MOQR, Anyone? Learning by Evaluating

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.)

MOOC Lessons

Planning and giving a university-level MOOC (massively open online course), as I did this fall, requires a complete re-evaluation of what it means to “teach” at that level. (Or any level, come to that, but university teaching is all I have first-hand experience of. My knowledge of K-12 teaching is limited to some familiarity with current theories of learning and some widely – but not universally – accepted principles of good pedagogy. Lots of theory, but no practice.)

I’ve always felt that the focus in university mathematics education should be on student-led learning, not teacher determined instruction. The key ability the student needs to develop is being able to take a novel problem and figure out a solution. That is, after all, what professional mathematicians do! As far as I can tell, I share this model of university mathematics education with the vast majority of my colleagues in the professoriate.

That’s not to say we – at least some of us – don’t reflect on what we do in the classroom, nor that we don’t attend courses, workshops, webinars, and presentations on educational technique. But my sense is that those of us that do this end up spending far less time providing “well crafted instruction”, and putting more of our effort into creating an environment in which our students can learn for themselves, and stimulating and encouraging them to do so.

In adopting this approach we capitalize on a hugely important factor you find at university but typically not at school: we are professionals who love our subject with a passion and have devoted our lives to its pursuit. When we stand in front of a class and write on a blackboard (mathematicians still prefer a blackboard to a whiteboard), we are not giving instruction so much as providing an example of how a pro thinks.

For, at heart, contact with the pros is what university education is about. For the vast majority of students, university is the first time in their lives they come shoulder-to-shoulder with the disciplinary experts. Those disciplinary pros do not have the pedagogic content knowledge required of a good K-12 teacher – at least nothing like to the same degree – but that is compensated by something that I think is far more important at that more advanced stage of a student’s development: learning by up-close observation of, and interaction with, a domain expert.

For sure, you will find university professors who have a different overall philosophy than the one I just sketched, but as I noted already, I think most of my colleagues have a similar view to mine.

Certainly, the celebrated physicist Richard Feynman, in the Preface to his 1963 book Six Easy Pieces, wrote:

The best teaching can be done only when there is a direct individual relationship between a student and a good teacher – a situation in which the student discusses the ideas, thinks about the things, and talks about the things. It’s impossible to learn very much by sitting in a lecture, or even by simply doing problems that are assigned.

A key element of operating in the fashion I am advocating (as is Feynman) is, then, the person-to-person interaction that takes place between a student and a professor (admittedly, often limited to the few short weeks of a university term). When a professor tries to port a course to a MOOC, however, that personal interaction goes out the window.

The primary issue is not the second O in MOOC – “online”, with the professor and students in different physical locations. That may be a significant factor, but just how significant is not yet known. (With the availability of rich social media, I think only empirical research will tell us the answer. Intuition is no longer a reliable guide to the importance of physical co-presence, if indeed it ever was.)

Rather, the key factor is that initial M – “massively”. In an online class with twenty-five students, the professor may be able to interact regularly with each student. But when there are 65,000 students, scattered around the Information Superhighway, there can be no meaningful interaction. The flow is asynchronous and entirely one way, from the professor to all those students.

That means the student becomes totally responsible for his or her learning. There can be one-on-one interaction, but it has to be student-to-student, perhaps within small study groups.

The task of the professor is then to design a course that can succeed as a result of student-student and small-student-group interactions.

As I was planning, and even more so when I was giving, my first MOOC, I felt very much like the conductor of a 65,000-player orchestra. I got to choose the pieces the orchestra will perform, I controlled when to start each piece and when to stop, and to some extent I dictated the tempo. I observed and occasionally commented on the overall group’s performance. I sometimes gave hints and advice. But each one of those 65,000 members of the orchestra did the actual playing. In principle, by pulling together, they should have been able to complete the current piece tolerably well, if I had suddenly been taken ill and had to put down the baton.

In fact, for the first time in my career, I was able to conduct a class the way I’d always wanted to: as an experienced guide who helps the committed learner in a minimal way, only when absolutely necessary.

