The Twelve Weeks of Differentiation MOOC

Course: Calculus 1A: Differentiation
Length: 13 weeks,6-10 hrs/wk
School/platform: MIT/edX
Instructor: David Jerison, Jen French, Stephen Wang
Quote:

How does the final velocity on a zip line change when the starting point is raised or lowered by a matter of centimeters? What is the accuracy of a GPS position measurement? How fast should an airplane travel to minimize fuel consumption? The answers to all of these questions involve the derivative.

I’ve taken three different calculus moocs in the past few years, and they’re all terrific in their own way. This was my second time through this one; I didn’t pass last time so I wanted to try again.

What I particularly like about the MIT courses is how they set up each topic with a series of lead-in questions. By the time you get to the actual instruction video, you’ve already seen a lot of what goes into the process, so it’s a natural extension of what they keep calling “intuition”. I’m not sure I’d call it that; I don’t think I have that much intuition about math, certainly not about calculus. There’s probably a sophisticated pedagogical term for this. Whatever it is, it helps. And yes, I managed a solid passing score this time.

It also helps that the two most frequently heard (if rarely seen) instructors, Jen and Steve, have a speaking style I like: calm, just the right speed, and with enough personality to forestall the “audio textbook” aura so many moocs have. I got to know Jen a little on the forums last time, and was very impressed with her patience and willingness to help us through questions. This time, the forums were primarily handled by a different instructor, Hanson, who was equally great to work with. Good people + good material = great class (some math is easy).

Though it’s 12 weeks long, this is only the first part of a series of three moocs designed to prep high schoolers for the AP Calculus exam. Integration starts in November, and Series/Sequences in the Spring.

The course starts with Week 0, a sort of optional orientation/prep week. No grades are recorded. There’s a set of prereq exams to gauge readiness, and a unit on limits for anyone who wants to get back into gear, as well as the opportunity for new users to get used to the platform.

The four content units are released every three weeks, a nice compromise between self-paced and scheduled; a missed week isn’t a catastrophe (and every deadline ended up extended anyway). Lots of questions and practice exercises are scattered in with the videos, and each unit has a final quiz with a part A – “nuts and bolts”, they call it – and part B, more application oriented. The timed final exam had a 48 hour window, which is a lot less stressful than requiring it all be done in one sitting. Each of these elements has a different impact on the ultimate grade.

The material consists of introductory intuition questions, videos by Jen and Steve, and occasional in-class lectures by Dr. Jerison. He’s a lot less warm and fuzzy about it all, but I’ve come to appreciate his style. After he goes through a step, he’ll pause, move to the side, and look around the room at the in-class students. I’m not sure if he’s checking for blank WTF faces, or just to see if most of them are done writing things down (or, for that matter, just catching his breath and finding his place in his notes), but it makes a nice rhythm that helps me to keep up. In terms of filming, I greatly appreciate that he gets out of the way of the board, allowing the live camera holder to adjust angles and zoom to incorporate everything. These are silly little logistical details that have nothing to do with math, but make it so much easier to follow.

I found the course far easier this second time around. I really don’t know if that’s because it was modified, or if something sank in over the past two years. I haven’t been working on calculus at all (though I do some math every day and have taken several math moocs and science moocs involving significant math, including just a whiff of calculus) , so I’m not sure what that would’ve been. One thing I’m pretty sure they added this time are “Recitation videos” explaining individual problems in great detail. These are part of the older OCW series; I found them extremely helpful, particularly those by Christine Breiner. They’re all available on Youtube or through the OCW site.

Though I’m feeling pretty good about doing so well, I realize that by this time I should be able to do this stuff in my sleep. The next course on integration will be a real challenge, since I’ve never been that comfortable with it and it was extremely difficult last time. IIRC, It’s also a lot less hand-holdy, with a lot more reliance on the in-class lectures by Dr. Jerison. But’s what’s next, so I’d best get to it. I did some review before this section, using Khan to refresh my memory on certain points; that’s probably more important for the integration course, so I should get started. I should. I should.

