Friday, August 1, 2014

Most Math Problems Do Not Have a Unique Right Answer

One of the most widely held misconceptions about mathematics is that a math problem has a unique correct answer.

(Some of those who hold that view also think that there is just one correct way to get that answer. A far smaller group, to be sure, but still a worryingly large number. Still, my focus here is on the first false belief.)

Having earned my living as a mathematician for over 40 years, I can assure you that the belief is false. In addition to my university research, I have done mathematical work for the U. S. Intelligence Community, the U.S. Army, private defense contractors, and a number of for-profit companies. In not one of those projects was I paid to find "the right answer." No one thought for one moment that there could be such a thing.

So what is the origin of those false beliefs? It's hardly a mystery. People form that misconception because of their experience at school. In school mathematics, students are only exposed to problems that (a) are well defined, (b) have a unique correct answer, and (c) whose answer can be obtained with a few lines of calculation.

But the only career in which a high school graduate can expect to continue to work on such problems is academic research in pure mathematics—and even then (and again speaking from many years of personal experience), cleanly specified problems that have (obtainable) "right answers" are not as common as you might think.

Since the vast majority of students who go through school math classes do not end up as university research mathematicians, whereas many do find themselves in careers that require some mathematical ability, it's reasonable to ask why their entire school mathematics education focuses exclusively on one tiny fraction of all possible mathematics problems.

The answer can be found by looking at the history of mathematics. Starting with the invention of numbers around 10,000 years ago, people developed mathematical methods to solve problems they faced in the world: arithmetic and algebra to use in trade and engineering, geometry and trigonometry for building and navigation, calculus for scientific research, and so forth.

While some of that mathematics was required only by specialists (e.g. calculus), arithmetic and parts of algebra in particular were essential for everyday living. As a consequence, mathematicians wrote books from which ordinary people could learn how to calculate. From the very earliest textbooks (Babylonian tablets, Indian manuscripts, etc.), two kinds of problems were presented: algorithm ("recipes") problems that showed the steps to be carried out to do a particular kind of computation, presented without any context, and word problems, designed to help people learn how to apply a particular algorithm to solve a real world problem. Ancient and medieval textbooks had many hundreds of such problems, so that a trader (say) could find a problem almost identical in form to the one he (and back then use of mathematics was primarily a male activity) actually wanted to solve in his business. If he were lucky, all he would have to do is substitute his own numbers for those in the book's worked word problem. In other cases, the book might not provide an exact match, but by working through five or six problems that were close in form, the individual could learn how to solve his real problem.

For the majority of people, that was enough. Life simply did not require anything more. The problems they faced in their everyday activities for which mathematics was needed were simple and routine. The mathematical word problems that today seem so unrealistic were by and large remarkably similar to the problems ordinary citizens faced every day.

"When do I need to leave home in order to catch that train?" There wasn't an app to tell you the answer; you had to calculate it yourself. That word problem about trains leaving stations in your math class showed you how.

Arithmetic, in particular, was an essential, basic life skill that remained so until the development of devices that automated the process in the 1960s. I am a member of the last generation for whom the question "What do I need arithmetic for?" simply did not arise. (We asked it about other parts of mathematics.)

But that computer technology that eliminated the need for people to be good calculators led to a world in which there is a huge demand for higher order mathematical skills, starting with algebra. I wrote about this change in this column back in 1998, in a piece titled "Forget 'Back to Basics.' It's Time for 'Forward to (the New) Basics.'" Looking back at what I wrote then, I am amazed at just how much things have changed in the intervening 16 years. In September of that year, Google was founded, and the Web became a dominant force in our lives and our work.

Today, we have instant access to vast amounts of information and to unlimited computing power. Both are now utilities, much like water and electricity. And that has led to a revolution in the mathematics ordinary citizens need in order to lead a fulfilling, productive life. In a world where procedural (i.e., algorithmic) mathematics is available at the push of a button, the need has shifted to what I and others have been calling mathematical thinking.

