Will Cantor’s Paradise Ever Be of Practical Use?

We really have no way of knowing what early humans thought when they gazed up at the sky. Since everyday practical experience is, by definition, limited to a very small region of space and time, it requires considerable cognitive sophistication to conceive of something – say the night sky – “going on for ever,” let alone to ponder whether that means it is “infinite,” or indeed what “infinite” actually means.

What we do know is that the ancient Greeks made what may have been the first substantial attempt to analyze the notion of infinity, with Zeno of Elea (ca 490-430 BCE) of particular note for his discussion of a number of (seeming) paradoxes that arise from the assumption that space and time are (or are not) infinitely divisible.

Archimedes’ (ca 287-112 BCE) calculations of areas and volumes made implicit use of infinity, and from today’s perspective can be recognized as the forerunner of integral calculus.

Skillful formal – though by modern standards not rigorous – use of the infinitely large and the infinitely small was made by Isaac Newton and Gottfried Leibniz in their development of modern infinitesimal calculus in the seventeenth century, though it was not until the nineteenth century when Bernard Bolzano, Augustin-Louis Cauchy, and Karl Weierstrass finessed the lurking problems of infinity by means of the famous (and for many a first-year mathematics major, infamous) epsilon-delta definitions of limits and continuity.

But none of these developments was about infinity as an entity; the focus rather was on the unending nature of certain processes, starting with counting. It was Georg Cantor (1845 – 1918) who really tackled infinity head on. His proof that the set of real numbers cannot be put into one-one correspondence with the natural numbers, and hence is of a larger order of infinitude, led to a series of papers, published in a remarkable ten-year period between 1874 and 1884, that formed the basis for modern abstract set theory, including the development of a fully formed arithmetical theory of infinite numbers (or “cardinals”).

Reactions to Cantor’s revolutionary new ideas ranged from outraged condemnation to fulsome praise. Henri Poincaré called Cantor’s work a “grave disease” that threatened to infect mathematics, and Leopold Kronecker described Cantor as a “scientific charlatan” and a “corrupter of youth.” Ludwig Wittgenstein, writing long after Cantor's death, complained that mathematics had become “ridden through and through with the pernicious idioms of set theory,” a theory he dismissed as “utter nonsense,” “laughable,” and “wrong.”

At the other end of the spectrum, in 1904, in the UK the Royal Society awarded Cantor its highest award, the Sylvester Medal, and in Germany David Hilbert declared that “No one shall expel us from the Paradise that Cantor has created.”

Having devoted the early part of my professional career to work in (infinitary) set theory, starting with my Ph.D. in “large cardinal theory,” completed in 1971, and moving on to work on alternative universes of sets (a particularly hot topic after Paul Cohen’s introduction of the method of forcing in 1963), in the early 1980s my interests started to shift elsewhere, to questions about information, communication, and human reasoning.

I found myself temporarily back in the world of set theory and the arithmetic of infinite numbers recently, when I was approached by the organizers of the World Science Festival to moderate a panel discussion on the topic of infinity and a more in-depth follow-up the following day.

Both discussions raised the question as to whether study of infinity – in particular the hierarchy of larger infinities that Cantor bequeathed to us – would ever have any practical applications. As panelist Hugh Woodin remarked at one point in the discussion, it is a foolish mathematician who declares that a particular piece of mathematics will not find applications. For instance, G. H. Hardy’s famous statement (in his book A Mathematician’s Apology) that his work in number theory would never find practical application, proved to be spectacularly wrong less than a century later, when number theory became the foundation for internet security.

Hardy’s observation was based on his familiarity of the world he lived in, a world in which the World Wide Web was not even a dream. Today, we cannot know what the world of tomorrow will look like. On the other hand, whatever our children and grandchildren will take for granted, their world will surely be finite, which makes it unlikely that Cantor’s theory – and the almost a century of development in set theory since then – will have practical use.

Or does it? What about calculus? Infinitesimal (!) calculus not only has applications in the modern world, but much of the science, technology, medicine, and even financial structure the underpins our world depends on calculus for its very existence. Applications don’t get more real than that.

True, but the dependence on infinities you find in calculus is essentially asymptotic. What really drives calculus is the unending nature of certain processes on the natural numbers. Talk of “infinitely large” or “infinitely small” is little more than a manner of speaking. Indeed, the epsilon-delta definitions (which do not involve infinities or infinitesimals) are a way to formalize that manner of speaking, effectively eliminating any actual infinite or infinitesimal quantities.