That approach to university “teaching” can be done in a traditional class setting, but it takes an unusual individual and an even more unusual environment in which to do it. R L Moore is the most famous example of a mathematician who “taught” that way. (It’s so unusual, I need to put quotes around the key verb.) (See my MAA columns from May 1999 and June 1999.)

I tried the Moore Method, as it is called, a few times in my career, but it never worked well. I have enormous respect for my colleagues who have made it work – and some have. But, faced with teaching a MOOC (better make that “teaching”), I had to rely on one (but by no means all) major element of the Moore Method: the students would have to figure things out for themselves.

Moreover, I was of necessity relieved of the factor that has always led to my abandonment (or severe weakening) of the Moore method whenever I tried it: students who can’t handle the approach drop out.

In a physical class of maybe twenty-five students, I always felt a responsibility to do the best I could for each one. Particularly problematic were the ones who had gotten to university by virtue of “good teaching,” who could jump through all the templated hoops that were placed before them, but were floored when presented with a totally novel problem. After all, it was not their fault they were disadvantaged by “good teaching.” I felt it was my job to rescue them as best I could.)

With a volunteer student body of tens of thousands, on the other hand, you can’t avoid losing a few thousand, and you can afford to do so. There will still be many thousands of students who remain. Indeed, the “end of course evaluation” is bound to be overall positive, because the ones who don’t like, or cannot cope with, your approach simply drop out along the way.

In short, a MOOC is very much a survival-of-the-fittest affair.

At this early stage, MOOCs are being developed and offered very much in an experimental mode. But if, and when, they become an accepted part of the global educational landscape, then it’s not just higher education that will change, but society, as the international playing field gets truly leveled, with the most talented and ambitious people from everywhere in the world competing on merit alone.

Facing that possible future, maybe we need to ask ourselves if we do the best for our own students here in the US by being “too helpful.” And if the answer is “no” at university level, maybe it should be “no” in the high school as well.

For further discussion of my MOOC, see my blog MOOCtalk.org.

What is mathematical thinking?

What is mathematical thinking, is it the same as doing mathematics, if it is not, is it important, and if it is different from doing math and important, then why is it important? The answers are, in order, (1) I’ll tell you, (2) no, (3) yes, and (4) I’ll give you an example that concerns the safety of the nation.

If you had any difficulty following that first paragraph (only two sentences, each of pretty average length), then you are not a good mathematical thinker. If you had absolutely no difficulty understanding the paragraph, then either you are already a good mathematical thinker or you could acquire that ability pretty quickly. (In the former case, you most likely pictured a decision tree in your mind. Doing that kind of thing automatically is part of what it means to be a mathematical thinker.)

Okay, I had my tongue firmly in my cheek when I wrote those opening paragraphs, but there is such a thing as mathematical thinking, it can be developed, and it is not the same as doing mathematics.*

In my last column, I talked about my decision to self-publish a really cheap textbook to accompany my upcoming MOOC (massively open online course) on Mathematical Thinking. At the time of writing this column, just shy of 40,000 students have registered – and there are over two more weeks before the class starts.

As a result of sending out a number of tweets, chronicling my experiences in developing my MOOC in a blog MOOCtalk.org, and posting some videos about the upcoming course on YouTube, I’ve already received a fair number of emails asking for details about the course. (At one point, so many so I had to temporarily shut off comments on MOOCtalk.org, lest WordPress closed me down under the assumption that with so much traffic it must be a porn site.)

In this column, I’ll answer one question that came up a number of times: What is mathematical thinking? In fact, I’ll do more, I’ll answer the four questions I opened with.

To people whose experience of mathematics does not extend far, if at all, beyond the high school math class, I think it’s actually close to impossible for them to really grasp what mathematical thinking is. I used to try to convey the distinction with an analogy. “K-12 mathematics is like a series of courses in digging trenches, pouring concrete, bricklaying, carpentry, plumbing, electrical wiring, roofing, and glazing,” I would say. And then, after a brief pause, I would continue,  “Mathematical thinking is the equivalent of architecting. You need all of those individual house-building skills to build a house. But putting those skills together and making use of them requires a higher-order form of thinking. You need someone who can design the building and oversee its construction.”

