Mathematics Content
Introduction
On this site I was asked to include more mathematics, so here I plan to give a description of various problem solving techniques and other ideas in mathematics and which can help a student extend their classroom knowledge of mathematics into more interesting areas, particularly topics which the Australian Mathematics Trust has considered useful in developing problem solving skills and structural knowledge. The subject of mathematics is so big that even quite talented students can extend their knowledge and remain within their school years, allowing what would be called enrichment, as compared with acceleration, where students merely get promoted to the following year.
The object of this part of the site is to directly introduce students to some of these topics, without relying on resources which may or may not exist in their local regions. These are the topics which will help students in competitions, certainly in events such as the Australian Mathematics Competition and others as run during the period of history covered here, but inevitably also into the future. But problem solving is as much an evolving branch of mathematics as research topics and sometimes new methods are developed, and sometimes methods can be generalised and simplified.
Most countries do have structures in place which enable students to access this mathematics, and generally these are via people running competitions, not commercial ones (which in Australia and elsewhere are many and most offer nothing more than tests), but normally organisations with programs starting at school and leading to international Olympiads, and which provide considerable backup resources. The Australian Mathematics Trust, through the events discussed on this web site, is the definitive organisation providing these services in Australia (and some neighbouring countries) for example.
The idea is that participation should be open to all, but voluntary, and students participate up to the level they wish.
The mathematics is often accessible to students from their classroom experience, often just systematising ideas which may be intuitive, but invariably is much more interesting than mathematics in the classroom, and can be as challenging as the student needs.
Students who master some of the ideas here can become more independent learners, not just learning for a particular exam outcome, and this can equip them much better for higher study and life experience.
In the Examples given in the various subpages here, it is strongly recommended that the reader try to solve the problem before reading the solutions. Some of them look at first glance exceptionally difficult, although in the end none will involve complicated calculation or particularly advanced mathematics.
Hopefully the reader will find the section to be my version of a journey through problem solving, generally beyond classroom syllabi but hopefully normally accessible. Emphasis is on ideas, structure and logic rather than complicated calculation. Many of these problems have been used in competitions whose problems and solutions can be found in detail in books published by the Trust on its bookshop, which can be found on its web site www.amt.edu.au.
Concluding comments
The common feature of all these problems is that no difficult calculations are needed anywhere, which helps to ensure that students in a normal class can reach out to the problem from their normal experience. All of the problems require the discipline of clear thinking, which will enhance the general problem solving capacity of the student.
Major advances in our society often emerge from the disciplined solution of apparently simple problems. For example, Euler’s analysis of the bridge problem led to a higher level of knowledge which is central to modern technology today. For him to have been able to develop an idea beyond what was known is similar to the challenges which people continually face on a day-to-day basis.