In contrast, much of the work on infinity (more precisely, infinities) carried out in the second half of the twentieth century (when I was working in that area) focused on properties of sets that made their cardinalities super-infinities of different orders: inaccessible cardinals, Ramsey cardinals, measurable cardinals, compact cardinals, supercompact cardinals, Woodin cardinals, and so on. Infinities which dwarfed into invisibility the puny cardinality of the set of natural numbers. Indeed, each one in that sequence dwarfed all its predecessors into invisibility. How could that work find an application?

I’ll lay my cards on the table. I think the chances are that it won’t. But I don’t think it is impossible. Indeed, I began to suspect a possible application in the very domain I worked in after I left set theory.

[This may of course be nothing more than a reflection of having at my disposal a large hammer which made everything look a bit like a nail. But let’s press on.]

The post 9/11 world saw me involved in a series of Defense Department projects the first being improving intelligence analysis (and the others essentially variants of that).

In today’s information rich world, the major nations can be assumed to have access to all the information they need to predict (and hopefully thence prevent) the majority of terrorist attacks. The trouble is, the few data points which must be identified and connected together to determine the likelihood of a future attack are just a tiny few in an overwhelming ocean of data. Even in the era of cloud computing, identifying the key information is analogous to using the naked eye to find a handful of proverbial needles in a non-proverbial field of haystacks.

To all intents and purposes, the available data is infinite. The only hope is to impose some structure on the data that makes it possible for humans and computers to work together on it, narrowing down the focus to the regions more likely to be of significance. Though modern computing systems can sift through massive (finite) amounts of data in a relatively short time, they need to be programmed, and writing those programs (at least, some kinds of them) will require some structure on those large sets of data. A possible place to find the appropriate structure(s) is infinitary set theory. In other words, to develop the appropriate structures, assume the data is infinite. View the infinite as a theoretical simplification of the very large finite. (Economists sometimes make a similar simplifying assumption about economies.)

Do I think this is likely? Frankly, no. But then, neither could Hardy conceive of any practical application of his work in number theory. [Incidentally, like Hardy, I don’t think mathematics needs applications to justify itself. It’s just that the question of application is what this article is about!]

The discussion about large cardinals you will find in those panel discussions at the World Science Festival might seem impossibly abstract and far removed from the everyday world. Indeed, it is. But the questions being discussed all resulted from a process of rigorous, logical investigation that arose directly from late nineteenth century attempts to understand heat flow. History tells us that what begins in the real world, very often ends up being used in the real world.

Prediction is hard, particularly about the future.

Incidentally, how did I end up working on a project for the DoD? They asked me. I might not be the only person to speculate about a possible use of Cantor’s paradise. This is your taxpayer dollars at work.

The Mother of All NCTM Addresses

This month’s column is short, but I am asking you to set aside 51 minutes and 36 seconds to watch the embedded video. It is a recording of the Iris M. Carl Equity Address given on Friday April 19 at this year’s NCTM Annual Conference in Denver, Colorado. The title of the talk is “Keeping Our Eyes on the Prize” and the speaker is Uri Treisman, professor of mathematics and of public affairs, and director of the Charles A. Dana Center, at the University of Texas at Austin.

I was not able to be at NCTM, but on the recommendation of several colleagues, I watched the YouTube video. I simply cannot write a column on mathematics or mathematics education in the same month as Treisman’s immensely more important, profound—and powerfully articulated—words became part of mathematics education history. As a community, we now have our own “I have a dream” speech. Thank you, Uri.

PDFs of Treisman's presentation slides are available here.

Only in Silicon Valley

ADDED MAY 1: NOTE THAT THIS COLUMN WAS POSTED ON APRIL 1, "ALL FOOLS DAY" IN THE USA AND SEVERAL OTHER COUNTRIES.

One of the benefits of being at a university like Stanford is that we occasionally get the opportunity to see up close the emergence of an amazing mathematical talent—someone who may turn out to be the next Euler or Gauss.

Just over 18 months ago, Avril Wan was, to all appearances, just another bright fourteen-year-old living in Taiwan, where her father Yewful Wan runs a large shipping company and her Welsh-born mother Melanie Wan is a university mathematics professor (and a former student of Timothy Gowers in Cambridge).

