MOOC update #1

I'm near the end of week 2 of Dr. Rob Ghrist's Calculus MOOC , so I thought it would be a good time for an update on how this experiment is going.  There are 38 UF students enrolled in this course with me.  I really don't know how many of them are keeping up with the class, doing the homework, etc.  That is one thing that I'm most interested in--how many people will actually complete the course? 

I created a poll to find out how much math each person has taken.  All 20 students who responded have already completed at least Calculus III at the university.  So this course is definitely review for all of us.  That's a general occurrence with MOOCs at this stage--many of the participants already know the material.  I suppose it would be a more interesting experiment for me to take a course on a subject in which I am not an expert, but I thought it might be better to start with something that would not require a lot of extra work on my part.   

Anyway, here are my initial impressions of the MOOC experience.  Fifteen-minute lectures are nice.  My attention doesn't really wander because there isn't time for that.  Rob's videos are particularly good because he draws well; the screen is always colorful and attention-grabbing.  I'm not surprised by this since Rob has a reputation for his careful attention to detail and seemingly boundless energy (fueled by Monster, I think).  His approach to the material is one I like--begin with Taylor series as definitions for functions.  Of course, this assumes that students have seen calculus before and know how to differentiate and integrate basic functions, but this idea leads to a more solid conceptual understanding of what's going on.  It also allows us to calculate things effectively and efficiently.  Here's one of the (challenge) homework questions: 

\[ \lim_{x\to 0^+} \frac{\sin (\arctan (\sin x))}{\sqrt{x} \sin 3x + x^2 + \arctan 5x} \]

A first-semester calculus student  would cringe at this before plowing ahead with L'Hopital's Rule.  The derivative of the numerator is a bit of a nightmare and the potential for algebraic errors is enormous.  But, if you know about the series expansions of trig functions, this becomes much easier.  For \( x \) close to \( 0 \), both \( \sin x \) and \( \arctan x \) are approximately equal to \( x \); therefore, the numerator above is about \( x \).  The denominator is approximately \( 3x\sqrt{x} + x^2 + 5x \) .  The limit is therefore \( 1/5 \) and we arrived at that answer pretty easily (indeed, I did it in my head).

So, the pedagogy is sound, but what about the interface?  As I said, the videos are great, but there is a fundamental problem which has no solution--there is no ability to ask questions of the instructor in real time.  Yes, there are discussion groups at the coursera site and they can be very useful.  That sort of peer instruction is valuable, and in this case one of the participants found a flaw in one of the homework exercises that could only be solved using complex analysis.  Rob himself was impressed by this, as was I, because no one had noticed a problem with the exercise in two runs of the course.  I certainly didn't catch it.  So I've definitely learned a couple of new things.  Still, if I were learning calculus for the first (or second) time via this platform, I'm not sure I would necessarily like it.  Students definitely have to be self-motivated.  That's true for traditional lecture courses at any university, of course, but in the coursera environment it is easier to let things slide.  That may be a function of the cost--free at coursera versus pricey at the university.

A word about the homework:  there aren't that many problems, but I think that's appropriate.  We often assign too much homework to our students; this course walks the line correctly in my view.  Each lecture has two homework sets attached, the core and the challenge.  The former reinforces the material in the lecture while the latter goes a bit deeper.  The problem I gave above is a challenge problem, of course, because it goes to the more conceptual aspects of Taylor series.  I will confess to making occasional errors on the problems, mostly small arithmetic mistakes because I tend to do them quickly (duh, I know this stuff already so surely I don't need to spend any time on the problems, right?).   

This is proving to be an interesting experience.  I'll be back with more updates in the coming weeks.  I still don't know what I think of the whole MOOC idea at this stage, but I'm glad I will be able to make a more informed decision about them in the future.