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.

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.

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.

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.

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!)

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.

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

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

The short, narrow descent |

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Here is our brief basking point.

The inflection point. One more detail to resolve. |

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

The summit of the climb |

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

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

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

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

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

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

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

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.'”

**What is Multiplication?**

**Mathematical Thinking**

**MOOCS**

**More about MOOCs**

*HUFFINGTON POST*

__http://www.huffingtonpost.com/dr-keith-devlin/why-moocs-remain-silicon-_b_4289739.html__

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.

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

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?

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

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

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

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.)

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 classroom—often 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 goal—the motivation for and purpose of that
practice—is 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 favor—without 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.

In last
month’s column, I reflected on how modern technology enables one person—in my case an academic—to 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 division—a binary operation on rationals that yields a
rational number for a given pair of integers or rationals—and divisibility—a 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 problem—to figure it out for themselves—it 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 500^{th}
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 product—something 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.

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