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Activity and Sign Michael H.G Activity and Sign Michael H.G. Hoffmann Johannes Lenhard Falk Seeger (Editors) Activity and Sign Grounding Mathematics Education Springer Michael H.G. Hoffmann, Georgia Institute of Technology, U.S.A. Johannes Lenhard, University of Bielefeld, Germany Falk Seeger, University of Bielefeld, Germany Library of Congress Cataloging-in-Publication Data Activity and sign: grounding mathematics education / Michael Hoffmann, Johannes Lenhard, Falk Seeger (editors). p. cm. Includes bibliographical references and index. ISBN 10: 0-387-24269-4 ISBN 13: 9870387242699 (acid-free paper)— ISBN 10: 0-387-24270-8 ISBN 13: 9870387242705 (E-book) 1. Mathematics—Study and teaching. 2. Mathematics—Philosophy. I. Hoffmann, Michael (Michael H.G.) II. Lenhard, Johannes. II Seeger, Falk. QA11.2.A28 2005 510’.71—dc22 2004066226 © 2005 Springer Science+Business Media, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if the are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 9 8 7 6 5 4 3 2 1 SPIN 11375494 springeronline.com TABLE OF CONTENTS ACKNOWLEDGEMENT VII MICHAEL HOFFMANN, JOHANNES LENHARD and FALK SEEGER / Grounding Mathematics Education - Michael Otte's contribution 1 MICHAEL OTTE / Mathematics, Sign and Activity 9 A. SIGN PROCESSES PAUL ERNEST / Agency and Creativity in the Semiotics of Learning Mathematics 23 SUSANNA MARIETTI / The Semiotic Approach to Mathematical Evidence and GeneraUzation 35 MICHAEL HOFFMANN / Signs as Means for Discoveries. Peirce and His Concepts of "Diagrammatic Reasoning," "Theorematic Deduction," "Hypostatic Abstraction", and "Theoric Transformation" 45 WILLIBALD DORJT^ER / Diagrammatic Thinking. Affordances and Constraints 57 FALK SEEGER / Notes on a Semiotically Inspired Theory of Teaching and Learning 67 B. SIGN PROCESSES IN THE MATHEMATICS CLASSROOM MARIA G. BARTOLINI BUSSI, MARIA ALESSANDRA MARIOTTI AND FRANCA FERRI / Semiotic Mediation in the Primary School: Diirer's Glass 77 HEINZ STEINBRING / Do Mathematical Symbols Serve to Describe or Construct "Reahty"? - Epistemological Problems in Teaching Mathematics in the Field of Elementary Algebra 91 NORMA PRESMEG / Metaphor and Metonymy in Processes of Semiosis in Mathematics Education 105 ANNA SIERPINSKA / On Practical and Theoretical Thinking and Other False Dichotomies in Mathematics Education 117 LUIS RADFORD / The Semiotics of the Schema. Kant, Piaget, and the Calculator 137 C. MATHEMATICS EDUCATION AS A SCIENCE KENNETH RUTHVEN / Towards a Normal Science of Mathematics Education? 153 GUY BROUSSEAU / The Study of the Didactical Conditions of School Learning in Mathematics 159 ROLAND FISCHER / The Formal, the Social and the Subjective: Variations on a Theme of Michael Otte 169 D. CROSSING BOUNDARIES BERND FICHTNER / Reflective learning - Problems and Questions Concerning a Current Contextualization of the Vygotskian Approach 179 RAINER BROMME / Thinking and Knowing about knowledge: A Plea for and Critical Remarks on Psychological Research Programs on Epistemological Beliefs 191 THOMAS MIES / The Cognitive Unconscious. Recalling the History of the Concept and the Problem 203 E. HISTORY OF MATHEMATICS AND MATHEMATICS EDUCATION HANS NIELS JAHNKE / Hilbert, Weyl, and the Philosophy of Mathematics 215 THOMAS MORMANN / Mathematical Metaphors in Natorp's Neo-Kantian Epistemology and Philosophy of Science 229 KARL-NORBERT IHMIG / Newton's Program of Mathematizing Nature 241 MIRCEA RADU / Did Hermann and Robert GraBmann Contribute to the Emergence of a Formal Axiomatics? 263 GERT SCHUBRE^G / A Case Study in Generalization: The Notion of Multiphcation 275 CIRCE MARY SILVA DA SILVA DYNNIKOV / Some German Contributions to Mathematics Research in Brazil 287 F. MAKING PHILOSOPHY OF MATHEMATICS RELEVANT ANDREAS DRESS /Data Structures and Virtual Worlds: On the Inventiveness of Mathematics 299 HERMANN DEs[GES / Variables, in Particular Random Variables 305 JOHANNES LENHARD / Deduction, Perception and Modehng: ITie Two Peirces on the Essence of Mathematics. 313 C. ULISES MOULINES / Models of Data, Theoretical Models and Ontology: A StructuraUst Perspective 325 MARCO PANZA / Some Sober Conceptions of Mathematical Truth 335 JEAN PAUL VAN BENDEGEM / Can There Be an Alternative Mathematics, Really? 