It is a great analogy. I felt sure it would convey the essence of mathematical thinking. But many conversations and email exchanges over the years eventually convinced me it was not working. Saying A is to B as C is to D works fine when the recipient has good understanding of A, B, and C and some understanding of D. But if they have not even a clue about D, or even worse, if they believe that D actually is C, then the analogy simply does not work. It’s one of those analogies that is brilliant if you are sufficiently familiar with all four components, but hopeless as a way to explain one in terms of the other three.

Once I realized that, I set out to find a better way to describe it. It took me most of a whole book to do it. Not the ultra-cheap textbook I mentioned above. That has a different purpose. Rather, my book on using video games in mathematics education.

Below, in about 850 words, is the nub of what I say in that book in about 75 pages. (Yes, that’s quite a compression ratio. Clearly, it’s lossy compression!) After the quote, I’ll give you a specific example of mathematical thinking from my own past involvement in national security research. (Don’t worry, my part was not classified. You can read it without me having to kill you.)

BEGIN QUOTE [pp.59–61]:

[Mathematical thinking is more than being able to do arithmetic or solve algebra problems. In fact, it is possible to think like a mathematician and do fairly poorly when it comes to balancing your checkbook. Mathematical thinking is a whole way of looking at things, of stripping them down to their numerical, structural, or logical essentials, and of analyzing the underlying patterns. Moreover, it involves adopting the identity of a mathematical thinker.]

[For instance] like most people, when I am doing something routine, I rarely reflect on my actions. But if I’m do ing mathematics and I step back for a moment and think about it, I see myself [not just as someone who can do math, but] as a mathematician.

“Well, duh!” I hear you saying. “You are a mathematician.” By which I assume you mean that I have credentials in the field and am paid to do math. But I have a similar feeling when I am riding my bicycle. I’m a fairly serious cyclist. I wear skintight Lycra clothing and ride a $4,000, ultralight, carbon fiber, racing-type bike with drop handlebars, skinny tires, and a saddle that resembles a razor blade. I try to ride for at least an hour at a time four or five days a week, and on weekends I often take part in organized events in which I ride virtually nonstop for 100 miles or more. Yet I’m not a professional cyclist, and I would have trouble keeping up with the Tour de France racers even during their early morning warm-up while they are riding along with a newspaper in one hand and a latte in the other. […] Being a bike rider is part of who I am. When I am out on my bike, I feel like a cyclist. And you know, I’d be willing to bet that the feeling I have for the activity is not very different from [the professional bike racers].

It’s very different for me when it comes to, say, tennis. […] I don’t have the proper gear, and I have never played enough to become even competent. When I do pick up a (borrowed) racket and play, as I do from time to time, it always feels like I’m just dabbling. I never feel like a tennis player. I feel like an outsider who is just sticking his toe in the tennis waters. I do not know what it feels like to be a real tennis player. As a consequence of these two very different mental attitudes, I have become a pretty good cyclist, as average-Joe cyclists go, but I am terrible at tennis. The same is true for anyone and pretty much any human activity. Unless you get inside the activity and identify with it, you are not going to be good at it. If you want to be good at activity X, you have to start to see yourself as an X-er  – to act like an X-er.