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It only hurts when I LAFF: Linear Algebra MOOC

Course: Linear Algebra – Foundations to Frontiers
Length: 12 weeks
School/platform: UTAustin/edX
Instructors: Maggie Myers, Robert van de Geijn
Quote:

Students appreciate our unique approach to teaching linear algebra because:
       • It’s visual.
       • It connects hand calculations, mathematical abstractions,
                 and computer programming.
       • It illustrates the development of mathematical theory.
       • It’s applicable.
What you’ll learn:
       • Connections between linear transformations, matrices,
              and systems of linear equations
       • Partitioned matrices and characteristics of special matrices
       • Algorithms for matrix computations and solving systems of equations
       • Vector spaces, subspaces, and characterizations of linear independence
       • Orthogonality, linear least-squares, eigenvalues and eigenvectors

I’ve never taken a linear algebra course before, though I’ve had some very basic work on geometric vectors, working with matrices, and Gaussian elimination through a variety of algebra and precalcs. I was looking forward to this. But, as sometimes happens (especially with math), it didn’t quite work out.

In brief: The course is set up as a series of lectures with embedded exercises, an additional set of problems at the end of the week, and four exams scattered throughout. A temporary license for Matlab is included, ending when the course is over. Staff coverage of the forums was excellent. A PDF of some material is provided, but they presuppose viewing the videos, and as usual with any math course, the video transcripts aren’t all that helpful without the videos. Disclaimer: I only made it through the middle of Week 8.

I quite enjoyed, and seemed to be doing very well at, the first three or four weeks. I think I learned a lot about linear combinations and transformations, what they have to do with matrices, and I had a lot of fun smashing Timmy Two-Space all over his grid. I saw a little hint of another point of all this with a (very primitive) weather prediction system, and that was pretty exciting. But it went downhill from there. I gave up in week 8, about halfway through. It wasn’t impossibly hard, but as time went on it had grown impossibly tedious; I just got seriously bored with slicing and dicing matrices for purposes that weren’t all that clear to me. We did have the option to skip over the Matlab algorithm exercises, but I had trouble telling where they began and ended. I completely lost the thread of “what am I doing and why am I doing it?” as calculations – small calculations, just adding and multiplying really but the stuff of nightmares for me – took over my life. I know there was something I was missing, but I never really understood what.

Let me say that I have no doubt at all that the material is essential to those who need linear algebra, and that those who are more comfy with math and computer programming would probably find it a great course. If I want to get to the point where I “know” linear algebra, I’ll probably have to take it again, but it wasn’t the right entry point for me. Of course, how would I know, since I’m still a bit hazy on what linear algebra is for.

I think one of the problems for me was that this was taught by computer science instructors, with a view towards optimizing algorithms as well as teaching linear algebra. Hence, memops and flops (which I actually understand, but don’t care about). Loops and indices. If those sound like music to your ears, this is the course for you, but as for me, STFU and leave me alone.

I’ve been hearing so many mathy people talk about how cool linear algebra is, and the course description includes “It’s visual” as a selling point. Other than Timmy, and a brief graphical description of two-rotation transformations, the only visuals I saw were printouts of algorithms and matrices, endless matrices to partition, multiply, transform. Maybe it got more visual in week 8, but I just didn’t want to do any more.

The instructors were very involved on the forums, promptly answering questions with humor, warmth, and encouragement. Prof. Myers told me about a very cool children’s book about basic combinatorics, Socrates and the Three Little Pigs; why kids that young would be learning combinatorics, I don’t know, but I spent a couple of nice hours figuring out how to fit three pigs into five houses under various conditions. Her videos of detailed proofs and exercise solutions were very helpful. And a mysterious image turned out to be computer wallpaper made from a beautiful image of a stained glass window from Prof. van de Geijn’s grandfather’s house in the Netherlands. These are great people! So I’m kind of puzzled about this: they seem to have gone out of their way to strip all that humor and warmth out of the course material itself. As a result, it was a “I’m going to read a textbook to camera and you watch the low-contrast, slightly out-of-focus slides” kind of course.

I’ve never thought of myself as someone who needs to be entertained in order to engage, but maybe I am, more than I’d like to admit, at least where math is concerned. And I admit I am somewhat spoiled by the truly exceptional moocs I’ve been fortunate enough to take. It’s also possible I no longer have the attention span for a longer course, especially one that requires so much of my time and fully focused attention over an extended period, since I was quite content for several weeks. I can sometimes skim through a philosophy or history lecture, but I have to pay attention to every detail when it comes to math, and it’s hard to sustain, even when I’m into it. And, of course, it’s very possible that, contrary to the Howling Stanfordtoids and their growth mindset, I’m just stupider than I think I am.

