Somewhat related, I’ve been slowly working my way through Sean Carroll’s The Biggest Ideas in the Universe: space, time, and motion. I know a lot of you got this info watching his video series during the pandemic. I did watch a few of those, but never had the time back then to just sit through all of it. So now I’m reading about it instead, which is fine since it’s a mode I learn better with anyway.
Carroll’s main endeavor in this book, and for the second one coming in May on quanta and fields, is to provide a relatively gentle introduction to the math involved in physics. The idea is to get you familiar with the mathematical structure of the theories, without having to learn to actually solve anything with them, that is, without having to become a practitioner. He’s not the first to take this approach, but the first I’ve seen be comprehensive about it.
Like many books in this genre, it does start off pretty gentle. Carroll doesn’t assume you know calculus or trigonometry, but he does assume a comfort level with algebra. And like many others, at a certain point, the book accelerates and feels far less gentle. Often steps get skipped, with the assumption that they’re obvious. (One benefit of reading rather than watching is being able to linger on these cases.) Despite my historical challenges with math, I’ve been able to follow along, at least until the chapters getting into spacetime geometry, which are turning into a struggle.
Among the hints about life I wish I could transmit to my younger self, one is the importance of repetition for learning anything, particularly something like math. I always found math homework agonizing, and so as a boy did the minimum possible. Unfortunately my foundational math classes happened during a period when math teachers weren’t keen on making students turn in their homework. I did eventually learn my lesson, you have to get your repetition in by doing the homework, but not before the damage was done.
Which isn’t to say I’m completely helpless. My education went through introductory calculus, trigonometry, statistics, and other basics. When my back is to the wall, I can usually get by. A career in computer programming helped somewhat. One of my issues with plain math is how abstract and seemingly unconnected it is with anything in the world. Learning to read equations as a sort of mini-program that reality executes helps me deal with them in scientific papers, or in books like Carroll’s.
This seems to be in line with a lot of discussions I’ve read over the years about finding a better way to teach math. Many of my teachers seemed to delight in the abstract nature that I found so unappealing, and so taught from that perspective. They saw a beauty in those abstractions, and seemed unable to grasp that most people simply don’t. Math education that starts and stays grounded in practical problem solving is probably a better avenue for most students.
So yes, the keys to understanding the nature of reality is math, and the keys to understanding math is doing the homework. As adults, those of us who didn’t get a great launch with it have to get by as best we can.
What do you think? Any tips and techniques you’ve found useful for dealing with math? Or for learning to appreciate its beauty in the way so many math teachers do?