# Calculated progress

Those of us actually teaching in the classroom recognise some of the pitfalls, as well as the advantages of calculator use. Just two examples illustrate where over-reliance on a calculator may be a serious drawback to understanding. First, in any conceptual understanding of rational and irrational numbers, it is essential for students to do some long division for themselves, so that they can recognise the character of such numbers and the distinction between them. Second, maths teachers, especially those working with better students, need to be concerned at the all-too-ready use of a calculator to obtain an approximate result when an exact result would be more appealing and more useful, as well as a failure to understand the difference between the two.

But it is a mistake to think that maths should stand still while technology progresses. Calculators able to draw graphs are readily available, soon they will be capable of manipulative algebra, including much of A-level calculus. They are getting cheaper all the time - we cannot ignore their presence, nor can we stop our students using them when doing mathematics.

More importantly, the calculator can be exploited to good effect. Take for example the 16-19 Mathematics course developed by the School Mathematics Project. The programmable graph plotting calculator is integrated fully into the development of the course, and this, combined with the use of spreadsheets and other technology, makes a challenging diet for sixth-form mathematics students.

When teaching the Newton Raphson method for the solution of equations for example, we are able to do so in a much more interesting way than we did some years ago. Students develop the theory as ever, then "number crunch" some solutions by hand. They are then able to program calculators to solve the equation, and finally set up a spreadsheet programme to solve it.

This provides for a much better treatment of the topic than we were able to give before. It is more interesting for the students, teaches a number of extremely valuable and useful skills, and generally encourages a great deal of worthwhile mathematical activity. The same is true for numerical methods in integration, and the 16-19 Mathematics course has developed a whole new approach to the initial development of the calculus directly as a consequence of the graphical calculator.

Naturally other aspects of the conventional school mathematics programme are treated in a traditionally rigorous way. In the 16-19 course, for example, there is a whole text on complex numbers and two full books on differential equations, with calculators and computers playing a significant role in both. A second unit on differential equations explores the modelling aspect of mathematics, now an important part of the new A-level common core. The complex numbers unit covers all of the traditional material that we have been teaching for years - it simply does it in a rather more exciting way (it is also very demanding).

The final chapter, entitled "Towards Chaos", has the student considering orbits and Julia Sets. Many students are able to write programmes to draw the Mandelbrot Set, and various Julia Sets, on the Casio FX 7000 calculator! Others have produced excellent coursework as part of their A-level studies, including a computer simulation of the movement of the planets, investigations of projectile motion.

The more traditional and obviously important topics and ideas are certainly not overlooked. For example, the 16-19 text on Mathematical Structure provides an excellent starting point for students to consider the special requirement in mathematics for rigour and proof.

An area where calculators have had a major impact on maths in the sixth form is in graph plotting. Students still need to be familiar with basic curve sketching techniques and to recognise the shape of the graphs of common functions or they simply will not recognise when their calculator, or perhaps they themselves, have made a mess of things. Classic examples are joining the separate parts of a graph across a discontinuity, or getting completely the wrong shape to a graph because the calculator is in the wrong range or mode. Students must be able to recognise these shortcomings for themselves.

Many of our critics would suggest that these skills are no longer taught in schools, while we know of course that they are. The difference now is that they are less practised than they once were, as students have a ready alternative to hand to do the work for them. So it is always worthwhile having sessions in which calculators are forbidden, to ensure that basic skills are practised. But we cannot blame calculators for students not being able to curve sketch. We must teach them to use the calculator as a ladder rather than a prop.

Chris Belsom is the head of maths at Ampleforth College, a chief examiner in A-level maths, and a former GCSE chief examiner