I wrote about this in my September 2012 Devlin's Angle. Broadly speaking, mathematical thinking is a way of approaching problems that is based on classical mathematics, but takes account of the fact that computation (both numeric and symbolic) can be readily done by machines.

In practical terms, what this means is that people can now focus all their attention on real-world problems in the form they are encountered. 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.

In the 1960s, we got used to the fact that the arithmetic part of solving a mathematical problem could be done by machines. Now we are in a world where almost all the procedural mathematics can be done by machines.

Of course, this does not mean we should stop teaching procedural mathematics to the next generation, any more than the introduction of pocket calculators meant we should stop teaching arithmetic. But in both cases, the reason for teaching changes, and with it the way we should teach it. The purpose shifts from mastering procedures—something that was necessary only when there were no machines to do that part—to understanding the concepts sufficiently well to make good use of those machines.

Though this change in emphasis has been underway for some years now, it did not garner much attention in the United States until the rollout of the Common Core State Standards, which are very much geared towards the mathematical thinking needs of the 21st century. The degree to which many parents were shortsighted by the shift was made clear when some of them took to social media to complain about the kinds of homework questions their children were being asked to do. While some of those questions were truly, truly awful, others garnering a lot of critical SM comments were actually extremely good.

What was particularly ironic was that many parents, faced with being unable to assist their child with elementary grade arithmetic homework, did not draw the obvious conclusion: "Gee, if I cannot understand something as basic as integer arithmetic—however it is done—there must have been something really lacking in my own education." Instead, they jumped to the totally off-the-wall conclusion that the current educational system must be wrong.

That's like waking up in the morning to find your car won't start and saying, "Oh dear, the laws of physics don't work." The smart person says, "I need to replace the battery."

I'll tell you something. I was taught math the "old-fashioned way" too, and some of those student arithmetic worksheets were new to me when I first saw them. But regardless of any views I might have as to how it is best taught in today's world, it didn't take a lot of effort to figure out what those kids were doing on those worksheets posted on Facebook. It was just whole number arithmetic for heavens sake! Anyone who understands the basic ideas of whole number arithmetic can figure it out.

It was not my training as a professional mathematician that helped me here. It was the simple fact that I understand whole number arithmetic, something that goes back to my early childhood, when I did not even know there was such a thing as a professional mathematician, let alone aspire to be one. Unfortunately, many Americans were never taught to understand arithmetic, they were just trained to execute procedures. It's not their kids who are being short-changed. They—the parents—were!

Breezing into this fray is University of Wisconsin mathematics professor Jordan Ellenberg, with his new book How Not To Be Wrong. I knew I would find a kindred spirit when I read the book's subtitle: “The Power of Mathematical Thinking.” With a Stanford MOOC and an associated textbook both called Introduction to Mathematical Thinking, how could I not?

Ellenberg's title is superb. In one fell swoop, it casts aside that old misconception that mathematics provides "right answers," replacing it with the far more accurate description that it is a great way to stop you being wrong. For, like me, he focuses not on the internal activities of pure mathematics, rather on how mathematics is used in today's real world.

To be sure, also like me, Ellenberg has devoted a lot of his career to working in pure mathematics, so he loves searching for those "right answers," and he enjoys the subject in its own terms. We both know that there are eternal truths within mathematics (a better term would be "tautologies") and have experienced the thrill of going after them. But we both realize that what we do as pure mathematicians is a very specialist pursuit. The society that supports us when we do that does so largely because of the payoff in terms of the benefits that emerge when mathematical thinking is applied to real world problems.

Ellenberg's book is chock full of examples of those benefits, from many walks of life, presented with a delightfully light touch. He grabs the reader's attention with his very first example, taken from the Second World War. The U. S. military chiefs wanted to reduce the number of warplanes that were being shot down. The obvious solution was to add more armor to protect them. But armor adds weight, which limits the distances that can be flown and the duration of the mission, as well as increasing the production cost. So the question was, where is the most effective place to put that extra protection?

