Circling the spiral
One of the important ideas attributed to the psychologist Jerome Bruner is the notion of a "spiral curriculum". In this model important concepts within a discipline are visited and revisited at different levels of sophistication during a child's schooling. The learner begins at the bottom of the helix and is gradually helped by teachers to progress around and up.
A similar helical structure could be used to model the educational debate in this country. Simply replace the subject-specific concepts on Bruner's spiral with a selection of the following issues: mixed-ability teaching, comprehensive schools, the 11-plus, Plowden, calculators, group work, A-levels, mental arithmetic, real books, discovery methods, testing, and more. However, a much more appropriate model in this case would be that of a circle rather than a spiral. Topics are revisited but each time we return to a subject, the level of sophistication of the debate remains the same.
One mathematics issue which has featured quite prominently in the press over the past few months is "the calculator". Inevitably, the views expressed are polarised, and as ever, the "truth" can probably be found somewhere in the middle. The following paragraphs juxtapose various viewpoints gleaned from recent press cuttings and from official publications.
The Government's chief curriculum adviser, Nick Tate, warns us (The Times 7.12.95) that calculators are being overused in primary schools, whereas a recent Office for Standards in Education publication on the teaching and learning of number in primary schools berates teachers for neglecting calculator skills, and informs us theyt were being used in only one in 10 lessons observed by inspectors.
We have a Cambridge don writing an article entitled "When Numbers Count" on page 25 of the Daily Telegraph (1.11.95) - despite the fact that an earlier page refers readers to a non-existent article on page 24 entitled "Why Numbers Count"! In this piece John Casey - a "self-confessed maths duffer" - argues that we are educating a whole generation of children who are growing up to be totally dependent on their calculators. The OFSTED publication Recent Research in Mathematics Education 5-16 contradicts this view and argues quite forcibly that open access to calculators does not lead to dependence.
Incidentally, I do wonder whether mathematician Joe Bloggs, a self-confessed English duffer, would ever be afforded the opportunity to speak about the quality of English teaching.
Science teacher, James William (TES 29.12.95), argues that reliance on calculators has rendered children poorer at deploying the mental skills of estimation and approximation. However, Peter Reynolds (TES 2.2.96) reminds us of the international acclaim accorded to the Calculator Aware Number curriculum project which found, among other things, that a group of 116 project pupils who took a standardised mathematics test for eight-year-olds out-performed or at least matched a similar-sized non-CAN group of children on a majority of the test items.
Primary teachers are advised by concerned parent Madeleine McDonald (TES 15.12.95) to ban the use of calculators in primary classrooms. She feels their use does not allow children to build up a commonsense approach to numbers. Unfortunately, these teachers are obliged by the national curriculum to teach key stage one children how to use a calculator both as a means for exploring numbers and as a tool for calculating with realistic data.
The arguments of those against using calculators usually stem from ignorance. The case appears to be premised on a limited view of children sitting at their desks using calculators to work through pages of "sums" day after day. If that were indeed a true picture, then almost all of us involved in mathematics education would have reservations about their use.
It is also the case that the pro-calculator lobby tends to exaggerate the positive aspects, quoting specific case studies, but not necessaily ones we can make generalisations from, and choosing to ignore conflicting evidence. It is simply not true that using calculators "will stop children using their brains". Nor is it the case that allowing free access to calculators will automatically lead to the development of number skills and concepts. All those involved need to accept that there is some truth in both sides of the argument. We need to make a case for calculator use which capitalises on the strengths and strives to eliminate weaknesses.
How should this case be made? The first irrefutable point is that calculators are here to stay, and they will continue to get cheaper and more powerful. Consequently, attempting to ban them would appear to be neither a sensible nor a plausible way forward. If we look both at the research literature and the many publications containing practical classroom ideas we will find excellent examples of calculator activities which can help children develop a better understanding of their number bonds, multiplication facts, relationships between mathematical operations as well as a more thorough understanding of place value.
Having used calculators with schoolchildren, student teachers and primary school teachers for about 20 years I have no doubts about the positive contribution calculators can make to the teaching of mathematics. The fact that the calculator provides instant feedback facilitates the development of "trial and improvement" strategies in a problem such as "find two numbers which when added together make 29 and when multiplied together make 204." The calculator display provides useful, non-threatening information to help the children decide whether they are right or wrong: there is no need to wait for the busy teacher to get round to the child's desk before he or she moves on to the next stage of the solution.
People who argue strongly for the use of calculators have to face the fact that if children's sole experience of using the instrument is as a device for finding solutions to arithmetical calculations then they are unlikely to develop their number awareness given this limited experience. There is some research and much anecdotal evidence to suggest that children given unrestricted access to calculators come to rely on them for calculations they could more quickly work out in their heads.
The implications appear to be that we need to spend much more time, as they do in other European countries, on developing children's mental calculation strategies. We also need to develop the confidence of pupils so that they are able to decide when and if to use a calculator. There is evidence to suggest that children believe the calculator to be infallible; that they reveal a lack of awareness of potential keying errors, and that they often write down all eight digits that appear in the display after a division calculation. We obviously need to spend more time on the concept of accuracy.
I feel very strongly that calculators can make a valuable contribution to the teaching and learning of mathematics, but I also feel that using them can have adverse effects which need to be addressed. We must put greater emphasis on certain skills which we currently simply pay lip-service to. Among these skills I would expect to find estimating, approximating, checking, rounding, decision making and, most importantly, a recognition of the need to deploy these skills and an awareness of when to use them. These are not trivial skills, nor are they particularly easy to teach. I am afraid that making calculators freely available for children's use does not make maths teaching easier!
Ian Thompson is a lecturer in the department of education at The University of Newcastle-upon-Tyne