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.)

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.

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.