349 G. CODA ROLAND FISCHER / An Interview with Michael Otte 361 INDEX 379 ACKNOWLEDGEMENT This book would not have been successfully completed without the work of many hands and heads and hearts. The editors would like to thank especially Christel Biere and Herta Ritsche for their work on various versions of the chapters, Jonathan Harrow and GUnther Seib for their efforts to make translations from non-English speaking authors into sounding like they were from native speakers, and Roland Fischer for his brilliant idea to do an interview with Michael Otte. MICHAEL H. G. HOFFMANN JOHANNES LENHARD FALK SEEGER GROUNDING MATHEMATICS EDUCATION Michael Otte 's contribution Mathematics education has a long past, but only a relatively short history as an insti­ tutional effort. Even though already Plato's "geometry" or medieval arithmetic books have been exceptionally "didactical" in their approach to present themselves to the reader, it was only in the 1960s that the institutionalization of mathematics education as a scientific discipline started on a larger scale. Following major changes of the role and place of science in society, the universities began at that time to change their organizational structures or added new, often interdisciplinary, or­ ganizational units focusing on applied or basic problems of research. In a surprisingly consonant manner, this development in mathematics education was from the beginning an international one, crossing the then still existing bounda­ ries between East and West. Without disregarding previous work of didacticians of mathematics in the 19^ and early 20^ century and their influence, this can certainly be viewed to a larger degree as the result of the work of the International Commis­ sions on Mathematical Instruction (ICMI) inspired and chaired by Felix Klein work­ ing from 1908 to the twenties on a survey comparing mathematics education in many countries of the world. The International Commissions prepared the ground for a worldwide attempt to reform and revolutionize mathematics education. The reform movement of the early seventies in a way took up the international spirit of the International Commissions - particularly in the form of the ICME-conferences taking place every four years with the 1972 Exeter conference being a signal for a new beginning. Most of the contributors to this volume have played a role in this historical period - some more central, some more peripheral. Michael Otte's scientific life and career is situated in this field of historical forces and developments. It was a founding period for systematic research and a new disciplinary self-image of mathematics education. Also in Germany, the institution­ alization of basic research in mathematics education within university departments and faculties began. At that time the "new math" movement which also had been a truly international affaire just had run aground, and the Volkswagen foundation had called for a proposal to establish a central research institute in Germany to reach a deeper scientific understanding of the disastrous failure of the new math approach in German primary schools. The underlying idea was as deceptively simple as attrac­ tive: finding out the reasons and reconstruct them theoretically would also provide a 1 M. H. G. Hoffmann, J. Lenhard, F. Seeger (Eds.), Activity and Sign - Grounding Mathematics Education, 1-7. 2 M. HOFFMANN, J. LENHARD, F. SEEGER platform or a ground to anchor a new idea of how mathematics teaching and learn­ ing should look like in the future. The original design for the Institute included five chairs furnished with opulent financial means, in terms of manpower as well as other facilities. Michael Otte was to become appointed as one of the three chairs directing the InstitutfUr Didaktik der Mathematik of Bielefeld University in the next 25 or so years. ^ Very soon after the Institute began its work, two of his important books ap­ peared. The first one was Mathematiker Uber die Mathematik [Mathematicians on Mathematics] in 1974, and Mathematik, die uns angeht [Mathematics Concerning Us] in 1977. Two ensuing volumes were on mathematics and learning from text (see Keitel, Otte and Seeger 1980), and on epistemology, history and science (see Jahnke and Otte 1981). These four volumes like landmarks claim the big territory Michael Otte is covering in his scientific efforts, while the programmatic volume on Das Formale, das Soziale und das Subjektive. Eine EinfUhrung in die Philosophie und Didaktik der Mathematik [The formal, the social and the subjective: An introduction into the philosophy and pedagogy of mathematics, see Otte 1994], in a sense, is try­ ing to sum up what had been important
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