In 1986 Hilary Shuard gave her presidential address to the Mathematical Association conference the title "Primary Mathematics Towards 2000". How much of her vision has come about? At the time she was also director of the Schools' Curriculum and Development Council PrIME (Primary Initiative in Mathematics Education) Project, a large part of which was devoted to the development of the CAN (calculator aware number) curriculum. So it is hardly surprising that much of her address questioned the need for continued emphasis on the teaching of written calculation methods:
"As we move towards the year 2000, the needed rethinking of the primary mathematics curriculum must take account of the fact that digital watches, calculators and computers ... are familiar tools to very many primary children ... Nobody any longer needs to learn the pencil-and-paper algorithms for computation of the four rules for use."
In 1987 she was succeeded as president of the MA by Anita Straker, who in her presidential address was even more adamant that the dominance of written methods needed to be questioned (published by the MA in an abridged version in 1987 as The Challenge to Change). She compared the use of a calculator in maths lessons with the use of an electric iron in what was then known as "home economics". She argued that if something goes wrong with an electric iron pupils are not taught how to use a "flat iron" but instead to "do the sensible thing, to borrow an iron or to go and buy a new one, or to put off ironing until the iron is mended. It is possible to calculate manually, using a pencil and paper method. The method is slow, and prone to error, because the techniques are difficult. For a very long time we have had aids to help us to calculate, ranging from the abacus, to Napier's rods, to logarithm tables. We now have two others: the electronic calculator and the microcomputer". So, she asked, "why do we continue with the 'flat iron' curriculum in mathematics?" It may be that Hilary Shuard saw written calculations as having a role in helping children understand the number system. This same attitude can now be detected in some of the approaches advocated towards written methods in the National Numeracy Strategy. However, her emphasis on not needing them "for use" is consistent with seeing calculators as the main calculating tool. It is unfortunate that this view was not encouraged by Department for Education and Employment statements at the launch of the National Numeracy Strategy (under the directorship of Anita Straker) suggesting, as a cornerstone of the strategy, the banning of calculators.
A second strand of Hilary Shuard's address in 1986 drew on research that showed the dominance of a "transmission" approach to teaching maths, where children were seen as "empty vessels" that the teacher must fill. Her suggested approach was to base maths lessons much more on discussion.
The numeracy strategy's emphasis on more oral maths and a large amount of interactive questions and answers would certainly seem to be in line with this view, but we need to research how teachers manage this in practice. At King's College we are in the fourth year of the Leverhulme Numeracy Research Programme, a five-year longitudinal study of teaching and learning primary maths. In many lessons we have seen teachers encouraging pupils to explain and share with each other methods for carrying out calculations. Usually these discussions come about when pupils have succeeded in reaching a right answer. It is interesting to observe that when a substantial nmber of children have got the answer wrong, teachers often resort to showing a method of solution, rather than exploring how the different answers were arrived at and discussing which must be correct. As Hilary Shuard pointed out: "Research on discussion in the primary maths classroom shows just how difficult many teachers find it to conduct true discussion."
Her emphasis on not treating children as "empty vessels" was mirrored in her attitude towards teachers. While the PrIME project is probably best remembered for the CAN curriculum strand, we should not forget the wealth of work carried out by teacher groups co-operating throughout the country. These groups came together to work on developing "their teaching styles towards incorporating more discussion, practical work, problem-solving and investigational work, and to make full use of the new technology - especially computers and calculators - to support these teaching styles", she said.
The then National Curriculum Council published a set of training materials so that schools could explore such issues in similar ways. This "bottom-up" approach to teacher development stands in contrast to the "top down" model of the numeracy strategy. While a top-down approach is possibly unavoidable in a nationwide initiative, we need to be wary of overtones of teachers being "empty vessels" when it comes to maths teaching, since there is a danger of this being carried over into their approaches with pupils.
Of course the most dramatic change in primary maths since 1986 has been in the increased prescription of the curriculum content. At that time, apart from an increased emphasis on the use of calculators, Hilary Shuard had little to say about the curriculum content. From the perspective of the numeracy strategy year-by-year framework for teaching maths, it is interesting to look back at Anita Straker's views in 1987 on an imminent national curriculum: "A national syllabus of defined age-related objectives based on ability to 'do sums' would only inhibit and constrain curriculum development in a field which constantly needs to move forwards."
So what have we gained over the past 15 years? It certainly seems that pupils' mental strategies, particularly in addition and subtraction, are improving greatly and many children and teachers are much more willing to treat maths as a subject to be talked about. Maths is becoming much more acceptable as a subject to admit to enjoy or even be good at. Research into effective teachers of numeracy carried out before the introduction of the numeracy strategy indicated that teaching was more effective in schools where the staff discussed their maths, and the strategy is encouraging such discussion. (Effective Teachers of Numeracy: report of a study carried out for the teacher training agency, by Mike Askew et al, Kings College London 1997.) Looking back over Hilary Shuard's and Anita Straker's addresses makes me wary of crystal-ball gazing but there appear to be aspects of maths teaching that we need to continue to work on. In particular, we still need to focus on pupils' applying their understanding to non-routine problems and the development of mental strategies for multiplication and division are two areas.
Most importantly, we need to keep the balance between attending to teaching and to learning and so, in Hilary Shuard's words, help children "realise their potential as active mathematical thinkers in the primary years".
* This article is based on the Hilary Shuard Memorial Lecture given to the Mathematical Association conference, Easter 2001 Dr Mike Askew is lecturer in mathematics education at King's College LondonE-mail: firstname.lastname@example.org