International Study Group on the Relations Between
HISTORY and PEDAGOGY of MATHEMATICS NEWSLETTER
An Affiliate of the International Commission on Mathematical Instruction: No. 44, November 2000
History and Culture in Mathematics Education
Report of Working Group for Action 13
Structure and themes
The Working Group concentrated on the following five major themes, which were identified in the call for papers. For each theme a keynote speaker was
invited. The keynote lectures were discussed in subgroups, in which further short presentations were given as well.
- Aspects of multidisciplinary work
Central question: How may mathematics education be improved by attending to the possibility of cross-disciplinary work with other subjects and teachers?
Both positive and negative aspects should be considered.
In the keynote lecture Mangho Ahuja (Southeast Missouri State University,
Cape Girardeau, MO 63701, USA) spoke about Traditional versus multidisciplinary teaching. He compared two teachers in their approach of the
Pythagorean theorem, one taking the traditional path and the other who introduced the topic through activity groups in a broad range of fields.
Teachers of other disciplines got involved via the questions that they received from the students, although they had been much reluctant to
cooperate when they were asked beforehand. The outcome of the multidisciplinary project was positive. Costs are high, certainly when the
curriculum does not provide incentives for this type of approach.
- Effectiveness of history in teaching mathematics
Central question: What evidence have we that using history or broader cultural dimensions in mathematics education improves the quality of that
education?
Karen Michalowicz (The Langley School, McLean VA, USA) spoke in the keynote lecture about Developing historical modules for use in the high school
classroom , a project funded by the National Science Foundation, which she is carrying out together with Victor Katz (University of DC, Washington DC,
USA). Six teams, each of three teachers and a university professor, have worked together during the last two years in order to produce resources for
classroom lessons, in a variety of fields. The modules are now distributed and are being field tested by an independent agency. The first impression,
from telephonic interviews with students, is that 'history works', especially with respect to the students attitudes, since students think that mathematics
taught in this manner is considerably more interesting.
- Probability theory and statistics
Problem definition: An important subject whose historical dimension has been too little attended to (except at a rather simple anecdotal level) is that of
probability and statistics. A fuller consideration of the contribution that its history could make to statistics education is overdue.
The keynote speaker was Arthur Bakker (Freudenthal Institute, Utrecht, NL). He discussed The history of early statistics and its didactical implications,
concentrating on a historical and then a didactical phenomenology of average values. These constitute a large family of notions that in early times were
not yet strictly separated. There are many parallels between history and the development of studentsí conceptions. Classroom observations indicate the
importance that students discover many qualitative aspects of average values before they learn how to calculate the arithmetic mean and the median. From
history, it is concluded that estimation; fair distribution and simple decision theory can be fruitful starting points for a statistical instruction sequence.
- The dance and poetry of mathematics
Problem definition: An aspect of mathematics education which historical-cultural studies are well able to support is its creativity, fun and beauty. Spelling out in more detail how this may be achieved will be a
useful service to teachers.
Hisato Kikuchi (Higashiyamagata Junior High school, Yamagata, JP) reported in his keynote lecture Sangaku as a teaching material about joint work with
Ikutaro Morikawa (Yamagata University, Yamagata, JP). Sangaku is one of Japanís indigenous mathematical customs from the Edo period. Many
mathematicians of this period would try to set original problems for themselves and solve them. Doing so, they produced plates with colourful
figures and dedicated them to a shrine or a temple. This custom showed not only appreciation for God, but also pride in oneís mathematical ability.
Sangaku appeared to be a fruitful medium for working with students, who studied constructing and solving problems through the making of Sangaku.
Studentsí appreciation of mathematics increased, as well as their confidence in problem solving.
- Culture
Problem definition: It is important to discuss the breadth of the idea of culture and to discuss how far it needs to be narrowed, and in what
directions, in order to make progress with bringing proposals for howmathematics teachers may be supported and encouraged.
The keynote lecture Mathematics education: cultural perspectives and underpinnings in the Indian context was given by Dilip K. Sinha
(Visva-Bharati, India). He reviewed a series of aspects of Indian mathematical culture, which ranged from early work discussed by Colebrook to
the fairly contemporary notes by Ramanujan. Although current mathematics education in India is predominantly shaped by western perspectives, one can
also recognise in it the essence of Indian culture, in that recent perspectives on mathematics education keep on developing with these three
categories: grassroot, esoteric and applicable.
Further presentations and discussion
After the keynote lectures further work was done in three subgroups.
The first group was chaired by Costas Tzanakis (University of Crete, Greece) and went on with the theme of multidisciplinary work. The discussion explored
what multidisciplinary work might be in the context of mathematics education. The conclusion was that there should be an emphasis on mathematics, and that
the teacher should adjust the work to the social context of the students. Important parameters are the educational level of the students, the subject,
the time available, and the teacherís own experience. Multidisciplinary work is possible in practically any subject. Examples signalled were: calculus,
differential equations, probability theory and statistics, combinatorics, vector analysis and functional analysis, but also subjects like number
theory, group theory and topology. The subjects may relate to non-mathematical topics such as: physics and natural sciences, philosophy,
music and arts, logic and linguistics, drama, literature and history.
