Proceedings of the HPM 2000 Conference History in Mathematics Education Challenges for a new millennium A Satellite Metting of ICME-9 Vol. I Edited by Wann-Sheng Horng & Fou-Lai Lin August 9-14, 2000, Taipei, Taiwan Department of Mathematics National Taiwan Normal University ORGANIZERS International Study Group on the Relations Between History and Pedagogy of Mathematics Department of Mathematics, National Taiwan Normal University, Taipei, Taiwan, ROC International Programme Committee Chair: Jan van Maanen, the chair of HPM, University of Groningen, The Netherlands Glen Van Brummelen, Bennington College, USA John Fauvel (the former chair of HPM), Open University, UK Gail FitzSimons, Monash University, Australia Lucia Grugnetti, University of Parma, Italy Wann-Sheng Horng, National Taiwan Normal University, Taiwan Victor Katz, University of the District of Columbia, USA Eleanor Robson, Oxford University, UK Chikara Sasaki, University of Tokyo, Japan Man-Keung Siu, University of Hong Kong, SAR, China Conference Venue and Secretariat Department of Mathematics, National Taiwan Normal University 88, Sec. 4, Ting Chou Road, Taipei, Taiwan, ROC Tel: 886-2-29312611 Fax: 866-2-29332342 Proceedings of the HPM 2000 Conference History in Mathematics Education： Challenges for a new millennium Editors： Wann-Sheng Horng Department of Mathematics National Taiwan Normal University Fou-Lai Lin Department of Mathematics National Taiwan Normal University Assistant editor： Jer-Nan Huang, Yi-Wen Su, Feng-Chu Chen, Wen-Pei Wang Tsang-Yi Lin, Chin-Yee Teo, Ching-Ju Chiou Cover design by Jovi Cheng & Jer-Nan Huang Published by Department of Mathematics, National Taiwan Normal University Printed by Sponsored by National Science Council, Taiwan National Youth Commission, Taiwan National Taiwan Normal University Chang Chau-Ting Memorial Foundation Chiu Chang Math. Press Yuan T. Lee Foundation Peng Wan-Ru Foundation Science Monthly Yuan-Liou Publishing Co., LTD. Copyrights of the Digital Edition: © 2016 left to the authors. Pagination has been adjusted for this edition. Table of Contents Vol. I I. Plenary Lecture 8/10 Two pots and one lid: 1 The first arithmetic textbooks in the Netherlands before 1600 Marjolein Kool 8/11 Introduction of Western Sciences in China, Japan and Korea: 11 A Comparative Comment Seong-Rae Park 8/12 Becoming a mathematician in East and West: 18 some cross-cultural considerations. Christopher Cullen 8/13 The Interplay between Algorithm and Proof in Ancient China Karine Chemla (The text is not published) 8/14 The Use of Technology in Teaching Mathematics with History 27 - Teaching with modern technology inspired by the history of mathematics - Masami ISODA II. Symposia S1：History of Asian and Pacific mathematics S1-01 Theoretical Research Tendency of Mathematics in Pre-Qin Times 35 Viewed from Suan Shushu Shuchun Guo S1-02 An investigation on Chang Qiujian Suanjing 41 Yan-Chyuan Lin i S1-03 The Notion of Volume in the Jiu Zhang Suan Shu, 43 and Japanese Mathematics Shigeru Jochi S2：Mathematics education before S2-01 A History of Calculus Education in Japan 55 Osamu Kota S2-02 The Brief Sketch of Mathematics Education for 64 Immediately Follow the World War II in Japan Naomichi Makinae S2-03 Mathematics Textbooks and Terminology: 72 The Impact of Calvin Winson Mateer's Work on the Transformation of Traditional Chinese Mathematics Education into a Modern One Yibao Xu S2-04 Teaching Fortification as part of Practical Geometry : 75 A Jesuit case Frédéric Metin S3：The effectiveness of history in teaching mathematics: empirical studies S3-03 An Episode in the Development of Mathematics Teachers' 83 Knowledge? The Case of Quadratic Equations Greisy Winicki-Landman S3-04 Teachers’ Teaching Beliefs and Their 89 Knowledge about the History of Negative Numbers Feng-Jui Hsieh S3-05 99 What Are Teachers’ Views of Mathematics ii An Investigation of How They Evaluate Formulas in Mathematics Chia-Jui Hsieh, Feng-Jui Hsieh S3-07 Building teaching, learning and curriculum of calculus following 113 the path of its historical development as well as along lines based on introspective reports of homogeneity in pattern of processes of invention or reinvention Ada Sherer S3-08 The Royal Observatory in Greenwich; ethnomathematics 133 in teacher training Wendy S. Troy S3-10 History of mathematics: why, and is that enough? 