A large part of becoming an X-er is joining a community of other X-ers. This often involves joining up with other X-ers, but it does not need to. It’s more an attitude of mind than anything else, though most of us find that it’s a lot easier when we team up with others. The centuries-old method of learning a craft or trade by a process of apprenticeship was based on this idea. [The video games scholar James Paul Gee, in his book What Video Games Have to Teach Us About Learning and Literacy, p. 18] uses the term semiotic domain to refer to the culture and way of thinking that goes with a particular practice – a term that reflects the important role that language or symbols plays in these “communities of practice,” to use another popular term from the social science literature. […]

In Gee’s terms, learning to X competently means becoming part of the semiotic domain associated with X. Moreover, if you don’t become part of that semiotic domain you won’t achieve competency in X. Notice that I’m not talking here about becoming an expert, and neither is Gee. In some domains, it may be that few people are born with the natural talent to become world class. Rather, the point we are both making is that a crucial part of becoming competent at some activity is to enter the semiotic domain of that activity. This is why we have schools and universities, and this is why distance education will never replace spending a period of months or years in a social community of experts and other learners. Schools and universities are environments in which people can learn to become X-ers for various X activities – and a large part of that is learning to think and act like an X-er and to see yourself as an X-er. They are only secondarily places where you can learn the facts of X-ing; the part you can also acquire online or learn from a book. […]

The social aspect of learning that goes with entering a semiotic domain is often overlooked when educational issues are discussed, particularly when dis cussed by policy makers rather than professional teachers. Yet it is a huge factor. […]

END QUOTE

In my blog MOOCtalk.org, I will explain what persuaded me to try to prove that the pessimism I expressed in the above passage about someone becoming an X-er through a remote experience like a MOOC might be misplaced, at least in part. But my focus here is describing mathematical thinking.

In many cases, the real value of being a mathematical thinker, both to the individual and to society, lies in the things the individual does automatically, without conscious thought or effort. The things they take for granted – because they have become part of who they are. This was driven home to me dramatically in the years immediately following 9/11, when I was one of many mathematicians, scientists, and engineers working on national security issues, in my case looking for ways to improve defense intelligence analysis.

My brief was to look at ways that reasoning and decision making are influenced by the context in which the data arises. Which information do you regard as more significant? How do you weight, and then combine, information coming from different sources. I’d looked at questions like this in pre-9/11 work – indeed that was the research that brought me from the UK to Stanford in 1987, and by the time the Twin Towers came down, I had written two research books and a number of papers on the topic. But that research focused on highly constrained domains, where the complexity was limited. The challenge faced in defense intelligence work is far greater – the complexity is huge.

I did not have any great expectations of success, but I started anyway, proceeding in the way any professional mathematician would. I could give you a list of some of the things I did, but that would be misleading, since I did not follow a checklist, I just started to think about the problem in a manner that has long become natural to me. I thought about it for many hours each day, often while superficially occupied with other life activities. I was not aware of making any progress.

Six months into the project, I flew to D.C. to give a progress report to the program directors. As I fired up my PowerPoint projection and copies of my printed interim report were passed around the crowded meeting room, I was sure the group would stop me half way through and ask me (hopefully politely) to get on the next plane back to San Francisco and not waste any more of their time (or taxpayers’ dollars).

In the event, I never got beyond the first content slide. But not because I was thrown out. Rather, the rest of the session was spent discussing what appeared on that one slide. I never got close to what I thought was my “best” work. As my immediate research report told me afterwards, beaming, “That one slide justified having you on the project.”

So what had I done? Nothing really – from my perspective. My task was to find a way of analyzing how context influences data analysis and reasoning in highly complex domains involving military, political, and social contexts. The task seemed impossibly daunting (and still does). Nevertheless, I took the oh-so-obvious (to me) first step. “I need to write down as precise a mathematical definition as possible of what a context is,” I said to myself. It took me a couple of days mulling it over in the back of my mind while doing other things, then maybe an hour or so of drafting some preliminary definitions on paper. The result was a simple statement that easily fitted onto a single PowerPoint slide in a 28pt font. I can’t say I was totally satisfied with it, and would have been unable to defend it as “the right definition.” But it was the best I could do, and it did at least give me a firm base on which to start to develop some rudimentary mathematical ideas. (Think Euclid writing down definitions and axioms for what had hitherto been intuition-based geometry.)

