Ask HN: How do I improve my command of mathematical language?

When I took calculus there were three things that really helped me with this:

1) A study partner that's competitive. when you have an intuition you're more likely to get the correct answer first because you can check what's going on quickly without having to recalculate things. Also when they get stuck you'll be expected to explain how you figured it out. Studying with someone like this will force you to grow an intuition.

2) Writing programs that use what you've learned. At least for me this helped a lot. I learned to program before I learned any interesting math so being able to rewrite what are often implicit ideas in an explicit language can really help you think things through. Like many people here I gained an intuition for integration early on by writing physics engines for games.

3) Just meditate on the ideas.

That’s just seems like bad writing. Out of math subjects I learned (discrete math, linear algebra, real analysis, statistics), only in the last one it may be unclear what’s going on, whether you are dealing with a random variable or its realization, and arguments of random variables are almost always omitted, so they don’t look like functions.

I suggest you find some maths videos on Youtube which explain maths you already know with lots of verbal detail narrating each step. Pay attention to the words and description (you already know the maths), then try talking through your own working in the same way. You could record yourself and watch it back (camera pointing straight down at paper is good, and phone camera held up on a ruler and stack of books is easy). As others have suggested, once you can do that to some extent, explaining things to others can be good practice, but if you're stumped as to how to describe anything, that's not going to be a good first step.

I teach university maths, and when I'm doing a problem in a lecture, writing down the steps to demonstrate, I am constantly talking, narrating everything I'm doing and why. E.g. "Okay, we need to cancel out that 3, so let's divide both sides by 3, and then we can use log rules to bring that 2 on the right inside the log as a power. Now to cancel out these logs we need to use an exponential on both sides, and if we move these terms over we'll have a quadratic to solve. We know how to do that! Hm, it doesn't seem easy to factorise, so let's complete the square. We need half of this coefficient to go inside the brackets, and don't forget the minus is part of it! Now if we think about expanding this out we'll get the x squared and minus 4 x that we want, but also a plus 4, so let's take that away and add on the 3 we need instead ..."

I'll also be referring to the context of the problem ("that makes sense, because ..." etc.) and earlier examples to compare and contrast with.

Try Playing with Infinity by Rózsa Péter. https://www.maa.org/press/maa-reviews/playing-with-infinity