Even though I chose not to complete the course, I did find it very worthwhile for initial material. I’m investigating several other linear algebra sources – 3Blue1Brown’s linear algebra playlist on Youtube (which takes visual to a whole other level), Pavel Grinfeld’s lemma unit on linear algebra, and a couple of OCWs (I have trouble with OCWs; I can never figure out how to navigate them, where all the pieces are), and I’m finding that the initial material from LAFF has helped enormously. And, by the way, I think I finally understand mathematical induction thanks to this course, or at least I understood its use in the cases encountered here. So I’m glad I did as much as I did, and I hope to some day pick it up again.

Algebraic smart-ALEKS MOOC

Course: College Algebra and Problem Solving
School/platform: Arizona State University/ALEKS/edX
Instructors: Adrian Sannier, Sue McClure
Quote:

[Y]ou will learn to apply algebraic reasoning to solve problems effectively. You’ll develop skills in linear and quadratic functions, general polynomial functions, rational functions, and exponential and logarithmic functions. You will also study systems of linear equations. This course will emphasize problem-solving techniques, specifically by means of discussing concepts in each of these topics.

A lot of students were effusive with their praise of this course, raving about how much they learned. The staff, from the top echelons down, are highly enthusiastic about its efficacy. It comes with the option to receive ASU credit, so if that’s the goal, or if you’re reviewing algebra and want to know what you don’t know, it might work out great.

My experience was a bit different (but then, I’m kinda weird, especially around math). Remember, I have no background or training in education, I’m a mathematical idiot, and I’ve been focusing on algebra in many different venues for over a year now, so it’s possible this was just the wrong class for me (rule 1: every mooc works for someone, and doesn’t work for someone else). But I was disappointed. I keep hoping for an algebra mooc that helps me understand algebra, and all I keep finding are mastery-based skill drills. Fifty shades of Khan Academy.

On the plus side: the forums were crawling with staff. Most of the questions early on were logistical (“I can’t get into ALEKS… I can’t get this to work… where is the course?”) and they were answered very promptly, usually with pictures showing exactly which button to click. Staff support is a huge issue, particularly when the course is technologically complicated (involving an off-site element, a coaching system separate from the discussion boards, and electronic procedures for proctoring credit-bearing exams), and they really had it covered. I also loved their Twitter icon.

As for the algebra instruction, that was outsourced to ALEKS, a Khan-like proprietary learning system now owned by McGraw-Hill (Bias alert: Just typing that makes me nervous). The first step was an evaluation test of 80 questions (I could be misremembering the number; it took me several sessions over a couple of days). Then the Pie shows up, and from there, it’s a matter of “learning” and “mastering” the 419 topics. For each topic, a problem would pop up; in most cases, a link to some off-site video demonstrating how the problem should be done (often a Khan Academy video, in fact) and/or a page of explanation, would be available if needed.

My evaluation results gave me credit for 75% of the skills, though I still had to do some reviews or knowledge checks or something; I didn’t bother to master the system’s lingo, I just logged in and did whatever it handed me. Took about 50 hours all together, but that’s log-in time, so that includes time spent looking for more information on, say, the graph of a quadratic-over-linear rational function, a few minutes to check Twitter here, a round of Weboggle there… Oh, yeah, like you work 100% of the time when you’re at your computer.

The system was loaded with something they probably think of as encouraging messages: Great Job, Student! Exceptional work, Student! Presumably, one’s name is supposed to be substituted in for “student” but the interface wasn’t ready for that. I thought it was a brilliantly ironic metaphor for the entireCapture “personalized” experience. Call me a curmudgeon, but it seemed like overkill to be so praised for so little. I also received frequent assurance that I only had 7 skills to go – or maybe it was 12. Turns out, one of the counters updates immediately, and the other doesn’t. More irony.

Given the title of the course, and the description emphasizing problem-solving, I’d hoped that would be part of the deal. But in this case, problem-solving seemed to mean: “Here’s a problem in solving a multi-step equation involving natural logarithms. Solve it.” After I’d “mastered” all 419 topics, the adorable @math_goat tweeted out an actual problem-solving problem (fresh from the pages of mathisfun.com) – and I had no idea what to do with it. Three years of math moocs, over a year focusing on algebra, I zipped through this course in a week without breaking a sweat, so why am I still so fucking stupid?!? (Turns out, lots of students posted the answer on the discussion forum, but no one – NO ONE, including staff – could give any rationale for solving it. I guess everyone googled it, and in this course, the right answer is all there is).  At any rate, I wish I’d taken the course that teaches how to do that problem, rather than the one that teaches 419 ways of looking at a polynomial. I’d still be stupid, but I’d have more fun in the meantime.

