Mathematics Enrichment Beyond 2000: The Australian Experience and the Role of WFNMC

This talk will start with some general observations about changes in the syllabus over the last twenty years. The Australian Mathematics Trust enrichment programmes will then be outlined, and an emerging role of these programmes will appear as part of the general curriculum, particularly as it applies to talented students. These will be presented to indicate, as an example, how we can have a wider role in mathematics education than in running mathematics competitions. Finally the World Federation's aims will be discussed and I will relate our experiences to these aims.

I became involved in mathematical enrichment programmes for two reasons. First, I had two good teachers. I believe most people who work their way through a mathematics degree can identify some form of inspiration in their lives, generally one or more outstanding teachers. In my case these teachers were able (probably not without much difficulty) that mathematics, a subject which could have been bland, was in fact a deep and challenging subject in which there were many fromtiers. So I believe it was necessary to have the teachers to get me to where I was, an academic at a rather fledgling University where research was discouraged (at the time my kind of University aimed at good quality vocational teaching).

If it had not been for my colleague Peter O'Halloran I probably would have written text-books. In fact I got involved working with the Open University in Britain and did write a couple of their texts. But Peter returned from study leave which he had spent in Canada (partly at Waterloo). He told me about a competition he had seen in Canada, and another in the US, both of which attracted many thousands of entries.

Peter believed that there was a demand for this in Australia, and was looking for volunteers to try out a similar project, with some modifications. I was very happy to give this a go. I had, with the inspiration of my own teachers, worked my way through both an undergraduate degree and a research degree. But there had been a competition at other schools in Adelaide, and I recalled that the students who had been to the schools doing the competition had an extra degree of maturity and confidence. I could see that these competitions had provoked much classroom discussion, giving the students a wider experience at the school level. Here was a chance to give benefit to a wide range of students.

When Peter called the first meeting in Canberra in April 1976 I was there, and found myself permanently involved. I might note that another colleague who is here, Warren Atkins, who was in a different Faculty (Education), was also at that meeting.

When I was at school, geometry was a strong part of the syllabus. We learnt to prove things. The basis of all our teaching was quite theoretical. This was fine for most of the higher achieving students, who enjoyed the challenge of solving a difficult problem.

The powers that be, in our country and elsewhere, argued successfully that this approach in teaching mathematics was lost on the average student. The average student did not appreciate the subject and saw no relevance in geometry, algebra and the like to everyday life. Much of the geometry, much of the logic, much of the rigour was taken out of the syllabus. Mathematics was now related to the real world. There was more statistics and data analysis. Students learnt through discovery and experimented with numbers. So the focus moved to the needs of the average student.

This opinion seems to be also supported by Mathews [2], who, from an American perspective states

Most of us who are teaching collegiate mathematics will find very little in the experience of Great Britain that is helpful in assessing how we should teach our classes. Their "reform" has been concerned with implementing a national curriculum for schoolchilden up to age 16. Most of the reform with which we have experience neither is nationally mandated nor deals with schoolchildren. Still, if one wishes to find some lessons learned, it is most probably in the way in which England's reform curriculum has failed to meet the needs of the top students.

Mathews is replying to an article by Andrews [1], in which reform in the first instance refers to the reform Calculus, a quite different topic than what I want to address here. The point he makes about England, however, I do wish to address. It may be a little narrow. I would suggest that the English school syllabus reforms are similar to those in many Western countries, but agree that it may be to the deteriment of the top students. Results of Western countries in the recent TIMSS study (measuring results of average students) place them typically behind Asian countries, and IMO results (measuring performance of elite students), for various reasons, show Western countries struggling against those of Asia and Eastern Europe (where special maths schools maintain some of the older traditions).

We also find continuing pressures, sometimes successful, to reduce the actual number of hours taught in the classroom. This is achieved by pressures to "broaden" the student's education, so seemingly large consumers of time, such as Mathematics, are the ones to be reduced.

Attitudes of course advisers also tend to encourage students into the "softer" sciences, where jobs are (debatably) claimed to be easier to find.

I might note at this point that we have a large amount of data on recent effects, which have been reported on by colleagues in the research section of this conference. It indicates healthy figures in terms of participation and performance at the year 7 to 10 level. However, we are finding in Australia that for whatever reason, both participation and performance at year 11 and 12 levels are down. University academics in Science and Engineering Faculties are also reporting increasing difficulties getting students with adequate backgrounds.

As I will outline, we (in the Australian Mathematics Trust) see a need to fill the gap for talented students, i.e. those in at least the top 10% and probably a lot more also. We believe that our programs, which I will describe, are providing this alternative.

