If we really want to understand mathematics, we have to get to grips with the language first.
The debate on teaching mathematics depends crucially on two variables. What kind of beings are involved in its teaching and learning, and what kind of entity is mathematics?
Biologists are giving us a fairly clear picture of the nature of the human race, and curiously it is remarkably close in some respects to the biblical account. We began as hunter-gatherers in Africa in a world (garden?) full of food, fruit, roots and scavengeable material. We shared this world with many other primates, to many of which we were very close relatives. It would not have been clear 150,000 years ago that one group was going to become wholly dominant.
But something happened. A genetic twist perhaps or divine intervention. Probably it was connected with the development of language between 50,000 and 100,000 years ago. The effect was swift: within a biological twinkling of an eye we became masters of our planet and masters of the world of knowledge. But it has been done with "software" which, while of potentially enormous sophistication, sits in "hardware" mainly designed for wandering around Africa collecting food. This then is the material we work with: a being perfectly designed for GNVQ berry picking.
What of mathematics? There are many competing models for the nature of the subject, but I would argue the case for one of the most ancient, the Platonic. Here mathematical truth exists in a realm of "forms" which are idealised and perfect. The essential point is that mathematical knowledge is in some way "other"; it exists absolutely and independently of ourselves. Even if this model is rejected in principle, it is still how mathematics appears to the learner. The theorem of Pythagoras is a piece of abstract knowledge independent of his or her existence.
Cardinal Newman insisted 150 years ago that man is not a rational being. He took the view that knowledge comes to reside in the brain as "idea", which is different to a Platonic "form". It is fluid, develops, flows and changes as it is transmitted from mind to mind and from one generation to another. The mind must organise the ideas with imagery and language.
In learning mathematics therefore we are confronted by forms which are cold, austere, perfect and remote. They cannot be learned as such. The brain must build up an idea or image linked to the form. It will do so as it explores the form with language, and builds up a mental picture. In this way the perfect "form" can become the hunter-gatherer's "idea".
This is why language and a rich classroom environment of language is crucial to teaching. It is also why investigation is so important - not because mathematics is an inductive process (it is not) but because the brain needs time to build up its "idea".
Understanding requires effort, even with the best teachers and computer packages. We must not fob people off with "application of number" or the latest euphemism for pseudo-mathematics. The greatest long-term transferable skill is to understand the language in which our society is increasingly constructed.
A-level gives us no "gold standard" here because its specialisation ensures that few study mathematics post-16. No other advanced nation tolerates such a situation.
Robert Barbour is head of mathematics at Hagley Roman Catholic High School, Worcestershire