Every four years a meeting is organised somewhere near the location of that year's ICME (International Congress on Mathematics Education), preferably in a different country, on the relations between history and pedagogy of mathematics (HPM). In 1996 the HPM meeting took place in Braga, Portugal (a valuable book of papers from that conference has recently appeared, edited by
Victor Katz: Using history to teach mathematics: an international perspective, Washington: MAA 2000). This year the ICME was in Japan, and the HPM meeting was held in Taiwan, at the National Taiwan Normal University, Taipei, where there was a tremendously warm welcome for foreign delegates (from nineteen countries and all continents) from Taiwanese students and
teachers.
The general pattern of each day was to lead off with a plenary lecture, followed by a mid-morning break involving cake, fruit and various teas; then
some talks in parallel sessions, followed by lunch. The afternoon sessions were again interspersed with fruit, cakes, juices and teas, and the evenings
were sometimes free for delegates to wander the nearby night markets and sometimes occupied with conference dinners and karaoke. Two of the
conference dinners, all consisting of an apparently endless selection of wonderful and delicious courses, were prepared and served by students at the
catering college which was one of the sponsors of the HPM meeting. The karaoke aspect of such evenings were no less impressive: unlike western
karaoke which is a solitary singer stumbling nervously through a dreary song, Taiwanese karaoke is a social event in which anyone volunteered receives
immediate backing, vocal and calisthenic, from audience and waiters alike; every song an opportunity for imaginative social supportiveness.
The conference expeditions took place on two afternoons, one to climb a volcano in a typhoon before a communal hot springs bath, the other to visit
the greatest assemblage of Chinese art in the world, the Imperial collection once housed in the Forbidden City, Beijing, and now held in Taipeiís National
Palace Museum.
These varied and well-judged social events formed an admirable context within which the academic content of the HPM meeting could flourish. The five
plenary lectures, given by Marjolein Kool (The Netherlands), Park Seong-Rae (Korea), Christopher Cullen (UK), Karine Chemla (France) and Masami Isoda
(Japan), provided a range of background studies against which various themes of the conference could be played out in symposia, workshops, round tables
and panels. The two-volume proceedings issued in advance, edited by Wann-Sheng Horng and Fou-Lai Lin, provided an invaluable aid for delegates to
study (before, during or afterwards) papers whose verbal delivery might be in an unfamiliar language. And of course the publication of a full range of
papers in advance makes it possible to become informed about the clashing sessions one could not attend. Looking through the papers in the proceedings
might provide interesting evidence for cultural nuances in the modern HPM research world: papers written by Japanese or Taiwanese researchers tend
towards looking more ìscientificî in the sense of having fuller statistical backup, while occidental researchers tend to report on their work in a more
purely verbal way. (But this observation may reflect a host of other factors such as what kind of researchers from different countries were able to
attend, as well as what kind of reports are easier to write in English.)
There was the usual rich mixture of contributions which is one of the strengths of HPM meetings: papers unveiling new historical research, and
overviews of historical themes to inform teachers and others of recent developments in historical understanding, as well as papers in educational
research and describing ways of incorporating history in the mathematics classroom. In the opening plenary address, Marjolein Kool (Netherlands)
struck exactly the right tone by showing how problems from sixteenth-century Dutch arithmetic books could enthrall and stimulate todayís pupils, and in
her classroom had done so: the moral being not that Taiwanese teachers should use old Dutch problems particularly, but that old texts introduce students to
issues such as different problem-solving strategies, the humanity of mathematics, its universality (as similar problems are found in many lands
and cultures), and above all generate an enthusiasm which not all mathematics education achieves.
Eastern mathematics, and issues around the meeting of eastern and western mathematics, played of course a welcome and major role in the programme. The
second plenary lecture was Park Seong-Raeís masterly survey of the introduction of western mathematics to China, Japan and Korea. It was a
revelation to many delegates that the history of modern (western-influenced) mathematics in Korea is pretty much a phenomenon of the past half-century.
Among the many other studies in eastern mathematics it is worth drawing attention to papers by Wann-Sheng Horng (Taiwan) on pedagogical aspects of a
nineteenth-century Korean mathematical text); Shigeru Jochi (Japan) on the influence of the Jiu Zhang Suan Shu on Japanese mathematics (the conclusion
seems to be that its influence was rather limited); Yan-Chyuan Lin (Taiwan) on a fifth-century Chinese computational canon; Osamu Kota (Japan) on the
history of calculus education in Japan; Naomichi Makinae (Japan) on post-1945 mathematics education in Japan.
The most impressive dimension of this HPM (of a meeting with many impressive dimensions) was the reports from Taiwanese teachers about how they have
incorporated history of mathematics in their teaching. Yu-Yi Lin, mathematics teacher at the catering college whose students prepared the
conference dinners, shared the way she uses history to help students overcome ìmath anxietyî, arousing their motivation with the bold claim ìLet students
find the meaning of life through mathematics historyî. Yen Fu Ming described using a range of historic proofs of Pythagorasí theorem, from several
cultures, to enable Junior High School students to construct the knowledge for themselves and begin to feel themselves to be mathematicians. Hui-Yu Su
reported on using the so-called Pascalís triangle in high school mathematics lessons, for students to learn actively about connections between different
mathematical concepts.