ASU added on a coaching system, separate from the edX discussion forums. At first, I thought this was a fantastic idea, and eventually it might be (this is the first run of the course, and they do seem interested in making improvements, as you’ll see…). I received prompt and helpful responses to questions, but when I tried to request further info, I discovered there was no way to reply to a coaching message. I asked about this on the forums, and was told the coaching system “is not meant to be for conversational purposes…. is set to simply be for a question and a response.” Conversational purposes? It isn’t like I was chit chatting about Beyonce. I was assured I could copy material into a new question and add an inquiry, but why complicate communication when this is supposed to be the star of this personalized system?

In one case, I’d asked my coach (the coaches don’t have names; could these be more bots?) a very specific question: “Here are the steps I took; I have the correct procedure now, and the right answer, but why was my first procedure incorrect?” I got a lovely reply showing me how to do the problem, complete with a video working it out for me, but no clue as to what was wrong with the incorrect method. And since conversation was discouraged… well, my buddy Purgy was generous enough to answer questions via email, so I found out what I was doing wrong, which involved a fairly important misinterpretation on my part about what it means to “do this to both sides of the equation” so it’s a good thing I’m motivated to go beyond getting the correct answer, even if that’s all the course requires.

And what of the discussion forums? Early on, I asked a question about one of the problems and was told to use the coaching system. That’s fair; you don’t want to be revealing answers to everyone. It meant that the discussion forums were almost entirely about logistical questions. No one wanted to talk math… except for one amazing student who asked some very detailed, in-depth questions about various topics, going way beyond the course material. That was a lot of fun for a while, but eventually he went over my head (or under my feet, really, since he was going deeper and deeper into “but how do you know a^{1/2} = a^{3/6}?” Aside from this one student, there was almost no discussion about math. I started a thread titled “I’m lonely” and tried to get some interest going by posting Numberphile videos, math blog posts, and the like, but no one was interested.

On a more positive note, I (and about 30 others) received an email from the course instructor Dr. Adrian Sannier, Chief Technology Officer of ASU (and it’s worth noting the chief instructor is not a mathematician, but a computer scientist who specializes in learning systems). We were among the first to pass some milestone of completion, so he asked our opinion of the course. So I… gave him my opinion. Opinions. Lots of them, flanked by disclaimers (I don’t know anything about teaching, and I’m not their target demographic). I’m not sure he was ready for all my opinions, but he was quite gracious and responded with detailed comments. Neither of us convinced the other of anything, but it was nice that someone cared enough to ask. They seem to be genuinely making an effort. I’m not sure I’m crazy about what it is they’re making an effort to do, but I can still appreciate effort.

Once I realized this wasn’t the course I’d hoped it would be, I finished up as quickly as possible and went back to emailing Purgy my questions. I keep hoping to find an algebra mooc that’s as engaging as the calculus and mathematical thinking courses I’ve taken (the details of which can be found elsewhere on this blog under the “MOOC” category or “math” tag), or at least as productive and interesting as AOPS or as enlightening and adorable as Mike Lawler’s work with his preteens.

The more I think about it, the more this course strikes me as the other side of the coin from the UT-Austin Discovery Precalc I took last year. That was all concept, no nuts-and-bolts, with hands-off staff. This was all nuts-and-bolts, no concept, and loaded with staff.  Maybe someday someone will put the two together; it’d be awesome.

Maybe I’m shopping for a lawnmower in a shoe store: maybe algebra is exactly what these moocs make it to be: a group of skills, like melting butter and separating eggs, and while I can beat an egg for all it’s worth, I just don’t have the chops for cake-baking class. Or, to stop switching metaphors, the shoe store only carries Manolos in 6AAA and I can’t wear anything but Easy Spirit in 8WW. I’m not sure what else to try other than whatever’s out there. And what’s out there are mastery-based skill sets. And Purgy, bless his heart, closest thing I’ve got to a lawnmower.

