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.

The Power of Dots

On June 29, the New York Times ran a story about the Common Core Mathematics Standards. If ever you wanted proof of the dismal mathematics education most Americans have been provided, you will find it in the story’s “human interest lede,” which described one mother’s response to seeing her daughter’s homework. By taking the daughter out of school to teach her herself “the old fashioned way” she herself had been subjected to, this well-meaning parent was ensuring that, as had clearly been the case for the mother, the daughter too would not be exposed to real mathematical thinking—the kind that in today’s world is a key to the most attractive jobs. Instead she would be subjected to the same, dreary, rote-skills-drills inflicted on previous generations—a process designed to train people for routine work in the pre-computer era, but so hopelessly inadequate for the 21st century that parents are un-equipped to figure out for themselves the simple (albeit unfamiliar) math homework their children are assigned.

Surely, if mathematics education should achieve one thing, it is develop the ability to figure things out for yourself. We’re not talking the Riemann Hypothesis here; the focus is basic school arithmetic, for heaven’s sake.

To continue with the Times article, arrays of dots seemed to figure large in this parent’s dislike of the Common Core. She felt it was pointless to spend time drawing and staring at arrays of dots.

True, it would be possible—and I am sure it happens—to generate tedious, and largely pointless, “busywork” exercises involving drawing arrays of dots. But the image of a Common Core math worksheet the Times chose to illustrate its story showed a very sensible, and deep use of dot diagrams, to understand structure in arithmetic. Much like the (extremely deep) dot array at the top of this article, which I’ll come to in a moment.

To the girl’s parent, mathematics is about numbers, but that’s just a surface feature. It’s really about structure. And throughout the ages, mathematicians have used the most simple symbols possible to bring out and understand that structure: namely, dots and lines.

The Times’ parent, so dismissive of time spent drawing and reflecting on dot diagrams, would, I am sure, think it a waste of time to devote any effort trying to make sense of the dot diagram I used to open this post. She would, I have no doubt, find it incomprehensible that an individual with a freshly-minted Ph.D. in mathematics would spend many months—at taxpayers’ expense—staring day-after-day at either that one diagram, or seemingly minor variations he would start each day by sketching out on a sheet of paper in front of him.

Well, I am that mathematician. That diagram helped me understand the framework that would be required to specify an infinite mathematical object of the third order of infinitude (aleph-2) by means of a family of infinite mathematical objects of the first order of infinitude (aleph-0). The top line of dots represents an increasing tower of objects that come together to form the desired aleph-2 object, and each of the lower lines of dots represent shorter towers of aleph-0 objects. In the 1970s, a number of us used those dot diagrams to solve mathematical problems that just a few years earlier had seemed impossible.

That particular kind of dot diagram was invented by a close senior colleague (and mentor) of mine, Professor Ronald Jensen, who called it a “morass.” He chose the name wisely, since the structure represented by those dots was extremely complex and intricate.

In contrast, the simple, rectangular array implicitly referred to in the New York Times article is used to help learners understand the much simpler (but still deep, and far more important to society) structure of numbers and the basic operations of arithmetic, as was well explained in a subsequent blog post by mathematics education specialist Christopher Danielson. The fact is, dot diagrams are powerful, for learners and world experts alike.

The problem facing parents (and many teachers) today, is that the present student generation is the one that, for the first time in history, is having to learn the mathematics the professionals use—what I and many other pros have started to call “mathematical thinking” in order to distinguish it from the procedural skills so important in past times.

The reason for that is that in the world today’s students will graduate into, computation is as plentiful as water or electricity. The smartphone we carry around with us is much faster, and more accurate, in carrying out mathematical procedures than any human.

In a single generation, society’s need for mathematical mastery has gone from procedural computation, to being able to make effective and reliable use of an effectively unlimited amount of automated computation. To put it bluntly, mastery of computational skills is no longer a marketable asset. The ability to make good use of computational power is where it’s at in math today.

For almost all the three thousand years of mathematical development, the focus in mathematics was calculation (numerical, symbolic, or geometric). Learning mathematics meant learning how to perform those calculations, which boiled down to achieving mastery of various procedures. Mastery of any one procedure could be achieved by rote learning—doing many examples, all essentially the same—leaving the only truly creative mental task that of recognition of which procedure to apply to solve which problem.

