Course: Calculus: Single-Variable
School: University of Pennsylvania via Coursera (free)
Instructor: Robert Ghrist
Calculus is one of the grandest achievements of human thought, explaining everything from planetary orbits to the optimal size of a city to the periodicity of a heartbeat. This brisk course covers the core ideas of single-variable Calculus with emphases on conceptual understanding and applications. The course is ideal for students beginning in the engineering, physical, and social sciences.● Are there any prerequisites for taking this course?Students are expected to have had prior exposure to Calculus at the high-school (e.g., AP Calculus AB) level. It will be assumed that students:▫are familiar with transcendental functions (exp, ln, sin, cos, tan, etc.);
▫are able to compute very simple limits, derivatives, and integrals; and
▫have seen slope and area interpretations of derivatives and integrals, respectively.
● Is this course hard?Yes, it is! Like, really hard. But it’s a satisfying-kind-of-difficulty not unlike running a race or climbing a mountain.
● Am I ready for this course?Take the diagnostic exam, which opens on Day 1 of the course: that should help you decide.
(Addendum: This course has been converted to Coursera’s new platform so the content may have changed and the experience will be quite different from that described here. The videos are available as standalones on Youtube)
This is the first Coursera MOOC I’ve outright flunked. Twice. It’s also one of the best courses I’ve taken. But take that warning above seriously: it is, indeed, like, really hard.
Much is made of the approach: everything you ever wanted to know about Taylor series, but couldn’t ask since by the time you got to Taylor series in Calc 1 you were fried. By Week 4 I was doing Taylor series in my sleep. I still have my cheat sheet taped to the wall over my computer. I could take it down, but I still think I’m going to try this again some day. Besides, I got to like Taylor series (even if I still get binomial and geometric series confused a lot).
Each week included an ungraded homework assignment of 6 to 10 questions; this wasn’t the “do the odd problems 1-25 on page 63” homework, each problem tested a different concept or approach. Lots of student did the homework in a half hour. It usually took me a couple of hours. Each “chapter” – a topically-focused group of about four weeks – ended with a graded quiz of another 10 questions or so. The quizzes, however, total only 20% of the overall grade, with 80% coming from the final exam. The timed final. Needless to say, I never got anywhere near that far, but even if I had, I wouldn’t have stood a chance. I would’ve liked to have been in a position to try, though.
And if the material isn’t hard enough for you: each week has a “bonus” video (my favorite was on applications of differentiation like the boundary operator and lists, but I can’t say I got past the “oh, cool” stage), and a “challenge” quiz available for those who dare, so it’s designed to take on those with a firmer grasp of calculus, as well. I even managed some of the Challenge questions in the first four weeks.
I wish I could continue to outline the whole course, but I never got beyond the Ordinary Differential Equations of Week 5. The spirit was willing, but the brain. Just. Could. Not. But it’s a calculus course, it’s got the stuff you’d expect – derivatives, integrals, differentials, optimizations, applications – at a deeper level than Calc 1.
So let’s talk about the art instead.
Art in a calculus class? That’s one reason this was so much fun. I like colors. This blog theme is black so the colors stand out. And the colors stand out in these videos as well. I mean, LOOK at this stuff. That isn’t a standard font, by the way; the font, the diagrams, the animation is all individually crafted – yes, crafted – for this MOOC.
Prof. Ghrist is quoted in another student blog as explaining each video took about 20 hours to make using PowerPoint. Now, I made a few short PowerPoint videos – nothing anywhere near this elaborate or high-quality, just using text, basic animation and sort-of-sync’d voiceovers – when I was in my Vidpo phase (I have a couple more I want to do, I’ve just been doing other things, like taking calculus moocs over and over) and I have no trouble believing each of these 15-minute lecture videos involved at least 20 hours of work. Since there are 60 lectures, that’s about 1200 hours. And that’s just producing the videos.
Every once in a while, a video went beyond “ooh, cool” and just knocked me out. Chaucer, Milton and Shakespeare showed up in an exercise modelling language change over time. Broke my heart that I couldn’t get that – my apologies, gentlemen, for not doing you justice. But just seeing you there made me happy.
Then there was the water faucet in the related rates video. The purpose was to compute the rate of change of the stream’s radius with respect to time, but what fascinated me was the bubbles (arrows added). A variety of bubbles. Some started on the left; some started on the right; some came in pairs; some were faster than others. I’m pretty sure there’s no Powerpoint button for “insert random bubbles” so the animation has to be designed with care. They were awesome bubbles.
Ok, so the bubbles were extraneous to the problem – what about the water leaking out of the tank video? Look at this time-lapse compilation of clips: the water really drips out of the bottom, sure, that’s nice. And the stream across the floor changes, that’s cool. But what’s really amazing is that, if you look at the timestamps (or take the course and see the actual animation), the level of the liquid in the tank drops faster as the volume decreases – which is the point of the problem (I think; remember, this is the part I flunked). I never really got to the point where I could quantify this. But the changing rate of the level of water as the tank narrowed, I got that – and again, I don’t think there’s a PowerPoint shortcut button to do that.
This is what it looks like when someone puts 1200 hours into a MOOC. Every minute shows. And that’s just the videos.
But wait, there’s more!