To answer this question, the chiefs brought in a team of mathematicians to analyze the evidence and determine what parts of the aircraft were most likely to be hit. They examined the fuselages of all the damaged planes that had flown back after being hit to see where the most damage was. It turned out that the engines had an average of 1.11 bullet holes per square foot, the fuel system had 1.55, the fuselages 1.73, and the rest of the plane 1.8.

So where was the optimal place to add extra armor? According to the data, the fuselages took a lot of hits, while engines suffered the least damage. So an obvious suggestion was to add armor to the fuselages. But that was not what the mathematicians suggested. Their solution was to add the armor to the engines, the part that had fewer hits when the planes got back.

And they were right. I'll leave you to figure out why that is the best solution. It's a great example of mathematical thinking. After you have convinced yourself why adding armor to the engines was the best strategy, you should buy a copy of Ellenberg's book and gain some understanding of just what mathematical thinking is, and why it is a crucial ability in today's world.

(My own book on mathematical thinking is more of a "how to" guide, as is my MOOC. Another, excellent book on mathematical thinking, that is somewhere between Ellenberg's and mine, is Burger and Starbird's The 5 Elements of Effective Thinking.)

Finally, and to some extent switching gears (and definitely switching media), I want to draw your attention to a new video game, DragonBox Elements, by the Norwegian-based educational technology company WeWantToKnow. The company made a splash with its first game, DragonBox (Algebra) a couple of years ago.

Unlike my own work in educational videogames, through my company BrainQuake, which is very strongly focused on real-world mathematical thinking, the DragonBox folks are seeking to enhance and strengthen school mathematics.

When I first played the new Elements game, I was initially confused, since I approached it with a Geometer's Sketchpad expectation. But Elements is not a geometry construction/exploration tool. The focus is on the importance of providing justification for steps in a proof. Knowing why something is true. And that is not only a key feature of GOFM (“Good Old Fashioned Math”), as was taught for two thousand years, it's one of the aspects of mathematics that is characteristic of mathematical thinking (as used in the real world). Euclid, the author of the first Elements (the book), would surely have approved.

The modern world has not made GOFM redundant. What has changed, and drastically, is the way GOFM fits in with the rest of human activities. Unless you are going to make a career for yourself in pure mathematics research, GOFM today is simply an amazingly powerful tool for acquiring one of the most important cognitive capacities in the 21st century: mathematical thinking.

In today's world, most of the important problems are complex and multi-faceted. There are few right answers. As Ellenberg demonstrates, mathematical thinking can help you choose better answers—and avoid being wrong.


Rick said...

Very nice read!

This reminds me very much of what I discussed with my friends in my last year in high school here in the Netherlands.

I'm afraid that the educational system in the Netherlands is still very much focused on repetitive problems (that indeed only have 1 correct answer!) and not at all on the ideas and concepts behind the problems (what am I actually doing when I calculate this primitive?).

It's frustrating to realize that: yes, I'm able to do the calculation, but if you were to ask me what I'm actually doing I'd be almost clueless. On the one hand I have all this knowledge on how certain systems work and how number relate to each other, but the isomorphism between the numbers and the real world is missing.

Isabel Gibson said...

This is great. I remember being really annoyed when a calculus class in university showed me that every series had an infinite number of correct next entries (or something very like that - it *was* back in the early 14th century, after all). I was still smarting from generating so many wrong answers in grade-school exercises designed to elicit one, and only one, answer. I thought then and think now that kids would be able to examine the different solutions they came up with and learn to see the different degrees of elegance in them - to learn why some answers were better than others. I hope this new approach is trying to teach kids to think.

Unknown said...

This is a great read!
I grew up at a time when calculators are not allowed for basic computations and students are forced to memorize the multiplication table. I finished my studies with great math grades but all because I was able to memorize algorithm and formulas. I never understood fully what the Pythagorean Theorem is for one. I am so used in memorization that once I forgot a formula, then I am stumped and cannot move on.
Hopefully with the onset of the Common Core State Standards students would be encouraged and trained to use their critical thinking and mathematical thinking. That they would be able to deeply explore, justify and prove why one thing is true.