Tuesday, April 1, 2014

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.

Saturday, March 1, 2014

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.

Saturday, February 1, 2014

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 enoughbecause 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 climbinto smaller, simpler problems, and solve those one at a time. Classic Polya.
But it's Polya with a twistand 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 upjust 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 seenand for a while I did watch my more experienced buddies negotiate this slope to get some cluesI 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, Ilike many riders I have seenwas on many occasions simply unable to make it either over the root or circumventing it on one sidethough all three options would clearly be possible with fresh legs. And on the few occasions I did make it, I felt I just got luckyI 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 comevery suddenlyto 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.

Friday, January 3, 2014

23 and Me. Play it again, Sam

It’s one of the most famous lines from one of the most famous movies of all time, Casablanca. Except it’s not what Ilsa, played by Ingrid Bergman, actually said, which was “Play it once, Sam, for old times' sake . . . [NO RESPONSE] . . . Play it, Sam. Play 'As Time Goes By.'”

This month’s column is in response to the emails I receive from time to time asking for a reference to articles I have written for the MAA since I began on that mathemaliterary journey back in 1991. (Yes, I just made that word up. Google returns nothing. But it soon will.)

I first started writing monthly articles for the MAA back in September 1991 when I took over as editor of the Association’s monthly print magazine FOCUS. When I stepped down as FOCUS editor in January 1996, the MAA launched its website, and along with it Devlin’s Angle.

During that time, in addition to moving from print to online, the MAA website went through two overhauls, leaving the archives spread over three volumes:

January 1996 – December 2003

January 2004 – July 2011

August 2011 – present

Throughout those 23 years, I’ve wandered far and wide across the mathematical and mathematics education landscape. But three ongoing themes emerged. None of them was planned. In each case, I simply wrote something that generated interest – and for one theme considerable controversy – and as a result I kept coming back to it.

I continue to receive emails asking about articles I wrote on the first two of those three themes, and the third is still very active. So I am devoting this month’s column to providing an index to those three themes.

I’ll start with the most controversial: what is multiplication? This began innocently enough, with a throw-away final remark to a piece I wrote back in 2007. I little knew the firestorm I was about to unleash.

What is Multiplication?

September 2007, What is conceptual understanding?

June 2008, It Ain't No Repeated Addition

July-August 2008, It's Still Not Repeated Addition

September 2008, Multiplication and Those Pesky British Spellings

December 2008, How Do We Learn Math?

January 2009, Should Children Learn Math by Starting with Counting?

January 2010, Repeated Addition - One More Spin

January 2011, What Exactly is Multiplication?

November 2011, How multiplication is really defined in Peano arithmetic


Mathematical Thinking

I first started making the distinction between mathematics and mathematical thinking in the early 1990s, when an extended foray into mathematical linguistics and then sociolinguistics led to an interest in mathematical cognition that continues to this day.

April 1996, Are Mathematicians Turning Soft?

October 1996, Wanted: A New Mix

September 1999, What Can Mathematics Do For The Businessperson?

January 2008, American Mathematics in a Flat World

February 2008, Mathematics for the President and Congress

October 2009, Soft Mathematics

July 2010, Wanted: Innovative Mathematical Thinking

September 2012, What is mathematical thinking?


MOOCS

No introduction necessary. MOOCs are constantly in the news. Though I was one of the early pioneers in developing the Stanford MOOCs that generated all the media interest in 2012, and I believe the first person to offer a mathematics MOOC (Introduction to Mathematical Thinking), the idea goes back to a course given at Athabasca University in Canada, back in 2008.

May 2012, Math MOOC – Coming this fall. Let’s Teach the World

November 2012, MOOC Lessons

December 2012, The Darwinization of Higher Education

January 2013, R.I.P. Mathematics? Maybe.

February 2013, The Problem with Instructional Videos

March 2013, Can we make constructive use of machine-graded, multiple-choice questions in university mathematics education?

September 2013, Two Startups in One Week


More about MOOCs

In addition to the MOOC articles listed above, I have also written articles about the topic in my own blog MOOCtalk.org and for the Huffington Post. Here are the references:

MOOCTALK

An irregular series of posts starting on May 5, 2012

HUFFINGTON POST

December 2013, MOQR, Anyone? Learning by Evaluating

March 2, 2013, MOOCs and the Myths of Dropout Rates and Certification

March 27, 2013, Can Massive Open Online Courses Make Up for an Outdated K-12 Education System?