Numerical and symbolic calculation (arithmetic and algebra) are so simple and routine that we can program computers to do it for us. That is possible because calculation is essentially trivial. Perceiving and understanding structure, on the other hand, is something that (at least at the present time) requires human insight. It is not trivial and it is difficult. Dot diagrams can help us come to terms with that difficulty.

When movie director Gus Van Sant was faced with introducing the lead character, Will Hunting (played by Matt Damon) in the hit 1997 film Good Will Hunting, establishing in one shot that the hero was an uneducated (actually, self-educated) mathematical genius, the first encounter we had with Will showed him drawing a dot diagram on a blackboard in an MIT corridor.

You can be sure that when an experienced movie director like Gus Van Sant selects an establishing shot for the lead character, he does so with considerable care, on the advice of an expert. By showing Will writing a network of dots on a blackboard, Van Sant was right on the button in terms of portraying the kind of thing that professional mathematicians do all the time.

The one bit of license Van Sant took was that the diagram we saw Matt Damon writing was not the solution to a problem that had taken an MIT math professor two years to solve. (Unless MIT math professors are a lot less smart than we are led to believe!) It was a real solution to a real math problem, all right. I am pretty sure it was chosen because it fitted nicely on one blackboard and looked good on the screen. It absolutely conveyed the kind of (dotty) activity that mathematicians do all the time—the kind of (dotty) thing I did in my early post-Ph.D. years when I was working with Prof Jensen’s morasses.

But it’s actually a problem that anyone who has learned how to think mathematically should be able to solve in at most a few hours. Numberphile has an excellent video explaining the problem.

So, New York Times story parent, I hope you reconsider your decision to take your daughter out of school to teach her the way you were taught. The kind of mathematics you were taught was indeed required in times past. But not any more. The world has changed dramatically as far as mathematics is concerned. As with many other aspects of our lives, we have built machines to handle the more routine, procedural stuff, thereby putting a premium on the one thing where humans vastly outperform computers: creative thinking.

Those dot diagrams are all about creative thinking. A computer can understand numbers, and process millions of them faster than a human can write just one. But it cannot make sense of those dot diagrams. Because it does not know what any particular array of dots means! And it has no way to figure it out. (Unless a human tells it.)

Next month I’ll look further into the distinction between old-style procedural mathematics and the 21st-century need for mathematical thinking. In particular, I’ll look at an excellent recent book, Jordan Ellenberg’s How Not to be Wrong.

The book’s title is significant, since it recognizes that the vast majority of real-world mathematical problems do not have a unique right answer, and that the real power of mathematical thinking is making sure you are not wrong. (The book’s subtitle is “The power of mathematical thinking.”)

I’ll also look at a new mathematics video game that also focuses on mathematical thinking, this time, school-room Euclidean geometry. It’s called DragonBox Elements.

You might want to check out both.

Déjà vu all over again: Fibonacci and Steve Jobs — Part 2

This month’s column is the second of a two-part video presentation of a public address I gave recently at Princeton, where I have been spending this semester as a Visiting Professor.

The talk was based on my 2011 e-book Leonardo and Steve, which itself was a supplement to my print book The Man of Numbers, published the same year.

Both the e-book and my presentation show how Jobs’s introduction of the Macintosh computer in 1984 was an almost exact replay of Leonardo of Pisa’s (Fibonacci) 13th Century introduction to Europe of Hindu-Arabic arithmetic.

Part 1 appeared last month.

Déjà vu all over again: Fibonacci and Steve Jobs

This month’s column is the first of a two-part video presentation of a public address I gave recently at Princeton, where I have been spending this semester as a Visiting Professor.

The talk was based on my 2011 e-book Leonardo and Steve, which itself was a supplement to my print book The Man of Numbers, published the same year.

Both the e-book and my presentation show how Jobs’s introduction of the Macintosh computer in 1984 was an almost exact replay of Leonardo of Pisa’s (Fibonacci's) 13^th Century introduction to Europe of Hindu-Arabic arithmetic.

