miyomiyo

Oof - "good math joke" is an oxymoron, but here's a pretty lame one...

A mathematician wakes up in the middle of the night and finds her kitchen on fire. She calls the fire department, they rush to her house and after an hour they manage to put out the fire. A few nights later she wakes up again and finds her toaster on fire. So, she quickly works to spread the fire until it consumes the rest of her kitchen, and announces "There - the situation has been reduced to a previously solved problem," and goes back to bed.

Ok I warned you it was bad

I love these questions! "Who decided? What makes that TRUE?" is really the central topic in the philosophy of math, and I think it's a shame they don't talk about this stuff in school. There's a section in my book dedicated to these questions, drawing from questions exactly like this I get from students all the time.

In terms of videos, the YouTube series 3Blue1Brown is fantastic, and the creator definitely likes to focus on intuition and reasoning. And for math enthusiasts who really want to roll up their sleeves and do some math, the Art of Problem Solving books (vol 1 and 2) are the best of the best. They never give a formula without explaining the reasoning that gets you there.

Hmm... I think it would be cool if people had a better understanding of proof techniques- how to prove something is true, carefully and methodically, not just by (I don't know) yelling and saying "I'm smart so I must be right." Working with mathematical proofs definitely gives you some humility, because it just gives you a lot of experience being wrong and having to work hard to understand something.

For sure! Whether or not you have diagnosed dyscalculia I think a lot of people have trouble with numbers and mental calculation-- that's just not the way human brains are built to work. The point of this book is to show that "math" means a lot more than just trying to do what a calculator does - math is about patterns, structures, and just a way of looking at the world and solving problems.

Ugh! I hate when people teach robotic techniques without explaining what's going on. I give a better answer to this question in my book (Chapter 5 - the continuum!) but here's a really rough quick answer:

Integration is basically just adding infinitely many things together. For instance let's say you're trying to find the length of a curvy line, but you only have a straight ruler. You can break the curve up into a bunch of little segments, measure them approximately with a ruler, and add them together. The smaller you make the segments, the better your approximation. Integration takes this to logical extreme: You break the line up into infinitely many infinitely-small segments and add them together. Somehow (magically) that gives you a real, finite answer.

The specific "integration techniques" you learn in a Calc 1 class are just this, with functions and symbols. Modern math uses a particular language (sine! log! square root!) to describe certain types of common curves, and these have been studied really well so we know how to integrate them automatically/methodically. (But - to be honest - these days most mathematicians just use WolframAlpha or Mathematica to do integration & derivatives... you don't really need to have this stuff memorized.)