Apply your beautiful mind and maths can be as easy as Pi

Your years of study will have given you the tool kit with which to tackle your maths exam. Follow the basic rules and you will succeed. Dr John Evans proves the point

Distracted by the Oscars and so on, you probably missed (or have already forgotten) an observation made by the heads of schools of science of the institutes of technology about the modern student of mathematics, who, it appears, is not so good at mental concentration over extended periods of time.

Is this you? Are you that student? Do you take Polaroid photographs during class and write down such messages as "This is X, my maths teacher. She will help you out of pity", or, "This is Y, my applied maths teacher. Don't believe his lies"?

Mental concentration by students seems to be helped if interest can be stirred up by the teacher. The teacher should show enthusiasm, though care should be taken to ensure that the students do not mistake this for mental illness (A Beautiful Mind, Pi, etc).

Some scribblers have suggested that the giants of maths might be presented in a more attractive light - for instance, the republican activities of Evariste "Gerry" Galois and his death in a duel at the age of 21. Also, the recent initiative taken by the music industry to give names such as Six to pop bands is particularly helpful. More prosaically, I think students might identify with Galois as a patron saint because of the frequency with which absent-minded teachers lost his homework copies.

This business of concentration and memory can perhaps be clarified by studies of mnemonists (persons with perfect recall).

The material recited by the mnemonist often consists of randomly-constructed lists of nonsense syllables, numbers and sounds and (obviously) bears no resemblance (whatsoever) to student attempts at proofs of trigonometric identities, for example. One mnemonist relied on converting lists of numbers into visual images in which each digit acquired a personality: 1 - a proud man, 2 - a high-spirited woman, 3 - a gloomy person, 6 - a man with a swollen foot, 7 - a man with a moustache, 8 - a sack. A series of digits could then be converted into a story: 8, 7, 3, 6 and so on might become: "A sack containing a man with a moustache was discovered by a gloomy person who owned a pop band".

Of particular interest to us, however, is the following. When presented with 1 2 3 4 2 3 4 5 3 4 5 6 4 5 6 7 etc, the mnemonist engaged in his customary intense effort of concentration before proceeding to recall the entire series. He did not see the pattern in the numbers. Most of us non-mnemonists, however, have to rely on some kind of narrative or story related to the "meaning" of the steps in the proof in order to recall it. For example, on the higher-level course, you might be asked to provide a proof that cos(A-B)=cosAcosB+sinAsinB. I try to get my students to associate the following items with this proof: The diagram (a great way to condense information).

Get | pq | using coordinate geometry.

Get | pq | using the Cosine Rule.

Set the two versions of | pq | equal to each other.

Get the required result.

In order to be able to use this kind of structure, you must have mastered the basic skills referred to in the "story" of the proof. The detailed outline of the mathematics exams which follows here frequently gives lists of the basic skills and concepts which you must have mastered as a kind of tool kit with which to tackle exam questions, as well as mentioning specific dos and don'ts which have turned up in chief examiner's reports.

Preparing for the exam, higher-level students should make sure the following points form part of their revision programme: handling fractions; factorising; handling inequalities; getting the square root of a complex number; recognising a recurring decimal as a geometric series; proving the product rule from first principles; differentiating inverse trig functions; applying differentiation to problems involving parallel tangents; finding the co-ords of the max point of a log function; applying integration to area problems; evaluating limits; reducing surds to simplest form; squaring algebraic expressions.

Ordinary-level students should make sure that they know all the formulae for co-ordinate geometry; sequences and series; roots of a quadratic; sin, cos, and tan ratios in a triangle.

Foundation-level students should practice rounding off answers to given numbers of decimal places and significant figures.

Setting Out Your Work

- Start each new question at the beginning of a new page.

- Do not write in pencil.

- Show all your work.

- Careful with those calculators.

- Write down the sum before you do it on the calculator.

- If the calculation is very complicated, break it up into steps.

- Check your work!

- Get someone to explain what the DRG button does.

- Use a calculator you are familiar with in the exam.

Good luck!

• Dr John Evans sometimes remembers to teach maths and applied maths at Mount Temple Comprehensive School in Dublin. He can be contacted at evans@mimesis.net