Part 2 will appear next month.

What good is math and why do we teach it?

This month’s column comes in lecture format. It’s a narrated videostream of the presentation file that accompanied the featured address I made recently at the MidSchoolMath National Conference, held in Santa Fe, NM, on March 27-29. It lasts just under 30 minutes, including two embedded videos.

In the talk, I step back from the (now largely metaphorical) blackboard and take a broader look at why we and our students are there is the first place.

Download the video here.

How Mountain Biking Can Provide the Key to the Eureka Moment

Because this blog post covers both mountain biking and proving theorems, it is being simultaneously published in Devlin’s more wide ranging blog profkeithdevlin.org. 

In my post last month, I described my efforts to ride a particularly difficult stretch of a local mountain bike trail in the hills just west of Palo Alto. As promised, I will now draw a number of conclusions for solving difficult mathematical problems.

Most of them will be familiar to anyone who has read George Polya’s classic book How to Solve It. But my main conclusion may come as a surprise unless you have watched movies such as Top Gun or Field of Dreams, or if you follow professional sports at the Olympic level.

Here goes, step-by-step, or rather pedal-stroke-by-pedal-stroke. (I am assuming you have recently read my last post.)

BIKE: Though bikers with extremely strong leg muscles can make the Alpine Road ByPass Trail ascent by brute force, I can't. So my first step, spread over several rides, was to break the main problem—get up an insanely steep, root strewn, loose-dirt climb—into smaller, simpler problems, and solve those one at a time.

MATH: Breaking a large problem into a series of smaller ones is a technique all mathematicians learn early in their careers. Those subproblems may still be hard and require considerable effort and several attempts, but in many cases you find you can make progress on at least some of them. The trick is to make each subproblem sufficiently small that it requires just one idea or one technique to solve it.

In particular, when you break the overall problem down sufficiently, you usually find that each smaller subproblem resembles another problem you, or someone else, has already solved.

When you have managed to solve the subproblems, you are left with the task of assembling all those subproblem solutions into a single whole. This is frequently not easy, and in many cases turns out to be a much harder challenge in its own right than any of the subproblem solutions, perhaps requiring modification to the subproblems or to the method you used to solve them.

BIKE: Sometimes there are several different lines you can follow to overcome a particular obstacle, starting and ending at the same positions but requiring different combinations of skills, strengths, and agility. (See my description last month of how I managed to negotiate the steepest section and avoid being thrown off course—or off the bike—by that troublesome tree-root nipple.)

MATH: Each subproblem takes you from a particular starting point to a particular end-point, but there may be several different approaches to accomplish that subtask. In many cases, other mathematicians have solved similar problems and you can copy their approach.

BIKE: Sometimes, the approach you adopt to get you past one obstacle leaves you unable to negotiate the next, and you have to find a different way to handle the first one.

MATH: Ditto.

BIKE: Eventually, perhaps after many attempts, you figure out how to negotiate each individual segment of the climb. Getting to this stage is, I think, a bit harder in mountain biking than in math. With a math problem, you usually can work on each subproblem one at a time, in any order. In mountain biking, because of the need to maintain forward (i.e., upward) momentum, you have to build your overall solution up in a cumulative fashion—vertically!

But the distinction is not as great as might first appear. In both cases, the step from having solved each individual subproblem in isolation to finding a solution for the overall problem, is a mysterious one that perhaps cannot be appreciated by someone who has not experienced it. This is where things get interesting.

Having had the experience of solving difficult (for me) problems in both mathematics and mountain biking, I see considerable similarities between the two. In both cases, the subconscious mind plays a major role—which is, I presume, why they seem mysterious. This is where this two-part blog post is heading.

BIKE: I ended my previous post by promising to

"look at the role of the subconscious in being able to put together a series of mastered steps in order to solve a big problem. For a very curious thing happened after I took the photos to illustrate this post. I walked back down to collect my bike from…where I'd left it, and rode up to continue my ride.

It took me four attempts to complete that initial climb!

And therein lies one of the biggest secrets of being able to solve a difficult math problem."

BOTH: How does the human mind make a breakthrough? How are we able to do something that we have not only never done before, but failed many times in attempts to do so? And why does the breakthrough always seem to occur when we are not consciously trying to solve the problem?

