Bless the hated calculating machine
One of the biggest controversies surrounding the attempt by the School Curriculum and Assessment Authority to revise the subject core for AS and A-level mathematics has concerned the proposal that calculators should be barred from some element of the assessment.
The debate has been confused by the fact that the sort of calculator was not made explicit in the proposals. By next year, the earliest that examinations with a new core could take place, it is likely that significant numbers of A-level students will be using extremely powerful calculators such as the Texas Instruments TI92, which came on to the market early last year.
In a few years, such machines, along with their even more sophisticated successors, will be within the reach of all students. This poses a challenge to those who design and assess A-level courses. Is the availability of such immense calculating power a threat, or an opportunity to the world of mathematical education?
Some see it simply as a threat. They say that such machines have no place in a mathematics course at this level, or that they should be used only after some basic skills and understanding have been acquired.
But I would suggest that students will still buy these calculators and use them to do their homework whatever happens in the classroom. If TI92s are banned from the classroom, the credibility of sixth-form mathematics will be reduced in students' eyes as they see their teachers ignoring what to them is an obvious technological aid.
Furthermore, if we fail to help students to use TI92s intelligently, they will become dependent on them in the way that many students have become reliant on more simple calculators. We rightly deplore a situation where good students resort to a calculator even to work out something like 12 x 2.5, because we feel that this should be so familiar to them that it is quicker to do it mentally. The same sort of feeling will arise when we find students resorting to their TI92s to expand (x + 1) squared or to factorise x-squared - 4. The danger is that mental laziness will undermine fluency and hinder understanding of vital ideas and skills.
Calculators have two roles in learning and doing mathematics. The first is the obvious one as a labour-saving tool for handling awkward calculations and mani-pulations.
These often distract from the main task in problem-solving of thinking out the steps that are required to find a solution, which is usually a harder task than actually performing the calculations. A calculator, no matter how sophisticated, does not decide the steps for you - it simply does the routine mechanical parts. But these new calculators provide an opportunity for releasing our energies to concentrate on the challenge of problem-solving. We should seize it with relish.
The second role of the calculator is as a learning tool for exploring new ideas and relationships. For instance, suitably chosen expressions can be transformed in ways that are surprising to the novice and used to develop understanding. The figures on the right shows some examples of this.
As far as assessment is concerned, there may well be a good case for an element in which calculators like the TI92 are not allowed. But that will not in itself ensure that students learn to use their calculators intelligently, nor will it prevent them becoming dependent upon them in inappropriate ways.
Textbook writers, examiners and teachers need to seize the opportunity to seek imaginative ways of using these new tools to extend students' understanding and skills, and to help them use their knowledge in problem-solving.
Doug French is a lecturer in education at the University of Hull
HOW A POWERFUL CALCULATOR CAN HELP EXTEND UNDERSTANDING
Fig 1 shows what happens when three particular algebraic functions are entered on a TI92. Many students have difficulty when simplifying such fractions. A request to "explain what is happening here" and to "find some more examples which behave in the same ways" may be a useful complement to the usual routine exercises.
Fig 2 When a TI92 is operating in approximate mode x, it shows a set of uninteresting decimal values when the cosines of various angles in degrees are evaluated.
Fig 3 shows the same examples when the calculator is operating in exact mode. This time the same set of similar-looking expressions give results that differ greatly and look much more interesting. There are plenty of other examples of how a TI92 can be used to generate discussion across the whole range of A-level maths topics.