141 Andy Begg S3-11 Lifelong Learning and the History of Mathematics: 148 An Australian Perspective Gail E. FitzSimons S3-12 The Quarrel between Descartes and Fermat to 156 introduce the notion of tangent in high school Michèle Grégoire S3-13 Using Mathematics History and PCDC Instruction Model to 163 Activate Underachievement Students’ Mathematics Learning Ya-Hui Hsiao, Ching-Kuch Chang S3-14 On the effectiveness of history in teaching 173 probability and statistics Ewa Lakoma S3-15 The Use of History of Mathematics To Increase 180 Students’ Understanding of the Nature of Mathematics Po-Hung Liu S3-16 Epistemological complexity of multiplication and division 191 iii from the view of dimensional analysis Yutaka OHARA S3-17 Method of Indivisibles in Calculus Instruction 198 Vrunda Prabhu, Bronislaw Czarnocha S3-18 Using History to Enliven the Study of Infinite Series 206 Bob Stein S3-20 217 Using mathematical text in classroom： the case of probability teaching Yi-Wen Su Vol. II II. Symposia S3：The effectiveness of history in teaching mathematics: empirical studies S3-23 The Practical Use of the History of Mathematics 1 and Its Usefulness in Teaching and Learning Mathematics at High School: The development of materials in teaching calculus (not in these Proceedings) Kumilo Tsukahara S3-26 The Development of Mathematics in Sub-Saharan Africa: 17 Challenges for the new millennium Luckson Muganyizi Kaino S3-27 The Introduction of History of Mathematics 27 in Norwegian schools Bjørn Smestad S3-28 Justification in mathematics and Procedures on Which 31 It Is based: A historical approach for didactical purposes Constantinos Tzanakis, Michael Kourkoulos iv S3-30 Several Characteristics of 52 Taiwan's high school mathematics curriculum Jya-Yi Wu Yu S3-31 F&B Mathematics Teaching in Vocational School: 61 A Team Work with HPM perspective Yu-Yi Lin S3-32 Using Mathematics Text in Classroom: 67 the case of Pythagorean Theorem Yen Fu Ming S3-33 Theories of ratio and theoretical music: an education approach 83 Oscar João Abdounur S3-35 The antiphairesis: a site of an educational dialogue 94 among Mathematics, History and Philosophy Ernesto Rottoli, Gianstefano Riva S3-36 103 ~~： A teaching report of “ Using ancient mathematical text in classroom”: Pascal’s triangle Hui-Yu Su S4：West and East, Contrast and transfer of mathematical S4-02 Multiculturalism in history: voices in 19th century 115 mathematics education east and west John Fauvel S4-04 On Computing the Volume of Sphere in the East and West: 121 A comparative study from the educational perspectives Yiu-Kwong Man, Yin-Kue Lo v S6：History of science and science education (especially for local participants) S6 From Atayal World to Science World: 130 A Pilot Study on a Science Class for Atayal Junior High Students Li-Yu Fu III. Workshop Antipodean Fibonacci Originals 142 Coralie Daniel IV. Round-Table RT-01 Why study values in mathematics teaching: 149 contextualising the VAMP project? A. J. Bishop, P. Clarkson, G. FitzSimons, W.T. Seah RT-01 Methodology Challenges and Constraints in the VAMP Project 157 Philip Clarkson, Alan Bishop, Gail FitzSimons, Wee Tiong Seah RT-01 Conceptions of Values and Mathematics Education held by 165 Australian Primary Teachers: Preliminary Findings from VAMP Gail E. FitzSimons, Wee Tiong Seah, Alan J. Bishop, Philip C. Clarkson RT-01 Score-ism As Their Pedagogical Value of Two Junior 174 High Mathematics Teachers Ching-Kuch Chang RT-01 An Elementary Teacher’s Pedagogical Values in Mathematics 180 Teaching: Clarification and Change Yuh-Chyn Leu, Chao-Jung Wu V. Panel P1-01 198 Ptolemy’s theorem and chord table vi Cheng-Mei Hsu P1-03 199 The Use of Dynamic Geometry Software and History of Mathematics in Teaching Conic Sections Jer-Nan Huang, Tai-Yih Tso P2-02 — 211 Ancient Mathematical Texts used in the Classroom—Quadratic Equations Hui-I Lauo P2-03 Using History of Mathematics in Teaching Radical Root Numbers 223 Ching-Ju Chiou vii viii Two pots and one lid： The first arithmetic textbooks in the Netherlands before 1600 Marjolein Kool Hogeschool Domstad, Utrecht A historical arithmetic problem Somebody has two silver pots and one silver lid. The price of the lid is 16 guilders. If he places the lid on the first pot, this combination will cost 4 times more than the second pot. If he places the lid on the second pot, this combination will cost three times more than the first pot. What will be the value of each pot? If you have tried to solve this problem, you probably wrote down two equations: x + 16 = 4y and y + 16 = 3x and had got the answer quite soon. I found this problem in a Dutch arithmetic manuscript of 1568. The author didn't use equations, letters and arithmetical symbols to solve this problem. These mathematical aspects, we are so familiar with, were hardly known in the sixteenth century Netherlands. Sixteenth century arithmeticians could solve problems like this one, just by using words, numbers and a few lines, as you can see in figure 1. They used the so called 'regula falsi' or 'rule of false position'. This rule is used to find the required unknown number with the help of two arbitrarily chosen numbers. Let me explain this. The value of the first pot with the lid is 4 times the value of the second pot. Imagine the first pot costs 4 guilders, than the second pot will cost 5 guil- ders. The second pot with the lid will cost three times the first pot. Three times 4 is 12, but here we have 21, so we have 9 guilders too much.