Successive British governments have assumed that learning simple arithmetic is a prerequisite for mathematical understanding. The underlying idea seems to be that once "fluency" in arithmetic facts is achieved, understanding follows automatically. Documentation sent to teachers about the National Numeracy Strategy states: "An ability to calculate mentally lies at the heart of numeracy." Almost all the advice to teachers of 5- to 11-year-olds concerns methods of calculation. Our latest research at University College London, however, shows that even students whom our educational system has turned into excellent calculators do not understand the principles of basic mathematics.
Most people can multiply 8 by 7 to get 56. They can add 3 to 2, and subtract 2 from 4. You probably believe that you understand how the four operations combine, how multiplication combines with addition and subtraction. You know that 8+3 multiplied by 7+2 is more than 8 times 7. You know that because you believe that if you add a number to each side then the product will be larger. Similarly, you know that 8-3 times 7-2 is less. You don't need to work that out, either. You will also be confident that working out the products (99 and 25) will confirm your deductions. But do you understand the principles on which such a deduction is based? Try to solve the following problem: Is 8+3 multiplied by 7-2 greater or less than 8-2 multiplied by 7+3?
Did you carry out the calculation or solve it from first principles? Most people taught in the traditional way, like you, will probably solve it by calculating the exact solution for the two parts and then comparing them. Given that the numbers are small, this makes crunching your way to the solution easy. Now try the next question, one from a set of problems we gave to 42 university students. We asked students not to calculate the exact solution but to solve it by reasoning, and we made the numbers large enough to make it relatively difficult. In the test, they had to select one of four possible answers.
Paul's garden is 36 ft by 170 ft.
Anna's garden is 46 ft by 160 ft. 1 Paul has the larger garden
2 Anna has the larger garden
3 Their gardens are equally large
4 Without calculation one can't know the answer
All the students would have been able to solve the problem by calculating the areas of the two gardens and comparing them. This would have meant they could remember and could use the recipe for long multiplication, but it would have told us little about their understanding of the relationship between multiplying, adding and subtraction.
We set 24 questions as problems about area (as in the example), as problems about money, as problems about arrays (eg rows of seats in a theatre), also bare arithmetical problems with no context. One of the things that we wished to find out was whether well-educated subjects are affected by the way in which the question is worded, or whether they are able to extract the mathematical bones and work on them.
All our subjects were university students with maths GCSE grade C or better as an entrance requirement, and they were divided into two groups: those who were taking science degrees and had taken A-level maths, and at least one university-level maths course, and those taking arts degrees.
The arts students scored just 38 per cent correct. Picking an answer at random would have achieved 25 per cent, but they weren't guessing. Overwhelmingly they picked answer 3 (both gardens of equal size). We call this the "false compensation" error. The structure of these problems, as you have probably noticed is this: Paul's garden is 36 x (160+10), while Anna's garden is (36+10) x160.
If this were an addition problem, (36 + (160+10) vs (36+10)+160) then the two expressions would be equivalent. The confusion between the laws that apply to addition and those that apply to multiplication was evident from the explanations they offered after the test. For example: "The differences between two sets of multiples averages out for the whole sum." Only one arts subject offered a correct explanation of the structure of the problem.
But since they are multiplications, the two areas are different: Paul's garden is (36x160)+(36x10), and Anna's is (36x160)+(160x10), and it's easy to see that Anna's is the larger.
The other striking finding was that arts students were deeply affected by the way the problem was phrased, scoring 54 per cent on money problems, but only 30 per cent on area problems.
The science students, as you may expect, did much better. Their maths training meant they were able to disregard the wording of the problems and get to the maths, and they didn't fall into the false compensation trap more than either of the two other errors (picking Paul's garden or saying that you needed to calculate). Despite this, they still only scored 60 per cent on average, and only one-third of them was able to offer a correct explanation.
Perhaps the most depressing finding was this. Each of the subjects attempted 24 of these problems, and you may have thought that after they had seen the trick in one of them, they would have little trouble with the remaining problems. However, only three of the subjects showed any improvement in the course of the experiment. The rest showed no evidence of learning from the previous problems, nor of applying the correct principles that formed their explanations: they seemed completely rigid in their approach.
The subjects in this experiment were all educated in British schools, and had reached university. The problems we set them did not involve any advanced maths or complex calculations, just the basic principles to a problem that has numbers but doesn't involve number crunching, may reflect our national obsession with endless drills of numerical facts and numerical recipes from the age of 5 (or even 4, if Mr Blunkett has his way). If they had been taught to use their heads instead of their pencils, perhaps they would have responded more flexibly and intelligently to problems that can be solved without calculation.
We need to re-think our approach to mathematics education and get away from drills; after all, what 10- year-old needs to know how to do long multiplication outside of school? Above all, we need to encourage children to think about the logic, and also about the beauty, of numbers.
Brian Butterworth is Professor of Cognitive Neuropsychology at University College London. His book, 'The Mathematical Brain', Macmillan, £12, is published in paperback this monthReuse content