Posts Tagged 'mathematical thinking'

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 …

How to design video games that support good math learning: Level 5

Procedures or thinking?
Part 5 of a series 

The vast majority of video games that claim to teach mathematics do not actually do that. Rather, what they do is provide a means for students to practice what they have already been taught. For the most part, the focus is on basic computational skills.

A good example is the first-person shooter Timez Attack. Mastery of the multiplication bonds (times tables in parent-speak) is an extremely useful thing to achieve, and the sooner the better. All it requires is sufficient repetition, and I know of no better way to achieve that than with an entertaining video game.

Such games are the low hanging fruit for the math ed video game designer, and like most low hanging fruit, it has pretty well all been picked, leaving game designers coming into the math ed space having to look elsewhere for a useful application of their talents. The good news is, since repetitive practice of basic computational skills is a tiny part of learning mathematics — albeit an important part, in my view (some educators disagree) —  most of the fruit in the math ed orchard is still waiting to be picked. The bad news is, that fruit is a lot higher up, and thus more difficult to reach.

The difficulty hits you as soon as you decide to go for more than mastery (ideally to fluency) of already taught basic computational skills. Are you going to approach mathematics as a collection of procedures or as a way of thinking? These are not completely separate classifications; indeed, the latter is in many ways a  broader conception than the former. But they do tend to cash out in very different forms of pedagogy. (Spoiler: instruction versus guided-discovery.)

This distinction is to a great extent relatively recent. Until the nineteenth century, mathematicians viewed the discipline as a collection of procedures for solving various kinds of problems. Originally, the problems studied arose in the world. Then, in due course, the focus widened to include more abstract problems arising within mathematics itself. Proficiency in math meant being able to carry out calculations or manipulate symbolic expressions to solve problems.

By and large, high school mathematics is still very much based on that earlier tradition, so few people outside the professional mathematical community are aware that in the middle of the 19th century, a revolution took place.

Working in the revolution’s epicenter, the small university town of Göttingen in Germany, the mathematicians Lejeune Dirichlet, Richard Dedekind, and Bernhard Riemann pioneered a new, broader conception of mathematics, where the primary focus was not performing a calculation or computing an answer, but formulating and understanding abstract concepts and relationships. This was a shift in emphasis from doing to understanding.

For the Göttingen revolutionaries, mathematics was about “Thinking in concepts” (Denken in Begriffen). Mathematical objects were no longer thought of as given primarily by formulas, but rather as carriers of conceptual properties. Proving was no longer a matter of transforming terms in accordance with rules, but a process of logical deduction from concepts.

Of course, during the course of this conceptual thinking, mathematicians still made use of procedures. What changed was the primary emphasis. The reason for the change? An increase in complexity, in science, technology, business, society, and, derivatively, within mathematics itself. In a simple world, a few well-practiced procedures can generally get you by. But when things get more complex, you need understanding in order to select from a variety of different procedures, to fix old procedures that no longer work, and to develop new ones.

I give this somewhat lengthy detour through recent mathematical history not because it has a direct bearing on how we teach K-12 mathematics. The one attempt to modify K-12 education to take account of the 19th century shift in mathematics as practiced by the professional mathematicians, the “New Math” movement of the 1960s, was so badly bungled that even a professional mathematician, Tom Lehrer, satirized it. (It was also hardly “new math” at the time, being already a century old.) Rather, I am stressing the distinction between math-as-procedures and math-as-thinking because it is now extremely relevant to the way we educate our next generation of citizens. The complexity of 21st Century life is such that ordinary citizens now need to upgrade their mathematical knowledge and abilities the same way the professional mathematicians did in the mid nineteenth century. The changes in society, and in particular technology and the way we do business, that were made possible by the newer, richer, and more powerful mathematics that developed in the 20th Century, now affect us all in the 21st.

I discussed the growing importance of “mathematical thinking” in a “Devlin’s Angle” column for the MAA back in 2010, and summarized those arguments in a more recent article in the Huffington Post. My purpose here is not to argue for any one approach to the design of video games to help students learn math. Heavens, the medium is so new, and there are so few games of any real educational merit, there is scope for a wide variety of approaches. Give me any video game that plays well and helps students learn math and I’ll applaud, whatever the pedagogy.

What distresses me is that the medium offers so much promise for good mathematics learning, it is a waste of time, effort, and money to focus on the lowest level — repetitive practice of the basic, procedural, computational skills. We’ve done that. Let’s move on.

Step 1 for the math ed video game designer today is, to recap, deciding whether to develop a game to help students master mathematics procedures or to develop powerful mathematical thinking capacities. As readers of my book will already know, I favor the latter, in large part because mastery of mathematical thinking capacity carries mastery of procedures along with it, just as the person who sets out to build a house will have to develop skills in bricklaying, carpentry, plumbing, and so on, along the way. But as I said a moment ago, the challenge we face in K-12 mathematics education is so great, and video games offer such potential, hitherto largely untapped, I’ll settle for any approach that works.

It’s your call which view of mathematics you take, pre-1850 or post. Both have strong track records. But you do have to make that call, as it will affect every design choice you make from then on. Engineers who set out to build a bicycle and then act as if they are building a car tend not to succeed, even though both are transportation devices. I’ll try to make this blog series helpful whichever way you make the call.


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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