Then, in September 2011, Stanford computer science professor Sebastian Thrun and Google researcher Peter Norvig offered what turned out to be the first of what is now a flood of Massively Open Online Courses (MOOCs), which make advanced university courses available to the entire world over the Internet. Ms. Wan enrolled for that first MOOC, in artificial intelligence, and was the only student to score a perfect 100% for the course.

When initial investigations made it clear that Ms. Wan’s performance was legitimate, Thrun moved quickly, and arranged for Stanford to offer her a place in Stanford’s famed Symbolic Systems Program (which has produced a whole string of graduates who have founded and led successful Silicon Valley companies, such as Reid Hoffman, who founded LinkedIn, and Marissa Meyer, an early employee of Google and the new—and controversial—CEO of Yahoo!).

By the time Wan arrived at Stanford, Thrun had left to form Udacity, a Silicon Valley start-up offering free online courses to the world, and the newly arrived student, who had just turned 15 (and was accompanied by her mother), was assigned to the educational care of another famous Stanford mathematics professor, Persi Diaconis, known for his ability to see familiar problems in novel ways.

In late spring of 2012, there was a buzz across the Palo Alto campus when it seemed that, under minimal guidance from Diaconis, the young Ms. Wan had solved the notorious P = NP problem, but Ron Graham of the University of California at San Diego quickly found an error, pointing out that she had implicitly assumed the existence of a complete, two-valued measure on the power set of the natural numbers—a question first raised by the famous (Second World) Wartime mathematician Stan Ulam.

Meanwhile, Ms. Wan’s mathematics blog had started to attract attention back in her home country, making her somewhat of a Taiwan celebrity. In particular, motivational videos she had posted on YouTube to encourage young Taiwanese girls to study mathematics, eventually came to the attention of News Corporation’s Rupert Murdoch, who pledged $5M to make her videos available throughout the developing world.

But then, online tech journalist Dan Gillmor posted an article pointing out that Murdoch’s funding was contingent on the distribution being restricted to streaming to tablets supplied by his own, for-profit company Amplify. If so, that would surely have killed the deal, since Ms. Wan recognizes the value of free educational resources to the development of the less affluent countries of the world.

At that point, events started to unfold at the kind of breakneck speed that only happens in Silicon Valley. Ms. Wan, still just 15 years old, remember, and technically without even a high school diploma, found herself inside the Palo Alto offices of the famed venture capital company Greylock Partners, which was willing to commit $100M to fund the establishment of a global, free, online mathematics education platform, tentatively called “Wan World.”

With Greylock having been early stage funders of some of the most successful start-up companies in recent years, most of which required several years before anyone had the faintest idea how they would make money, that interest was all it took to unleash the floodgates. Within a few days, Ms. Wan (or rather, the group of advisers her father quickly assembled to cope with the interest) had been approached by Apple, Google, and Facebook, each of which wanted to develop the platform on which Wan World would run, and by McGraw Hill, Pearson Education, and Amazon, who wanted to own the content.

Meanwhile, despite all this frenzy, Ms. Wan herself seems remarkably unfazed by the sudden changes in her life. Speaking to an unusually full room in a recent meeting of Stanford’s Education’s Digital Future lecture/discussion series (which is where I first met her), she concluded her presentation by admitting to her fellow students, “Like you, right now, I just want to graduate.”  

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

A good metaphor for the current state of MOOC education is provided by this historical video. But when you look at those images, please remember what those events led to. Unless you are able to keep that history in mind, you should not at this stage get into the MOOC business. For there be only dragons.

With the second edition of my Stanford MOOC Introduction to Mathematical Thinking starting this weekend on Coursera, I have once again been wrestling with the question of the degree to which good, effective mathematics learning can be achieved at scale, over the Internet.

Once I had made the decision to try to take (elements of) my 35-year-old mathematics transition course into the then emerging MOOC format—less than a year ago!—I was immediately brought face-to-face with the necessity of making use of two educational devices I had loathed (and never used) throughout my entire career in higher education:

1. machine-graded pop quizzes
2. machine-graded multiple-choice questions

For MAA readers, I don’t think I need to explain my dislike for either of these über-simplistic devices, which can surely be justified in a regular classroom only in terms of making life easier for the instructor.