The fairly large group of really smart academics, defense contractors, and senior DoD personnel in that meeting room spent the entire hour of my allotted time discussing that one definition. Not because they were trying to decide if that was the “right” definition, or the best one to work with. In fact, what the discussion brought out was that all the different experts had a different conception of what a context is, and how it can best be taken account of – a recipe for disaster in collaborative research if ever there was.

What I had given them was, first, I asked the question “What is a context?” Since each person in the room besides me had a good working concept of context – different ones, as I just noted – they never thought to write down a formal definition. It was not part of what they did. And second, by presenting them with a formal definition, I gave them a common reference point from which they could compare and contrast their own notions. There we had the beginnings of disaster avoidance, and hence a step towards possible progress in the collaboration.

As a mathematician, I had done nothing special, nothing unusual. It was an obvious first step when someone versed in mathematical thinking approaches a new problem. Identify the key parameters and formulate formal definitions of them. But it was not at all an obvious thing for anyone else on the project. They each had their own “obvious things.” Some of them seemed really clever to me. Others seemed superficially very similar to mine, but on closer inspection they set about things in importantly different ways.

“Your work is not classified, so you are free to publish your results, if you wish,” the program director told me later, “but we’d prefer it if you did not make specific reference to this particular project.” “Don’t worry,” I replied, “I have not done anything that would be accepted for publication in a mathematics journal.” Which is absolutely the case. I had not done any mathematics in the familiar sense. I had not even taken some mathematical procedure and applied it. Rather, what I had done was think about a complex (and hugely important) problem in the way any experienced mathematician would.

I’ve had a number of similar experiences over the years, and though they appear on the surface to be widely different (from analyzing children’s fairy stories to looking at communication breakdown in the workplace to trying to predict the endings of movies like Memento to trying to make sense of the modern battlefield), at their (mathematical) heart they all have the same general pattern.

That then, is mathematical thinking. How do you teach it? Well, you can’t teach it; in fact there is very little anyone can teach anyone. People have to learn things for themselves; the best a “teacher” can do is help them to learn.

The most efficient domain to learn mathematical thinking is, perhaps not surprisingly (though it’s not such a slam-dunk as you might think) mathematics itself. Particularly well suited parts of mathematics for this purpose are algebra, formal logic, basic set theory, elementary number theory, and beginning real analysis. These are the topics I have chosen for my MOOC. Other topics could serve the same purpose, but would require more background knowledge on the part of the student. But it’s not about the topic. It’s the thinking required that is important.

---

*One of the features of mathematical thinking that often causes beginners immense difficulty is the logical precision required in mathematical writing, frequently leading to sentence constructions that read awkwardly compared to everyday text and take considerable effort to parse. (The standard definition of continuity is an excellent example, but mathematical writing is rife with instances.) The opening paragraph is a parody of such writing. This comment was added a day after initial publication, when a letter from a reader indicated that he missed the fact that the opening was a parody, and complained that he found it difficult to read. That difficulty was, of course, the whole point of the opening, but that point is lost if readers don't recognize what is going on. So I added this remark.

The future of textbook publishing is us

In my May column, I announced my intention to give a free online course (a MOOC) this coming fall, and asked for assistance from the mathematical community.

That course, a school-to-university transition course, titled Introduction to Mathematical Thinking, is going ahead, with the first lecture on September 17. There is a brief description of the course, together with a short promotional video, on the Coursera website.

I have started a blog, MOOCtalk.org, to chronicle my experiences working with this new format and to provide a platform for feedback and discussion once the course gets going.

I also wrote a short textbook to accompany the course, Introduction to Mathematical Thinking. Though the textbook is not required for the course, some of my Stanford colleagues who gave the first generation of “Ivy League MOOCs” – just a few months ago, so fast has this new movement taken off – told me that many students want an old-fashioned physical book. On the other hand, all the transition textbooks I am familiar with are fairly pricey, which would put them well beyond many of the students who are likely to enroll. Moreover, none of them are designed to accompany a MOOC. So I decided to write one.

