended publications
  • Letter from Descartes to Desargues 1 (19 June 1639)
    Appendix 1 Letter from Descartes to Desargues 1 (19 June 1639) Sir, The openness I have observed in your temperament, and my obligations to you, invite me to write to you freely what I can conjecture of the Treatise on Conic Sections, of which the R[everend] F[ather] M[ersenne] sent me the Draft. 2 You may have two designs, which are very good and very praiseworthy, but which do not both require the same course of action. One is to write for the learned, and to instruct them about some new properties of conics with which they are not yet familiar; the other is to write for people who are interested but not learned, and make this subject, which until now has been understood by very few people, but which is nevertheless very useful for Perspective, Architecture etc., accessible to the common people and easily understood by anyone who studies it from your book. If you have the first of these designs, it does not seem to me that you have any need to use new terms: for the learned, being already accustomed to the terms used by Apollonius, will not easily exchange them for others, even better ones, and thus your terms will only have the effect of making your proofs more difficult for them and discourage them from reading them. If you have the second design, your terms, being French, and showing wit and elegance in their invention, will certainly be better received than those of the Ancients by people who have no preconceived ideas; and they might even serve to attract some people to read your work, as they read works on Heraldry, Hunting, Architecture etc., without any wish to become hunters or architects but only to learn to talk about them correctly.
    [Show full text]
  • Desargues, Girard
    GIRARD DESARGUES (February 21, 1591 – October 1661) by HEINZ KLAUS STRICK, Germany GIRARD DESARGUES came from very wealthy families of lawyers and judges who worked at the Parlement, the highest appellate courts of France in Paris and Lyon. Nothing is known about GIRARD's youth, but it is safe to assume that he and his five siblings received the best possible education. While his two older brothers were admitted to the Parisian Parlement, he was involved in the silk trade in Lyon, as can be seen from a document dating from 1621. In 1626, after a journey through Flanders, he applied to the Paris city council for a licence to drill a well and use the water from the well. His idea was to construct an effective hydraulic pump to supply water to entire districts, but this project did not seem to be successful. After the death of his two older brothers in 1628, he took over the family inheritance and settled in Paris. There he met MARIN MERSENNE and soon became a member of his Academia Parisiensis, a discussion group of scientists including RENÉ DESCARTES, GILLES PERSONNE DE ROBERVAL, ÉTIENNE PASCAL and his son BLAISE. (drawing: © Andreas Strick) The first publication by DESARGUES to attract attention was Une méthode aisée pour apprendre et enseigner à lire et escrire la musique (An easy way to learn and teach to read and write music). In 1634 MERSENNE mentioned in a letter to his acquaintances that DESARGUES was working on a paper on perspective (projection from a point). But it was not until two years later that the work was published: only 12 pages long and in a small edition.
    [Show full text]
  • 1 H-France Forum Volume 14 (2019), Issue 4, #3 Jeffrey N. Peters, The
    1 H-France Forum Volume 14 (2019), Issue 4, #3 Jeffrey N. Peters, The Written World. Space, Literature, and the Chorological Imagination in Early Modern France. Evanston: Northwestern University Press, 2018. vii + 272 pp. Figures, notes, and Copyright index. $34.95 (pb). ISBN 978-0-8101-3697-7; $99.95 (cl). ISBN 978-0- 8101-3698-4; $34.95 (Kindle). ISBN 978-0-8101-3699-1. Review Essay by David L. Sedley, Haverford College The Written World has on its cover an image from La Manière universelle de M. Desargues, pour pratiquer la perspective (1648). This book, written and illustrated by Abraham Bosse and based on the projective geometry of Girard Desargues, extends the theories of perspective codified by Leon Battista Alberti and his followers. [1] Alberti directed painters to pose a central point at the apparent conjunction of parallel lines in order to lend depth and coherence to their compositions. Desargues reinterpreted and renamed Alberti’s central point (and other points like it) as a point at infinity. Consequently, the convergent lines of a visual representation could be taken to indicate the infinite more emphatically than before. As an illustration of the art of putting objects in a perspective that emphasizes their connection to infinity, Bosse’s image suits Peters’ book to a T. Peters represents his objects of study—mainly a series of works of seventeenth-century French literature—with an eye to showing their affinity with the infinite. He frequently discusses infinity through chora, the Ancient Greek term used by Plato and adopted by Jacques Derrida to denote the space underlying all finite places and place- based thought.