Simply putting a class online does not require the use of either device, of course. Technologies such as video conferencing and screen sharing can make learning at a distance almost as good as traditional classroom learning, and in some circumstances can make it better in some respects. But making a class available to tens of thousands of students online changes everything. With such large numbers, the “class” dynamics change dramatically. But it’s not all for the worse.

The first thing to realize is that a MOOC is in many ways like radio or TV. Though both of those familiar features of modern life are referred to as “mass media,” they are in fact highly individual. The newsreader on radio or TV is not addressing a large audience; she or he is talking to millions of single individuals. The secret to being good on the radio or TV is to forget the millions and think of just one (generic) person. After all, the listener or viewer is not in a room with millions of other people; in fact, if the broadcast is successful, that listener or viewer is cognitively in a room with just the presenter. The really successful radio and TV newsreaders and presenters are the ones who can do that really well. They create that sense that they are talking just to You.

In my own case, I already knew that from many years of occasional media work, but I think all MOOC instructors come to that realization very quickly. When your voice, with or without your face, is in someone’s living room, there is a direct human connection that in important ways is far more intimate than is possible in a lecture hall filled with anything more than a handful of students.

Once you realize this feature of the MOOC medium, the underlying pedagogic model is obvious. It’s one-on-one teaching/learning—something that in the traditional academy is (of necessity) reserved only for doctoral students.

At which point, the appropriate use of both pop quizzes and multiple-choice questions starts to look feasible. (They ought to; doctoral advisers use both extensively, and to great positive effect, though they do not refer to them as such, and there is no machine-grading!)

Of course, in a MOOC it remains the case that the student cannot communicate directly with the professor, nor can the professor see and comment on an individual student’s work. That means two further techniques have to be used as well:

3. peer tutoring
4. peer evaluation 

In the first version of my MOOC, last September, I built the course around the doctoral-student education model, deliberately setting out to create the experience of a student sitting alongside me at my desk. (There is a low resolution example here.)

But as a result of a career-long dislike of the first two and a deep suspicion of the fourth, I used all but the third of those auxiliary devices reluctantly and as little as possible. (The one I did embrace, peer tutoring, did not work well the way I set it up. See below for details of Attempt Two.)

Because of my caution, I think I avoided a fate reminiscent of NASA’s first attempts to launch a rocket into space. But that was a first, exploratory experience, and I wanted to live to try again. This time around, based on what I learned, I am going to use all four much more aggressively, but in ways I think might work.

I’ll be describing how I’ll be using them in a series of posts to my blog MOOCtalk.org. For a brief—and decidedly limited—foretaste, check out this video excerpt of a conversation my MOOC TA Paul Franz and I had recently with radio and TV personality Angie Coiro, host of the syndicated radio and television interview show In Deep.

The goal of Version 2 of the course is not to reach the Moon. Chances are high that we’ll crash and burn. The goal is to at least get off the ground before we do, and, if we are lucky, maybe even reach the upper atmosphere. For sure, there will still be a long way to go.

If you want to live dangerously and be part of this huge experiment, and if you have a Ph.D. (or pending Ph.D.) in mathematics and several years of college teaching behind you, I am still looking for well qualified volunteers to act as “Community TAs” for the course, to answer students' questions on the course discussion forums. So far I have 14 volunteers, comprising 5 college professors, 3 Ph.D. students, 3 individuals currently working in the software industry, a K-12 education consultant, a research laboratory scientist, and a stock analyst. If you want to volunteer, and have the requisite experience, please drop me an email at devlin@stanford.edu. (There is no payment for doing this—that includes me!) But being part of a large and truly global community, who come together for several weeks for the sole purpose of learning how to think mathematically (the course carries no college credit), is truly a wonderful experience.

The Problem with Instructional Videos

With the second offering of my MOOC Introduction to Mathematical Thinking about to go live on March 2, I am once again asking myself if the current MOOC structure is the best way to make effective, quality higher education available in a cost-effective way on global scale, making use of the existing technology.

The two words that inhibit my confidence that we’ll ever achieve what I and my fellow first-generation MOOC instructors are trying to do, are “effective” and “quality.”