My two main criteria were: it had to be short (no more than 100 pages) and cheap (less than $10 in the US). The only option was to self-publish in print-on-demand format with Amazon’s CreateSpace service. I did not quite hit my page-limit; when I include the front material, the tally comes out at 102 pages, but that’s close enough. But I figured I’d cover my costs if I set the retail price at $9.99, just below my target.

The procedure is so ridiculously straightforward, I can see no reason why anyone should ever publish another textbook a different way, given the huge expense of textbooks. We authors have been typesetting our own manuscripts ever since Don Knuth first released TeX in 1978, and all that is required to produce a book with CreateSpace is to generate a PDF file that fits the page-size you select.

CreateSpace does not provide TeX support (by way of a style file), but they do provide sample pages for authors submitting manuscripts in Word, and I just played around with the page parameters in LaTeX until the output matched their samples for both odd and even numbered pages, which I checked by printing out copies of both, putting my output on top of theirs, and holding the two up to the light. (Low tech, but effective.)

In my case, I decided to produce my book in the standard 6 in x 9 in format, and the key LaTeX parameters I came up with are (for the record)
\oddsidemargin 1 in \evensidemargin .55 in
\marginparwidth .75in \marginparsep 7pt
\topmargin -.5in \headheight 12pt
\headsep .25in \footheight 12pt \footskip .35in
\textheight 7.5in \textwidth 4.95 in

When I submitted my final PDF file, CreateSpace’s automated checking system flagged the manuscript as possibly not being correctly formatted, but I pressed forward, since the next stage is that one of their employees examines the manuscript, and indeed that individual accepted it, confirming my suspicion that I was probably off by a millimeter or two, something that could upset an automated checking system but is close enough to pass a human eyeball test.

The point is, the whole process is so well designed, there is no reason why anyone who can use LaTeX should do anything other than self-publish from now on. With a very small number of exceptions, no one who writes a university-level textbook does so to make money. Our goal is to get material in front of students as quickly and cheaply as possible. If there were a way to do so that can save the students money, I am sure we would all want to do so, the more so given the way textbook costs have skyrocketed in recent years. With modern print-on-demand technology, we now can do just that.

You don’t need to know anything about publishing to do this. CreateSpace does for book publishing what TurboTax did for filing your tax return, and it does it in much the same way, by taking you through the entire process in a simple, step-by-step fashion, including cover design, securing an ISBN code, and selecting marketing channels.

For sure, the finished product is not quite as good as would be achieved with the professional expertise of a good publishing house. But to my mind, for a textbook, it’s close enough, especially when the resulting book can be sold for as little as a tenth the price a publisher would charge.

The one thing I paid someone else to do was copy-editing. I have written enough books to value highly the services of an experienced copy editor. (You might also want to pay an indexer. I did my own, but I have done so for several of my previous books.)

Of course, even with good copy editing, occasional errors creep through. Not long after my book was on the market, I was looking through one of my author’s copies (you have to buy them, but at an even lower price than the retail mark), and spotted a couple of small typos. A few minutes editing the LaTeX file, followed by a quick upload of the replacement PDF, and the correction was made, ready for the next person to buy a copy.

Returning to the MOOC now, let me re-iterate the request I made in my May column. I am giving my MOOC in the early fall to coincide with the many transition courses offered at colleges and universities across the US, in the hope that instructors of such courses will incorporate my MOOC in their courses in some way. My reason for this is that I think the only way to make a transition course MOOC work is to have enough participants who either are already familiar with the material (such as instructors) or else have direct access to such expertise (e.g., their students in a transition course). I see no other way for students struggling to understand the material to get the help, advice, and feedback they will need to progress. Social media provides various platforms for students to interact, to ask questions of one another and to comment on others’ work. But there has to be a mechanism for mathematical truth to find its way into the discussions!

So the key to making something like this work is, I think, to build up a Wikipedia- like community of instructors who, for five weeks each year, will make available their expertise to the thousands of students around the world who are taking advantage of a MOOC to obtain an education they would otherwise not have access to.

