“Will this (mathematics) be of any use?”

Readers who follow me on Twitter will have noticed many tweets on the recent revelations about illegal NSA surveillance. Here is why I think that none of us in mathematics and mathematics education can ignore that debate.

There’s a popular conception that mathematicians are unworldly, and that mathematics is, at its heart, walled off from the real world, its pursuit a form of escapism that takes the pursuer into a realm of pure, abstract thoughts.

Certainly, that’s a general sense of mathematics that I held for many years. Yes, like all my fellow mathematicians, I always knew that mathematics – all of it – arose, directly or indirectly, from real world problems, and that any branch of mathematics having any discipline-internal significance almost always turns out to have real-world applications. But neither of those was why I did mathematics. For most of my life as a mathematician, I simply did not care about the history or application of what I was doing. It was all about the chase – the search for new knowledge in a beautiful domain.

Early on in my career, when more politically active colleagues urged me to boycott conferences and workshops funded by NATO (a big issue back in the 1970s), or to avoid applying for research funds from commercial or military sources, I essentially turned a deaf ear to what they were saying, and got on with the work that interested me.

As a mathematician working in axiomatic set theory, with particular foci on the properties of sets of large infinite cardinality and on undecidability proofs, I felt fairly confident that nothing I did would ever find practical application, so for me the issue was purely one of where the money came from to support my research. I felt “clean,” and not under any moral pressure regarding potential unethical uses being made of my work.

True, I was aware that the famous early twentieth century mathematician G. H. Hardy had made the same claim about his work in number theory, yet in the mid-1970s his work found highly significant application in the design of secure cryptographic systems. But I felt that a similar outcome was unlikely in the case of infinitary set theory. (I am no longer quite as sure about that; I speculated about possible applications of Cantor’s set theory in my June column.)

I think we all have to address the morality-of-possible-applications question about our work as mathematicians at one time or another. Some, from Archimedes to Alan Turing, have actively engaged in military research; others try to avoid any direct contact with commercial or warfare-related activities.

The rise of math-based corporations such as Google that form a large and influential part of today’s global world, and the closely related growth of the modern, math-driven security state, as iconicized by the NSA, make it impossible to maintain any longer the fiction (for such it always was) that we can pursue mathematics as a pure activity, separate from applications, be they good or ill.

The uncomfortable fact is, we are in no different a situation than manufacturers of sporting guns who deny any agency when their product is used to kill people. (Yes, people pull the trigger, but as comedian Eddie Izzard pointed out, “the gun helps.”)

If we want to be able to maintain that our work will not play a role in someone’s death, torture, or incarceration – or in someone else achieving enormous wealth and power – our only option is to not go into mathematics in the first place. The subject is simply way too powerful as a force – for good or for evil.

Shortly after September 11, 2001, I was asked to join a research project funded by the U.S. intelligence service. For me, that was my crunch time. The work that led to that invitation was an outgrowth (described in my 1995 book Logic and Information) of my earlier research in mathematical logic and set theory. Like it or not, I was already in deep. To say no to that invitation would have been every bit a positive action as to say yes. Sitting on the fence was not a possibility. I was a mathematician. I’d already made the gun.

As the Google founders Larry Page and Sergei Brin eventually discovered, “Do no evil” is a wonderful grounding principle, but the power of mathematics renders it an impossible goal to achieve. The best we can do is try to make our voice heard, as many mathematicians and nuclear physicists did during the Cold War, who spoke publicly about the massive scale of the danger raised by nuclear weapons.

Finding out (as I have over the past few weeks) that the work I’d done over the past twelve years – for various branches of the U.S. government (intelligence and military) and for commercial enterprises (in my case, the video game industry) – was part of a body of research that had been subverted (as I see it) to create a massive global surveillance framework, I felt I could not remain silent.

Not because I felt that I, as an individual, did anything of significance. I worked on non-classified projects, and made no major breakthroughs. I was a very tiny cog in a very big machine. (If “they” are keeping an eye on me, they are definitely wasting our tax dollars!)

But I did take the money and I did do the work. I don’t regret doing so. The fact is, I’d made the crucial choice long before 2001; back in my youth when I decided to become a mathematician.

Those of us in mathematics education have always told our students that math is useful. In today’s world more than ever, we cannot at the same time pretend it is free of moral issues. Agnosticism is not an option (if it ever really was). To say or do nothing is inescapably a positive act, just as significant as saying or doing something.

We humans have created our mathematics, and used it to help shape our world. Now we have to live in it. Not only are we the ones who bear a large responsibility for that world, we are also, by our very expertise, the ones who (in many fundamental ways) understand it best. (It often seems that only the mathematically sophisticated really appreciate that an American is more likely to die in his or her bathtub than from a terrorist attack, and that more people died on the roads due to increased traffic during the time after 9/11 when all flights were grounded than did in the Twin Towers attack.)

So, to return to the question implicit in my title, “What is mathematics used for?” Douglas Adams provided the answer: “Life, the universe, everything.” With such reach and power comes responsibility. 

FOOTNOTE: For a more personal take on the above issues, see the interview I did on June 21 on Shecky Riemann’s Math Tango blog.

“It Only Takes About 42 Minutes To Learn Algebra With Video Games”

When tech folk dabble in education (and tech writers cover it), the excess of hype is sometimes matched only by their breathtaking lack of knowledge about education. Even so, the above headline to the July 1 post by Forbes contributor Jordan Shapiro must rank as one of the most stupid and ignorant statements in human history.