The task gets a whole lot easier if you set your sights really, really low. Say, “Pass the standardized course test that comes at the end.” But that’s equivalent to the goal of engineers who set out to build something routine, like a software package or a bridge. Does the software do what was intended? Does the bridge meet the specifications? It’s also a meaningful goal of human training, where people want to acquire a new skill.

But no one, surely, would make passing a standardized test the goal of higher education, or even a significant metric thereof. The purpose, after all, is to build more capable thinkers. No, the thought that anyone would make that kind of mistake seems so unlikely, I’ll move on without giving it any more attention, and get back to my main theme: the videotaped lecture.

I’ve commented on a number of occasions (for example, in my MOOCtalk.org blog) that I think the videotaped lecture is, from a learning perspective, the least important constituent of a MOOC, and that, for me at least, MOOCs seemed to offer the possibility of scaling (at least some elements of) higher education because they can draw on our experience with Facebook, rather than YouTube.

One huge problem with a videotaped lecture is that we know that instructional videos about science (and other disciplines where the learner starts with some beliefs, including mathematics) simply do not work.

In true MOOC fashion, we are now far enough in to my column that I should give you a multiple choice quiz. Here it is:

QUIZ: Where do trees get most of their mass?
1. From nutrients in the soil.
2. From the water
3. From the energy coming from the sun
4. From the air

When you have made your selection, take a look at this video to get the answer.

Done that? Did you notice the way the video was put together. Most of the video was devoted to the presenter (Derek Muller, who got his Ph.D. at the University of Sydney, Australia, a few years ago on the effectiveness of science videos) discovering people’s misconceptions. That certainly makes for “good television,” but does it have a place in an educational video? You bet it does.

The reason is, what is arguably the main finding of Muller’s research: that the principal effects of a well made, clear, instructional, science video are (1) to reinforce the viewer’s existing belief, whatever it is, and (2) to make that viewer even more confident in that belief. Nothwithstanding the fact that the video might present information that flatly contradicts the belief.

Muller summarized those findings in a critique of Khan Academy a couple of years ago, which is how I first came across his work. Anyone thinking of giving a MOOC should spend the eight minutes it takes to watch that video.

Since completing his doctorate and critiquing Khan, Muller has gone on to make a number of science videos. He is, clearly, still experimenting with the format (and I for one hope he continues to do so), and as a result, the videos are of varying quality. But a consistent theme is to begin with common misconceptions and force people to confront those erroneous beliefs.

Sure, this means getting people to say wrong things on camera, which can make some viewers feel uneasy. This has led to some criticism – though anyone you see on the final video has agreed to be shown, of course. He addresses this issue in an amusing fashion in another video. But the real point is that learning does involve confronting – and then correcting – our misconceptions. One of the most crucial abilities of a good teacher is to tell people they are wrong, and help them correct the error, without making them feel small or stupid.

The fact is, the experts make mistakes all the time. Indeed, an expert only achieves that status by having learned how to capitalize from being proved wrong, over and over again. In a sequel to the tree-mass video, Muller made another film about the mechanism trees use to acquire that mass, and in that video (which is truly amazing) you see three experts give the wrong answer.

So if videotaped instruction doesn’t work, how can we achieve learning in a MOOC? Well, there are not many things available. Other than the lecture videos, some screen-readable or downloadable course readings, and a few online quizzes, the only other possible source of learning within a MOOC is the body of other students. (In a physical class, the professor herself can play a role, but for a MOOC class of 60,000 or more, that’s clearly out of the question.)

That’s why I think MOOCs are more Facebook than YouTube, and why I think the key to making them anything more than just textbooks-on-steroids – an approach we know won’t work – is to learn how to structure them to encourage and support group collaborative work.

R.I.P. Mathematics? Maybe.

Is mathematics about to die? More precisely, are we rapidly approaching a time when progress in mathematics effectively comes to an end? I posted some thoughts on this issue in my answer to the latest Edge Question, an annual online event organized by the literary agent John Brockman. See if you agree with me.

Every year Brockman manages to get many leading scientists and intellectuals to contribute essays, for free, by the effective strategy of putting together a list of contributors from which no one wants to be left out – no matter how challenging the question he proposes.

The professions most heavily represented in the list are physicists, computer scientists psychologists, cognitive scientists, and journalists. Mathematicians fare much less well. In fact, the only other mathematician in the club besides myself is Steven Strogatz. If we include people who have written expository books on mathematics, you can add three more to the list: Mario Livio, Clifford Pickover, and Charles Seife.

