Teaching and learning of knot theory in school mathematics

198 Pages·2012·6.845 MB·English

by  Akio Kawauchi, Tomoko Yanagimoto (auth.), Akio Kawauchi, Tomoko Yanagimoto (eds.)

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in School Mathematics

Akio Kawauchi • Tomoko Yanagimoto
Editors
Teaching and Learning
of Knot Theory
in School Mathematics
Editors
Akio Kawauchi Tomoko Yanagimoto
Professor Professor
Osaka City University Osaka Kyoiku University
3-3-138 Sumiyoshi, Sugimoto 4-698-1 Asahigaoka

Osaka City, Osaka 558-8585 Kashiwara, Osaka 582-8582
Japan Japan
ISBN 978-4-431-54137-0 ISBN 978-4-431-54138-7 (eBook)

DOI 10.1007/978-4-431-54138-7

Springer Tokyo Heidelberg New York Dordrecht London

Library of Congress Control Number: 2012942161

© 2012 by OMUP (Osaka Municipal Universities Press)
1-1, Gakuen-Cho, Naka-Ku, Sakai City, Osaka 599-8531 Japan
TEL: [phone] FAX: [phone]
All right reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted,
in any form or by any means, except the prior permission of the publisher.

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)
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Foreword
Each level of mathematical knowledge has to be understood as an elaboration of
knowledge at a lower level and at the same time as a steppingstone to knowledge at a
higher level. This genetic principle applies to both the historical development of
mathematics as a scientific discipline and to the learning of individuals. Over centuries
the elementary parts of arithmetic, algebra, geometry and analysis have served as the
iron basis of mathematical curricula. The enormous progress of the mathematical
sciences in the past decades has shown the need to broaden the elementary layer in
favor of new topics. Stochastics is the most prominent example. Its elementary parts are
nowadays taught already at the primary level. However, even the classical parts of
elementary mathematics, in particular geometry, cannot remain untouched. In the past
elementary geometry was restricted to Euclidean geometry and to the analytical
treatment of lines and planes. Here a broadening of the scope towards “curved objects”
is badly needed.
The book is the result of a joint venture of Professor Akio Kawauchi, Osaka City
University, well-known for his research in knot theory and the Osaka study group of
mathematics education founded by Professor Hirokazu Okamori and now chaired by his
successor Professor Tomoko Yanagimoto, Osaka Kyoiku University.
The seven chapters address the teaching and learning of knot theory from several
sides. The reader will find an admirably clear and concise introduction into the
elementary parts of knot theory, a survey of curricular developments in Japan, and in
particular a series of teaching experiments at all levels which not only demonstrate the
creativity and the professional expertise of the members of the study group, but also
give a lively impression of students’ learning processes. In addition the reports show
that elementary knot theory is not just a preparation for advanced knot theory but also
an excellent means to develop spatial thinking.
The book can be highly recommended for several reasons:
First of all, and that is the main intention of the book, it serves as a comprehensive text
for teaching and learning knot theory. Moreover it provides a model for a cooperation of
mathematicians and mathematics educators based on substantial mathematics. And
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vi
finally it is a thorough introduction into the Japanese art of lesson studies – again in the
context of substantial mathematics.
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Erich Ch. Wittmann
Technical University of Dortmund
Faculty of Mathematics
Germany
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Preface
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It was in July, 2004 that a project to study ‘Teaching Knot Theory in Mathematics
Education’ was formed, which was made possible only when Akio Kawauchi of Osaka
City University and Hirokazu Okamori, a professor emeritus of Osaka Kyoiku
University met together. Okamori is a researcher of mathematics education. Kawauchi
is an expert of knot theory and then he was the leader of “Constitution of wide-angle
mathematical basis focused on knots,” one of the 21st Century Center of Excellence
Programs in Japan from April 2003to March 2008.
In recent years, knot theory has been making a rapid progress and contributed to
elucidating problems of various fields as a science. Knot theory experts are hoping to
get across its fun and importance widely to pupils and students in their early years. Also,
our cherished hope is that by so doing, we will pick out students with high levels of
interest and learning ability in knot theory and bring them up to be excellent experts on