There’s no textbook for the course – though there’s a very good Wiki available that includes basic explanations and examples – but there is a Funny Little Calculus Text. And yes, it’s funny; it’s downright amazing. I cadged a clip on Archimedes’ last words for my semi-secret Blogging Euclid project (something else that’s languishing while I’ve been moocing myself to death this fall). I brought a Polyphemus clip in to my Greek Mythology course last summer. I’m not sure I’d be able to learn any calculus from it, but it’s a treasure hunt, with all the digressions (and I do love a good digression), puns and “pretentious literary references” (the only literary references worth reading).
The most important part of any math MOOC, of course (for me at least) are the discussion forums, where I can go crawling for help, comfort, and the ever-popular communal whine. The Summer session was far more active than Fall, but both were sufficient. While my questions were at the level easily handled by other students Prof. Ghrist would crop up at the unlikeliest times, to congratulate a student who’d made a particularly astute observation, to shepherd the adventurous through more advanced applications – or to reply to a random comment I made in a “venting” thread (as opposed to a more topic-oriented thread) with an incredibly kind reply. This means a lot – and so much for the impersonal, automated MOOC the haters keep talking about. This doesn’t happen in every course, but it tends to happen in math courses most often, I’ve found. Then again, I’m usually at my goofiest in math courses, since, unable to pay back the favor of helping students with the math, I turn to entertainment to earn my keep.
And there was the student, a fluent but not native English speaker, who was such a wonderful companion through the Summer session. He was operating at a far more advanced mathematical level than I, but we had a great time anyway, discussing such fine points of language as “click bait” and the perils of unintended idiomatic meaning in the use of the verb “to suck”. We ran into each other in another course, in fact, though I didn’t recognize him (sometimes people go incognito) and got into a rollicking discussion of the possible reasons a computer grading system would accept 7 ½ and 7.50, but not 7.5. I still think it has to do with significant digits, but I don’t have the foundation or the confidence to argue effectively – but winning the argument wasn’t the point.
When I said I’d flunked this course twice, I wasn’t counting the first time I signed up, since I only lasted 13 minutes before un-enrolling (to be honest, it was probably more like a week, but 13 minutes feels truthier). I hated it. I hated everything about it: the videos were stupid, the voice-overs were horrible (even in retrospect, I have to admit they take some getting used to; the prosody is a little odd, and the vocal fry would’ve earned Prof. Ghrist the cover of a NYTMagazine cover if he were a woman), and who the hell is this guy who insists on being referred to by title? In reality, it was just too far over my head, and I needed more background, but as a typical student, I blamed everything in sight rather than blame myself. I did, however, re-take Calc1 among other things.
When it became apparent early this year that this was the course to take next (it was highly recommended by seriously mathy people I’ve come to admire), I went looking for a way to feel better about it, and found a Youtube video of a talk by Prof. Ghrist in which he starts talking about Milton, and then uses the Divine Comedy to talk about the topology of Dante’s afterworld (a lecture, by the way, that I found more interesting than most of the material in the edX Dante course – and, by the way, no trace of vocal fry). Yep, that’ll do it.
Then I found the Funny Little Calculus Text, and realized – there’s a sense of humor there. How’d I miss that first time around? Besides that I was so depressed at being out of my depth, I guess. Once I turned it into Steven Colbert Teaches Calculus (there’s a passing resemblance for those of us with facial recognition deficit, to whom all white guys with short dark hair and glasses look the same), I started looking forward to it. And I ended up loving it enough to flunk it twice. I’d flunk it again, but I can’t take the heartbreak (people think I’m kidding when I talk about crying over math), and it’s not good for the completion stats all the MOOC haters love to quote.I keep trying to be a success story, but I think it’s my destiny to instead be a cautionary tale: teach your children well, or decades later they’ll still be banging their heads against a brick wall, trying to find a way in.
All of my favorite math MOOCs – oddly, just math – end up with theme songs; I’m not sure why. Jim Fowler’s Calc1 had “Still Alive” from the video game Portal; Keith Devlin’s Mathematical Thinking had the weird and incomprehensible “Hand-Made” set against a mathematically modellable murmuration of starlings, and now, SVCalc has Coldplay’s The Scientist“: “I was just guessing at numbers and figures / pulling your puzzles apart… Tell me your secrets, ask me your questions… running in circles, chasing our tails, coming back as we are; / Nobody said it was easy, no one ever said it would be this hard; / I’m going back to the start” (I have to skip over most of the lyrics or it gets a little creepy – it is after all a love song, and Edward Frenkel’s the only one who can get away with that sort of thing.)
I did, in fact, go back to the start, three times, but I think I need to go farther back before I try to go forwards again. I’ve got some high-school level moocs on edX coming up in 2015. And there’s still Mike Lawler’s kids (who, at 8 and 10 are way ahead of me in their grasp of what numbers do; they spent last weekend’s Family Math figuring out the last 4 digits of Graham’s number), who teach me something every day. Most importantly, they’re teaching me to be mathematically fearless. I have a ways to go.
Based on what I saw happening with other students in this MOOC, I think those who have a better grasp on math than I do may find this course difficult, but productive, and passable. And for those who aren’t highly invested in grades, but are looking for a way in, it might just do that, as well.
Or you can just admire the art, and come up with your own theme song. You may learn something in spite of yourself.