Today (August 1, 2014) Keith Devlin published a nice article touching on both mathematical ideas and math education:

Most Math Problems Do Not Have a Unique Right Answer

If you have any interest at all in math or math education this article is well worth the 5 minutes it’ll take to read. I want to extend one of the points in the piece a little, though. Maybe it was my own poor reading of what he wrote, but I felt that one of the ideas near the middle of the article did not come through as clearly as I would have liked:

“Knowing how to solve an equation is no longer a valuable human ability; what matters now is formulating the equation to solve that problem in the first place, and then taking the result of the machine solution to the equation and making use of it.”

I think that Devlin has condensed a lot of information into the phrase “formulating the equation” so I’d like to un-condense it a bit. I worry that it is easy to think (as I did at first) Devlin’s comment means that you just write down some equations, head off to the computer, get your results, and charge full speed ahead.

The mathematical thinking that Devlin focuses on lies in understanding the results, not in simply writing down the equations. Rather than discussing this idea in the abstract, though, I want to give some fairly concrete examples from my own life and development in understanding this process.

The first example comes from my undergraduate thesis in college. In the spring of 1992 researchers (who would go on to win the Nobel prize in physics for their work) released the first detailed map of the cosmic microwave background radiation. I sat in the auditorium for this presentation and couldn’t help getting caught up in the excitement. After the talk I asked my adviser, Professor Ed Bertshinger, if I could try to study some piece of these results for my undergraduate thesis. He liked the idea and mentioned that he’d been wondering how the microwave background radiation would be distorted if the entire universe was rotating. Seemed like a fun problem – especially since my interest in physics was more math-y than hands on physics-y, so I dove in.

After 6 months or so I presented my findings to Professor Bertshinger. I was quite happy to have both an equation and some computer modelling. Though we had spent plenty of time talking over the course of the year, this was the first time that he saw the final results. Upon seeing the equation he drew a couple of diagrams on his chalkboard and eventually arrived at a picture nearly identical to the one drawn by my little computer model. It was stunning for me to see – six months of work for me and he drew the damn picture in about a minute just by waving his hands! Given enough time I could write down the equations, but he understood them. Seeing first hand the difference between those two acts was a powerful lesson for me.

The next examples are two fun billion dollar prize promotions that I’ve have come across my desk at work. The first was Pepsi’s “Play for a Billion” in 2003 and the second in Quicken’s billion dollar bracket game from this past spring.

In the Pepsi game 1,000 contestants took a guess at a six digit number from 000000 to 999999. So, 1,000 guesses at a number selected at random from one million possibilities. The chance of someone guessing correctly was 1,000 / 1,000,000, or more simply 1 in 1,000. As long as you’ve got the right security to prevent cheating, the math behind this game is not particularly hard (or even particularly interesting). This is a case where understanding the formula and solution does not take much time.

The Quicken promotion involved contestants trying to predict the outcome of 63 basketball games. Since you don’t know the precise chance of picking the outcome of any one game, the math you need to use to understand this promotion is pretty different than the math you needed for the Pepsi promotion. A purely formulaic approach is going to have a variety of problems – not the least of which is small changes in your assumptions lead to large changes in the estimate of the odds of someone winning this promotion. Worse, it is not at all obvious what the correct assumptions should be to begin with! Nonetheless, given the amount of money on the line, you need to be confident that you have analyzed the problem correctly and that’s where the mathematical thinking comes into play. Many of the articles about this promotion assumed that each game was a 50/50 chance, and thus sadly missed an opportunity to write a really neat and math-related article. Working through the various mathematical ideas behind this promotion (even the “birthday paradox” came up!) was one of the most interesting problems that I’ve ever worked on.

I guess the last example that’s not too hard to understand comes from problems in the financial markets. Roger Lowenstein’s incredible book “When Genius Failed” talks through the problems which arose when just one hedge fund got into trouble in 1997. More recently, we are all probably way too familiar with the problems that arose in the financial markets in 2008. Not all, but many of these problems came from groups of people using mathematical models and formulas that they did not fully understand.

The financial crises, and all of the terrible consequences that resulted, probably explains better than anything else why I wanted to clarify Devlin’s point. When Devlin writes about mathematical thinking, he isn’t talking about just writing down equations. Anyone – from undergrads writing about physics, journalists writing about basketball promotions, to derivative traders playing with other people’s money can write down equations and play with computer models. If we don’t understand the equations, solutions, or limitations, we get into trouble. That trouble can range from writing a dull senior thesis to causing the collapse of the world’s financial system. When we learn a little bit about mathematical thinking, hopefully we’ll do neither!