    [Show full text]
  • Mathematics and Arts
    Mathematics and Arts All´egoriede la G´eom´etrie A mathematical interpretation Alda Carvalho, Carlos Pereira dos Santos, Jorge Nuno Silva ISEL & Cemapre-ISEG, CEAFEL-UL, University of Lisbon [email protected] [email protected] [email protected] Abstract: In this work, we present a mathematical interpretation for the mas- terpiece All´egoriede la G´eom´etrie (1649), painted by the French baroque artist Laurent de La Hyre (1606{1656). Keywords: Laurent de La Hyre, \All´egoriede la G´eom´etrie",baroque art, mathematical interpretation, perspective. Introduction The main purpose of this text is to present a mathematical interpretation for the masterpiece All´egoriede la G´eom´etrie, from a well-known series of paint- ings, Les 7 arts lib´eraux, by the French baroque artist Laurent de La Hyre (1606{1656). Figure 1: All´egoriede la G´eom´etrie (1649), oil on canvas. Recreational Mathematics Magazine, Number 5, pp. 33{45 DOI 10.1515/rmm{2016{0003 34 allegorie´ de la geom´ etrie´ Laurent de La Hyre painted the series Les 7 arts lib´eraux between 1649 and 1650 to decorate G´ed´eonTallemant's residence. Tallemant was an adviser of Louis xiv (1638{1715). The king was 10 years old at the time of the commission. According to the artist's son, Philippe de La Hire (1640{1718), writing around 1690 [5], (. .) une maison qui appartenoit autrefois a M. Tallemant, maistre des requestes, sept tableaux repres´entant les sept arts liberaux qui font l'ornement d'une chambre. Also, Guillet de Saint-Georges, a historiographer of the Acad´emieRoyale de Peinture et de Sculpture, mentioned that it was Laurent's work for the Ca- puchin church in the Marais which led to the commission for the \Seven Liberal Arts" in a house [2].
    [Show full text]
  • PAINTING and PERSPECTIVE 127 the Real World, That the Universe Is Ordered and Explicable Rationally in Terms of Geometry
    PAINTING AND PERSPECTIVE 127 the real world, that the universe is ordered and explicable rationally in terms of geometry. Hence, like the Greek philosopher, he be­ lieved that to penetrate to the underlying significance, that is, the reality of the theme that he sought to display on canvas, he must reduce it to its mathematical content. Very interesting evidence of x the artist's attempt to discover the mathematical essence of his sub­ ject is found in one of Leonardo's studies in proportion. In it he Painting and Perspective tried to fit the structure of the ideal man to the ideal figures, the flquare and circle (plate VI). The sheer utility of mathematics for accurate description and the The world's the book where the eternal sense philosophy that mathematics is the essence of reality are only two Wrote his own thoughts; the living temple where, Painting his very self, with figures fair of the reasons why the Renaissance artist sought to use mathematics. He filled the whole immense circumference. There was another reason. The artist of the late medieval period and T. CAMPANELLA the Renaissance was, also, the architect and engineer of his day and so was necessarily mathematically inclined. Businessmen, secular princes, and ecclesiastical officials assigned all construction problems During the Middle Ages painting, serving somewhat as the hand­ to the artist. He designed and built churches, hospitals, palaces, maiden of the Church, concentrated on embellishing the thoughts cloisters, bridges, fortresses, dams, canals, town walls, and doctrines of Christianity. Toward the end of this period, the and instru­ ments of warfare.