The Australian Mathematics Trust now provides enrichment programs at three levels:

1. Australian Mathematics Competition (500,000 students of all standards, about one in three enrolled secondary students),
2. Mathematics Challenge for Young Australians (which caters for about 15,000 talented students and has a formal course-work component), and
3. The Olympiad Program (about 100 elite students).

We do many other things, including

• Publication of Enrichment material, principally the mathematics enrichment series of books, which include national materials of a number of countries and pedagogical material,
• The Australian Mathematics Teacher Enrichment Project, on which Steve Thornton has reported separately at this conference,
• Other mathematical events (International Mathematics Tournament of Towns, Singapore Mathematical Olympiad for Primary Schools, Mathematics Days),
• Research in Mathematical Education, and
• Informatics (this raises some interesting issues).

We are also showing an increased interest in the primary school situation, where our expertise is not so strong.

This competition was inspired from the Canadian and US models after Peter O'Halloran's visit in 1973. The Competition is our most successful activity. In 1997 it had about 530,000 entries from 36 countries. The Australian entries, about 450,000 in number, represented about one in every three of students enrolled in secondary schools.

The competition has broad aims, which include

1. To give the average student a chance to achieve in mathematics (this being enhanced by the certificate structure),
2. To highlight the importance of mathematics in the curriculum (this being helped by a policy of setting questions in scenarios with which the students can relate),
3. To provide teachers with resources, and
4. To promote mathematics as an enjoyable activity.

The paper is carefully graded, along the principle that every student will find a challenge somewhere. The first questions are very easy. They become progressively more difficult until even the most elite students are challenged at the end of the paper. Significantly, every year we have had in Australia alone at least one perfect score each year but no more than five (of the 450,000 entries). (Later note: the 1998 paper proved an exception, producing 9 perfect scores in Australia.)

From the point of view of the Trust this competition has given it the resources to extend its activities and achieve wider aims. Because the Trust is a non-profit activity in which its resources can only be expended in accordance with its aims this has in fact been guaranteed.

The Competition has in turn become a model for competitions in other disciplines in other countries.

Elsewhere in this conference I have given a talk on this activity on behalf of the Director, Bruce Henry. The Challenge, commenced in 1991, after we were successful in obtaining government funding to support the Olympiad program, is divided into three stages.

The first stage, the Challenge stage, comprises 6 in-depth problems. Students have 3 weeks to solve these problems. These problems may be discussed with teachers (preferably) but obviously anybody, including family. Teachers are provided with extra support materials, such as further discussion and extension problems, which can provide useful for the exceptonal student. Sometimes a question can thereby introduce a student to a new branch of mathematics. About 13,000 students (between years 7 and 10) enter the Challenge stage annually.

The Enrichment Stage is our formal extension course. This course, in four sequential versions designed for years 7 to 10, provides formal instruction in many theoretical branches of mathematics, such as geometry, discrete mathematics, algebra. logic and number theory. This course material, with further notes available to the teacher, takes about 6 months of the academic year. Students eventually receive certificates based on their success in the course problems. Interaction with the teacher is stressed. Unfortunately teachers in Australia often feel stressed, with lower salaries and conditions than in professions with comparable training and community responsibility, so it is always an issue with us to persuade the teachers to take on the commitment. Whereas most of our activities are stable in numbers, the Enrichment Stage saw a considerable increase in numbers during 1998. Well over 6,000 students, many of them the same as in the Challenge Stage, have participated.

Finally in the Challenge there is the Australian Intermediate Mathematics Olympiad (AIMO) which attracts about 600 students. This year we have restructured this more along the lines of the American AIME and have set up a prize structure.

We feel that this program, which probably involves about 15,000 students each year, considerably enhances the preparation of Australian youth to undertake university study, particularly in the Scientific, Engineering and Technological areas.

The best 100 students, generally identified by state directors, from performance in the earlier activities, participate here in an advanced programme of competitions, training schools and mentoring, with the best six students selected to represent Australia at IMO.

The main activity is our Enrichment Series of Books. We now have a total of 13 books in the series, with six further books in various stages of preparation. These include problems and solutions of various national and international competitions. More significantly they contain pedagogical material. Maybe the key version of this is the Toolchest, an informal syllabus for Olympiad training, but there are other examples.

We also regard the Challenge Enrichment Student notes as published material.

The data base for the AMC is one of the largest of its types in the world. It contains every response to each of thirty questions over a period of 15 years. Each student is identified by name, school, date of birth, gender, and school year.

With this data base we have been able to study gender differences, trends over time (as reported by colleagues at this conference) and risk-taking activity. We have been able to extend these projects into educational psychology and modelling gifted students.

There is no reason why we will not in the future be able to work in collaborative projects with other countries, provided the samples are similar.