August 19, 2013, MOOC Mania Meets the Sober Reality of Education

November 18, 2013, Why MOOCs May Still Be Silicon Valley's Next Grand Challenge
http://www.huffingtonpost.com/dr-keith-devlin/why-moocs-remain-silicon-_b_4289739.html

Monday, December 2, 2013

MOQR, Anyone? Learning by Evaluating

Many colleges and universities have a mathematics or quantitative reasoning requirement that ensures that no student graduates without completing at least one sufficiently mathematical course.

Recognizing that taking a regular first-year mathematics course—designed for students majoring in mathematics, science, or engineering—to satisfy a QR requirement is not educationally optimal (and sometimes a distraction for the instructor and the TAs who have to deal with students who are neither motivated nor well prepared for the full rigors and pace of a mathematics course), many institutions offer special QR courses.

I’ve always enjoyed giving such courses, since they offer the freedom to cover a wide swathe of mathematics—often new or topical parts of mathematics. Admittedly they do so at a much more shallow depth than in other courses, but a depth that was always a challenge for most students who signed up.

Having been one of the pioneers of so-called “transition courses” for incoming mathematics majors back in the 1970s, and having given such courses many times in the intervening years, I never doubted that a lot of the material was well suited to the student in search of meeting a QR requirement. The problem with classifying a transition course as a QR option is that the goal of preparing an incoming student for the rigors of college algebra and real analysis is at odds with the intent of a QR requirement. So I never did that.

Enter MOOCs. A lot of the stuff that is written about these relatively new entrants to the higher education landscape is unsubstantiated hype and breathless (if not fearful) speculation. The plain fact is that right now no one really knows what MOOCs will end up looking like, what part or parts of the population they will eventually serve, or exactly how and where they will fit in with the rest of higher education. Like most others I know who are experimenting with this new medium, I am treating it very much as just that: an experiment.

The first version of my MOOC Introduction to Mathematical Thinking, offered in the fall of 2012, was essentially the first three-quarters of my regular transition course, modified to make initial entry much easier, delivered as a MOOC. Since then, as I have experimented with different aspects of online education, I have been slowly modifying it to function as a QR-course, since improved quantitative reasoning is surely a natural (and laudable) goal for online courses with global reach—that “free education for the world” goal is still the main MOOC-motivator for me.

I am certainly not viewing my MOOC as an online course to satisfy a college QR requirement. That may happen, but, as I noted above, no one has any real idea what role(s) MOOCs will end up fulfilling. Remember, in just twelve months, the Stanford MOOC startup Udacity, which initiated all the media hype, went from “teach the entire world for free” to “offer corporate training for a fee.” (For my (upbeat) commentary on this rapid progression, see my article in the Huffington Post.)

Rather, I am taking advantage of the fact that free, no-credential MOOCs currently provide a superb vehicle to experiment with ideas both for classroom teaching and for online education. Those of us at the teaching end not only learn what the medium can offer, we also discover ways to improve our classroom teaching; while those who register as students get a totally free learning opportunity. (Roughly three-quarters of them already have a college degree, but MOOC enrollees also include thousands of first-time higher education students from parts of the world that offer limited or no higher education opportunities.)

The biggest challenge facing anyone who wants to offer a MOOC in higher mathematics is how to handle the fact that many of the students will never receive expert feedback on their work. This is particularly acute when it comes to learning how to prove things. That’s already a difficult challenge in a regular class, as made clear in this great blog post by “mathbabe” Cathy O’Neil. In a MOOC, my current view is it would be unethical to try. The last thing the world needs are (more) people who think they know what a proof is, but have never put that knowledge to the test.

But when you think about it, the idea behind QR is not that people become mathematicians who can prove things, rather that they have a base level of quantitative literacy that is necessary to live a fulfilled, rewarding life and be a productive member of society. Being able to prove something mathematically is a specialist skill. The important general ability in today’s world is to have a good understanding of the nature of the various kinds of arguments, the special nature of mathematical argument and its role among them, and an ability to judge the soundness and limitations of any particular argument.

In the case of mathematical argument, acquiring that “consumer’s understanding” surely involves having some experience in trying to construct very simple mathematical arguments, but far more what is required is being able to evaluate mathematical arguments.

And that can be handled in a MOOC. Just present students with various mathematical arguments, some correct, others not, and machine-check if, and how well, they can determine their validity.

Well, that leading modifier “just” in that last sentence was perhaps too cavalier. There clearly is more to it than that. As always, the devil is in the details. But once you make the shift from viewing the course (or the proofs part of the course) as being about constructing proofs to being about understanding and evaluating proofs, then what previously seemed hopeless suddenly becomes rife with possibilities.

