When reality confronts ritual
Routine practice based on exam questions won't inspire A-level students, says Hugh Burkhardt. Life is going to demand further skills in self-directed approaches to problem solving.
These two new series make you wonder where A-level maths is heading over the next few years. Will it remain untouched by time, despite the changing backgrounds of the students and needs of a changing population? Is it going to follow the GCSE into the mainstream of development of mathematical education worldwide, with its emphasis on people learning to use maths more powerfully and flexibly? Or will it be the next political football - a domain for unsupported controversy, and for imposing the untested opinions of politicians and populists on long-suffering teachers and their students?
The authors and publishers of these two series had some unenviable choices. They have gone firmly for the safe short-term solution. The books are exclusively focused on the current syllabuses of the examination boards, which are close to those I took in the 1950s and saw through my children's eyes in the 1970s. Indeed the texts are overwhelmingly collections of imitative exercises based on the examination questions.
There is remarkably little explanation of the methods, let alone the concepts behind them - the briefest of outlines leads straight into the illustrative examples and the exercises. This is an old tradition that goes back over half a century. The teacher will explain; the book is for the exercises. Its success depends on students having good teachers, who know their stuff. They also have to know, not just how to "get it over" but, much more difficult, how to enable their students to learn it effectively. With 35 per cent of the age group now heading for university, this means A-level courses now need to serve a much wider range of students.
The learning model here is "practice makes perfect" - it seems reasonable, at least as long as "they" don't ask you to tackle problems not quite like those you have seen. Since "they" won't until after you have your A-level results, this may seem a reasonable approach. It is certainly the standard one. In fact, research suggests that it does not work very well. Why? Essentially because people do not long remember exactly how to do things that they have practised - they remember roughly what to do. Effective performers have the skill to "debug" their own procedures, checking and correcting themselves as they go along. (To get a feeling for this phenomenon, try working out the procedure for extracting the square root of a number from the following hints - set it out like a long division sum, but "bring down" two digits at once, and it must involve (a+b)2 in some way. This is the state most examinees are in when trying to handle the range of demands at A level).
Because, let us be clear, A-level mathematics is no joke. A reasonably able person may cope with GCSE on five years' osmosis in class and a couple of months of hard revision, but A level is two years' hard work, even for good students. These books provide the grist for that mill. The class that works through them diligently will have practised each type of problem many times, checking their answers and having their mistakes corrected by the teacher. They will meet no surprises in the exams. They will not have received much encouragement to think about what they are doing - to be aware of "the tools in their mathematical toolbox", their uses, strengths and limitations.
There is one partial exception - each of the books on applied maths (mechanics and statistics) begins with a section on mathematical modelling, which describes how the mathematics is used to understand better the practical situation being "modelled". This is the essence of "mathematical power" over the real world, whether it be estimating the financial value of a university education, putting an astronaut on the moon or understanding why music keyboards are not perfectly "in tune". But in these books once the ritual chapter is over there is no further reference to the modelling process, crucial though it is in understanding the what and the why of the other problems that fill the book. A key opportunity is missed to deepen students' understanding and, crucially, to help them over their difficulties.
The distinctions between the two series are less important than their similarities, and less significant than the variations within each. On the whole, I preferred the explanations in Heinemann Modular Mathematics, where the experienced examiner-authors have also ensured a very close grounding in the syllabuses. Perhaps because the National Extension College is more aware of students who are working with limited or no teaching, the Advanced Modular Mathematics authors provide rather better help with the challenges of learning. But these books, too, are basically collections of practice exercises. Some of them have nice twists of context or of detail that help to lift the spirits, but it is basically the standard stuff that has ignited the interest of so few.
The effects on students who have just emerged from a good GCSE will be unfortunate. They have to move from real achievements on substantial pieces of self-directed work, and less-routine exam questions that require them to think how to tackle the problem, back to technically difficult routine imitative exercises.
We must hope we shall move to assessment that is better balanced between strategic and technical demands, even if it means universities have to revise their first-year courses. They will have to anyway, for several reasons - the larger cohort, the reduced popularity of science and engineering among the ablest students and, particularly, the revolution that has come from computers in the way maths is actually done and understood in the world outside education. This technology is a boon - it offers more power with less pain to more students. But we shall need to develop good courses at all levels that seize the opportunity in a way that is accessible to current teachers - a major task. Until that is done, we are likely to continue with books and syllabuses like these.
Hugh Burkhardt is currently director of the Balanced Assessment Project based at the University of Nottingham and California at Berkeley.