**In Part 4, I describe how I try to avoid the educational danger of a well taught class.**

*This post continues the previous three in this series.*

We’re still on the problems caused by the expectations students bring to a course – problems I hope can be alleviated (to some extent) by that new introductory video I talked about in my last post.

Unfortunately, working against any positive effect my pep talk may have is the fact that my MOOC looks like a traditional, instruction-based course.

So too does the regular classroom instantiation of the course, whenever I give it. It’s not.

Neither version of the course is instruction-based. When the twenty-five or so students in a physical class are glowering at me from a few feet away, because I have not shown them in advance how to solve the problems on the assignment, I can patiently explain that their expectations have led them astray, and I can coach them into seeing my lectures for what they are: *motivational examples*. It’s not about learning how to solve a particular kind of problem, I can tell them. It’s developing the ability to set about solving a completely novel problem. And doing that inevitably involves a lot of failure. For solving novel problems frequently boils down to acquiring the ability to respond positively and constructively to failure.

That kind of direct feedback is not possible in a MOOC. There’s the difference.

Yes, I do use those lectures to provide some initial coverage of some standard material (as well as to do some sample worked solutions). But it’s not pretty. It’s WYSIWOSG – “What You See Is What Our Students Get.” When I record one of those lectures, I imagine a student has come into my office asking for help, and we sit down together at the desk and work through whatever is giving them grief. It is a simulation of two people working together, rather than one giving instruction.

It’s not scripted, I record “as live,” and I leave any mistakes there on the page. The technology is kept deliberately at the low end. (I do sometimes ramp up the speed of my handwriting at the editing stage to match my voice, but interestingly many students tell me they had not noticed that until they read or heard me point it out.)

Because it’s not about the content, you see. It’s all about the thinking. Those lectures are really about the *how*, not the *what*.

For anyone who wants to see a polished presentation of the *what*, the basic factual content, beautifully laid out, there is a short, ultra cheap, self-published (optional) textbook (more accurately, a course companion book) I wrote to accompany the course. Or the student can reconstruct their own polished account based on my lectures, basing them either on brief notes they take during the lecture or by printing off the screen after I have finished the page.

It’s important to see my lectures for what they are: *examples* of mathematical thinking. They are certainly not provided so the student can learn to replicate me. Each student has to develop mathematical thinking for themself, and it doesn’t have to be identical to mine in all respects.

Of course, the nature of mathematics places significant limitations on what constitutes mathematical thinking. There is an intrinsic limit to how far they can deviate from me or any other mathematician. But in many cases, the range can be much broader than most people realize – certainly so wide that the idea of a “model solution set” makes no sense.

If I were to provide a model solution to a given problem, as students frequently ask for, I am depriving them of one more well constructed novel problem on which to develop their ability to solve new problems. I will have shown them my way of solving it, and for ever after they will be that bit more likely to approach any similar problem my way, instead of developing their own ability.

It’s the same when I record a video of myself solving an assignment problem after I have given the students time to attempt it themselves. (I call those “tutorial videos.”) The actual solution I produce is not important – there are usually other ways to solve the problem, or at least other ways to express the solution.

Notice that I said I solve problems *after* I have asked the students to try them. I don’t assign problems that are slight variants of ones I have demonstrated. I give them an opportunity to try – and fail – to solve the problems I assign, before I demonstrate one of the possible solutions – again as an example, not “*the* solution.”

Of course, the whole thing *is* planned out. It could not be otherwise. But the planning is at a high level, well above the details. A casual viewer would not see my performance as a “well taught lesson.”

At least, I hope they would not, since “well taught lessons” rarely lead to good results, as has been demonstrated by a number of research studies, starting with Alan Schoenfeld’s much-cited 1988 paper When Good Teaching Leads to Bad Results: The Disasters of “Well-Taught” Mathematics Courses.

For students who seek the comfort of a well-presented coverage of the core content, there is my book. Though cheap, it is not free, and that too is deliberate. It is not a part of the course. It’s something that is there for anyone who wants it. There is a (token) cost for getting it. It is a carefully constructed, stand-alone artifact, of value in its own right. Take it or leave it. Consulting books when required is a valuable part of mathematical thinking. But my book is not part of the course; it is not really a course textbook as such, rather it is optional, *supplemental* *reading*. A course companion, or a reference, if you will.

The course is about *thinking*. About *doing*. About *action*. In my lectures I am demonstrating what it is like to *do* mathematics – to *generate it* within one’s own mind. Reading it is something else.

The videos are an attempt to take the student inside my mind, in real time, to get a sense of what it means to think mathematically. Again, let me stress, not to replicate me. Just to provide an example.

It should be obvious that having pre-prepared pages or slides in the videos, either hand-written or typed, with or without staged reveals, will not work. That approach might help a student learn how to read and comprehend mathematics, but not how to produce it.

That people can in fact acquire the ability to think mathematically from such limited exposure to another person doing so is, in many ways, remarkable. But it’s hardly unusual; countless people have learned this way over thousands of years.

The ingredients of mathematical thinking are already present in the human brain. (See my 2000 book The Math Gene, an account of the evolutionary development of human mathematical capacity.) It just takes a series of suitable triggers to bring those ingredients together. My role as instructor is to make those triggers available. But the student has to pull them and to have a reasonably good aim when doing so.

This is the only way I know how to “teach” mathematics (i.e., mathematical *thinking*). It’s how I learned.

It’s possible mathematical thinking may not be achievable for everyone, even with a personal tutor. But making the attempt is achievable.

An absolutely key ingredient to any level of success, however, is accepting that failure is part of the process, the focus of Part 2 in this series.

That’s why, having observed how three MathThink MOOC classes have responded to the three versions of the course I have offered, I have decided to make awarding of the course Statement of Accomplishment in future dependent entirely on course completion, not level of performance. I’ll pick up that theme next time.

Step by step I am moving away from features common in traditional, classroom courses. I am also constantly asking myself what exactly is a course, and why do we so often package learning into courses. There too, student expectations play a significant role. Clearly, this blog is not close to ending. I, for one, am curious to know where it will go next. (Seriously. Like my MathThink course and MOOCs in general, this blog is a work in progress. A living document. Think of it as reflective lab notes.)

After reading this blog entry, “Let’s teach the world” ought to be modified to “World, let’s learn!” I took your MOOC this fall, I learned a lot, & I enjoyed every instance of it. Thank you & keep up the good work of helping people dance with genuine thinking.

Have you considered taping an actual in-person tutorial with one, two, or maybe three students? It’s an option used sometimes with lit classes (you should talk to Al Filreis, the two of you, plus the MOOCulus folks, are the hardest-working and most thoughtful MOOC-adapters I’ve encountered so far as a student). With “real” students, of course, it can get out of control really fast, so it’s very tricky (technically, too – sound, video, etc), but it’s also interesting to see the misconceptions people get (something I’ve been seeing a lot of lately in connection with science).

That’s definitely something I’ve thought about. It would take a fair amount of time and some funding to do it, and so far I have neither. But if/when both are available, I might give it a try. The Veritasium videos I mentioned in one post make excellent use of capturing people’s misconceptions and confusions in science, and a similar approach should work for mathematics.