Course: Discovery Precalculus
School: University of Texas at Austin via edX
Instructors: Dr. Mark Daniels
Quote:Discovery Precalculus is not like any other mathematics course you’ve ever taken. Our classroom for this course will be the University of Texas at Austin inside the Blanton Museum of Art. I can’t think of a better classroom to learn mathematics in a creative way than to be surrounded by a creative environment. It is your creative energy that will make the course work, and your creativity can be put into mathematics.
Sounds great, doesn’t it? Well, that art museum stuff is pure bait-and-switch; true, each of the seven units started with a one- to two-minute video filmed in an art museum, but that was about the extent of it. Maybe I shouldn’t have had such high hopes.
And I did have exceptionally high hopes. Given how much I complained about the last Precalc I took being 100% skill-drill, this seemed made to order. In fact, this could’ve been a terrific class. But it wasn’t. And I get cranky when a course that could’ve been terrific, is seriously flawed by administrative decisions and/or poor execution.
Now, I admit, I’ve never taught anything, and the people who designed this course are experts in math education. And: I’ve always admitted, I’m a mathematical idiot. But I do have the benefit of, after a lifetime of mathphobia, nearly three years of working on math in a wide variety of online settings from scheduled MOOCS to following blogs and homeschooling videos to working through self-paced sequences to clawing my way through books. And I’ve been a student in a ridiculous number of MOOCs in a wide variety of fields using different teaching styles on five different platforms. So while I know nothing about classroom teaching, and while actual teachers (and some students as well) may view my ravings as somewhere between naive and downright ignorant, I have some opinions about what helps in a MOOC (or at least what helps me), and what gets in the way.
The material itself wasn’t the problem. I liked that it was truly aimed at preparation for calculus. While it included the usual trig identities and algebra review, it progressed to far more interesting places. I recognized something very close to Riemann sums and integrals. Limits and simple derivatives were part of the course – and not just in a superficial way. While I haven’t done any multivariable calc, I think I noticed some material in matrices and polar coordinates that presaged that as well.
Though some of the material went by me (regressions, for instance), I had a great time with other units. Before things got quiet, a little knot of three or four of us had a blast figuring out what kinds of bottles would result from a variety of volume/height graphs, inventing bottles-within-bottles and magical pressure-sensitive spigots. I loved the more hypothetical questions about functions – can there be a function that’s both even and odd? Is there a function that can’t be inverted, even if the domain is restricted to one point? This is way more fun than one of those “how to find the equation of a parabola given the focus and directrix” precalcs.
I also had a lot of fun with the unit on alternate coordinate systems: polar roses, vectors, parametric equations, topics I was barely aware of before. But: I had fun because I found material on youtube and in pdfs all over the internet that helped me understand what was going on; if I’d relied on the course material, I don’t think I would’ve made much progress at all. And it was a very lonely journey, since by then, forum posts were few and far between.
So in spite of my complaints – and I’ll get into more details about that in a second – I’d recommend to course to someone who’s very comfortable figuring things out, who doesn’t mind working without much direction. In other words: if you understand how IBL works – or if you can learn math by reading a textbook – this could be the course for you. The problem is: most students who can learn math by reading a textbook, don’t need MOOCs to begin with. As for the rest, they need a course to teach them how to take the course.
I tend to gravitate towards strugglers; they’re my peeps. On day 1, I noticed a lot of students asking things like, “Where are the lectures?” and “Is anyone going to teach anything, or is this just a collection of problems?” I tried to help – I answered questions, I modeled how to work on some of the first problems, I gave suggestions for how to approach material – but a lot of students gave up immediately (and, by the way, IBL is supposed to teach you that the one thing you can never, ever do, is give up, but it seems the course assumed students had already learned that lesson). By the end of the first week, posts had slowed to a trickle (not a good sign in a pedagogy that’s built on communicating ideas to others and working together to develop solutions), and by week 3, I doubt there were 10 posts a week (other than mine). It’s generally understood that only a very small percentage of MOOC students ever post to the boards, but here, where people were posting and then were never heard from again, I suspect it was more of an indication of a sky-high dropout rate.
Why did I stay?
I had some advantages going in. That’s unusual for me; I usually start math courses with one hand tied behind my back, one foot in a bear trap, and a headache. But in this case, I’d already encountered a lot of formative assessment, so seeing questions on material that hadn’t been covered didn’t strike me as strange. And, most importantly, I’d taken a couple of IBL-based MOOCs, both of which prepared me in very different ways.
