Archive for the 'problem solving' Category

How mountain biking can provide the key to the Eureka moment

Because this blogpost covers both mountain biking and proving theorems, it is being simultaneously published by the Mathematical Association of America in my Devlin’s Angle series. 

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 blogpost 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 synch 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 “reptilean 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 reptilean 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 did 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 by the Mathematical Association of America in my Devlin’s Angle series.

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

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

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?

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.

The short, narrow descent

The short, narrow descent

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.

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.

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

Steep. 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.

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 nobbly 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

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 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.

AD8

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.

AD9

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, close to 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 …


I'm Dr. Keith Devlin, a mathematician at Stanford University, an author, the Math Guy on NPR's Weekend Edition, and an avid cyclist. (Yes, that's me cycling on the Marin Headland.)

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