Calculated advantage

Paul Lukas demonstrates the best way to use advanced calculators without destroying basic numeracy

Graphing and algebraic calculators offer virtually limitless opportunities to take school maths much further than previous generations thought possible. Yet neither of these types of calculator is in widespread use in schools nor allowed in all examinations. Even more surprisingly, many universities restrict their use. Why is this so when the personal computer is not generally considered controversial?

The scientific calculator was the first readily available electronic aid in the classroom. This plastic box quickly became a familiar classroom companion and as a result a generation of children were able to tackle, for example, trigonometry questions without performing logarithmic computations at the same time. Pupils tackled more trigonometry, but did less arithmetic. Generally, mathematics gained but computation lost out.

If generalisation is possible, it could be said that while many maths teachers and pupils were content with the delegation of "sums" to their scientific servants, parents and politicians were less happy and perceived a reduction in numerical competence.

Even as scientific calculators became more advanced, the types of questions asked in textbooks and examination papers did not change significantly. Numbers used in newer books were supposed to be more "realistic", but calculators were used somewhat indiscriminately; the mathematical notion of exactness more or less disappeared as the degree of accuracy of data, working out, and answers became a more topical issue.

With the wisdom of hindsight, what was needed was a more sophisticated approach which took account of our newly acquired assistants. Whereas it is appropriate to do most problems in trigonometry with a calculator to hand, it is not as appropriate to do most of the arithmetic, in most of the problems, with most other topics, on a calculator.

When it became clear that children had lost numerical fluency because they were not doing much arithmetic, we introduced somewhat artificial numeracy units; this is reminiscent of the notion of being encouraged to take vitamin pills to supplement a diet lacking in essential nutrients.

It is useful to think of scaffolding. It is a temporary structure that is removed once a building is completed.

We could use a similar idea in maths: use the calculator to bypass the arithmetic while learning new ideas but remove the machine and use sensible numbers for a good proportion of the consolidation exercises.

But most of our books assumed that the exercises would be done with the help of a calculator, and this is the way we ended up doing mathematics most of the time.

Of course, in trigonometry our practice was appropriate and here we should perhaps accept that our use of the technology is a permanent and largely suitable "bridge" over the arithmetic.

In the same way as the scientific calculator quickly became a classroom tool, we are likely to see graphing and algebraic calculators similarly becoming an everyday tool.

Much has been written about their enormous potential in teaching, but what happens when our pupils settle down, armed with the equivalent of a mathematical Exocet, to do the next 10 questions in the book?

How do we avoid loss of skills when the calculator can reduce the technical part of each question to a mere button pushing exercise?

We need two things: a curriculum model, and then suitable questions to fit the model. The following model, tried with pupils, is a starting point and can be used to construct units of work using an advanced calculator.

* Calculator free: non-calculator sections; work to develop numerical competency; consolidation; work where the numerical component is not such that important concepts are obfuscated by computations.

* Calculator friendly: work in which a calculator is obviously useful. This may be numerical, graphical, or experimental, and could be done by traditional methods but would be tedious and time consuming.

* Calculator focused: work in which a sophisticated calculator makes certain kinds of problem approachable which would otherwise be beyond reach. This is technology dependent work, and encourages its creative use.

There is something for everyone here, though the hard work begins in constructing suitable questions. In discussing advanced calculators in the classroom, we should consider "how" and not "whether". The question we should ask is "what questions should we ask?"

Paul Lukas is head of the secondary mathematics department at Vienna International School Email: Casio runs a range of free and paid for courses on the use of graphical calculators in the classroom.Tel: 020 8208

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