In between, there is the plethora of topics which have been developed to use within mathematics itself to explore new mathematics, and also to use in science, engineering, economics, and so on.
At the school level, we would be thinking of trigonometry, solving equations, calculus, geometry, matrices, for instance. Some of these are called applied mathematics at university, though they are often called pure mathematics in schools. There is no clear cut line between pure mathematics and applied mathematics; but in both cases, mathematics is characterised by its own internal logic and the language and notation used to express this logic. The power and compactness of this language and notation is one of the problems for the student and a part of its compelling attraction for mathematics teachers and mathematicians.
I want to draw a distinction between the development of numeracy and spatial awareness, and the rest of mathematics. The development of the former is essential for everyone and is not mathematical; its aims are connected with a pupil's ability to function in a technological society and such a course could almost be described as a life-skills course.
These aims are rather different from the aims of teaching the rest of mathematics where we are (for 11 to16-year-olds) at the beginnings of a subject which teaches the foundations of logical thought and which gives the pupils a firm foundation for further study.
It is the foundations of logical thought that are so important for the individual; mathematics offers pupils the chance to exercise their powers of deductive thinking, albeit at an elementary level, and demonstrates the importance of precision of thought and expression.
All pupils should be exposed to this in varying degrees depending on their aptitudes, and not just the high fliers. At least 50 per cent and probably even 70 per cent can benefit. To deny children this opportunity to try out their powers of deductive thinking is to deny them an important element of their education. Mathematics contributes to their developing intellect in a way that no other subject does.
How can we include deductive thinking in our courses? Even this question itself is a sad reflection on what has happened to the course that is known as GCSE mathematics. How can we envisage a subject called mathematics that does not include some deductive thinking? But in a recent higher-level GCSE paper, I found just one question out of 45 that was multi-step and therefore required the pupil to work deductively from one step to the next. All the other questions were "one-liners" where the answers are either "known" or "not known" (see example below left).
It seems that we are making assumptions about children's abilities that are totally false. It is not true that children cannot cope with deductive thinking; some children are quite capable of handling multi-step problems and many more would be successful given the right encouragement, so we need to develop these problem-solving powers.
All children need the chance to try to develop these powers of deduction otherwise we are denying them an important part of their education.
In summary, we have two distinct aims in mathematical education: * developing numeracy and spatial awareness, * teaching mathematics as a subject in its own right.
In addition, mathematics must include more opportunities for deductive thinking and multi-step problems for everyone. Our mathematical aims are very important for the success of the course. They need to be focused and clear.
I believe we are failing in the second aim because we are trying to do too much in the same course and at the same time. GCSE mathematics tries to be a numeracy course for all at the same time as a mathematics course for the more able. I suggest that we should offer two GCSEs in mathematics, one principally about numeracy and spatial awareness, and the second focused specifically on the subject itself.
Mathematics (and numeracy) would be offered in three tiers as it is now. This course would be compulsory for everyone and it would aim to give everyone what is necessary in order to function effectively in an increasingly technological world. The mathematics could include trigonometry and geometry much as now, and also mechanics and statistics. Everyone should be given the opportunity to do multi-step problems to varying degrees.
But "Extra Mathematics" would be offered to those who wantneed more mathematics. This course would aim to give a good education in mathematics as a subject and a second foundation for further studies. Naturally it would include deductive thinking and multi-step problems. I would expect all those who go on to study science and mathematics after GCSE to take this subject.
The recent Dearing Report, Review of Qualifications for 16-19 Year Olds, published in March, also suggested an additional GCSE papercourse to paper over the cracks between GCSE and A-level.
But what does that say about either or both of these examinations? I am suggesting a full GCSE in additional Mathematics so that we can focus our two aims more precisely in each of them and hence give our children a better Mathematics education in both of them.
Janet Jagger is a lecturer at Trinity and All Saints College, Leeds University