    [Show full text]
  • A Tale of the Cycloid in Four Acts
    A Tale of the Cycloid In Four Acts Carlo Margio Figure 1: A point on a wheel tracing a cycloid, from a work by Pascal in 16589. Introduction In the words of Mersenne, a cycloid is “the curve traced in space by a point on a carriage wheel as it revolves, moving forward on the street surface.” 1 This deceptively simple curve has a large number of remarkable and unique properties from an integral ratio of its length to the radius of the generating circle, and an integral ratio of its enclosed area to the area of the generating circle, as can be proven using geometry or basic calculus, to the advanced and unique tautochrone and brachistochrone properties, that are best shown using the calculus of variations. Thrown in to this assortment, a cycloid is the only curve that is its own involute. Study of the cycloid can reinforce the curriculum concepts of curve parameterisation, length of a curve, and the area under a parametric curve. Being mechanically generated, the cycloid also lends itself to practical demonstrations that help visualise these abstract concepts. The history of the curve is as enthralling as the mathematics, and involves many of the great European mathematicians of the seventeenth century (See Appendix I “Mathematicians and Timeline”). Introducing the cycloid through the persons involved in its discovery, and the struggles they underwent to get credit for their insights, not only gives sequence and order to the cycloid’s properties and shows which properties required advances in mathematics, but it also gives a human face to the mathematicians involved and makes them seem less remote, despite their, at times, seemingly superhuman discoveries.
    [Show full text]
  • Projective Geometry Seen in Renaissance Art
    Projective Geometry Seen in Renaissance Art Word Count: 2655 Emily Markowsky ID: 112235642 December 7th, 2020 Abstract: For the West, the Renaissance was a revival of long-forgotten knowledge from classical antiquity. The use of geometric techniques from Greek mathematics led to the development of linear perspective as an artistic technique, a method for projecting three dimensional figures onto a two dimensional plane, i.e. the painting itself. An “image plane” intersects the observer’s line of sight to an object, projecting this point onto the intersection point in the image plane. This method is more mathematically interesting than it seems; we can formalize the notion of a vanishing point as a “point at infinity,” from which the subject of projective geometry was born. This paper outlines the historical process of this development and discusses its mathematical implications, including a proof of Desargues’ Theorem of projective geometry. Outline: I. Introduction A. Historical context; the Middle Ages and medieval art B. How linear perspective contributes to the realism of Renaissance art, incl. a comparison between similar pieces from each period. C. Influence of Euclid II. Geometry of Vision and Projection A. Euclid’s Optics ​ B. Cone of vision and the use of conic sections to model projection of an object onto a picture. Projection of a circle explained intuitively. C. Brunelleschi’s experiments and the invention of his technique for linear perspective D. Notion of points at infinity as a vanishing point III. Projective geometry and Desargues’ Theorem A. Definition of a projective plane and relation to earlier concepts discussed B.
    [Show full text]
  • Girard Desargues, Maître De Pascal Eishi KUKITA
    Girard Desargues, maître de Pascal Eishi KUKITA La vie et l’œuvre de Girard Desargues (1591-1661), savant et ingénieur du XVIIe siècle, sont aujourd’hui presque inconnues1. Pourtant les plus lucides de ses contemporains, dont Descartes et Pascal, tous deux grands géomètres qui ont marqué chacun à sa manière l’aube des sciences naturelles modernes, lui trouvaient une intelligence hors du commun. Une lecture attentive des correspondances de Descartes nous permettra de constater que le philosophe du cogito n’exagérait point en écrivant à Mersenne, au sujet de ses Méditations (1641) : Je serai bien aise que M. Desargues soit aussi un de mes juges, s’il lui plaît d’en prendre la peine, et je me fie plus en lui qu’en trois théologiens2. Quant à Pascal, publiant à l’âge de seize ans son premier article Essai pour les Coniques en 1640, il rendit un hommage vibrant à celui qu’il considérait comme son maître : Nous démontrerons aussi cette propriété, dont le premier inventeur est M. Desargues Lyonnais, un des grands esprits de ce temps et des 1 La littérature sur Desargues est franchement pauvre. Autant que nous le sachions, seuls quatre livres en français ont été publiés depuis XIXe siècle. N.-G. Poudra, Œuvres de Desargues, 2 vol., Léiber, 1864, rééd. fac-similé, Cambridge Univ. Press, 2011 (sigle : Pou). R. Taton, Œuvres mathématiques de G. Desargues, PUF, 1951, 2e éd., Vrin, 1988 (sigle : Tat). J. Dhombres et J. Sakarovitch (dir.), Desargues en son Temps, Blanchard, 1994 (sigle : DS). M. Chaboud, Girard Desargues, Bourgeois de Lyon, Mathématicien, Architecte, Lyon, Aléas, 1996 (sigle : Cha).
