This book is the result of a joint venture between 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 perspectives. Readers will find an extremely clear and concise introduction to the fundamentals of knot theory, an overview 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 cooperation between mathematicians and mathematics educators based on substantial mathematics. And finally it is a thorough introduction to the Japanese art of lesson studies again in the context of substantial mathematics.
Preface to book
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 2003 to 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 education. 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 of mathematics held their respective meetings 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 teaching materials. We proceeded to discuss the exploration into teaching contents based on the following points:
. Making teaching knot theory something meaningful for pupils rather than something that just propagates knowledge.
. Grasping the real situations of their understanding in their respective years, to this end.
. 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 and teachers of mathematics in Japan. In 2006, our research results on knot theory with senior high school students from the SSH (Super Science High School) program, which was one of our experimental practices, won the highest 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 representing mathematics educations not only in Japan but also abroad showed interests in our research results. In November 2004, Erich Ch. Witttmann, professor of Dortmund University in Germany, came to visit Osaka and inspected our experimental classworks on mathematics education 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 encouragement and letters of research interchanges from Heinrich Besuden, professor of Oldenburg University in Germany and a former chairman of the 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 of Mathematics Education), lots of interests were directed to our teaching contents, children's reactions, the process developing teaching materials, and the way of proceeding on the research project. Thus, at this time, we have decided to publish our work as a book in English so that we can introduce our research results and its process to many people, including researchers of mathematics education around the world, knot theory researchers and 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 curriculum and actual examples of researches and practices, conducted at elementary, junior and senior high schools and in university general education. All the researches and practices have actually been conducted. This book introduces our guidance of knot theory that takes various forms of not only classworks but also experimental teachings with several students, club activity guidances 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-range evolutions to scientific objects" and the JSPS Grant-in-Aid for Scientific Research (A) No. 21244005, "Studies in Knot Theory."
Early Spring of 2011
Akio Kawauchi and Tomoko Yanagimoto
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