The first thing to note is that we never experience the process of making that breakthrough. Rather, what we experience, i.e., what we are conscious of, is having just made the breakthrough!

The sensation we have is a combined one of both elation and surprise. Followed almost immediately by a feeling that it wasn’t so difficult after all!

What are we to make of this strange process?

Clearly, I cannot provide a definitive, concrete answer to that question. No one can. It’s a mystery. But it is possible to make a number of relevant observations, together with some reasonable, informed speculations. (What follows is a continuation of sorts of the thread I developed in my 2000 book The Math Gene.)

The first observation is that the human brain is a result of millions of years of survival-driven, natural selection. That made it supremely efficient at (rapidly) solving problems that threaten survival. Most of that survival activity is handled by a small, walnut-shaped area of the brain called the amygdala, working in close conjunction with the body’s nervous system and motor control system.

In contrast to the speed at which our amydala operates, the much more recently developed neo-cortex that supports our conscious thought, our speech, and our “rational reasoning,” functions at what is comparatively glacial speed, following well developed channels of mental activity—channels that can be built up by repetitive training.

Because we have conscious access to our neo-cortical thought processes, we tend to regard them as “logical,” often dismissing the actions of the amygdala as providing (“mere,” “animal-like”) “instinctive reactions.” But that misses the point that, because that “instinctive reaction organ” has evolved to ensure its owner’s survival in a highly complex and ever changing environment, it does in fact operate in an extremely logical fashion, honed by generations of natural selection pressure to be in sync with its owner’s environment.

Which leads me to this.

Do you want to identify that part of the brain that makes major scientific (and mountain biking) breakthroughs?

I nominate the amygdala—the “reptilian brain” as it is sometimes called to reflect its evolutionary origin.

I should acknowledge that I am not the first person to make this suggestion. Well, for mathematical breakthroughs, maybe I am. But in sports and the creative arts, it has long been recognized that the key to truly great performance is to essentially shut down the neo-cortex and let the subconscious activities of the amygdala take over.

Taking this as a working hypothesis for mathematical (or mountain biking) problem solving, we can readily see why those moments of great breakthrough come only after a long period of preparation, where we keep working away—in conscious fashion—at trying to solve the problem or perform the action, seemingly without making any progress.

We can see too why, when the breakthrough (or the great performance) comes, it does so instantly and surprisingly, when we are not actively trying to achieve the goal, leaving our conscious selves as mere after-the-fact observers of the outcome.

For what that long period of struggle does is build a cognitive environment in which our reptilian brain—living inside and being connected to all of that deliberate, conscious activity the whole time—can make the key connections required to put everything together. In other words, investing all of that time and effort in that initial struggle raises the internal, cognitive stakes to a level where the amygdala can do its stuff.

Okay, I’ve been playing fast and loose with the metaphors and the anthropomorphization here. We’re really talking about biological systems, simply operating the way natural selection equipped them. But my goal is not to put together a scientific analysis, rather to try to figure out how to improve our ability to solve novel problems. My primary aim is not to be “right” (though knowledge and insight are always nice to have), but to be able to improve performance.

Let’s return to that tricky stretch of the ByPass section on the Alpine Road trail. What am I consciously focusing on when I make a successful ascent? 

BIKE: If you have read my earlier account, you will know that the difficult section comes in three parts. What I do is this. As I approach each segment, I consciously think about, and fix my eyes on, the end-point of that segment—where I will be after I have negotiated the difficulties on the way. And I keep my eyes and attention focused on that goal-point until I reach it. For the whole of the maneuver, I have no conscious awareness of the actual ground I am cycling over, or of my bike. It’s total focus on where I want to end up, and nothing else.

So who—or what—is controlling the bike? The mathematical control problem involved in getting a person-on-a-bike up a steep, irregular, dirt trail is far greater than that required to auto-fly a jet fighter. The calculations and the speed with which they would have to be performed are orders of magnitude beyond the capability of the relatively slow neuronal firings in the neocortex. There is only one organ we know of that could perform this task. And that’s the amygdala, working in conjunction with the nervous system and the body’s motor control mechanism in a super-fast constant feedback loop. All the neo-cortex and its conscious thought has to do is avoid getting in the way!