I started to make this shift with the last session of my MOOC this fall, and though there were significant teething troubles, I saw enough to be encouraged to try it again—with modifications—to an even greater extent next year.

Of course, many QR courses focus on appreciation of mathematics, spiced up with enough “doing math” content to make the course defensibly eligible for QR fulfillment. What I think is far less common—and certainly new to me—is using the evaluation of proofs as a major learning vehicle.

What makes this possible is that the Coursera platform on which my MOOC runs has developed a peer review module to support peer grading of student papers and exams.

The first times I offered my MOOC, I used peer evaluation to grade a Final Exam. Though the process worked tolerably well for grading student mathematics exams—a lot better than I initially feared—to my eyes it still fell well short of providing the meaningful grade and expert feedback a professional mathematician would give. On the other hand, the benefit to the students that came from seeing, and trying to evaluate, the proof attempts of other students, and to provide feedback, was significant—both in terms of their gaining much deeper insight into the concepts and issues involved, and in bolstering their confidence.

When the course runs again in a few week's time, the Final Exam will be gone, replaced by a new course culmination activity I am calling Test Flight.

How will it go? I have no idea. That’s what makes it so interesting. Based on my previous experiments, I think the main challenges will be largely those of implementation. In particular, years of educational high-stakes testing robs many students of the one ingredient essential to real learning: being willing to take risks and to fail. As young children we have it. Schools typically drive it out of us. Those of us lucky enough to end up at graduate school reacquire it—we have to.

I believe MOOCs, which offer community interaction through the semi-anonymity of the Internet, offer real potential to provide others with a similar opportunity to re-learn the power of failure. Test Flight will show if this belief is sufficiently grounded, or a hopelessly idealistic dream! (Test flights do sometimes crash and burn.)

The more people learn to view failure as an essential constituent of good learning, the better life will become for all. As a world society, we need to relearn that innate childhood willingness to try and to fail. A society that does not celebrate the many individual and local failures that are an inevitable consequence of trying something new, is one destined to fail globally in the long term.


For those interested, I’ll be describing Test Flight, and reporting on my progress (including the inevitable failures), in my blog MOOCtalk.org as the experiment continues. (The next session starts on February 3.)

Monday, November 4, 2013

The Educational Power of Elementary Arithmetic

The trouble with writing about, or quoting, Liping Ma, is that everyone interprets her words through their own frame, influenced by their own experiences and beliefs.

“Well, yes, but isn’t that true for anyone reading anything?” you may ask. True enough. But in Ma’s case, readers often arrive at diametrically opposed readings. Both sides in the US Math Wars quote from her in support of their positions.

That happened with the book that brought her to most people’s attention, Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States, first published in 1999. And I fear the same will occur with her recent article "A Critique of the Structure of U.S. Elementary School Mathematics," published in the November issue of the American Mathematical Society Notices.

Still, if I stopped and worried about readers completely misreading or misinterpreting things I write, Devlin’s Angle would likely appear maybe once or twice a year at most. So you can be sure I am about to press ahead and refer to her recent article regardless.

My reason for doing so is that I am largely in agreement with what I believe she is saying. Her thesis (i.e., what I understand her thesis to be) is what lay behind the design of my MOOC and my recently released video game. (More on both later.)

Broadly speaking, I think most of the furor about K-12 mathematics curricula that seems to bedevil every western country except Finland is totally misplaced. It is misplaced for the simple, radical (except in Finland) reason that curriculum doesn’t really matter.  What matter are teachers. (That last sentence is, by the way, the much sought after “Finnish secret” to good education.) To put it simply:

BAD CURRICULUM + GOOD OR WELL-TRAINED TEACHERS = GOOD EDUCATION

GOOD CURRICULUM + POOR OR POORLY-TRAINED TEACHERS = POOR EDUCATION

I am very familiar with the Finnish education system. The Stanford H-STAR institute I co-founded and direct has been collaborating with Finnish education researchers for over a decade, we host education scholars from Finland regularly, I travel to Finland several times a year to work with colleagues there, I am on the Advisory Board of CICERO Learning, one of their leading educational research organizations, I’ve spoken with members of the Finnish government whose focus is education, and I’ve sat in on classes in Finnish schools. So I know from firsthand experience in the western country that has got it right that teachers are everything and curriculum is at most (if you let it be) a distracting side-issue.