The first, Intro to Mathematical Thinking (taught by Keith Devlin out of Stanford on Coursera) completely changed how I view math, taught me how to go from “I don’t know how to do this so I’m stupid and I give up” to “so how do I figure out what to do?” and emphasized that confusion, frustration, and failure are part of learning, too; that the point of a math class isn’t to get a good score, but to understand more about math; and to take the time I need – and multiple passes if necessary – to get to that understanding. Granted, MathThink involved its own complaining and kicking and screaming (the center spot on the Bingo card would be “I’ve been a math teacher/engineer/physicist for x years, and if I’d ever taught like this I’d be fired”), but I knew what to do in Discovery Precalculus because Keith Devlin taught me that knowing math isn’t about knowing what to do the instant I read the question, but knowing how to sit down and figure out what to do when I don’t know. I ended up recommending MathThink, which was about to start at the same time, to a lot of people who didn’t understand the approach. I’m sure that endeared me to edX and to UTA. Interestingly, another Discovery Precalc student, perhaps the most active after me, turned out to be currently enrolled in MathThink course (she wrote a beautiful proof of a logarithm law and I asked if she’d taken Keith’s class), and she agreed, MathThink helped her handle Discovery.
The second IBL course I’d taken was Effective Thinking Through Mathematics from another UTA professor, Michael Starbird. In order to prep for that course, I’d gotten hold of a used copy of his book The Heart of Mathematics which starts with a bunch of puzzles, and a lot of advice to just try to figure out what’s involved in them. What’s strange is that I felt more support, encouragement, and guidance from the pages of that book than I felt in this MOOC, and I don’t think that’s the way it should be. But the upshot was, I knew what to do when faced with a bunch of puzzles without lectures, puzzles without spaces for the answers, in the first week of Discovery. And I understood some problem-solving approaches (make it a simpler problem; retreat to what is already known; ask questions, make mistakes) that, along with my own penchant for drawing diagrams and using colors in equations, James Tanton’s dictum: “Do something,” and Richard Rusczyk’s addition, “If you can’t do something smart, do something stupid,” gave me some ideas for how to proceed.
I sure wish some of those lessons, and similar lessons, had been included in this course, instead of the art gallery shtick. They’re incredibly valuable in any case, and might’ve helped a few strugglers stick around a little longer.
So what went wrong? I can give you my purely subjective, non-expert opinion: a handful of administrative decisions meant what may be a great classroom experience didn’t effectively translate to the MOOC world. Such as:
…the decision not to include a detailed video introduction to IBL methodology, and some specific suggestions on how to deal with it. The only introduction was some high-concept vague rhetoric about art and pronouncements along the lines of “This is a different kind of course… you will construct your own knowledge…” but no hint on how someone should start on doing that. At the end of the first week, they added an FAQ page, but it was more high-concept, vague rhetoric. Too little, too late.
I’m from the humanities side of the aisle, so I love high-concept vague rhetoric, but it would’ve been nice if they’d also included some concrete advice like: “Play with the ungraded questions; you might not immediately know how to do them, but try to figure it out. Draw pictures, or think about concepts you already know; hypothesize and see what works and what doesn’t. Make a start, then post your attempt and see if someone else can add to it.” A single example showing someone figuring out a problem from a starting point of total confusion – a la the Starbird course – would’ve been a terrific modeling tool.
It still wouldn’t have been for everyone (most students want the score, the heck with the learning), and there would’ve been plenty of grousing, but I think more people would’ve stayed if they’d understood that some degree of confusion and not-knowing was inevitable and had practical tips on how to deal with it. Because IBL isn’t just about math, it’s about developing learning skills.
… the decision to run the course as self-paced, meaning some students were working on unit 3 on day 1, and some were in the final unit 7 on the first week. I understand this is non-negotiable (all MOOCs are moving in this direction, I’m sad to say), so it’s something they’re going to have to find a way around, because it’s not conducive to the kind of group effort and communication central to IBL. My own experience bore this out: I was a bit ahead of the curve, and after the first couple of weeks, when I posted for help or just to trade ideas on a particular topic that interested me, there was no one there. The people ahead of me never looked back, the people behind me weren’t there yet. I finished the coursework a few days ago, and I still check the message boards (of late, I’ve even started posting supplemental materials in the hopes of getting something going), because dammit, IBL is supposed to teach you about communicating math concepts, and I want to learn to “speak math,” but I can’t do that working by myself, so if by some miracle someone asks a question, I’m going to benefit from it for as long as the course is open. Then again, I’m kind of weird.
… the decision to hide answers. This seems to have been a conscious, pedagogy-based decision, since they kept defending it (“we want you to get out of the habit of immediately jumping to the back of the book for the answer when you can’t solve a problem on the first try”) and I don’t necessarily disagree with the concept. Here, it was problematic, because there were many places where you couldn’t be sure which of a group of questions you got wrong. And, because we weren’t permitted to discuss answers to graded questions (a completely valid prohibition), some of us never knew what we did wrong. In a “real” IBL setting, this might work because you could eventually go over an answer (or get it from another student, hopefully after the test), but here it meant a lot of us were just out of luck on some concepts.