Maybe we mathematicians feel uncomfortable going out publicly on a limb – for the goal is to stretch the boundaries of what we know – as reflected in the group’s title: Edge.

Since my Edge essay was inspired by the MOOC I gave recently, which I reported on in my December column, I’ll end by announcing that I am giving a slightly revised version of the same online course this spring, starting on March 4. I’m looking for college and university mathematics faculty and mathematics graduate students and postdocs to volunteer to act as “Community TAs” for the course, going onto the discussion forums every now and then and guiding the discussion threads in productive directions.

Last year, I put out a general call for volunteers for this role, and it did not work out. About 600 signed up, but only a handful of them actually had sufficient knowledge and experience to carry out the task. The vast majority were simply well meaning folks who wanted to help. Since the Community TAs are so designated when they post on the forum, this effectively rendered useless the TA designation.

There is no remuneration for doing this. (There’s none for me as instructor, either. I do this on top of my regular Stanford duties.) It’s all for the love of teaching and the drive to change the world. But it’s a lot of fun, and truly fascinating. For the length of the course, you are an active, contributing member of a genuine global community (North Korea excepted), who come together for a few weeks of intense interaction as they pursue a common goal.

If you want to give it a try, simply sign up for the course and then send me an email (to devlin@stanford.edu) giving me your Coursera login name so I can confer Community TA status to you. (I won’t repeat this request on the course site, since what I (or rather the 64,000+ students) really need is maybe 20 knowledgeable mathematicians wandering around the discussion forums – not several hundred well meaning non-mathematicians.)

Though, as I just noted, you won’t (currently) get paid for being a Community TA, one day soon it may help you get tenure or promotion. As the Coursera platform develops, we intend to introduce a mechanism for tracking forum TA activity, in terms both of frequency and positive impact as measured by recipient feedback. Once we have that, I suspect it won’t be long before a good record as a TA in a MOOC will become a submission item in a faculty tenure and promotion case. This has already occurred for Wikipedia contributors, another online volunteer activity.

For more background on my MOOC, and MOOCs in general, see my blog MOOCtalk.org.

BTW, in addition to my online course, last fall I gave a five-week survey course on mathematics and its applications in Stanford’s Continuing Studies program, which video-recorded the entire series to distribute for free on iTunes University.

The Darwinization of Higher Education

Stanford president John Hennessy has described the current changes in higher education initiated by technological innovations as an approaching tsunami. His remark was prompted largely by the emergence and rapid growth of MOOCs (massively open online courses), first from Stanford itself, joined soon afterwards by MIT and Harvard.

Are MOOCs going to initiate, or be part of, an educational tsunami? I think it’s too early to say. But in true mathematical fashion, I’m going to pursue the hypothesis that this is the case and examine what is happening. In doing so, I’ll draw on the insight into MOOCs I gained from giving my own a few weeks ago, which I wrote about in last month’s column, and have been blogging about regularly at MOOCtalk.org.

Hennessy's observation was widely interpreted as being about the structure and business of higher education, and that may indeed be what he had in mind. He does, after all, have the responsibility of ensuring the survival and continuing prosperity of one of the world's leading universities. (A task that, as someone who receives a Stanford paycheck every month, I wish him every success in fulfilling.)

But when you look a bit more deeply at the way MOOCs are developing, you see that the real tsunami is going to be a lot bigger than that. It's not just higher education that will feel the onslaught of the floodwaters, but global society as a whole.

Forget all those MOOC images of streaming videos of canned lectures, coupled with multiple-choice quizzes. Those are just part of the technology platform. In of themselves, they are not revolutionizing higher education. We have, after all, had distance education in one form or another for over half a century, and online education since the Internet began in earnest over twenty-five years ago. But that familiar landscape corresponds only to the last two letters in MOOC ("online course"). The source of the tsunami lies in those first two letters, which stand for "massively open."

Right now, the most popular MOOCs draw student enrollments of about 50,000 to 100,000. In this it’s not unreasonable to expect those numbers to increase by at least a factor of 10, once people realize what is at stake.

