School: UC-Irvine, through Coursera (free).
Instructors: Dr. Sarah Eichhorn, Dr. Rachel Cohen Lehman
Quote: This course is designed to prepare you for a college-level Calculus course. Through this course you will acquire a solid foundation in algebra and trigonometry. Emphasis is placed on understanding the properties of linear, polynomial, rational, radical, piece-wise, exponential, logarithmic, and trigonometric functions. You will learn to work with various types of functions in symbolic, graphical, numerical and verbal form.
[Addendum: This course has been converted to the new Coursera platform; it’s divided into two parts. Content may have changed, and the experience is likely to be different]
That might’ve been the intent… but for me, the experience was a lot different. This course reminded me that I hate math. I forgot for a while there.
It sounds churlish to bitch about something offered for free, but considering the other math MOOCs I’ve taken over the past year and a half – courses I loved – this was a disappointment. It’s ironic that my disappointment is partly due to the fact that I’ve had the good fortune to encounter some spectacular math courses through MOOCs (most of which I’ve talked about at length), with instructors who put thought and effort into their planning and execution. Granted, I struggled through some of them, and sometimes felt a little beat-up by the time they ended, but I always recognized their value. Like most silver linings, it comes with a cloud: ordinary math courses just don’t cut it any more.
This was a very ordinary math course. It isn’t what MOOCs can be. It isn’t even what math can be.
Things started out with great promise. Week 0 – a terrific idea – allowed everyone to review some concepts from prereq algebra, to get used to the (very clumsy, grr) system of answer entry (come on, Coursera, can’t you figure out how to integrate LaTex? And why so stingy with the Preview buttons?), and do the usual “Hi, I’m such-and-such from Hereabouts and I’m 16/30/75 years old and I love/hate/never took math” forum posts. I noticed a number of acquaintances from other math MOOCs, all of them seemingly beyond the need for pre-calculus (most of us met in various calculus classes, in fact) but someone unfamiliar with my difficulties with algebra probably thought the same thing about me. I was looking forward to the course, and to finding out what I’d been missing all along, whatever it was that was keeping me from understanding algebra enough to recognize what I encountered in other settings. I was also looking forward to being able to help out others; I love answering questions, giving hints, working on explanations, and I figured I could do more of that than I typically can in Calculus.
But things went downhill fast.
Each lecture video, delivered by a disembodied voice, started out with, “Let’s look at… solving rational equations/evaluating logarithmic expressions/using half angle identities” and ended with “And this is how we… solve/evaluate/use.” In between, one or two problems was worked out step by step in great detail. That’s great – and in one case, I discovered why I have so much trouble solving inequalities – but nothing related to anything else; no particular reason for looking at rational equations, or logarithmic expressions, or half angle identities, was given. It was back to 10th grade, when I thought what mathematicians did all day was solve problems out of a book, never even thinking about who wrote the problems in the book or why they needed to be solved; it was what math was about: here’s a problem, find the answer, next.
A PDF textbook was included in the course materials, and it seems the idea was: if you want to know why synthetic division, or the quadratic equation, or the half-angle identity works, go look it up. Now, there’s a lot to be said for doing personal research, but if I could learn math from a textbook, I wouldn’t be taking MOOCs. It seems to me even a few videos explaining key concepts would’ve gone a long ways. And, for pete’s sake, the MathIsFun website was used as a major resource. It’s not that I have any problems with the website (except the name) – I love their “interactive unit circle” – but it shows an attitude of “Why teach? Just link. We have better things to do.” Maybe that’s the idea behind the course: it’s not about increasing understanding, it’s about listing resources, and after that, you’re on your own. Just like real life.
The instructors, who seemed active on the boards in Week 0, disappeared completely after that. Now, that happens in lots of MOOCs (though less so in math courses), but usually staff or CTAs (community teaching assistants, students who took the class before and did well, and showed some ability and interest in helping other students on the forums) are on hand to provide expertise. Not so here. At various points, even the Coursera technical staff seemed to abandon us, and issues of missing videos, out-of-order videos, and inaccessible elements went unaddressed. Even the strangest element of the course – they announced the discussion forums would be “closed” once the exam was released – never happened; I’ve never heard of that being done before, it seems ridiculous to me, but to announce it and not do it just shows how unconnected the people running this are. It’s like they’ve converted this course to “remote control/self-paced” while keeping the time limits. I don’t like the self-paced approach (and, unfortunately, that’s where MOOCs are going), but even I recognize each approach has advantages and disadvantages; still, combining the worst of both makes no sense.
Some of us tried to expand beyond the “here’s a problem: do it” mentality. We had a rollicking discussion of positive and negative square roots, but I still could use some expert guidance on this; it seems sometimes the primary square root is always assumed, and sometimes it isn’t, and I don’t feel confident that I’ve nailed down the possibilities. I would’ve liked to have done a lot more work on logarithms and exponentials, one of the main reasons I took the class; I got more out of my random wanderings through AoPS and Khan than I did from this course, which covered how-to-do-it but not that elusive why-it-works. Trig identities was the biggest disappointment. Someone asked about the connection between the unit circle and the traditional Cartesian graph of trig values, and while I could point them to lots of interesting graphics, I realized I have no idea how to explain it. I should, at this point. I should be able to create those graphics (well, except for the programming part). So the takeaway is this: I took yet another trig class and all I got was a list of identities. I have that already. I wanted to understand them, how they fit together, why they work.