My colleague Steve Thornton has given a detailed talk on this project. It was developed because of our concerns that many teachers feel able to support the development of their talented students. In a major sense we can not help as teachers feel underpaid and under-resourced, so there is resistance in being involved in what they see as non-compulsory activities.

However they do not receive much training in handling the successful students and it is our aim to help teachers do this better. We have accredited a Graduate Certificate in Mathematics Enrichment which will also help a maths teacher to obtain a higher qualifications.

We have recently followed the lead of Ron Dunkley's Canadian group by setting up an Informatics project. The basic aim is to enter Australian teams in the International Olympiad in Informatics (IOI). However already we are finding some interesting issues arising as to what mathematics topics, such as Discrete Mathematics and some branches of Operations Research, which would help support Informatics.

The model for what became the Australian Mathematics Trust was principally developed by Peter O'Halloran. The principal elements he ensured that the Trust would be independent, particularly of an institution or professional society.

Independence has ensured the widest possible volunteer network and ensured that resources developed by the Trust have been free for use in developing and nurturing other worthwhile projects. The University, as Vice-Chancellor and Trust Board Chairman Don Aitkin was able to report at WFNMC2 in Bulgaria, has been able to take a benign but protective role, nevertheless gaining general kudos from the community.

Despite Peter O'Halloran's death, soon after WFNMC2 in 1994, the network has remained strong and the Trust structure will change with circumstances.

The Trust has developed its role from just running competitions. It is now, as I have indicated developed a more formal role in complementing a changing syllabus.

As a result we are working more closely with the professional societies. We have also been granted observer status in our own right at the Mathematical Sciences Council of Australia, the umbrella society for mathematics and the main government mathematics lobbyist.

We also exhibit our activities as a separate organisation at conferences to emphasise our status as an independent organisation.

Having explored our own profile I would now like to look at the aims and profile of the WFNMC. The WFNMC recently made significant advances when Ron Dunkley oversaw the development of a Constitution. This Constitution, among other things, spells out a number of aims.

Clearly, the WFNMC has three major current activities which are recognised in the Constitution. These include the Journal Mathematics Competitions which Warren Atkins has developed and edited from 1984, this Conference which has now been held for the third time and the Hilbert and Erdos Awards.

The preamble states: The [WFNMC] is a voluntary organisation, created through the inspiration of [Peter O'Halloran], that aims to promote excellence in mathematical education and to provide those persons interested in promoting mathematics education through mathematics contests an opportunity of meeting.

The Constitution then goes on to list aims. The first of these is To promote excellence in, and research associated with, mathematics education through the use of school mathematics competitions.

What is not yet clear among us is a further articulation of this aim. I would suggest that we ask ourselves questions such as Why are we, as members of WFNMC, interested in competitions? Do we see ourselves as having a wider role? How do we see competitions as assisting mathematics education? How do our activities relate to the formal teaching of mathematics? and attempt to answer these questions in writing.

As I have attempted to indicate using the Australian experience as an example, I think our interests are genuinely wider than purely in competitions, and we have seen this in some other countries. For example the Waterloo centre has a much wider interest than in mathematics competitions.

More specific questions can be raised as to what we see as a useful curriculum for talented students. Most people see the IMO as testing the pinnacle of some form of international syllabus. If we ask any IMO team leader we would get a rather definitive reply as to whether a certain problem could be set at IMO, even though no formal attempt has ever been made to write out an IMO syllabus.

The Tournament of Towns is less inhibited. It sets problems which go well outside the IMO bounds, not only going into some "forgotten" areas of geometry but also using contemporary methods of problem solving. At a higher level the Tournament of Towns runs summer conferences, where secondary students work on what we might be called research problems, certainly problems whose solutions are unknown even to the problem creators.

Indeed, here we are training the new breed of mathematician, who can lead tecnological development into the next generation. Hopefully I am suggesting that in addition to research based on competitions, where we have nominal aims in our relationship with ICMI (but not widely developed), we may even be able to take a more fundamental role with ICMI in relation to the curriculum for the talented student (i.e even more than 10% of the student population).

Within ICMI we are sometimes seen as curiosities or eccentrics by the broader group of mathematics educators. However as we evolve I think we will become conscious of our own wider, often not documented interests, and tangibly develop a much more significant contribution to mathematical education than just through running competitions.

Programs at future congresses should be sufficiently flexible to allow these interests of the organisation to develop.

To summarise I would therefore like to make two basic points:

• We should articulate why we see competitions as helping to develop mathematics education, and
• We should explore a wider role that we are well qualified to take in the area of mathematics education.

1. Andrews, George E, Mathematics Education, a Case for Balance, The College Mathematics Journal, 27, 5, pp341-348, November 1996.
2. Mathews, David M, Mathematics Education, A Response to Andrews, The College Mathematics Journal, 27, 5, pp349-353, November 1996.