I'm Milo, 25-year-old math wiz and author of MATH WITHOUT NUMBERS. AMA

My daughter was a math hater until we adopted a "why before how" approach. Why is the volume of a sphere represented by this formula? Who decided? What makes that TRUE? Once we learned that there were ways to derive these things and that they MEAN something beyond an arbitrary arrangement of symbols in a formula, she really became fascinated. She is now a jr in high school, finishing Calc BC, and interested in pursuing math and engineering degrees.

Our favorite eureka moment came from Welsh Labs' Imaginary Numbers are Real series. Can you recommend some other "behind the math" resources for math enthusiasts?

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.

Would your book be helpful for people with dyscalculia? Have you heard of dyscalculia?

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.

What's the most practical advanced math everyone should know for regular life?

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.

Wasn't expecting this answer. I was expecting maybe statistics or personal finance calculations so this is a refreshing answer.

What is your favorite proof technique?

Proof by induction is really cool and powerful. A kind of random one, I really love proofs by coloring, because they're so unexpected. Here's an example.

I did really good in Calculus 1 but I don't actually understand what a derivative or integral is.

Whats a good way that you would describe them thats easy to understand?

Like when a professor says "we integrate this" i just follow integration steps but i don't actually know in my mind what im changing.

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.)

Got any good math jokes?

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 like what you say about pattern recognition and answering questions about every day things. I love math but sometimes it's hard to explain why people might enjoy it or use it in the real world. What do you say when people ask that question?

It's so frustrating, right?? People have always asked me why I like math and that's why I ended up writing this book. One thing that's helped for me is stressing that when mathematicians say "math" they're not talking about the stuff you learn in school (computation, tools, etc) but more this kind of puzzling, creative problem-solving, thinking about abstract concepts, stuff. Who doesn't want to know about infinity, and the fourth dimension, ...?

Any recommendations on teaching math to a little one?

Yes! Young people are my favorite to teach math to, because they haven't been told yet that they're supposed to be scared of math.

My number one recommendation for teaching math to kids is to focus on building curiousity and interest in the subject - with riddles, games, puzzles, etc. I really love the puzzle book 1000 Playthinks, and the fiction book The Number Devil, for that purpose.