I also brought in a couple of goofy “how would you solve this” puzzles from other sources. In both cases, that led to wonderful explorations with one or two other students. Most of it didn’t have much to do with material in the course (though it was the first time I’ve ever been motivated to actually use logarithms by anything other than a math test), but it was enjoyable; maybe, for me, learning to “enjoy” math is the most important lesson. There wasn’t much interest in this, however (only one other person ever joined in), so I stopped doing it.
Another great experience was in helping another student, through email. He’d missed a week way back, and was struggling with a few questions. Going through his work and figuring out 1) what the answers should be, and exactly why, and 2) where he went wrong, felt like a very profitable use of what turned into a significant chunk of time. The old “you don’t understand something until you can explain it to someone else” is very true. The discussion forums provide some opportunity for this, but mostly people are looking for answers. Also, I’m so slow in coming up with explanations, I’m usually too late, and while I’m working out the details (and discovering what I don’t understand about the underlying principles), someone else has answered the question and everyone’s moved on. I’ve always been too slow for real time – even the real time of a message board. Discussions tended to dead-end without any feedback from the original questioner. Math course message boards are usually terrific (I still refer to old Calculus and Mathematical Thinking posts occasionally) but not here; I’m not sure why. The students seemed younger; lots of high schoolers, maybe that had something to do with it.
Maybe I have unrealistic expectations. Maybe I’m missing something so obvious to everyone else, no one needs this stuff. Maybe I’m lazy and I should continue to research it myself (which hasn’t, to now, been a screaming success, and is laden with misconceptions that don’t always become evident until a unique set of circumstances exposes them – at which point I’m back to square one). Maybe this course was too easy for me – though that strikes me as a ridiculous notion. Maybe the course I want doesn’t exist, or I’m looking in the wrong place. But I expected a lot more in the way of understanding, and instead got a lot of “this is how we graph parabolas.”
I did well on the weekly quizzes, score-wise, which surprised me. Nearly every quiz, I was shocked when I scored 4/4, 5/5 on the first attempt. Here’s what still concerns me: if I don’t know whether or not I’ve got the right answer, does it count?
Then there was the final exam. The timed final exam.
I’ve always said I don’t care about grades, and to a large degree that’s true; at my age, I’m over grades. But with this course, it was a matter of pride; if I’m going to slam something, it doesn’t look good to flunk. Also, since I’ve been taking Calculus for a year and a half (and I’ll keep taking it for the next year and a half, until I feel like I understand it), I should be able to do pre-calc. So I felt some pressure to pass. But I don’t do math quickly, and the time limit worked out to about 4 minutes per question. That’s barely enough time for me to set things up so I’m ready to do the math. See, I work in multiple media: I’ve become quite adept at using the Word Equations function for algebraic calculations, which eliminates handwriting mistakes (but allows typos; nothing’s perfect). But sometimes I use a whiteboard, for drawing unit circles or graphs or just putzing around diagramming a number line or graph. I also have piles of paper, which is nice if I want to work standing at the window instead of sitting at my computer. Sometimes I start in one medium, then realize another is better suited. Sometimes I just use the completely wrong approach, and don’t realize it until halfway through; I have to start over. And sometimes (often), I make “bone-headed mistakes” – drop a minus sign, calculate 4*8=36, that sort of thing. Sometimes I have to stop and think about adding and subtracting negatives and positives. Sometimes I need a walk around the block, or a cup of coffee, or just a rest break. This all adds up to a lot more than 4 minutes per question.
Suddenly it became all about getting a grade – a pat on the head, approval, performing the tricks I’ve been trained to do – instead of about doing math. I realized: this is why I always hated math classes. And this is why I’ve loved the math MOOCs I’ve been taking, even when I didn’t do well on tests: I was still learning something, failing at something worth doing, something worth trying for again (which is why I take so many math classes more than once).
For the record: two attempts at the final were allowed. On the first, I only got to 24 of 35 questions, got 2 of them wrong, for an overall score of about 60. I went through every test question (some of the questions I’d skipped because they looked scary turned out to be quite simple, if I’d just taken the time to actually read them and think about them), checked a few procedures (I know how to find the inverse of a function, I just don’t always remember that I know), lined up my ducks in a row (do I have my whiteboard? Calculator? Coffee? Teddy bear? Half-angle identities cheat sheet? Because, no, I’m not going to memorize that). And I ended up with a perfect score. Yes, a few of the ones I’d already seen were repeated, but most were new.
But did I learn anything?
Well, of course I did – I got some much-needed practice in trig identities, for instance. But mostly, I learned what I wanted to be doing instead.
I identified some concepts I want to understand better. The primary square root, for example, versus the square root function. Derivations of trig identities. One question from the final intrigued me: it turns out tan^2(x) – sin^2(x) = tan^2(x)sin^2(x). How can the difference of two functions, equal the product of those same two functions? I know the identities; I can solve the problem – I got the question right, so I know the procedure – but there’s a relationship there that I don’t grasp just from knowing tan = sin/cos… What is it? But it was the final, I had four minutes; so I saved it, and I looked at it more closely once the timer stopped ticking away – but I’m a mathematical idiot. I’ll look again.
During the introductions of Week 0, it seemed to me that the course was taken by more first-time MOOCers than most. I felt like going around apologizing to them, telling them, “This isn’t what a MOOC can be.” But who am I to decide for someone else? Maybe it was exactly what they needed. It wasn’t what I was hoping for, but maybe I can spin a silk purse out of a sow’s ear anyway. Maybe that’s what I needed to learn.