It would be somewhat less ludicrous, though still open to debate, if the headline had said “learn some algebra.” But “algebra”? All of it?

Almost certainly, Shapiro himself did not write the headline—writers rarely do. In fact, the article itself is fine. I have no problem with what Shapiro wrote. But the fact that the ludicrous headline had not been changed 24 hours later indicates that Forbes’ editors feel happy with it. Sigh.

What the article itself reports is that, on average, students who played a particular video game (DragonBox, of which more later) completed a sufficient part of it in 42 minutes. Since the game itself is based on algebraic principles, they could, therefore, be said to have engaged in algebraic thinking. (I would be inclined to say just that, though with any kind of machine learning—and human teaching if the instructor is not paying close attention—one should always be on the lookout for an instance of Benny’s Rules.)

Whether such performance in a video game justifies saying that the students learned some (!) algebra in 42 minutes depends on what metric you use to determine what learning has taken place.

Of course, if you define algebra to be (or to include) symbolic manipulation, then successful completion of any video game is not going to count as “doing algebra.” That is why I used the term “algebraic thinking” a couple of paragraphs back. (See my previous blog post What is Algebra? for a discussion of the distinction.) But is that the appropriate measure? What do we want K-12 students to learn under the title “algebra”?

[ASIDE: There is another definitional question as to the classification of DragonBox as a video game. Game developers have different views as to what constitutes a video game. Some would describe DagonBox as an entertaining, interactive, digital app, but would stop short at classifying it as a game.]

Before I go any further, I should give some disclaimers. First, as readers of my blog profkeithdevlin.org (or my book Mathematics Education for a New Era) will be aware, I am a strong proponent of the use of video games in mathematics education. In fact, I advocate an approach to the design of math ed video games that definitely includes DragonBox. I’ve met the developer, Jean-Baptiste Huynh, and one of the co-founders of his company WeWantToKnow, and I used their game as an example in a feature article on math ed video games I wrote for American Scientist in March of this year. I am about three-quarters of the way through the second, greatly expanded version of the game, DragonBox2. Among the designs for math ed video games that my own company, InnerTube Games, has been working on for several years, are a couple that have much in common with DragonBox. (We are due to release our first one, Wuzzit Trouble, this summer, but chose one based on arithmetic and number theory to be our initial release, with algebra-based games to come later.) So I am not a dispassionate outsider here.

For his Forbes article, Shapiro interviewed Jean-Baptiste Huynh, and everything the DragonBox designer says, I agree with 100%. Here is my take on the benefits of playing DragonBox (besides the fact that is it fun).

A student who plays through the new, greatly expanded version of the game will undoubtedly engage in a substantial amount of (contextualized) algebraic thinking focused on the solution of linear equations in one variable. The score they obtain in the game will provide a good measure of how well they have mastered that form of thinking (i.e., solving single-variable linear equations).

Does that mean the student can then sit down and ace a standard written algebra exam? Not at all. Even though the later stages of DragonBox and DragonBox2 involve on-screen manipulations of the standard symbolic representations of equations, the step from physically moving digital objects to manipulating symbolic expressions on a page is a much harder cognitive challenge than one might first think. The human mind simply finds it very difficult to reason in a purely abstract fashion. (In my book The Math Gene, published in 2000, I investigated the reasons for that difficulty.)

At issue is the notorious transfer problem, which, roughly speaking, is the difficulty humans face in taking something that has been learned in one context and applying it in another.

Huynh is of the opinion that it requires a human teacher to help the student take the difficult step from completion of his game to mastery of symbolic algebra, and I agree with him. I suspect that not everyone will be able to make the transition, no matter how good the teaching, but many will.

There is certainly a lot to be gained from mastery of symbolic algebra. First of all, learning at that level of abstraction is readily applicable to any specific domain. Second, being able to reason free of the complexities of any application domain is extremely powerful.

On the other hand algebra (or, more accurately, algebraic thinking) was successfully used in commerce for many hundreds of years before the modern, symbolic variety was introduced in the sixteenth century. So acquiring useful algebra skills is not totally dependent on mastery of symbolic algebra.

A major question is, will playing DragonBox increase the likelihood that a student will be able to master symbolic algebra, compared with a student who does not have that game experience? There is good reason to assume the answer is “Yes,” but that remains to be fully tested—something that can be done only now the game (and others like it) is out. (The analogous question remains to be answered for my own company’s forthcoming games.)

My reason for suspecting that playing video games like DragonBox is highly beneficial in learning symbolic mathematics—the kind that is tested in our school system—is perhaps best explained by an analogy from Hollywood. In the 1984 movie The Karate Kid (I can’t bring myself to watch the 2010 remake) and its sequel (KK2), martial arts instructor Mr Miyagi prepares his young pupil Daniel for Karate tournaments by getting him to polish a car, sand a floor, catch a fly with chopsticks, and paint a fence, all of which develop the reflexes and muscle memory required for key Karate moves, which Daniel uses to great effect later in the movies.

True, this is not sound educational theory, though many teachers (and most athletic coaches) adopt a similar approach. (This is a blog, remember, not a research journal.) But until we have something more concrete, the analogy works for me. Indeed, I am betting my company on it—as is Jean-Baptiste Huynh.

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.