I was in that category myself for many years. I think this was at least partly the result of bad teaching. My earliest memory of maths at school is of being a member of a class, aged perhaps six or seven, chanting the multiplication tables in chorus. Rote learning of this kind has its place, I suppose, but only provided the pupils understand what they are learning.
I don't remember how old I was when it dawned on me that these arcane incantations weren't magic spells to be memorised and invoked according to fixed rules to "do sums", but were simply the result of sequential additions. Presumably my teachers thought that this was too obvious to need stating, but it would have made my life much easier if it had been pointed out early on. We could so easily have been shown how to construct the tables for ourselves.
Throughout my early years at school mathematics was synonymous with arithmetic. We were drilled in long multiplication and long division and the all-important thing was to get the right answer. A near miss was treated as a complete failure, whereas in fact an intelligent approximation should have been reason for congratulation.
At that time Britain still used pre-decimal currency (pounds, shillings and pence) and Imperial units of measurement, which provided rich opportunities for fiendishly difficult arithmetical problems, such as how much it would cost to buy seven yards of cloth at a price of two pounds thirteen shillings and ninepence three-farthings a yard.
Later, I was introduced to geometry, algebra and trigonometry. I found geometry relatively easy, because it was visual. And although I dislike algebra —much memorising of rules—I saw solving equations, including even those of the simultaneous and quadratic variety, as a kind of logical puzzle-solving that I could understand and almost enjoy. But it seemed a completely pointless exercise.
By now I'd decided I much preferred words to numbers and was probably constitutionally unsuited to maths, so I put it out of my mind as much as possible. At the age of sixteen I had to take the School Certificate exam (forerunner of the GCSE "O" level), which included papers in mathematics. These I failed; in fact, I remember I scored an almost incredible 3 per cent in one of them (probably arithmetic). This didn't trouble me at all; I assumed I had done with maths for good.
I failed the exams at the end of the year, and rather than retake them in the autumn I decided to leave and do my compulsory two years' National Service in the Army. I can't say I found this useful or enjoyable in most respects but it did lead indirectly to a radical revision of how I thought about maths, so I have to thank the Army for that.
I spent my National Service as a sergeant instructor in the Royal Army Educational Corps. This no longer exists as a separate organisation in the Army but at that time it was tasked, among other things, with continuing the education of military personnel; this had become compulsory for regular servicemen who were not up to scratch in that respect.
At that time I was lodged in quarters used by the Royal Marines. One of them, a warrant officer named Bill Nelson, was also interested in improving his mathematics and we started to work through the problems in the book together. I was getting on so well by this time that I volunteered to teach a maths class for the troops to prepare them for the GCSE. Then I thought I might as well take the exam myself, which I did, passing easily with a score of over 60 per cent.
This experience cured my former mathematical anxiety. I no longer feared the subject, quite the contrary. A couple of years after I left the Army, I re-entered medical school and did sufficiently well in the first year's physics and chemistry exams that I was asked to be one of the student demonstrators for the next years' students.
I've often wondered why this change in my view of mathematics happened. Possibly it was due to a delayed maturation in the wiring in my brain, but more probably it was the result of a change in my attitude to the subject. I've always tendendeg to prefer self-education over formal instruction (it's how I learned Spanish when I was in my late 'teens.) Perhaps it was both together. Anyhow, it changed my life.
What would that entail? Presumably calculus, which was obviously important but about which I knew next to nothing. In the 1960s and 1970s there was no easy way for me to investigate calculus, but things are different now by at least an order of magnitude, thanks to the internet, so I've embarked, slowly, on an exploration of the subject—not with the ambition to learn it formally but at least to get a feeling for what it consists in. I thought it would be worth posting a brief description of my explorations so far in case it's of interest to anyone thinking of doing the same.
I started by reading Infinite Powers: The Story of Calculus, by Steven Strogatz. This takes a mainly historical approach, starting with Archimedes and going on to Newton and Leibnitz. It's really a book of two halves. The first half contains a fair amount of mathematics and is demanding but rewarding; the second half looks at more recent applications of calculus and doesn't have much about the theory. But even in the first part Strogatz doesn't expect his readers to do any actual calculus.
It isn't necessary to know how to do calculus to appreciate it, just as it isn't necessary to learn how to prepare fine cuisine to enjoy eating it. I'm going to try to explain everything we need with the help of pictures, metaphors, and anecdotes. I'll also walk us through some of the finest equations and proofs ever created, because how could we visit a gallery without seeing its masterpieces?I'm not fully convinced by this analogy; I think that music may be better. It isn't necessary to play an instrument in order to appreciate music but if you try to do so, no matter bow incompetently, you will get an insight into the process which probably isn't available without it. I won't say I felt short-changed by Strogatz but he left me with a desire to actually try calculus for myself—which isn't a bad thing, of course.
So I next tried Kalid Azad's website, Calculus Better Explained , which seems to be aimed at people like me who want to have hands-on experience of calculus but are starting completely from scratch.
Learn the essential concepts using concrete analogies and vivid diagrams, not mechanical definitions. Calculus isn't a set of rules, it's a specific, practical viewpoint we can apply to everyday thinking.I've worked through most of Azad's chapters, with diversions to remedy my ignorance of things like the binomial theorem, Pascal's triangle, polynomials, and the natural logarithm.
I'm also reading Seventeen Equations that Changed the Modern World, by Ian Stewart. Like Strogatz, Stewart takes a historical approach but he includes more mathematics than Strogatz does. I find I need to take his book slowly but it's certainly rewarding.
I'm writing this at a moment (April 2020) when the whole human world is undergoing a huge metamorphosis caused by the corona virus Covid-19. How we will emerge from it, no one knows; I don't even know if I shall emerge at all, and meanwhile my wife and I are effectively in lockdown at home. But I find that trying to enhance my knowledge of mathematics even by a small margin brings a sense of enlargement, rather like learning a new language—which in fact it is, complete with vocabulary, syntax, and script.
The nearest analogy I can find is writing a novel. This gives one a new kind of writing experience; producing fiction and dialogue is liberating in a different way from any other kind of writing, and this is true whatever the intrinsic quality of what results. The writer and critic F.L. Lucas has remarked on the same thing.
The philosopher Galen Strawson has said that his work gives him something like what meditation is said to do (he hasn't had much success with meditation itself).
I think philosophy really does change one over time. It makes one's mind large in some peculiar manner. It seems to me that the professional practice of philosophy is itself a kind of spiritual discipline, in some totally secular sense of "spiritual"; or at least that it can be and has been for me. It would be very surprising if intense training of the mind couldn't change the shape of the mind as much as intense training of the body changes the shape of the body.Unlike Strawson, I'm not a professional philosopher, but I think I understand what he's saying here. One can't practise any intellectual discipline, including mathematics at even a very basic level, without being changed by it, At any rate, I hope so.