These days, in the case of Alpine Road, now I have “solved” the problem, the only things my conscious neo-cortex has to do on each occasion are switching my focus from the goal of one segment to the goal of the next. If anything interferes with my attention at one of those key transition moments, my climb is over—and I stop or fall.

What used to be the hard parts are now “done for me” by unconscious circuits in my brain.

MATH: In my case at least, what I just wrote about mountain biking accords perfectly with my experiences in making (personal) mathematical problem-solving breakthroughs.

It is by stepping back from trying to solve the problem by putting together everything I know and have learned in my attempts, and instead simply focusing on the problem itself—what it is I am trying to show—that I suddenly find that I have the solution.

It’s not that I arrive at the solution when I am not thinking about the problem. Some mathematicians have expressed their breakthrough moments that way, but I strongly suspect that is not totally true. When a mathematician has been trying to solve a problem for some months or years, that problem is always with them. It becomes part of their existence. There is not a single waking moment when that problem is not “on their mind.”

What they mean, I believe, and what I am sure is the case for me, is that the breakthrough comes when the problem is not the focus of our thoughts. We really are thinking about something else, often some mundane detail of life, or enjoying a marvelous view. (Google “Stephen Smale beaches of Rio” for a famous example.)

This thesis does, of course, explain why the process of walking up the ByPass Trail and taking photographs of all the tricky points made it impossible for me to complete the climb. True, I did succeed at the fourth attempt. But I am sure that was not because the first three were “practice.” Heavens, I’d long ago mastered the maneuvers required. It was because it took three failed attempts before I managed to erase the effects of focusing on the details to capture those images.

The same is true, I suggest, for solving a difficult mathematical problem. All of those techniques Polya describes in his book, some of which I list above, are essential to prepare the way for solving the problem. But the solution will come only when you forget about all those details, and just focus on the prize.

This may seem a wild suggestion, but in some respects it may not be entirely new. There is much in common between what I described above and the highly successful teaching method of R. L. Moore. For sure you have to do a fair amount of translation from his language to mine, but Moore used to demand that his students not clutter their minds by learning stuff, rather took each problem as it came and then try to solve it by pure reasoning, not giving up until they found the solution.

In terms of training future mathematicians, what these considerations imply, of course, is that there is mileage to be had from adopting some of the techniques used by coaches and instructors to produce great performances in sports, in the arts, in the military, and in chess.

Sweating the small stuff will make you good. But if you want to be great, you have to go beyond that—you have to forget the small stuff and keep your eye on the prize.

And if you are successful, be sure to give full credit for that Fields Medal or that AMS Prize where it is rightly due: dedicate it to your amygdala. It will deserve it.

Want to learn how to prove a theorem? Go for a mountain bike ride

Because this blogpost covers both mountain biking and proving theorems, it is being simultaneously published in Devlin’s more wide ranging blog profkeithdevlin.org.

Mountain biking is big in the San Francisco Bay Area, where I live. (In its present day form, using specially built bicycles with suspension, the sport/pastime was invented a few miles north in Marin County in the late 1970s.) Though there are hundreds of trails in the open space preserves that spread over the hills to the west of Stanford, there are just a handful of access trails that allow you to start and finish your ride in Palo Alto. Of those, by far the most popular is Alpine Road.

My mountain biking buddies and I ascend Alpine Road roughly once a week in the mountain biking season (which in California is usually around nine or ten months long). In this post, I'll describe my own long struggle, stretching over many months, to master one particularly difficult stretch of the climb, where many riders get off and walk their bikes.

[SPOILER: If your interest in mathematics is not matched by an obsession with bike riding, bear with me. My entire account is actually about how to set about solving a difficult math problem, particularly proving a theorem. I'll draw the two threads together in a subsequent post, since it will take me into consideration of how the brain works when it does mathematics. For now, I'll leave the drawing of those conclusions as an exercise for the reader! So when you read mountain biking, think math.]