    [Show full text]
  • Undergraduate Texts in Mathematics
    Undergraduate Texts in Mathematics Editors S. Axler K.A. Ribet Undergraduate Texts in Mathematics Abbott: Understanding Analysis. Chambert-Loir: A Field Guide to Algebra Anglin: Mathematics: A Concise History Childs: A Concrete Introduction to and Philosophy. Higher Algebra. Second edition. Readings in Mathematics. Chung/AitSahlia: Elementary Probability Anglin/Lambek: The Heritage of Theory: With Stochastic Processes and Thales. an Introduction to Mathematical Readings in Mathematics. Finance. Fourth edition. Apostol: Introduction to Analytic Cox/Little/O’Shea: Ideals, Varieties, Number Theory. Second edition. and Algorithms. Second edition. Armstrong: Basic Topology. Croom: Basic Concepts of Algebraic Armstrong: Groups and Symmetry. Topology. Axler: Linear Algebra Done Right. Cull/Flahive/Robson: Difference Second edition. Equations: From Rabbits to Chaos Beardon: Limits: A New Approach to Curtis: Linear Algebra: An Introductory Real Analysis. Approach. Fourth edition. Bak/Newman: Complex Analysis. Daepp/Gorkin: Reading, Writing, and Second edition. Proving: A Closer Look at Banchoff/Wermer: Linear Algebra Mathematics. Through Geometry. Second edition. Devlin: The Joy of Sets: Fundamentals Berberian: A First Course in Real of Contemporary Set Theory. Second Analysis. edition. Bix: Conics and Cubics: A Dixmier: General Topology. Concrete Introduction to Algebraic Driver: Why Math? Curves. Ebbinghaus/Flum/Thomas: Bre´maud: An Introduction to Mathematical Logic. Second edition. Probabilistic Modeling. Edgar: Measure, Topology, and Fractal Bressoud: Factorization and Primality Geometry. Testing. Elaydi: An Introduction to Difference Bressoud: Second Year Calculus. Equations. Third edition. Readings in Mathematics. Erdo˜s/Sura´nyi: Topics in the Theory of Brickman: Mathematical Introduction Numbers. to Linear Programming and Game Estep: Practical Analysis in One Variable. Theory. Exner: An Accompaniment to Higher Browder: Mathematical Analysis: Mathematics.
    [Show full text]
  • The Historiography of Perspective and Reflexy-Const in Netherlandish
    The Historiography of Perspective and Reflexy-Const in Netherlandish Art Sven Dupré Introduction In The Heritage of Apelles Ernst Gombrich famously drew a distinction between art North and South of the Alps along optical lines. Painters in fifteenth-century Florence, such as Domenico Veneziano, used different optical features than their contemporaries in Bruges, the likes of Jan Van Eyck, to create the illusion of space. Gombrich commented: „We all associate Florentine art with the development of central perspective, and thus with the mathematical method of revealing form in ambient light. The other aspect of optical theory, the reaction of light to various surfaces, was first explored in modern times by painters North of the Alps. It was there that the mastery of lustre, sparkle and glitter was first achieved, permitting the artist to convey the peculiar character of materials. Indeed, for a time, during the first decades of the fifteenth century, the two schools of painting appeared thus to have divided the kingdom of appearances between them.‟1 While Italian painters of the fifteenth century used perspective, Netherlandish artists studied and painted the reflection of light from surfaces of different textures and materials to create the illusion of space. For Gombrich, the point of contact between North and South was Leonardo da Vinci, „the greatest explorer of natural appearances‟, who must have been „a keen student of Northern painting‟.2 Here I am less interested in Gombrich‟s geography of optics, but more in how his juxtaposition of perspective and painterly light reflects and complicates the historiography of perspective. For reasons which fall outside the scope of this paper, since the seminal work of 1 Erwin Panofsky linear perspective has become a locus classicus of the study of artistic practice and science, and the history of perspective, which Panofsky in Die Perspektive als symbolische Form (1927) disconnected from optics, a field of research of its own, independent from the history of science and the history of art.