The only people for whom curriculum really matters are politicians and the politically motivated (who can make political capital out of curriculum) and publishers (who make a lot of financial capital out of it).

But I digress: Finland merely serves to provide an existence proof that providing good mathematics education in a free, open, western society is possible and has nothing to do with curriculum. Let’s get back to Liping Ma’s recent Notices article. For she provides a recipe for how to do it right in the curriculum-obsessed, teacher-denigrating US.

Behind Ma’s suggestion, as well as behind my MOOC and my video game (both of which I have invested a lot of effort and resources into) is the simple (but so often overlooked) observation that, at its heart, mathematics is not a body of facts or procedures but a way of thinking. Once a person has learned to think that way, it becomes possible to learn and use pretty well any mathematics you need or want to know about, when you need or want it.

In principle, many areas of mathematics can be used to master that way of thinking, but some areas are better suited to the task, since their learning curve is much more forgiving to the human brain.

For my MOOC, which is aimed at beginning mathematics students at college or university, or high school students about to become such, I take formalizing the use of language and the basic rules of logical reasoning (in everyday life) as the subject matter, but the focus is as described in the last two words of the course’s title: Introduction to Mathematical Thinking.

Apart from the final two weeks of the course, where we look at elementary number theory and beginning real analysis, there is really no mathematics in my course in the usual sense of the word. We use everyday reasoning and communication as the vehicle to develop mathematical thinking.

[SAMPLE PROBLEM: Take the famous (alleged) Abraham Lincoln quote, “You can fool all of the people some of the time and some of the people all of the time, but you cannot fool all the people all the time.” What is the simplest and clearest positive expression you can find that states the negation of that statement? Of course, you first have to decide what “clearest”, “simplest”, and “positive” mean.]

Ma’s focus in her article is beginning school mathematics. She contrasts the approach used in China until 2001 with that of the USA. The former concentrated on “school arithmetic” whereas, since the 1960s, the US has adopted various instantiations of a “strands” approach. (As Ma points out, since 2001, China has been moving towards a strands approach. By my read of her words, she thinks that is not a wise move.)

As instantiated in the NCTM’s 2001 Standards document, elementary school mathematics should cover ten separate strands: number and operations, problem solving, algebra, reasoning and proof, geometry, communication, measurement, connections, data analysis and probability, and representation.

In principle, I find it hard to argue against any of these—provided they are viewed as different facets of a single whole.

The trouble is, as soon as you provide a list, it is almost inevitable that the first system administrator whose desk it lands on will turn it into a tick-the-boxes spreadsheet, and in turn the textbook publishers will then produce massive (hence expensive) textbooks with (at least) ten chapters, one for each column of the spreadsheet. The result is the justifiably maligned “Mile wide, inch deep” US elementary school curriculum.

It’s not that the idea is wrong in principle. The problem lies in the implementation. It’s a long path from a highly knowledgeable group of educators drawing up a curriculum to what finds its way into the classroomoften to be implemented by teachers woefully unprepared (through no fault of their own) for the task, answerable to administrators who serve political leaders, and forced to use textbooks that reinforce the separation into strands rather than present them as variations on a single whole.

Ma’s suggestion is to go back to using arithmetic as the primary focus, as was the case in Western Europe and the United States in the years of yore and China until the turn of the Millennium, and use that to develop all of the mathematical thinking skills the child will require, both for later study and for life in the twenty-first century. I think she has a point. A good point.

She is certainly not talking about drill-based mastery of the classical Hindu-Arabic algorithms for adding, subtracting, multiplying, and dividing, nor is she suggesting that the goal should be for small human beings to spend hours forcing their analogically powerful, pattern-recognizing brains to become poor imitations of a ten-dollar calculator. What was important about arithmetic in past eras is not necessarily relevant today. Arithmetic can be used to trade chickens or build spacecraft.

No, if you read what she says, and you absolutely should, she is talking about the rich, powerful structure of the two basic number systems, the whole numbers and the rational numbers.

Will that study of elementary arithmetic involve lots of practice for the students? Of course it will. A child’s life is full of practice. We are adaptive creatures, not cognitive sponges. But the goalthe motivation for and purpose of that practiceis developing arithmetic thinking, and moreover doing so in a manner that provides the foundation for, and the beginning of, the more general mathematical thinking so important in today’s world, and hence so empowering for today’s citizens.

