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.)