    [Show full text]
  • TME Volume 6, Number 3
    The Mathematics Enthusiast Volume 6 Number 3 Article 18 7-2009 TME Volume 6, Number 3 Follow this and additional works at: https://scholarworks.umt.edu/tme Part of the Mathematics Commons Let us know how access to this document benefits ou.y Recommended Citation (2009) "TME Volume 6, Number 3," The Mathematics Enthusiast: Vol. 6 : No. 3 , Article 18. Available at: https://scholarworks.umt.edu/tme/vol6/iss3/18 This Full Volume is brought to you for free and open access by ScholarWorks at University of Montana. It has been accepted for inclusion in The Mathematics Enthusiast by an authorized editor of ScholarWorks at University of Montana. For more information, please contact [email protected]. TMME, vol6, no.3, p. 295 THE JOURNAL (WHEEL) KEEPS ON TURNING Bharath Sriraman The University of Montana The title of this editorial is a spinoff on the opening lyrics of a famous Lynyrd Skynyrd song (fill in the blank: Sweet Home__________). When the song came out in the 70’s the popular media misunderstood the song and took some of its lyrics to mean support for the (infamous) George Wallace’s governorship of Alabama, when in fact the band sarcastically boo’ed his segregative policies that scarred the South. The song goes In Birmingham, they love the governor (boo boo boo) Now we all did what we could do Now Watergate does not bother me Does your conscience bother you? These lines bring me to the theme of this editorial, which is namely: (1) What does it take to keep the journal’s wheel turning (running, moving, progressing), and (2) “Does your conscience bother you?” Issue #3 brings volume 6 for the year 2009 of the journal to an end.
    [Show full text]
  • Laura Branchetti, Teaching and Learning Mathematics
    Isonomia – Epistemologica Volume 7 TEACHING AND LEARNING MATHEMATICS SOME PAST AND CURRENT APPROACHES TO MATHEMATICS EDUCATION Volume 1 Il realismo scientifico di Evandro Agazzi Mario Alai (ed.) Volume 2 Complessità e riduzionismo Vincenzo Fano, Enrico Giannetto, Giulia Giannini, Pierluigi Graziani (eds.) Volume 3 Oltre la fisica normale Isabella Tassani (ed.) Volume 4 Mettere a fuoco il mondo Elena Casetta, Valeria Giardino (eds.) Volume 5 Metaphor and Argumentation Francesca Ervas, Massimo Sangoi (eds.) Volume 6 Forecasting the Future Stefano Bordoni, Sara Matera (eds.) Volume 7 Teaching and Learning Mathematics Laura Branchetti (ed.) ISONOMIA - Epistemologica Series Editor Gino Tarozzi [email protected] TEACHING AND LEARNING MATHEMATICS SOME PAST AND CURRENT APPROACHES TO MATHEMATICS EDUCATION Edited by Laura Branchetti © ISONOMIA – Epistemologica ISSN 2037-4348 Scientific director: Gino Tarozzi Editor in chief: Pierluigi Graziani Department of Foundation of Sciences P.za della Repubblica, 13 – 61029 Urbino (PU) http://isonomia.uniurb.it/ Design by [email protected] All rights reserved. No part of this publication may be reproduced, stored in a re- trieval system or transmitted in any form, or by any means, electronic, mechanical, photocopying, recording or otherwise without prior permission, in writing, from the publisher. Cover image by Antonello Ghezzi Table of contents LAURA BRANCHETTI, Some Past and Current Approaches to Mathematics Education ............................ 7 GIORGIO BOLONDI, Epistemology and Didactics in Federigo Enriques ............................................... 21 MARCO PANZA, ANDREA SERENI Platonism and Nominalism in Philosophy of Mathematics ................................... 35 JEAN DHOMBRES, A Debate on the Long Term and Reactions in Mathematics Teaching. Is Computation Just a Technique or Can it Be a Way to Learn Mathematics? ...
    [Show full text]