True, those numbers don't tell the whole story. In particular, roughly 90% of the students who sign up do not complete the course. But that leaves many thousands who do finish, many of them with near perfect scores. And when that tenfold increase kicks in, it will be tens of thousands that complete. Paradoxically, it's the high rate of dropouts that will generate the tsunami (if there is one).

A good analogy is Google. Before Stanford graduate students Sergei Brin and Larry Page came up with their search algorithm, finding information (on the Web or elsewhere) was a time-consuming, and often hit-or-miss affair. At heart, what makes Google work is the efficient way it discards almost every possible answer to your query. Occasionally, in so doing, it may throw away the one item you really should see. But, given the way the algorithm works, that happens very, very rarely. As a result, Google gives you answers that are good enough for your purposes, most of the time.

The ability to sift through a massive amount of data means that there is no need for precise identification in search; with enough data, "good enough" really is good enough. In information terms, it's survival of the fittest; the process has no respect for the individual, but overall is extremely effective.

Now, the same university that gave you Google has launched the truly massive, open online courses. (Earlier MOOCs were not really massive. Indeed, the really massive ones, with millions of students, are probably a year or two away - yes, it could be that short a timeframe.)

Right now, the media focus on MOOCs has been on their potential to provide (aspects of) Ivy League education for free on a global scale. But an educational system does more than provide education. It also identifies talent - talent which it in part helps to develop. That makes a MOOC the equivalent of Google, where it is not the right information you want to find but the right people.

And the world definitely wants to find the right people. Last year, the World Economic Forum and the Boston Consulting Group issued a report describing the scale of the increasing need for talented individuals in today's world, and the numbers are staggering. For instance, the report states, "The United States ... will need to add more than 25 million workers to its talent base by 2030 to sustain economic growth, while Western Europe will need more than 45 million." The educational systems of these countries are not coming anywhere close to meeting those needs.

At the level of the individual student, MOOCs are, quite frankly, not that great, and not at all as good as a traditional university education. This is reflected (in part) in those huge dropout rates and the low level of performance of the majority that stick it out. But in every MOOC, a relatively small percentage of students manage to make the course work to their advantage, and do well. And when that initial letter M refers not to tens of thousands but to "millions," those successes become a lot of talented individuals.

One crucial talent in particular that successful MOOC students possess is being highly self-motivated and persistent. Right now, innate talent, self-motivation, and persistence are not enough to guarantee an individual success, if she or he does not live in the right part of the word or have access to the right resources. But with MOOCs, anyone with access to a broadband connection gets an entry ticket. The playing field may still not be level, but it's suddenly a whole lot more level than before. Level enough, in fact. And as with Google search, in education, "level enough" is level enough.

Make no mistake about it, MOOC education is survival of the fittest. Every student is just one insignificant datapoint while the course is running. Do well, do poorly, struggle, drop out - no one notices. But when the MOOC algorithm calculates the final ranking, the relatively few who score near the top become very, very visible. Globally, talent recruiting is a $130BN industry (Forbes.com, 2.12.12). It's "Google search for people" in action.

For those of us in education, MOOC education requires a major adjustment in attitude. Most of us go into the profession because we care about the individual. We love to interact with our students. Moreover, universities have all kinds of structures in place to catch and help struggling students. But in a MOOC, all of that goes out the window.

Doubtless, some current higher educational institutions will step in and provide support for MOOC students who need it. But what they won't be able to do is make education a local affair, where it is enough to do better than most of your fellow students at University X, or even in country Y. The fight to hire top talent will be global. And for American students from even moderately affluent backgrounds, a lot of their competition will have far more to gain from doing well, with all the added motivation that will bring.

Yes, some organizations will make money from MOOCs, though it is unlikely to be the Ivy League course providers whose stellar faculty and exclusive brands make their courses so attractive. They cannot afford to lose their exclusivity. But new sources of revenue for some colleges and universities who can adapt to the arrival of MOOCs, and the possible death of those that cannot, is just a market adjustment. If we are going to witness a tsunami, it is likely to be the true globalization of higher education and talent search.

POSTSCRIPT ADDED DECEMBER 5: By chance, a day after this column appeared, Coursera sent out a mass email informing current and former students of their new talent placement service. (I have no connection to Coursera other than using their platform for my MOOC, and no knowledge of their business plans.) See my recent post at mooctalk.org for more details, where I also give a brief history of this column. (Yes, it has a curious past.)