This discussion group was run by an organizing team of three persons, Peter Taylor (University of Canberra) and Frédéric Gourdeau (Université Laval, Québec) as co-Chairs with Petar Kenderov (Bulgarian Academy of Sciences) making up the third member of the team. Andre Deledicq (France) had originally been appointed as co-Chair with Peter Taylor, and it should be recorded that he had a role in the design of the program, but he withdrew from this role before ICME and his place was taken by Frédéric Gourdeau. It should also be noted that Titi Andreescu (USA) had been appointed as a member of the organizing team, but did not take part in any of the planning or discussion.
The program had two two-hour sessions and a third session of a single hour. The first session, with invited contributions from Andrejs Cibulis and Dace Bonka (Latvia) and Peter Crippin (University of Waterloo), focused on the range of competitions and related activities which are available. The second session, with an invited introduction from Andy Liu (University of Alberta) discussed the relation between competitions (and related activities) and the teaching and learning process. The last session summarized the previous proceedings with a view to writing a final report.
It should be noted that the discussion group was well-attended, not only by regular participants in World Federation of National Mathematics Competitions activities but also by many different people, mainly teachers and educators from countries in Europe. About 40 people attended each of the first two sessions and about 12 attended the last session. It is estimated that between 60 and 70 people attended at least one of the sessions. The following report on competitions has been prepared by the organizing team based on the discussions and after providing all participants who left their email address a chance to comment.
The discussion group noted that in recent years the meaning of the word “competition” has become much more general than the traditional meaning of either a national Olympiad, or more broadly based multiple choice question exams which have become popular in a number of countries. The World Federation of National Mathematics Competitions, the principal international body comprising mathematics academics and teachers who administer competitions, has itself formally defined competitions as also comprising such activities as enrichment courses and activities in mathematics, mathematics Clubs or “Circles”, Mathematics Days, Mathematics Camps, including live-in programs in which students solve open-ended or research-style problems over a period of days,
It was also noted that Publication of Journals for students and teachers containing problem sections, book reviews, review articles on historic and contemporary issues in mathematics and support for teachers who desire and/or require extra resources in dealing with talented students were also important activities related to competitions.
It was noted that competitions themselves come in a number of categories, the elite national Olympiads, the broader and popular inclusive competitions usually involving (regretfully) multiple choice questions, and special themed competitions, which sometimes involve teams rather than individuals. In some cases, these teams are composed of whole classes, giving a very different feel to the competition.
In particular special note was made of project, or research based activities, in which students have a longer time frame to solve problems than normally permitted in an exam-based environment.
These activities all have in common the values of creativity, enrichment beyond the normal syllabus, opportunities for students to experience problem solving situations and provision of challenge for the student. Competitions give students the opportunity to be drawn by their own interest to experience some mathematics beyond their normal classroom experience.
It was noted that competitions are usually administered by teachers on a voluntary basis beyond their required duties and that the bodies which administer competitions are usually independent of the normal curriculum and assessment bodies.
It was considered there were many reasons for this. Competitions provide for example a focus on problem solving, sometimes giving students an opportunity to be associated with a cutting edge area of mathematics in which new methods may evolve and old methods be revived.
Competitions provide opportunity for creativity, as students often use different than envisaged solution styles in solving problems. The success of competitions over the years, particularly the resurgence in the last 50 years, indicates that these are events in which students enjoy mathematics. Different students derive different experiences, and it is exciting for students when they see a problem can reach the same solution by two quite different techniques.
Because competitions give students an opportunity to discover a talent which they can not normally demonstrate, they provide a stimulus for improving learning. There was also a feeling that competitions allow independent thinking by students.
There was a discussion about various attitudes towards competitions. Some present preferred individual competitions, others said it was positive for students to develop a competitive attitude. There was also a strong feeling present supporting the team competitions and those which involve interactivity.
There was some discussion about the creation of problems and the importance of creating problems with good structure which can capture the imagination.
It was noted that Paul Erdös had commented on competitions, noting the most important thing about them was the enthusiasm they generated. It was noted that for most participants in popular competitions, the aim was not to win, but to take part, taking up the challenge provided. Olympiads provided higher mountains for the more able students to climb.
A number of criticisms are often made of competitions. These include claims that competitions are only for the elite, they involve pressure, widen the knowledge gap, are a negative experience for many students, and favour boys over girls.
The discussion group did not engage in a detailed discussion of these criticisms for each of the competitions represented. Competitions are varied and have different objectives: for instance while some favor broad-based competitions with a high level of success, others aim to support more gifted individuals.
Some participants argued that for competitions to have a positive impact, the teachers must see the progress made by their students. In this view, the role of competitions is to develop a critical body of kids who can do problem solving: in a sense, this role is to get people interested. This suggests that a different look at competitions may be needed.
For some, the suggestion that doing mathematical competitions had a negative impact on many students was not borne out at all by their experience of broad-based mathematical competitions.
However, it is noted that International Mathematical Olympiad team members contained predominantly more boys than girls. (Apparently, evidence shows that average scores of boys and girls are similar and that boys show a greater standard deviation.) This could be an important subject to research and understand better. Certainly evidence from large, broad-based competitions indicated at least equal participation by girls, at least up to the age of about 15.
With relation to the other points, it was noted that entry in competitions was usually voluntary; they did not normally affect the student’s normal assessment, and if anything gave the student an opportunity to discover talent (as argued in the previous section). One teacher noted that in their experience, elite students in mathematics do not act elitely and that mathematics was an area in which there was much less social pressure than for instance in sport.
Finally there was much discussion on this item. It had been noted in the invited talk by Peter Crippin and elsewhere that competition organizers are now focusing increased attention on support for teachers. This takes place in various forms.
The competitions themselves, often available in well graded and classified form, provide vast resources for class room discussion. It was noted that the material available to the teachers should not just include problems and solutions, but also that such information should be well structured, with good information on practical use. Some competitions even provide didactical notes for the teachers so that they can know what type of solutions to expect and how to use these in their teaching. Many organizations which run competitions are now running more structured seminars and workshops for teachers.
13 September 2004