When is proof appropriate?

Got this question on twitter last night:

It was a nicely timed question because I had just talked through two problems that had given the boys trouble earlier in the day. Each of the problems essentially asked the kids to find the largest / smallest solution to a specific problem. Both times the boys were able to find a single solution, but weren’t really sure if the solution they had found was the largest / smallest.

I think that Cathy O’Neil’s piece about what math teaches you applies to the situation my kids were struggling with yesterday:

Mathematicians know how to admit they’re wrong

This piece helped me understand one way that mathematicians see the world that is different than the way most non-math people see the world. In fact, the idea she lays out here is probably one of the most important things that I’ve come to understand in the last couple of years:

To be a really good mathematician you need to be a skeptic and to walk around with a metaphorical gun to shoot holes in other people’s arguments. Every time you hear a reasoned explanation, you look for the cases it doesn’t cover or the assumptions it’s making.

And you do the same thing with your own proofs to help yourself realize your mistakes before looking like a fool. Because putting out a proof of something is tantamount to asking for other people to shoot holes in your argument.

I obviously don’t know the right age to introduce formal proof, but what O’Neil explains in the next paragraph is one of the beginning features of proof / mathematics that I’m trying to work on a little bit with my kids:

For that reason, every proof that one of these young kids offers up is an act of courage. They don’t know exactly how to explain their thinking, nor do they yet know exactly how to shoot holes in arguments, including their own. It’s an exercise in being wrong and admitting it. They are being trained to get shot down, to admit their mistake, and then immediately get back up again with better reasoning. The goal is to get so good at being wrong that it doesn’t hurt, that it’s not taken personally, and that it’s even fun to be wrong and to improve your argument.

So, with that introduction, here are the two problems.

First up my older son talking through Problem 23 from the 1997 AJHME

This problem asks you to find the largest numbers which has two specific properties. It is hard for him to see that any number than 17 meets the requirements and it takes a while, in O’Neil’s language, for him to see how to shoot holes in the 17 argument.

Next is my younger son talking through a problem from an old MOEMs test. This problem asks for the lowest possible value of a high score. Working during the day he found several different possible values of the high score, but struggled to find a way to determine the lowest value. Working through the problem in the evening, he finds a really nice way to explain the problem:

So, I don’t have the slightest idea what the right age to introduce proofs to kids, though I’m pretty sure it is older than 4th and 6th grade! I do, however, really like the ideas about proof from Cathy O’Neil’s post I linked above. Working on explanations and working on a little bit of skepticism about their own work seems like a nice way to help them start down the path to understanding mathematical proofs.

Texbook prices – I just don’t even . . . .

I haven’t thought about buying a textbook in probably 20 years. Recently some of the neat sports stats stuff that Christopher Long is sharing on Twitter made me want to brush up / dig a little deeper on probability and stats, so I went looking for a book.

This morning I happened to be over at MIT, so I popped into their book store to see what textbooks they were using in their prob and stats classes. What caught by eye, though, was the prices of the textbooks for the intro courses – h.ooooooooooo.l.y. crap. Here are some links with the prices from Amazon:

Simmons Calculus – $231

Apostol Calculus I and II together – $430

and one of the stats books:

Mathematical Statistics and Data Analysis by Rice

I’m sure these are all super duper great books, but I wasn’t prepared – even a little bit – for the sticker shock.

That said, here’s a little shout out to some of the authors of books that were not so expensive:

Special gold star to Arthur Mattuck (who also happened to give me my first teaching job) and Keith Devlin:

Introduction to Analysis – $15

Keith Devlin’s book on the Millenium Problems was also $15

Some books for the advanced courses did not seem particularly expensive. For example:

Steven Strogatz’s Nonlinear Dynamics and Chaos was $52


David Eisenbud’s Commutative Algebra was around $50, too


A few days ago, Steven Strogatz shared this study on Twitter:

I wonder if anyone’s ever looked at what, if any, impact the cost of the textbook has on a student’s enjoyment or performance in the course?

Anyway, I didn’t find a stats book that I wanted to buy, so that search will continue a little while longer. Instead I got this:


$20 to learn about quantum mechanics from Leonard Susskind . . . now THAT is a bargain 🙂