Here are just a few more of the memorable sessions that struck me. (Those who were lucky to attend other sessions will have their own list, of course.) Li
Yu Fuís fascinating explanation of the Bunun calendar that formed the conference logo: the Bunun people of central Taiwan are among the so-called
"aboriginal", that is, pre-Chinese, inhabitants of the island. Karine Chemla and Christopher Cullen enjoying a robust interchange over the deep
historiographical question of whether the same word ëproofí can be used to describe both what ancient Greek and ancient Chinese mathematicians did. In
a remarkable tour-de-force of technological as well as intellectual sophistication, Masami Isodaís plenary lecture on using technology in
teaching mathematics with history incorporating images, on his computer, from the conference itself, photographed a few hours earlier. The Botswana
delegate Luckson Kainoís account of the challenges facing mathematics education in sub-Saharan Africa in the years ahead. Coralie Danielís
Maori-inspired Fibonacci scarves. The Platonic solids (or rather hollows, being Chinese lanterns and the like) constructed from Coca-Cola cans, old
milk cartons and other symbols of western capitalism by the pupils of Chun-Chih Peng. And FrÈdÈric Metinís talk on seventeenth century
fortification proved so popular that he was asked by the Taiwanese students to repeat it.
There were two Round Tables at the conference, on values in mathematics education and on issues of transmission between and meeting of cultures. In
the bad old days (at other conferences in other disciplines) a "round table" used to mean a number of speakers successively reading their pre-prepared
texts at the audience until the time ran out, a thoroughly dispiriting event for all concerned. The round tables at HPM Taipei were quite different, full
of rich interest and leading to stimulating and genuine discussion. The values session, chaired by Alan Bishop (Australia) was to introduce the
Australian-Taiwanese team who are working on a three-year project enticingly called VAMP (Values in Mathematics Education). The cultural transmission
session -- "Culture meets culture: where next?" -- sought with the help of an international and multicultural panel, chaired by John Fauvel (UK), to
explore how to take forward into ones home community the insights and experiences of a conference such as this, with the aid of those who have
studied and observed other meetings of cultures in history and elsewhere.
The contribution made by Taiwanese teachers and students to the conference marked an important consolidation of a trend already noticeable in earlier
HPM meetings, in the strength of the home team. The Taiwanese school-teachers at the conference were already informed and enthusiastic
about HPM issues, having been trained at the Normal University in Taipei, and the students were currently studying there, often for master's degrees, under
the guidance of Wann-Sheng Horng and his colleagues. So there was already a strong base for fruitful interaction with the visiting teachers, historians
and educators, and a sense that the activities and approaches stimulated by the HPM meeting could and would continue afterwards. Thus the efforts put in
beforehand over several years, by the conference organisers, in their role as teachers at the Normal University, ensured that the HPM meeting was part of
the ongoing development of HPM studies in Taiwan as well as benefiting HPM activities world-wide.
In developing HPM activities further in the region, the hope was expressed for holding a series of regular future conferences, somewhat after the
fashion of the European Summer University, which could bring together students and teachers from many East Asian countries, notably Japan, Taiwan
and Hong Kong.
In this report I have stressed the group dynamics and social context of the meeting for a reason, in order to bring out the strength of HPM culture as it
is developing. For the sake of contrast, consider what apparently happens among our colleagues in science education. Last year's Fifth International
Congress on History and Philosophy of Science in Science Teaching, a science version of HPM held in Italy in September 1999, was reviewed in the BSHS
Education newsletter (no 30, February 2000, pp. 6-10) by a number of British participants at the conference. In a surprisingly critical set of reports,
Clive Sutton drew attention to the lack of discussion (discussion (in the sense of a real engagement between people with different perspectives) did
not occur) while Peter Ellis drew attention to the distance of the whole proceedings from classroom reality:
"What was sad about the conference was the lack of reports about actual experiences in the classroom and strategies for introducing and using
historical material. It was sad that some of the few school teachers who did speak were criticised for trivial errors or generalisations by their more
authoritative academic counterparts. [. . .] Unless the group can be more appealing to science teachers and science teacher trainers it does not seem
to be fulfilling its function of promoting the history and philosophy of science in science teaching."
(I suspect the writer half had in mind to describe the academics as ìauthoritarianî before substituting the less charged word "authoritative"!)
And Elspeth Crawford, another conference delegate, commented sardonically on the relation between the congress's activities and its title, "Science as
Culture", and the conference culture itself, in forthright terms:
"I am not sure if the title did anything other than allow people to come and talk about what they were going to talk about anyway. There was no attempt
that I heard to address the issue of science as culture, so I still do not know what it means to refer to science in this way. [. . .] Once again, I
find my report of a conference finishing with a plea that those who organise conferences pay attention to the group dynamics and the culture set up within
the conference itself. We need far more time spent working at particular topics and issues, in genuine discussion, and a much greater emphasis on how
to enable teachers within the systems which employ them."
The authoritarian image of the history and pedagogy of science congress emerging from these authoritative reflections could not be further in every
respect from the tradition of HPM meetings, not least in its latest splendid manifestation in Taiwan. The success of the Taipei meeting was due
especially to the enormous care of Wann-Sheng Horng and his colleagues to meet Elspeth Crawfordís last point, in effect, paying attention to the group
dynamics and conference culture. Everything that could be done was done to foster an inclusive atmosphere in which people from different cultures,
traditions, and locations within the educational world could all participate and contribute their insights to furthering the relations between history and
pedagogy of mathematics. Everyone knew, and cared, that what this meeting was ultimately about was what would happen in classrooms, and took on board
the absolute responsibility to support teachers and learners at the front line, in countries across the world.
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