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.
NPR had a piece this morning (12 Nov) comparing the Eastern learning paradigm of 'struggle' vs the Western view of 'intelligence'.
As a MOOC student in Prof Devlin's class, I saw the challenge faced by trying to expand the Western paradigm and allow for struggle in the learning process.
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.
Couldn't help but think of Dan Meyer's moniker "less helpful" when I read the above. It's not about less or more, it's about appropriate help. Everyone needs that. Students need help in learning how to learn. That conversation rarely takes place.
So, then, what are the consequences of turning public schools and universities into a "survival-of-the-fittest affair"?Teachers and professors are held responsible for "retention". Administrators contend that these "fittest" students never needed us in the first place, and it's precisely the folks who don't survive that society needs us to educate.
At MathFest 2012, Robert Ghrist gave an interesting talk about using algebraic topology to teach calculus to undergraduates. When an audience member asked about precalculus mathematics, he responded that as a university professor he had not given any consideration to the teaching of precalculus mathematics. For me that highlighted the vast disconnect between the Ivy League optimism about MOOCs and the reality that most math professors, especially most members of the MAA and attendees of MathFest, must spend a considerable amount of their careers working with students in precalculus mathematics, like it or not.
You have previously written about the overemphasis of symbolic mathematics and how the typical high school student who will not be a scientist my not need symbolic mathematics. Do you advocate a survival-of-the fittest approach to these students, as well, and if so what are the consequences for the students who are apparently not fit and their impact on society? I feel like the MOOC approach is not so much addressing any problem in education, but rather it is merely a means for top institutions to cast a wider net so that they can more easily disperse with unfit students. If this is the case, it certainly does not serve as a model example for the rest of us who deal with the castaways.
Whistler: Your comment strikes me as depressingly US-centric. The main driving force behind MOOCs, at least at Stanford, which hitherto has been leading the charge, is to make Stanford education available (at least the parts that can be provided on a MOOC platform) to the entire world, particularly the millions of people who would otherwise have no access to higher education of any kind. The many emails I and my fellow MOOCers have received from such individuals around the globe demonstrates that, even at this early stage, we are hitting that target.
Is MOOC education survival of the fittest? Absolutely it is. Given what that initial letter M stands for, the medium cannot offer anything else. If a student who grows up in the affluent US, and attends a college here, can’t hold his or her own in a MOOC environment against a young person from an impoverished third world nation, with no such advantages, and requires coddling, then frankly, I’ll focus my efforts on the latter. That maximizes the impact I can have, given my position. As you say, you and many of your colleagues can, and do, focus on the former. There’s demand and there is a supply to meet it. What’s the problem?
The fact is, the US student has many fallback options, including the ones you mention. The third world student has nothing else, and nowhere else to go. MOOCs are not as good as a traditional college education. They serve little purpose for a student who is able to attend college or university. But they (the Stanfiord MOOCs) are not designed for those students; they are for the millions around the world (including the millions of educationally disenfranchised people in the US) who have no access to quality education.
Stanford president John Hennessy has talked about a “tsunami” about to hit higher education. I think he is right in using that term, but not in the scenario he has in mind. What I see is the true democratizing of higher education on a global scale. And in today’s world – global village, Flat World, call it what you will – I think that is exactly what we (i.e., the entire world, not just the highly privileged US) will need. There is already an acute shortage of adequately skilled people to meet today’s needs, and a stroll around any high tech hub indicates that the need it being met by global trawling for talent. That practice will surely continue to expand. MOOCs open the door to millions who would otherwise have no access to the kind of job, and hence the kind of life, that many of us in the US take for granted. Are US students thereby being denied the same opportunities? Not at all. We are simply trying to level the playing field.
Incidentally, who ever said MOOCs were intended to be a model example for the rest of higher education?
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