Think Again (and again and again) MOOC

Course: Think Again: How to Reason and Argue
School: Duke via Coursera
Instructors: Dr. Walter Sinnott-Armstrong, Dr. Ram Neta
Quote:

Over the 12 weeks of Think Again, you will learn how to analyze and evaluate arguments and how to avoid common mistakes in reasoning. These important skills will be useful to you in deciding what to believe and what to do in all areas of your life. We will also have plenty of fun.

[addendum: Coursera has converted this course to their new platform; content may have changed, and the experience may be very different]

I’ll say this for them: they did indeed have a lot of fun, as evidenced by that screen shot from the final video of the course. These guys take logic so seriously, that when a student in a prior run of the course made a convincing argument that Walter should shave his head, he had no choice but to comply (and added in Ram shaving his beard as well; I’m not sure where painting his face blue came in, but in for a penny, in for a pound). There was also an argument convincing us to always carry sausages (to fend off wild dogs, of course), probability questions using pig dice (where boxcars and snakeyes were replaced by snouters and leaning jowlers), a great deal of friendly back and forth jibing (the two instructors work at different schools), impromptu video celebrations at the close of the week, and a real-life Ghost in the Machine mystery on the forums that turned out to be some kind of technical glitch.

And it’s a good thing, because as I’ve said before, at some point all logic classes turn into someone droning on about if p then not q and r or if q then p and r implies p or q. That wasn’t the case here, since propositional logic was only a small portion of the course, but then there was probability, Bayesian equations, dozens of fallacies, Venn diagrams, and speech acts. Because of the breadth of the material, the depth was minimal, but I think the idea was to give an overview of different ways of approaching reasoning and arguments.

The course was divided into four segments of three weeks each, with Walter and Ram teaching alternating quarters. Most lectures included a set of ungraded exercises, anywhere from two to ten questions. Each three-week segment concluded with a graded quiz. The format of the quizzes were a little unusual: four quizzes for each segment were available, each a bit different (it seemed to me they got harder but that might’ve been fatigue), with only the best grade counting for that segment. It’s a variation of the multiple-attempts-with-a-pool-of-questions; I preferred it for some reason.

Not surprisingly, I found the segments I was most familiar with – propositional logic and probability – to be the easiest, and the rest to be more difficult. What makes this interesting is that the “easier” sections were more mathematical, while the more difficult argument construction and fallacy analysis were more about analyzing text. I may have to stop claiming to be a words person. In fact, I found close analysis of an argument – finding markers, reconstructing the argument – to be the most difficult part of the course. I still don’t know that I ever “got” it. In fact – and I feel bad saying this, since these two profs seem like incredibly nice guys who really enjoy teaching – I wish some of the effort put into hijinks had been put into figuring a better way of teaching close analysis and fallacies.

The forums were uneven. I started off very active, but backed off pretty quickly when it seemed there were some students who took the whole “argument” theme to heart and wanted to argue every minor point. I came back with vengeance in the propositional logic and probability sections, since I felt a little more secure there. While there were no CTAs and the instructors rarely posted, it seems they did keep an eye on things; after giving an explanation to another student, I made an offhand comment to the effect that I wish he could get an explanation from the instructor, since I wasn’t sure I was making sense, at which point Walter appeared to reassure me that my explanation was “right on target. Thanks, Walter, I needed that.

This was quite the course to take during the current Presidential primary season. I discovered Donald Trump uses assurance – all three varieties, authoritative, reflexive, and abusive – more than anything else, including facts or policy. And by the way, I’ve realized how much I rely on guarding – all the little hedge words like “probably” and “most”. But that’s the way the world often is; very few things are definite.

I signed up for this course because somewhere along the line, someone recommended it highly. As much as I like these guys, I can’t be that enthusiastic about it; I think the Melbourne logic courses did a far better job on propositional logic (the problem being, they’re no longer available), and AOPS does everything that can be done with probability and expected value. Those are specialty classes, however, and this was a broader overview, so the purpose is different. If you’re looking for a smorgasbord of approaches to logic – including some that are more real-life than technical – this might work great for you. And it is a lot of fun.

Discovering A Lot Besides Precalc MOOC

Detail from James Drake’s “Brain Trash” exhibit

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 communication  Update: 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.

