As one who allegedly had some hand in helping our daughter Sarah gain a rather unexpected and unwanted prominence as a champion of mathematics, I have been asked to say something about Leaving Certificate mathematics.

The course with which I am most familiar is the higher-level course, having on occasion been called on by Sarah to help her subdue some particularly knotty problem from a past paper. This syllabus is similar to a slightly "heavier" one which I studied more than 30 years ago, but is all the better for being that bit lighter and is richer for the broader range of topics it offers.

Having lectured on mathematics for close on 25 years, I can see, from my perspective at third level, the rationale for most of the subject material which aims to cover topics essential to the pursuit of studies in science, engineering, business studies and computer science, to name but a few. I have no doubt that when this syllabus is taught well by an enthusiastic teacher, willing and receptive students can learn a great deal of mathematical technique which is sure to stand them in good stead in their immediate futures in the workplace or at college.

However, because of the manner in which it is currently being examined I am not sure that students can be expected to see the sense of the syllabus and they must often ask despairingly, "What is it all about?"

The questions that are being asked do not appear to me to be adequately meeting the challenge of testing the all-important "why" of the subject and so are not addressing one of the expressed aims of the syllabus, which is to give an appreciable sense of the relevance and applicability of mathematics. While students who successfully answer these questions, which place a heavy emphasis on technique, are likely to be skilled in the "how" of mathematics, they must rather disappointingly view the subject as simply "maths for math's sake".

Unfortunately the manner in which a syllabus is taught is often influenced by how it is examined, particularly when those who teach it do not have the privilege of examining it. Whatever the merits or otherwise of a single national examination taken by one and all at the end of a course of study, it is generally acknowledged that such examinations tend to settle down a short while after their inception to a predictable pattern.

The expectation, if not insistence, that they vary little from year to year can be a tyranny for teachers who would like to illustrate the power and scope of their subject. It must be disheartening not to venture on a mathematical excursion which might lend some depth to the treatment of a topic for fear of hearing, "But this stuff won't be examined."

It is detrimental to any proper teaching if what is not likely to be examined is not likely to be taught. Some rather alarming consequences of this "exam pragmatism" can be that even students who have scored well on such national tests often

have a very low awareness of the usefulness of mathematics;

are hazy on fundamental concepts;

possess little notion of what passes for a proof;

do not know how to derive simple but frequently used formulae;

use prose scantily if at all when answering questions and rarely use logical connectives, such as an implication sign, to give coherence to explanations.

I know that the explaining and examining of the "why" of mathematics is a much more arduous task than the teaching of the "how". I am also aware that like me, most teachers moved from one academic environment (which laid no great stress on the relevancy aspect of the subject) straight into another one without ever having plied their trade in the "real world". The result of having no tangible experience of the vitality of mathematics as a problem-solving discipline can make one feel ill-equipped to say with some measure of assurance where mathematics is used.

However, answering "Why are we doing this?" with the age-old "Because it helps you to reason and think clearly" just won't cut it any more as the sole reason for studying mathematics. While it is still a very valid and valuable reason, there are many others which convey the practicality of "maths in action" and are much more likely to engender students' enthusiasm and interest.

So without sacrificing what the current system achieves, the challenge is to invigorate the teaching of mathematics by displaying its essential relevance, while emphasising the playful element of exploration and discovery that is its vital driving force.

To achieve such ambitions may require that existing teachers be given comprehensive courses which illustrate a number of non-trivial but accessible examples of how mathematics in its various forms is used in the world beyond the walls of our academic institutions.

Knowledge of even a few genuine applications can make such a difference to the winning or losing of the battle of convincing young people of the importance of mathematics. The task of developing more flexible syllabi and examinations to assess this new emphasis will require much thought but will surely be worth the effort.

Some of the developments that have taken place over the past 10 to 20 years can be harnessed to aid in these endeavours. Recently many books have been published which discuss the deliberations and serious thinking done by other nations on the design and assessment of syllabi so as to improve the overall teaching of mathematics and make the subject more enjoyable.

There are also many excellent books of an expository nature which deal with many different aspects of mathematics such as its history, its new areas of application and the men and women who make careers in it. Powerful mathematical packages along with other software offer new ways for teachers to greatly enhance what can be done in the classroom, while the Internet, besides being a repository of a huge amount of information, mathematical and otherwise, seems poised to be one of the most exciting new learning environments of all.

David Flannery is Lecturer in Mathematics at Cork Institute of Technology. Sarah Flannery was awarded the 1999 Young Scientist the Year.