Mathematics for the Adventurous Self-Learner

I put it in context because it's not quite a direct comparison since I have been in greatly different situations and ages between studying math and CS, but putting that aside, I have to say I have greatly enjoyed the computer science means of teaching more than math, doubly when it comes to self-learning. Concepts in math are generally taught entwined with the means of proving those ideas. That's important if you're a grad student looking to be a math researcher, but (IMO) it is not so great if you're a newer student or learning on your own and trying to grasp the concept and big picture. A proof of a theorem can be (and too often is) a lot of detail that really doesn't help you grasp the concept the theorem provides or is used towards, often because it involves other ideas and techniques from higher levels or just different types of math, both of which are out of the scope of the student learning the topic. Worse yet, it is standard for a proof to be written almost backwards from how it would be thought out. Anyone from a math educational background has the experience in homework of solving a problem, then rewriting it almost in total reverse to be in the proper form to submit. This means not only is the proof of the theorem not useful towards conceptual understanding, reading the proof doesn't show you chronologically how you would discover it yourself. That is a lot of overhead cost to break through to get to real understanding, real learning. As you mention, notation as well is another thing you need to break through.

I have found computer science and related classes to be taught more constructively. Concept is given first, and then your job as student learner is to construct it. Coming from the ML field, I love comparing math and CS proofs of topics here. Explanations from CS people of back propagation, for example, are always visual, and books/courses will have you construct a class and methods to do the calculations. Someone with a bit of programming knowledge can follow along in their language of choice. Math explanations get into a ton of notation from Calc 3+, and it's going to take a lot of playing around and frustration to get a working system out of the explanation. Even the derivation section on Wikipedia is not something most people will understand and be able to turn into useful output.

The more I see other ways concepts are taught, the more I wish math had been taught a different way. There is a lot to break through in order to get to real understanding, just by the way it's formed and taught.

One thing I tell my high-school students: mathematics always looks harder than it actually is. One of the essential skills in succeeding in math is looking at a page of arcane "stuff" and having your reaction be, "Whoa! Can't wait to learn what this means," rather than, "Whoa! This looks so complicated!"

Mathematical is its own language that has developed across continents and millennia. It has its quirks and foibles, but overall, community consensus has guided its notation. Mathematicians want things to be simple and "make sense", especially the notation they use. It's never as terrible as it looks.

Sigma specifically is a Greek letter, but the notation is not Greek. Like a large amount of modern mathematical notation, the convention came from Leonhard Euler in the 18th century. It was a disambiguation choice because the letter S was overloaded.

Single-symbol identifiers are enormously popular in mathematics because mathematics is not computing. Because math is (even now) essentially a handwritten subject, its design plays to the strengths of handwriting. Line size, height, and character layout are essentially freeform. Character accents and modifiers are easy. discrete_sum would never fly in a handwritten world, just like ∑ wouldn't pass code review.

After I went back to school, I tutored in the math study center to pay the bills, which really helped cement not just the learning but also the notion that I could survive academia. I'd gone in with a plan to study engineering, but after I transferred to university, I kept dawdling on the math prerequisites and not taking the engineering courses that needed them. So it kinda gradually dawned on me that math was what I loved, and away I went.

I never strayed too far from computers. I'm a graph theorist, specializing in computation; had I known better I'd have gone into computer science because that's where I see the most progress being made.