"Well, it's all math," he says. "If the math works, why then you should be sure of yourself. That's the whole point of math." Neal Stephenson, Cryptonomicon
"When am I ever going to use this?" the question that math students have been asking since the time of Pythagoras, has always been a tough one for math teachers to answer because, also since the time of Pythagoras, math has been taught as a list of rules and procedures that are to be learned "because if you don't you are likely to get a C- in the class." (p. 24)
Jordan Ellenburg, a mathematician and professor of mathematics at the University of Wisconsin, takes different approach to the question.
According to Ellenberg, even those of us who never got much beyond pre-algebra in high school are engaged with the math everyday, even when "we don't call it math."
In his book How Not to Be Wrong, Ellenberg avoids the standard answer that "when you begin a career, you will see how math matters," and provides answers to three questions:
Ellenburg divides mathematics into four quadrants based on two x-axis dimensions. Math can be shallow and it can be profound. On the y-axis, the dimensions are that it can be simple as well as complicated.
School math occupies the shallow quadrants: simple and complicated. Professional mathematicians do most of their work in the profound and complicated quadrant. The readers (both those with limited math knowledge and those who majored in math) are concerned with the quadrant defined by Ellenberg as profound and simple.
(1) The hidden math is both profound and simple. It is profound in that it is embedded in the problems "that we think about every day--problems of politics, of medicine, of commerce, of theology--are shot through with mathematics. It is simple because it is woven into the way we reason; it's common sense on steroids and understanding it will give you access to insights accessible by no other means."
(2) The answer to the second question, how we learn about the hidden math, emerged for Ellenberg as he dove deeper and deeper in the northeast quadrant with very esoteric problems such as those that explored "residually modular Galois representations," "cohomology of moduli schemes," and "dynamical systems on homogeneous spaces," he began to notice how much math was going on "in the world outside" of the northeast quadrant. The ideas embedded in the hidden math were "simpler, older, and just as deep" as those in the northeast quadrant.
The ideas in the hidden math could be "engaged with directly and profitably" even if your math training ends with pre-algebra. They are not math facts but principles "whose application extends far beyond the things you're used to thinking of as mathematical."
Instead of numerals and calculations, the hidden math is about not being wrong. We use our common sense to avoid error: common sense helps us ignore the deal that is "too good to be true." But over the centuries of its development mathematicians have created a math that is "the science of not being wrong about things, its techniques and habits hammered out by centuries of hard work and argument." These techniques and habits can make your common sense more reliable.
(3) How do you learn them? According to Ellenberg, all you need is a coach, or even just a book to "teach you the rules and some basic tactics." He will be our coach. "I will show you how." (p. 15)
In the eighteen chapters of How Not to Be Wrong, Ellenberg connects us with a wide variety of problems--from politics, economics, medicine, health--and coaches us to see the mathematical thinking that will help us not to be wrong. (p. 28)
One of the first things we learn is that it's really easy to be wrong. Our common sense is pretty unreliable and easily misled, especially when numbers are involved.
Ellenberg tells us about how, during the second world war, the U. S. had a research bureau that worked on specific engineering problems such as how to best place armor plate on airplanes. (A difficult problem because too much armor makes the plane dangerously slow; too little makes it more vulnerable.) The bureau collected data about where the bullet holes were on American fighters and bombers when they returned from missions over hostile territory.
The finding was that there were more bullet holes in the fuselages than in the engines of surviving planes, so the engineers now saw the problem as how to place the armor on the fuselage to the best advantage. After all, the available data had been carefully analyzed.
That is, all the engineers except one, Abraham Wald, an Austrian mathematician and refugee from the Nazis. Wald had little experience with aircraft in wars but he did what a good mathematician does: to ask "what are the assumptions behind the investigation?"
Wald helped the engineers see that their assumption was flawed because they were missing important data. Their data was from planes that survived; but the planes that didn't return from the mission; the non-survivors provided additional data, the fatal bullet holes. The fact that there were few holes in the engines of the survivors, indicated that the missing holes were probably in the planes that didn't survive.
The lesson: when you have a set of data, you probably need to ask about the numbers you don't have. A mutual fund invests your money in a variety of investment funds. Over the years some of those funds thrive while others fail. When the mutual fund reports its 10 year record as being some large number, the investor whose common sense has been reinforced by the hidden math will ask: what about the data from the funds that didn't survive? The true success of the mutual fund would include the data from the investment funds that were failures as well as those that survived.
The classroom definition of math as calculation turns out to be very partial; Ellenberg's book shows how math as thinking is both fascinating and relevant.
Perhaps if more kids learned about both kinds of math, more of them wouldn't ask "when am I ever going to use this?"
Ellenberg, Jordan. (2014): How Not to Be Wrong: The Power of Mathematical Thinking. Penguin Books: New York.
Stephenson, Neal. (2000): Cryptonomicon, Harper Perennial, New York.
Dr. John Holton
Dr. John Holton joined the S²TEM Centers SC in July of 2013, as a research associate with an emphasis on the STEM literature including state and local STEM plans from around the nation.