Paradoxically Infinite MOOC

Prof. Rayo demonstrates the plus-sized replica of the 100%-gravity-powered Digi-Comp II built for MIT by Richard Lewis of Evil Mad Scientist Laboratories.

Prof. Rayo demonstrates the plus-sized replica of the 100%-gravity-powered Digi-Comp II built for MIT by Richard Lewis of Evil Mad Scientist Laboratories.

Course: Paradox and Infinity
School: MIT through edX (free; see below)
Instructors: Agustin Rayo
Quote:
 
   There is one kind of philosophy that I am especially interested in. And that’s, for lack of a better name, mathy philosophy. It’s very hard to say something more specific because really what unites the different problems that mathy philosophy is about is the fact that they interact with mathematics and, more generally, with the use of formal methods – so for example, the use of probabilities to model beliefs.
   So say that this is mathy philosophy. So if one is to talk about mathy philosophy, there are basically two options. Option one is to do it properly…. We would spend all semester doing the groundwork. And some of it would be brutally dull. And then, finally at the end, there would be this jewel. And it would be wonderful. But it would only be for the brave.
   So what I’ve decided to do is not do it properly.
   So there are all these delicious fruit within mathy philosophy. And what I’m going to do is I’m going to go straight for the fruit.…In some ways, that will make the class more difficult because, if you have the details, if you have the proof, you can go and check it.
   And the only thing between you and the fruit is work.

The short version: Great class. But if you don’t know what you’re getting into – if you think, “Oh, cool, time travel, science fiction!” – you might be disappointed by Week 2. Not to mention overwhelmed.

And what are you getting into? Like the man said: Mathy philosophy. Probabilities, omega sets, infinite series, Turing machines… yep. Mathy. I knew that going in. The course description and teaser video (which is pretty great, btw, worth watching just for fun) don’t quite make clear just how mathy it is. I’ve encountered many of these terms before, so I had some idea what was coming. Yet, as it turned out, I didn’t know the half of it. I had fun anyway. And I think I may have learned a few things.

The first week was the least mathy, with a look at time travel. I found it to be the least interesting of all the topics. Don’t get me wrong, I love me my Star Trek time loop episodes, but I’ve taken two philosophy courses that dealt with time travel and I still haven’t seen anything to equal Robert Heinlein’s “By His Bootstraps” in terms of dealing with the predetermination factor.

Then we spent some time on probability, one of my weakest areas (not that I have any mathematical strengths), which generates some interesting paradoxes. Monty Hall did not show up in this course, though his cousin Newcomb did. The connection to free will was tenuous, but I suspect the whole “time travel and free will” was marketing.

In the second three-week module, we ended up in set theory and abstract algebra, the “infinity” part of the course, which, it turns out, I love, with or without the melons (used to illustrate the conversion of the Cayley graph into the Banach-Tarski paradox). I’ve heard of Banach-Tarski before, but rather than just saying it exists, the objective here was to show how it works (Coincidentally, just as the class was concluding, VSauce put up a video explaining some of the topics in this unit). Much to my surprise, I kinda sorta get it. Heavy on the kinda sorta, since I’m still a little hazy on equivalence relations and cells. It was frustrating, difficult – and fun. It’s the area I most want to understand better.

The last three weeks were devoted to Turing machines, and while I got the basics of the beginning, I got lost in the middle and kind of gave up, so I never got to Godel’s Theorem. I was done. But overall, it was a course very much worth taking. And surprisingly, I “passed” – thanks to a very low bar, and an exam structure that allowed a basic understanding to suffice. I’d like to know more, and hope I can do some additional work on these topics. But given my capacity to screw up math, the pass was somewhere between a gift and a miracle. I’m not fooling myself, thinking I “know” anything – but it’s a start.

Worth special mention is the inclusion of several “Meet the Expert” videos – lectures, interviews, conversations, collaborations with a variety of professors with special interest in the topic at hand. These included physicist Alan Guth on time travel, epistemologist Susanna Rinard on a number of paradoxical probability examples, logician Graham Priest on dialetheism (my new word for the day, meaning statements can be true and false simultaneously – such as “This statement is false”), Steve Yablo on ω-sequences, and computer scientist Scott Aaronson on computational complexity. Some of these were advanced topics; others were very accessible.