The whole numbers and the rational numbers are perfectly adequate for achieving that goal. You will find pretty well every core feature of mathematics in those two systems. Moreover, they provide an entry point that everyone is familiar with, teacher, administrator, and beginning elementary school student alike.

In particular, a well trained teacher can build the necessary thinking skills and the mathematical sophistication and cover whatever strands are in current favorwithout having to bring in any other mathematical structure.

When you adopt the strands approach (pick your favorite flavor), it’s very easy to skip over school arithmetic as a low-level skill set to be “covered” as quickly as possible in order to move on to the “real stuff” of mathematics. But Ma is absolutely right in arguing that this is to overlook the rich potential still offered today by what are arguably (I would so argue) the most important mathematical structures ever developed: the whole and the rational numbers and their associated elementary arithmetics.

For what is often not realized is that there is absolutely nothing elementary about elementary arithmetic.

Incidentally, for my video game, Wuzzit Trouble, I took whole number arithmetic and built a game around it. If you play it through, finding optimal solutions to all 75 puzzles, you will find that you have to make use of increasingly sophisticated arithmetical reasoning. (Integer partitions, Diophantine equations, algorithmic thinking, and optimization.)

I doubt Ma had video game instantiations of her proposal in mind, but when I first read her article, almost exactly when my game was released in the App Store (the Android version came a few weeks later) that’s exactly what I saw.

Other games my colleagues and I have designed but not yet built are based on different parts of mathematics. We started with one built around elementary arithmetic because arithmetic provides all the richness you need to develop mathematical thinking, and we wanted our first game to demonstrate the potential of game-based learning in thinking-focused mathematical education (as opposed to the more common basic-skills focus of most mathematics-educational games). In starting with an arithmetic-based game, we were (at the time unknowingly) endorsing the very point Ma was to make in her article.

Tuesday, October 1, 2013

Math Ed? Sometimes It Takes a Team

In last month’s column, I reflected on how modern technology enables one person—in my case an academicto launch enterprises with (potential) global reach without (i) money and (ii) giving up his day job. That is true, but technology does not replace expertise and its feeder, experience.

In the case of my MOOC, now well into its third offering, I’ve been teaching transition courses on mathematical thinking since the late 1970s, and am able to draw on a lot of experience as to the difficulties most students have with what for most of them is a completely new side to mathematics.

Right now, as we get into elementary, discrete number theory, the class (the 9,000 of 53,000 registrants still active) is struggling to distinguish between divisiona binary operation on rationals that yields a rational number for a given pair of integers or rationalsand divisibilitya relation between pairs of integers that is either true or false for any given pair of integers. Unused to distinguishing between different number systems, they suddenly find themselves lost with what they felt they knew well, namely elementary arithmetic.

Anyone who has taught a transition course will be familiar with this problematic rite of passage. I suspect I am not alone in having vivid memories of when I myself went through it, even though it was many decades ago!

As a result of all those years teaching this kind of material, I pretty well know what to expect in terms of student difficulties and responses, so can focus my attention on figuring out how to make it work in a MOOC. I know how to filter and interpret the comments on the discussion forum, having watched up close many generations of students go through it. As a result, doing it in a MOOC format with a class spread across the globe is a fascinating experiment, when it could so easily have been a disaster.

My one fear is that, because the course pedagogy is based on Inquiry-Based Learning, it may be more successful with experienced professionals (of whom I have many in the class), rather than the course’s original target audience of recent high school graduates. In particular, I suspect it is the latter who constantly request that I show them how to solve a problem before expecting them to do so. If all students have been exposed to is instructional teaching, and they have never experienced having to solve a novel problemto figure it out for themselvesit is probably unrealistic to expect them to make that leap in a Web-based course. But maybe it can be made to work. Time will tell.

The other startup I wrote about was my video game company. That is a very different experience, since almost everything about this is new to me. Sure, I’ve been studying and writing about video game learning for many years, and have been playing video games for the same length of time. But designing and producing a video game, and founding a company to do it, are all new. Although we describe InnerTube Games as “Dr. Keith Devlin’s video game company,” and most of the reviews of our first release referred to Wuzzit Trouble as “Keith Devlin’s mathematics video game,” that was like referring to The Rolling Stones as “Mick Jagger’s rock group.” Sure he was out in front, but it was the entire band that gave us all those great performances.

In reality, I brought just three new things to our video game design. The first is our strong focus on mathematical thinking (the topic of my MOOC) rather than the mastery of symbolic skills (which is what 99% of current math ed video games provide). The second is that the game should embed at least one piece of deep, conceptual mathematics. (Not because I wanted the players to learn that particular piece of mathematics. Rather that its presence would ensure a genuine mathematical experience.) The third is the design principle that the video game should be thought of as an instrument on which you “play math,” analogous to the piano, an instrument on which you play music.