Alpine Road used to take cars all the way from Palo Alto to Skyline Boulevard at the summit of the Coastal Range, but the upper part fell into disrepair in the late 1960s, and the two-and-a-half-mile stretch from just west of Portola Valley to where it meets the paved Page Mill Road just short of  Skyline  is now a dirt trail, much frequented by hikers and mountain bikers.

Alpine Road. The trail is washed
out just round the bend

A few years ago, a storm washed out a short section of the trail about half a mile up, and the local authority constructed a bypass trail. About a quarter of a mile long, it is steep, narrow, twisted, and a constant staircase of tree roots protruding from the dirt floor. A brutal climb going up and a thrilling (beginners might say terrifying) descent on the way back. Mountain bike heaven.

There is one particularly tricky section right at the start. This is where you can develop the key abilities you need to be able to prove mathematical theorems.

So you have a choice. Read Polya's classic book, or get a mountain bike and find your own version of the Alpine Road ByPass Trail. (Better still: do both!) 

My mountain bike at the start of the bypass trail

When I first encountered Alpine Road Dirt a few years ago, it took me many rides before I managed to get up the first short, steep section of the ByPass Trail. 

What lies around that sharp left-hand turn?

It starts innocently enough—because you cannot see what awaits just around that sharp left-hand turn.

After you have made the turn, you are greeted with a short narrow downhill. You will need it to gain as much momentum as you can for what follows.

The short, narrow descent

I've seen bikers with extremely strong leg muscles who can plod their way up the wall that comes next, but I can't do it that way. I learned how to get up it by using my problem-solving/theorem-proving skills.

The first thing was to break the main problem—get up the insanely steep, root strewn, loose-dirt climb—into smaller, simpler problems, and solve those one at a time. Classic Polya.

But it's Polya with a twist—and by "twist" I am not referring to the sharp triple-S bend in the climb. The twist in this case is that the penalty for failure is physical, not emotional as in mathematics. I fell off my bike a lot. The climb is insanely steep. So steep that unless you bend really low, with your chin almost touching your handlebar, your front wheel will lift off the ground. That gives rise to an unpleasant feeling of panic that is perhaps not unlike the one that many students encounter when faced with having to prove a theorem for the first time.

If you are not careful, your front wheel will lift 
off the ground.

The photo above shows the first difficult stretch. Though this first sub-problem is steep, there is a fairly clear line to follow to the right that misses those roots, though at the very least its steepness will slow you down, and on many occasions will result in an ungainly, rapid dismount. And losing momentum is the last thing you want, since the really hard part is further up ahead, near the top in the picture.

Also, do you see that rain- and tire-worn groove that curves round to the right just over half way up—just beyond that big root coming in from the left? It is actually deeper and narrower than it looks in the photo, so unless you stay right in the middle of the groove you will be thrown off line, and your ascent will be over. (Click on the photo to enlarge it and you should be able to make out what I mean about the groove. Staying in the groove can be tricky at times.)

Still, despite difficulties in the execution, eventually, with repeated practice, I got to the point of  being able to negotiate this initial stretch and still have some forward momentum. I could get up on muscle memory. What was once a series of challenging problems, each dependent on the previous ones, was now a single mastered skill.

[Remember, I don't have super-strong leg muscles. I am primarily a road bike rider. I can ride for six hours at a 16-18 mph pace, covering up to 100 miles or more. But to climb a steep hill I have to get off the saddle and stand on the pedals, using my body weight, not leg power. Unfortunately, if you take your weight off the saddle on a mountain bike on a steep dirt climb, your rear wheel will start to spin and you come to a stop - which on a steep hill means jump off quick or fall. So I have to use a problem solving approach.]

Once I'd mastered the first sub-problem, I could address the next. This one was much harder. See that area at the top of the photo above where the trail curves right and then left? Here is what it looks like up close.

The crux of the climb/problem. Now it is really steep.

(Again, click on the photo to get a good look. This is the mountain bike equivalent of being asked to solve a complex math problem with many variables.)

Though the tire tracks might suggest following a line to the left, I suspect they are left by riders coming down. Coming out of that narrow, right-curving groove I pointed out earlier, it would take an extremely strong rider to follow the left-hand line. No one I know does it that way. An average rider (which I am) has to follow a zig-zag line that cuts down the slope a bit.

