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Mark Ellse: A grasp of arithmetic is vital to mastering advanced maths


In my last post I explained how ‘raising standards’ advocates have seriously damaged mathematics education for those who find maths difficult. Here we shall see how those same laudable but misguided views have not helped the maths education for the most able mathematicians.

For historical reasons, mathematics has been taught in English and Welsh schools as a single subject comprising arithmetic, algebra and geometry. In other countries, arithmetic is taught and examined separately. In Scotland, O grade arithmetic was examined separately until 1986. Then Scotland, too, lost its separate arithmetic exam.

It is interesting to compare arithmetic with other mathematical skills. Certainly moving from arithmetic to algebra is a conceptual step up. Almost all adults are comfortable with calculating 7 x 7; but many would regard solving, for instance, the equation x² = 49, or using sines and cosines to calculate an unknown angle, as much harder problems. With very good reason, arithmetic has gained the reputation of being easier than algebra or trigonometry, as indeed it is.

But, despite being easy, arithmetic is the foundation for all other mathematics. If one wants to solve any algebraic or trigonometric problem, one needs arithmetic. If one wants to multiply out (8x – 5)(7x – 9), one needs to be able to multiply 8 by 7, 8 by 9, 5 by 7 and 5 by 9. The algebra needs a bit of care but careful arithmetic is also necessary.

In response to the ‘raising standards brigade’, GCSE maths exams must be made ‘hard’. So the Higher GCSE papers, those designed for the students getting A* down to C grades, are largely devoid of simple arithmetic questions. Of course arithmetic is tested in these papers, but it isn’t their focus. The result is that GCSE maths tests some mix of skills, of which arithmetic is only a part. When a child gets a grade B in a maths GCSE, one cannot tell whether the child’s problem was not understanding the hard mathematical topics, or merely not being able to do the arithmetic accurately enough to get the right answer.

The influence of the Raising Standards Brigade on the National Curriculum has been profound. Key Stage 3 for mathematics specifies, amongst other things, that pupils must be taught to simplify and manipulate algebraic expressions to maintain equivalence by: collecting like terms, multiplying a single term over a bracket, taking out common factors and expanding products of two or more binomials. I suspect that the word mathematics in the first sentence, or perhaps algebra in the second paragraph may mean that only those comfortable with maths have read this far. It may well be that all those who remain with me can indeed expand the products of two or more binomials. But these are certainly rather hard skills which, according to government diktat, should be taught to every pupil before the age of 14. Of course it’s potty to teach these subjects to all children. Children should do arithmetic first so that their basic numeracy is good. Only then should they proceed to the more conceptual topics.

In the real world maths mainly means arithmetic. The Raising Standards Brigade have been obsessed with stuffing GCSE with ‘hard topics’. They have marginalised the essential core skill of arithmetic which has resulted in our children going out into the world, or into the sixth form, without fluent and accurate arithmetic. Scotland, a country in which arithmetic used to have prominence, has seen directly the error of dropping the separate arithmetic exam.

My own school, being independent, was free from the National Curriculum. We insisted that pupils pass Foundation GCSE Maths, the exam designed for weaker pupils at GCSE, before teaching them any of the harder mathematics content. The current Foundation Maths GCSE, before Michael Gove’s tinkering, has been an excellent test of basic arithmetic skills. To get a grade C in Foundation Maths candidates needed to get 80 per cent of their arithmetic right. If you tell good pupils that you expect 100 per cent, many will achieve it. Most children can take Foundation Maths very early. My own school found that all but a handful could pass the exam at the age of 13. For them it was a tremendous boost. Not only did they then have their first GCSE at 13, thereafter they had strong arithmetical skills which meant that they progressed much more rapidly. After Foundation Maths, our good mathematicians romped through higher maths at the age of 14 and then took A level maths alongside their GCSEs at 16. The mathematical performance of the very best mathematicians is very significantly improved by doing the basics very thoroughly and mastering these before going on further.

Girls in particular benefit from a slow and steady approach to maths. Girls are more likely than boys to worry if they don’t understand something and are much more reluctant to ask for help. When they get lost in maths, they often stay lost. One particular girl, from a local state school had, at the age of 13, a burning ambition to do well academically. She had been told by her school that she would never pass GCSE maths. She approached my own (independent) school requesting a scholarship place. When she joined us she was regarded by us as a remedial mathematician because most of her peers in our school had already passed Foundation Maths. She was given a diet of the simple, mainly arithmetic, Foundation GCSE Maths. She could understand that, of course, and passed Foundation Maths at the age of 14. It was a tremendous boost to her confidence. Higher maths followed at the age of 15. When she took A level maths she was only a few marks away from an A*. She is now studying veterinary medicine at university.

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Mark Ellse
Mark Ellse
Mark Ellse is a physicist and author. He is a former headmaster, independent school inspector and A level chief examiner.

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