The benefit to the students in the transition classes given by MOOC-participating instructors is that their learning will assuredly be enhanced by acting as tutors for the students who are not so privileged. Both because teaching others is a powerful way to learn – as most of us discover when we become TAs at graduate school – and because those students will surely feel much more incentivized to understand by playing such a feel-good role.

Stay tuned to my MOOCtalk blog for updates on the project. And if you are an instructor giving a transition course this fall, please consider getting involved.

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

Higher education as we know it just ended. Exactly what will take its place is not at all clear. All that can be said with certainty is that within a few short years the higher education landscape will look very different.

That is not to say that existing colleges and universities will suddenly go away, or indeed change what they do – though I think both will occur to varying degrees in due course. What is changing now is what classifies as higher education, who provides it, how they provide it, who will have access to it, how they will obtain it, and how it will be funded. Distance education, for many years the largely-ignored stepchild of the higher education system, is about to come of age.

This is not just my opinion. My own university, Stanford, recognizes what is going on, and is taking significant steps to lead and stay on top of the change, and a number of Silicon Valley’s famed venture capital firms, who make their fortunes by betting right on the future, have sunk significant funding into what they think may be key players in the new, higher ed world.

Last fall, Stanford computer science professor Sebastian Thrun used the Internet to open his on campus course in artificial intelligence to anyone in the world with Net access, and 160,000 students from 190 countries signed up. Some 22,000 of those students finished the course, receiving “certificates of completion” signed by Thrun (and co-teacher Peter Norvig of Google), but no Stanford credit. (For that, a student has to be on campus and officially registered; annual tuition is $40,050 and entry is fiercely competitive.)

Demonstrating the entrepreneurial spirit that Stanford faculty are famous for, Thrun promptly left Stanford to found a for-profit online university, Udacity. With Udacity receiving financial backing from a large Venture Capital firm, the MOOC – massive open online course – suddenly came of age. A short while later, two more Stanford computer science faculty, Andrew Ng and Daphne Koller, secured $16M of venture capital funding to launch a second Stanford spin-off company, Coursera, a Web platform to distribute a broad array of interactive courses in the humanities, social sciences, physical sciences, and engineering.

Initial courses offered on Coursera include, in addition to several from Stanford, offerings from faculty at the University of Michigan, the University of Pennsylvania, and Princeton. Stanford president John Hennessy appointed a blue-ribbon panel of Stanford faculty to develop a strategy for developing, and delivering, online courses. For free. To the world.

Yes, you read that correctly. The faculty, the universities, and the new platforms are making the courses available for free. All the funding is coming – for now – from for-profit investors and the private universities themselves. Why are they doing that? If you have to ask the question, you don’t really understand the Internet and how it changes everything. Think Napster and the music industry or Skype and the telephone industry. Like the settling of the American territories in the nineteenth century, the initial focus is on establishing a presence in the new land; monetization can come later – almost certainly in ways very different from today’s.

Computer-assisted, distance learning is not new, of course. Stanford was one of the universities that pioneered it the 1960s; many universities have for several decades offered adult professional education courses for a fee, largely to raise funds; and there are the for-profit online schools like the University of Phoenix. More recently, led by MIT, a number of universities started making recordings of their regular courses, together with course materials, available online for free. So what has changed now?

The answer is the platform and the target audience’s experience and expectations have changed. What has been missing so far is the active participation of the distant student in a learning community. Building on technology developed at Stanford to support flipped classroom experiences for its regular students, Udacity and Coursera have secured the major investments required to build scalable, robust platforms that can take the small learning seminar and create a similar experience across the Internet.

A generation that has grown up on the Web has taken to the new online medium like fish to water. During the term when Thrun made his AI course available online, most of the Stanford students enrolled in his class stopped attending his lectures and took their information delivery online, at times convenient to them.

Is this the beginning of the end of physical universities? I doubt it. Though online courses are excellent for in-career professional learning, the absence of being a member of a physical community makes them a poor substitute – arguably no substitute – for a traditional college or university when it comes to providing first-pass education. But what about the millions (make that billions) in the world who do not have access to a university education? “Let’s teach the world” is a buzz phrase you hear increasingly among the Stanford faculty these days. And Stanford is putting resources into making this attractive dream a reality.