Specific examples of actual implementations were presented by Oscar Joao Abdounur (Brazil), about Historical aspects of ratio and proportion in music
and mathematics education; Costas Tzanakis, about Elaborating on abstract algebraic concepts on the basis of physical ideas and concepts: special
relativity on the basis of elementary matrix algebra and group theory and Paul Manning (USA) on Intersections of mathematics and the humanities
discovered by accident: language, literature, philosophy.
A second group, chaired by Karen Michalowicz, went on with the themes Effectiveness of history in teaching mathematics and Probability theory and
statistics. Short presentations were given by Catherin Murphy (USA), about A historical course for teachers; Rebecca Kessler (USA), about A module about
Archimedes for the mathematics classroom; Osamu Takenouchi (Japan), about History and mathematics teaching in Japan; Phyllis Caruth (USA), about A
module about the history of combinatorics and statistics for the mathematics classroom, and Bernd Zimmermann (Germany) about Appealing geometrical
problems from Al-Sizji. The subsequent discussion was mainly about effectiveness. The conclusion was that there are many ways to implement
history, some of them needing special attention and care. For example, one should be critical when students use information that comes from the
Internet. History can have a function, it was agreed, either to enrich mathematics (e.g. if you know a subject already, to do it once more but in a
different manner), or to introduce a subject to students. It can be applied in order to develop a new learning trajectory; it can produce heuristics for
problem solving, and many more useful things. Historical games were also discussed as a positive contribution in mathematics lessons.
The third group, chaired by Florence Fasanelli (Washington DC, USA) and Jan van Maanen, worked on the broader cultural perspective, as reflected by the
final two keynote lectures. Short presentations were by Lawrence Shirley (USA) about Using costumes and connecting to local peculiarities and
Man-Keung Siu (Hong Kong) about his course Mathematics: a cultural heritage. A variety of aspects of culture came up for discussion:
- the clash between western and eastern mathematical traditions
- the influence that the prevailing culture may have on individual students, or on groups of students (e.g. the gender problem is closely linked to the
mathematical culture)
- paying attention to the specific culture of the region, or the culture of an ethnic subgroup of a mathematics class, can have a positive influence, for
example in increasing the self-confidence of the group
- cultural happenings (visits to a museum, drama, etc.) often attract criticism from colleague-teachers and parents, so one should be prepared for that. On
the other hand enthusiasm of students is one of the best and most convincing arguments for doing these types of activities, certainly with the parents.
- the relation between culture, history and mathematics education is under-researched, and is worth further research.
Looking back on this Working Group some general conclusions may be drawn. The first is that the relation between history and mathematics education is still
an area in which many developments take place. The systematic production and testing of historical modules, as described by Michalowicz and Katz, is one
example of a type of research with a practical outcome that is very important. The value of making a connection with local culture was brought
forward more than once, and with reports of positive results. Increasing confidence with students is one of the key-words connected with the positive
evaluation.
As always, positive results require input. The balance between cost and result was discussed, and although it was agreed that the costs are still
high, many participants appeared willing to invest in this manner. One of the reasons was their own pleasure in preparing historical material for students,
but the main reason was that they noticed many positive effects with students. In some countries the curriculum is not supportive of this work.
Further work has to be done on national levels.
More history at ICME-9
As an appendix I shall list here the other historical activities at ICME-9.
There were regular lectures by Niels Jahnke (Germany) about Historical sources in the mathematics classroom: ideas and experiences, by Osamu
Takenouchi (Japan) about Some characteristic features of Wasan, the Japanese traditional mathematics and by Ewa Lakoma (Poland) about History of
mathematics in educational research and mathematics teaching ó a case of probability and statistics.
Then there were two sessions of the International Study Group on the relations between History and Pedagogy of Mathematics (HPM), with the
following speakers: Bjørn Smestad (Norway) on History of mathematics in Norwegian textbooks, Peter Ransom (UK) on Teaching geometry through the use
of old instruments, Osamu Kota (Japan) on John Perry and mathematics education in Japan, Yoichi Hirano, Katsihusa Kawamura and Shin Watanabe
(Japan) on Mathematical exhibits at museums from viewpoints of mathematics education, Nobuki Watanabe (Japan) on A practice of the cultural history of
mathematics in elementary school. The second HPM-session was concluded by the installation of HPMís new chair for the period 2000-2004, Fulvia Furinghetti
(University of Genova, Italy).
And finally, the book History in mathematics education: The ICMI Study, edited by John Fauvel and Jan van Maanen, and published by Kluwer Academic
Publishes (Dordrecht 2000), was launched with presentations by several chapter-coordinators (Fasanelli, Jahnke, Michalowicz, Nagaoka, Siu and
Tzanakis).
Jan van Maanen (University of Groningen, Netherlands)
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