Like most riders I have seen—and for a while I did watch my more experienced buddies negotiate this slope to get some clues—I start this part of the climb by aiming my bike between the two roots, over at the right-hand side of the trail. (Bottom right of picture.)

The next question is, do you go left of that little tree root nipple, sticking up all on its own, or do you skirt it to the right? (If you enlarge the photo you will see that you most definitely do not want either wheel to hit it.)

The wear-marks in the dirt show that many riders make a sharp left after passing between those two roots at the start, and steer left of the root protrusion. That's very tempting, as the slope is notably less (initially). I tried that at first, but with infrequent success. Most often, my left-bearing momentum carried me into that obstacle course of tree roots over to the left, and though I sometimes managed to recover and swing  out to skirt to the left of that really big root, more often than not I was not able to swing back right and avoid running into that tree!

The underlying problem with that line was that thin looking root at the base of the tree. Even with the above photo blown up to full size, you can't really tell how tricky an obstacle it presents at that stage in the climb. Here is a closer view.

The obstacle course of tree roots that awaits 
the rider who bears left

If you enlarge this photo, you can probably appreciate how that final, thin root can be a problem if you are out of strength and momentum. Though the slope eases considerably at that point, I—like many riders I have seen—was on many occasions simply unable to make it either over the root or circumventing it on one side—though all three options would clearly be possible with fresh legs. And on the few occasions I did make it, I felt I just got lucky—I had not mastered it. I had got the right answer, but I had not really solved the problem. So close, so often. But, as in mathematics, close is not good enough.

After realizing I did not have the leg strength to master the left-of-the-nipple path, I switched to taking the right-hand line. Though the slope was considerable steeper (that is very clear from the blown-up photo), the tire-worn dirt showed that many riders chose that option.

Several failed attempts and one or two lucky successes convinced me that the trick was to steer to the right of the nipple and then bear left around it, but keep as close to it as possible without the rear wheel hitting it, and then head for the gap between the tree roots over at the right.

After that, a fairly clear left-bearing line on very gently sloping terrain takes you round to the right to what appears to be a crest. (It turns out to be an inflection point rather than a maximum, but let's bask for a while in the success we have had so far.)

Here is our brief basking point.

The inflection point. One more detail to resolve.

As we oh-so-briefly catch our breath and "coast" round the final, right-hand bend and see the summit ahead, we come—very suddenly—to one final obstacle.

The summit of the climb

At the root of the problem (sorry!) is the fact that the right-hand turn is actually sharper than the previous photo indicates, almost a switchback. Moreover, the slope kicks up as you enter the turn. So you might not be able to gain sufficient momentum to carry you over one or both of those tree roots on the left that you find your bike heading towards. And in my case, I found I often did not have any muscle strength left to carry me over them by brute force.

What worked for me is making an even tighter turn that takes me to the right of the roots, with my right shoulder narrowly missing that protruding tree trunk. A fine-tuned approach that replaces one problem (power up and get over those roots) with another one initially more difficult (slow down and make the tight turn even tighter).

And there we are. That final little root poking up near the summit is easily skirted. The problem is solved.

To be sure, the rest of the ByPass Trail still presents several other difficult challenges, a number of which took me several attempts before I achieved mastery. Taken as a whole, the entire ByPass is a hard climb, and many riders walk the entire quarter mile. But nothing is as difficult as that initial stretch. I was able to ride the rest long before I solved the problem of the first 100 feet. Which made it all the sweeter when I finally did really crack that wall.

Now I (usually) breeze up it, wondering why I found it so difficult for so long.

Usually? In my next post, I'll use this story to talk about strategies for solving difficult mathematical problems. In particular, I'll look at the role of the subconscious in being able to put together a series of mastered steps in order to solve a big problem. For a very curious thing happened after I took the photos to illustrate this post. I walked back down to collect my bike from the ByPass sign where I'd left it, and rode up to continue my ride.

It took me four attempts to complete that initial climb!

And therein lies one of the biggest secrets of being able to solve a difficult math problem.

To be continued ...

See "How Mountain Biking Can Provide the Key to the Eureka Moment"