…the decision to ignore the few ways edX message boards can be made more conducive to communication. I’m not sure if this was a decision, or just a lack of understanding of the edX platform. They’d do well to check out DelftX’s Calc001x, or, better yet, the superb MIT three-part Calculus series, which made the edX forums as usable as they get (
which still isn’t great, since the edX platform has at least three built-in roadblocks to effective forum communicationUpdate: edX has, as of May 2016, greatly improved the forums with the addition of one feature: “bumping” posts to the top when replies are made, a feature that’s standard on most message boards but somehow only now has made it to edX; I am rejoicing, as it’s a huge help): things like having a dedicated “introductions” thread, putting post windows in each topic so the post is automatically and correctly categorized and students can find posts on a given problem, and having staff pin general information threads so they don’t get lost. There were some other indications that the staff didn’t fully know what they were doing. Every once in a while, we were advised to check the Progress tab for the bar chart that indicates scores. This is fine, except that multiple exercises are included in each bar, so a perfect score on one set of exercises can yield a Progress Bar score of… 65% for the first part of the unit. Which freaks people out (trust me, it does). A great many errors in the auto-grader also cropped up over time, which brings us to…
…the decision to keep staff far away. I can appreciate that this, too, is non-negotiable, and in fact is part of the IBL “students teaching each other” thing. Though there were obvious problems with that here, I agree with the concept. But the lack of staff became particularly acute around posts indicating potential errors in the autograder. Sometimes they’d get fixed; sometimes staff would acknowledge the posts but not fix the error; sometimes they’d be acknowledged and never mentioned again so who knows if they’d been fixed; and sometimes they weren’t seen by staff at all. Every new course has errors; this one had too many, and too little attention was paid to fixing them promptly. I’ve seen this in other MOOCs, including those on other platforms: now that professors aren’t involved in the courses any more, and “distance learning specialists” are in charge of administering courses, no one really takes ownership. The educational experience suffers. MOOCs are turning into what people thought they were back in the beginning, when they weren’t that way at all.
I know I sound pretty negative about this course, but I really did try to cooperate. In the first couple of days, I answered a lot of “But where are the lectures?” and “What are we supposed to do?” questions. I posted my own thought processes on some of the ungraded questions, to model an approach that other students might find helpful and to get some discussion of the math going. When another student suggested we make a list of resources, and staff responded by opening up a course Wiki, I populated it with multiple resources, both general math sites and topic-specific videos (antithetical to IBL, but sometimes necessary) and web pages. I don’t think anyone ever used the Wiki – only two other students ever posted a resources – but I was so happy that staff made this effort to meet us halfway, I fell all over myself cooperating.
I don’t know why I took such ownership of this course. Maybe because I’m mentoring and CTAing other courses, so I’ve suddenly taken a fancy to shepherding and advising and liaising. By the way, it’s not lost on me that my efforts to help were often clumsy and ultimately ineffectual, if not counterproductive; after all, the course did still dwindle. But I tried, because it seemed to me someone had to try something. Or maybe because I so wanted it to work, for my own selfish reason: I’m still desperately trying to understand math – especially those nasty parabolas that smile and frown at me but won’t give up their secrets, those parabolas I now don’t truly understand in three different ways – and it still eludes me.
In fact, I tried so hard to help, I found an email in my inbox a few days into the course. I was nervous: were they kicking me out? I’d been pretty blunt with a staff member about some of the problems we were facing (“May I give you some feedback… .This is my opinion, and should not be taken personally, though it may feel that way; I just met you, I don’t know you well enough to want to yell at you. But after 2 1/2 days of this class, I want to yell at someone….), and equally blunt with other students (“We all have three options: Complain, quit, or see what we can get from it”). But they weren’t kicking me out, they were… well, I’m still not sure if they were trying to shut me up or co-opt me, but the Project Manager asked if I’d act as CTA for the course. Knock me over with a feather! I wrote back to say 1) I didn’t know if I would continue in the course (I did), 2) I’m a mathematical idiot (though, as it turned out, I did quite well score-wise), and 3) there’s more to CTAing than a badge that appears on posts. For the past couple of months I’ve been involved in Coursera’s Mentoring and CTA programs on a couple of different courses, and having other CTAs to confer with – and a private forum in which to confer – as well as having staff contact, is essential, and that wasn’t part of the deal. So I declined, but made some pointed suggestions.
This was, by the way, a unique experience for me; I’ve never been seen as a leader in a math course before. Sometimes I’m the class clown, sometimes I’m the cheerleader for the lost, and sometimes I’m just the slow kid in the class, but I’ve never really been in a position to be helpful before, and I’m grateful that I had that opportunity. I see the course is scheduled to start again in January 2016, and I’m almost tempted to go back as a CTA, just to liaise some more. But I think maybe I should leave well enough alone.
Maybe it’s the system they want, for whatever reason. Maybe its real purpose is use in classrooms, a popular thing right now; that might be a really great use, in fact, since a classroom teacher could oversee and shepherd the experience into more of IBL should be – but that means this isn’t a MOOC at all, it’s a curriculum or OCW. Maybe in the next session, a different group of students will complain less and participate more, and the forums will be rollicking. Maybe, in spite of the empty forums, thousands of students completed the course and are thrilled with it. Maybe a lot of things.
All I know is my experience, so that’s all I can report. It could’ve been great. It wasn’t. Doesn’t mean it was worthless, not at all – but it could’ve been so much better, for so many more students.