Like so much of what I write these days, the thoughts expressed below owe much to something I read elsewhere on Substack. In this case, the pen that produced the “post of origin” belongs to Noah Smith, who argues that, if only for the sake of the high-tech economy, America needs more people who are fully au fait with higher mathematics.
The people commenting on this article, as well as Mr. Smith himself, assumed that most (if not all) mastery of mathematics would result from formal instruction that took place in conventionally configured colleges and schools. In addressing this question, however, I would like to shift the focus from the teaching of mathematics to the learning of the same. To this end, I will tell the sad tale of a lad in whom the love of numbers was beaten to a bloody pulp by the aforementioned formal instruction.
As a child, I loved to calculate. The first object of my figuring was time and distance. (Once I had grokked miles per hour, I was able to give up the irritating habit of asking my long-suffering parents “are we there yet?”) Soon thereafter, I took up coin collecting, and thus such pastimes as counting the number of groats in a guinea.1 (Neither groats nor guineas found their way into my little collection of bronze pice and aluminum annas, but I had read about them in my little red coin collector’s handbook.)
At the age of twelve, I received, at Christmas, the princely gift of an electronic calculator. As I needed no help with simple sums and such, I spent many happy hours using this marvelous device to work out square roots. (In the days before the appearance of the √ key, this required several cycles of estimation and testing.)
Then, when I was in the eighth grade, the madman who ran the little school I attended decided that he was going to teach algebra.2 However, rather than actually doing this, he treated his students to a wide variety of insights on a wide variety of subjects, few of which had anything to do with the art of balancing equations. I therefore entered high school with “Algebra I” on my transcript, but very little algebra in my head.
This set the stage for a year of academic agony, in which I struggled to keep up with a course of instruction for which I was woefully unprepared. It did not help, moreover, that the teacher charged with guiding me through “Algebra II” knew little of the art of conveying complex thoughts. Thus, rather than teaching the subject to myself on my own schedule, and thus properly, I ran a race in which I found myself, more often than not, “a day late and a dollar short.”
While this experience wrought much injury upon my love of mathematics, it did not kill it. I am happy to report that, in the years that followed, I managed to restore my fondness for the subject to a reasonable state of health. As you might guess from the title of this post, this recovery owed much to well-worn copies of Mathematics for the Millions, Teach Yourself Trigonometry, and the non-fiction works of Isaac Asimov, but nothing at all to classroom instruction.
In days of yore, a groat was a silver coin worth 1/3 of a shilling and a guinea was a gold coin worth 21 shillings. Thus, there were 63 groats in a guinea.
The affliction in question, I suspect, was what today’s psychiatrists call “bipolar disorder.”
I'm currently an online student working towards a mathematics degree. I've always had a knack for math, but I've only recently started applying that knack.
So far, I've been coding my higher order mathematics problems in Python using a Google Colabs environment. I usually use the Sympy library, but now that I'm getting deeper into the weeds with Calc II and differential equations, I'm starting to use Numpy and Scipy as well. Also, my "online tutor" so far is GPT-4. It isn't perfect, and sometimes I have to go back and forth if there's something that it's having difficulty with, but it works great 85% of the time.
As for understanding the concept of mathematics itself, I've found the books of David Berlinski to be invaluable, specifically "A Tour of the Calculus", but all his books on mathematics are incredibly illuminating. I have yet to read something he puts out where I don't learn something or see a topic in a different light.
I second the recommendation for Isaac Asimov. His non-fiction was a crucial factor in my intellectual development, particularly his essay collections. Do you know if they're still available?
I just wrote a post with some of my own book recommendations, here: https://aetherczar.substack.com/p/reading-list-for-chapter-1-on-generation