it. Experts of mathematics education have a desire to keep developing teaching contents
from much broader perspectives beyond the bounds of traditional school mathematics in
order to cultivate children’s understanding of mathematics.
In pursuit of the realization of the hopes from these two positions, the research
project with a team work of researchers of mathematics and mathematics education in
university and elementary, junior and senior high school teachers of mathematics was
formed by Tomoko Yanagimoto and started discussions of introducing knot theory to
educations. All the members began to tackle such work by learning the basics of knot
theory. At the same time, we delved into its educational meanings from the viewpoint of
mathematics education. Elementary, junior and senior high school teachers held their
respective meeting at the school levels and developed teaching materials along with the
researchers of mathematics education. Also, based on this, all the members had
meetings once a month, where they discussed from their own standpoints under mutual
respect as equals and confirmed the systematization of the contents that can be used as
materials. We proceeded to discuss the exploration into teaching contents based on the
following points.
(cid:2285)Making teaching knot theory something meaningful for pupils rather than
something that just propagates knowledge.
(cid:2285)Grasping the real situations of their understanding in their respective years to this
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end.
(cid:2285) Developing effective teaching tools as needed.
The report of our research results has already been compiled in Japanese into three
issues, attracting the attention of mathematicians, researchers of mathematics education,
teacher of mathematics in Japan. In 2006, our research results on knot theory with
senior high school students among the SSH (Super Science High School) program,
which was one of our experimental practices, won the highest award in the four “first
prizes” of the sections of mathematics, physics, chemistry and biology in Japan. The ex
post facto evaluation as a part of the 21st Century Center of Excellence Programs of
2008 was of the highest. As one of the reasons for that evaluation, it was pointed out
that we had been steadily educating elementary, junior and senior high school students
on knot theory. Further, researchers of representing mathematics educations not only
in Japan but also in foreign countries showed interests in our research results. In
November 2004, Erich Ch. Wittmann, professor of Dortmund University in Germany,
came to visit Osaka and inspected our experimental classwork on mathematics
educations of knot theory conducted at Tennoji Junior High School and Tennoji
Elementary School attached to Osaka Kyoiku University and showed a deep interest in
it. Many times we received encouragements and letters of the research interchanges
from Heinrich Besuden, professor of Oldenburg University in Germany and a former
chairman of German Mathematics Education Society. Also, whenever we made
presentations of our research on teaching knot theory at the conferences of mathematics
education in Japan and abroad, including ICME(International Conference on
Mathematic Education), lots of interests were directed to our teaching contents,
children’s reactions, the process developing teaching materials, and the way of
proceeding a research project. Thus, this time, we have come to decide to publish our
work as a book in English so that we can introduce our research results and its process
to many people, including the researchers of mathematics education in the world, knot
theory researchers, school teachers of mathematics who have shown interests in our
activity.
This book contains an introduction of knot theory, up-to-date topics on it, the
educational meanings of knot theory from the perspective of the evolution of
mathematics education, the methodology for making knot theory into a curriculum and
actual examples of researches and practices, conducted at elementary, junior and senior
high schools and a university general education. All the researches and practices have
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Preface ix
actually been conducted. This book introduces our guidance of knot theory that takes
various forms of not only classwork but also experimental teachings with several
students, club activity guidance in concerted efforts between senior high schools and
universities, and others.
This work was supported in part by the Priority Research of Osaka City University
“Mathematics of knots and wide-angle evolutions to scientific objects” and the JSPS
Grant-in-Aid for Scientific Research (A) No. 21244005, “Studies of Knot Theory”.
(cid:3) (cid:3) (cid:3) Early spring of 2011
Akio Kawauchi and Tomoko Yanagimoto
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