In fact, I was not alone among the company co-founders in favoring the mathematical thinking approach. One of us, Pamela, is a former middle-school mathematics teacher and an award winning producer of educational television shows, and she too was not interested in producing the 500th animated-flash-card, skills-mastery app. (Nothing wrong with that approach, by the way. It’s just that the skills-mastery sector is already well served, and we wanted to go instead for something that is woefully under-served.) I may know a fair amount about mathematics and education, and I use technology, but that does not mean I'm an expert in the use of various media in education. But Pamela is.

And this is what this month’s column is really about: the need for an experienced and talented team to undertake anything as challenging as designing and creating a good educational learning app. Though I use my own case as an example, the message I want to get across is that if, like me, you think it is worthwhile adding learning apps and video games to the arsenal of media that can be used to provide good mathematics learning, then you need to realize that one smart person with a good idea is not going to be anything like enough. We need to work in teams with people who bring different expertise.

I’ve written extensively in my blog profkeithdevlin.org about the problems that must be overcome to build good learning apps. In fact, because of the history behind my company, we set our bar even higher. We decided to create video games that had all the features of good commercial games developed for entertainment. Games like Angry Birds or Cut the Rope, to name two of my favorites. Okay, we knew that, with a mathematics-based game, we are unlikely to achieve the dizzying download figures of those industry-leading titles. But they provided excellent exemplars in game structure, game mechanics, graphics, sounds, game characters, etc. In the end, it all comes down to engagement, whether the goal is entertainment and making money or providing good learning.

In other words, we saw (and see) ourselves not as an “educational video game company” but as a “video game company.” But one that creates video games  built around important mathematical concepts. (In the case of Wuzzit Trouble, those concepts are integer arithmetic, integer partitions, and Diophantine equations.)

Going after that goal requires many different talents. I’ve already mentioned Pamela, our Chief Learning Officer. I met her, together with my other two co-founders, when I worked with them for several years on an educational video game project at a large commercial studio. That project never led to a released product, but it provided all four of us with the opportunity to learn a great deal about the various crucial components of good video game design that embeds good learning. Enough to realize, first, that we all needed one another, and second that we could work well together. (Don’t underestimate that last condition.)

By working alongside video game legend John Romero, I learned a lot about what it takes to create a game that players will want to play. Not enough to do so myself. But enough to be able to work with a good game developer to inject good mathematics into such a game. That’s Anthony, the guy on our team who takes a mathematical concept and turns it into a compelling game activity. (The guy who can give me three good reasons why my “really cool idea” really won’t work in a game!) Pamela, Anthony, and I work closely together to produce fun game activities that embed solid mathematical learning, each bringing different perspectives. Take any one of us out of the picture, and the resulting game would not come close to getting those great release reviews we did.

And without Randy, there would not even be a game to get reviewed! Video games are, after all, a business. (At some point, we will have to bring in revenue to continue!) The only way to create and distribute quality games is to create a company. And yes, that company has to create and market a productsomething that’s notoriously difficult. (Google “why video game companies fail.”) Randy (also a former teacher) was the overall production manager of the project we all worked on together, having already spent many years in the educational technology world. He’s the one who keeps everything moving.

Like it or not, the world around us is changing rapidly, and with so many things pulling on our students’ time, it’s no longer adequate to sit back on our institutional reputations and expect students to come to us and switch off the other things in their lives while they take our courses.

One case: I cannot see MOOCs replacing physical classes with real professors, but they sure are already changing the balance. And you don’t have to spend long in a MOOC to see the similarities with MMOs (massively multiplayer online games).

We math professoriate long ago recognized we needed to acquire the skills to prepare documents using word processing packages and LaTeX, and to prepare Keynote or PowerPoint slides. Now we are having to learn the rudiments of learning management systems (LMSs), video editing, the creation of applets, and the use of online learning platforms.

Creating video games is perhaps more unusual, since it requires so many different kinds of expertise, and I am only doing that because a particular professional history brought me into contact with the gaming industry. But plenty of mathematical types have created engaging math learning apps, and some of them are really very good.

Technology not only makes all of these developments possible, it makes it imperative that, as a community, we get involved. But in the end, it’s people, not the technology, that make it happen. And to be successful, those people may have to work in collaborative teams.