What makes it fascinating to a faculty member, is figuring out how to take a learning experience that works in a small-group setting on a campus, and re-creating a similar – or equivalent – experience online. Having decided last December that I would offer a math MOOC this fall, I found myself at once faced with a number of challenges.

By far the greatest problem is how to provide the personal, expert feedback that is essential to good mathematics learning. Web delivery is fine for providing instruction, but that is just a part of learning, and a minor part at that, as I discussed in the March Devlin’s Angle. At first, it seemed an impossible task. But with Stanford and the now independent Coursera building innovative new platforms, I began to see the glimmer of opportunities. Over the coming months, I’ll use this forum to write about my progress. And hopefully get your assistance.

My focus for this first foray into this new educational landscape is the high school to university transition. As every university mathematics instructor knows, many students encounter difficulty going from high school math to college-level mathematics. Though the majority survive the transition, many do not. To help them make the shift, colleges and universities often have a transition course. I myself developed one of the first transition courses in the late 1970s, when I was teaching at the University of Lancaster in England.

Such courses typically comprise a mix of some elementary mathematical logic, proof techniques, some set theory through to an analysis of relations and functions, with a bit of elementary number theory and introductory real analysis thrown in to provide examples.

Given the problems students typically have when they meet this material for the first time, doing this at a distance is a challenge. Even if they did well at math in school, most beginning university students are knocked off course for a while by the shift in emphasis, from the K-12 focus on mastering procedures to the “mathematical thinking'' characteristic of much university mathematics. Clearly, offering such a course as a MOOC is a huge experiment.

This is where you come in. (I hope.) One of the things we’ve learned at Stanford from offering MOOCs, is that a key component is the creation of a strong online community. Learning is all about human interaction. The technology just provides the medium for that interaction. In offering my math transition MOOC at the start of the fall term, when many colleges and universities offer their own transition course, I am inviting any instructor who will be giving such a course, together with their students, to join me and my MOOC students online, making interaction with other students around the world a part of a much larger learning community.

The result could be a total failure. I won’t know until I try. On the other hand, anyone who joins me might just find themselves at the start of something major, new, and exciting. The online learning revolution is going to happen, and existing educational institutions are going to have to adjust to it, just as the music industry did to the iTunes revolution. Why not jump on the train as it is leaving the station?

I’m going to make my course just five weeks long, starting in early October. By incorporating participation in my Stanford course part of your students’ learning experience, everyone could benefit. For one thing, your students are likely to be inspired by being part of an educational revolution that for millions of less privileged people around the globe can quite literally be life changing.

Because they will be supported by being part of a physical learning community, with the personal support of you, their instructor, your students will be highly empowered, privileged members of that online community. They can take advantage of your support so that they can help others. And as we all know, there is no more powerful way to learn than to try to teach others.

For that student half way round the world, trying to improve his or her life through education – by learning to think mathematically – the potential benefit is, of course, far greater. Helping that unknown young (or not so young) person make that step might just help inspire your own students to put in that bit of extra effort to master that tricky new transition material. Everyone wins.

If my Stanford MOOC draws a student body in the tens of thousands, which it might, based on the experience of my colleagues here, there is no way I and a couple of graduate TAs can provide individual feedback to every student. But if instructors and their students across the US join me, then maybe we can collectively achieve something remarkable.

I am making my MOOC deliberately short, five weeks, so participation will leave most of the semester open for participating instructors to concentrate on giving their own course, perhaps using their students’ initial experience in the MOOC community as a springboard for the rest of the course.

By the time I post next month’s column, I hope to have more details available. In the meantime, I ask anyone giving a transition course this fall to consider joining me in this experiment. The only cost is our time. There is no need to make any advance commitment to me or to Stanford. At this stage, all I ask is that you consider joining me. I believe we will all benefit. Let’s teach the world.