One of the problems I had was that I needed more basic explanations than were included in the course materials. The lectures were from the actual MIT course Rayo teaches, and while it was fun to see all those MIT nerds (I use that term with great respect, btw) coming up with perfect explanations for why we can’t know whether Thompson’s lamp is on or off at midnight, there’s a lot missing from those video clips. And, sure, there was written material, but I have this problem reading math: by the time I figure out what all the symbols mean, I forget what I’m doing there in the first place. Fortunately, I was able to find more elementary materials. And, incredibly enough, I was able to get help through the edX forums.

I say “incredibly enough” because the edX forums are notorious for being less than user friendly, and it matters to me most in courses like this one where I often need a bit of advice, or just reassurance. Because “daily digest” notifications of replies are only sent every 24 hours (Coursera sends them nearly instantly), conversation requires a great deal of motivation on the part of participants; I’ve learned to check the forums for replies often during the day, but if my interlocutors don’t do the same, the lag time is extreme. Then there’s the opt-in setup for “follow” on replies, meaning no notifications on replies are ever received, and it’s very difficult to find a reply at a later time; if follows were simply changed to opt-out for replies, as they are for initial posts, it would help tremendously. But either the edX powers that be have decided this requires too much work/cost, or they aren’t interested in improving forum functionality. In any case, conversation, collaboration, and support is possible, but it takes people who are aware of system limitations and are willing to put in the extra steps to compensate.

But it worked here because I was lucky to encounter an old friend. I first met Purgatorio last summer in SVCalc – he was one of the inspirations for the Dante work I did this year, and I think it’s fitting that Purgatorio turned out to be my favorite canto (maybe I just like middles). Dear Purgy is rigorous about “mathematical truth” and has little patience for fakery or shortcuts. This sometimes causes a bit of friction for him. Fortunately, since I enter every math course with a white flag raised in surrender (I keep saying I can’t afford ego when it comes to math), we’ve struck up a friendship. He’s also very funny, sometimes unintentionally so (while highly educated and fluent, English is not his native language, so occasionally a colloquialism comes out a little sideways). He has also shown a great deal of patient willingness to walk me through concepts when I struggle, and though I don’t always get the whole point, I do always take away something of value. In this course, the combination of banter, and explanation of fine points of set theory, got me through what might otherwise have been a lost cause, and I’m very grateful.

Another aspect of this course worthy of note: the “human-graded” option. MOOCs depend on machine grading, of course – multiple choice tests, numerical answers. As restrictive as this is, there are some courses (and this is one of them) that manage to include some reasoning-intense questions in these formats. Some courses use essays graded via peer assessment by other students. The results, in terms of grades, are inconsistent, though I usually find the experience worthwhile in that I have to consolidate and organize my thoughts in order to write a coherent essay in the first place, and the papers I’m assigned to review often include some wonderful insights I hadn’t considered. This course, depending on multiple choice with a number of optional, non-graded questions sprinkled throughout, offered something else: a $300 human-graded option. That’s what it’s called, honest.

Would you like to have your work graded by humans? If you sign up for a verified certificate, you will be assigned problems that are graded by teaching assistants and given professional written feedback. This will bring your learning experience one level closer to that of residential students at MIT. And if you pass the class, you will receive an MITx certificate, in addition to edX’s Verified Certificate of Achievement.

This included some odd notations on the course forums; I’m not sure if human-graded students had their own forums or what, but some posts were specified as “Non-human graded cohort only” which seemed… a little creepy. I’m not sure if the optional questions were human-graded, which would make sense. I can see how this could have appeal to certain people in a position to take full advantage of it. I fully support MOOCs doing what they need to do to survive, though I sadly realize that eventually the “free” part of all this will end, and the door to all this education will slam shut in my face. That’s why I’m trying to cram it all in, while I can.

In spite of the difficulty level, I enjoyed this course. Rayo is an engaging lecturer, the material is interesting, and I can highly recommend it for those who are more prepared than I – that is, those for whom things like set theory, equivalence relations, and Turing machine processing are not completely new concepts, or who can easily acquire that material from rather cursory explanations. I would imagine math/CS students would love the opportunity to peek into an MIT classroom. As for the rest of us, I can still recommend it for those who are less prepared but are interested in the topics, have a high frustration tolerance and are willing to find more elementary material to supplement course materials.

Perfectly Logical MOOC

Course: Logic: Language and Information I and 2
School: University of Melbourne via Coursera (free)
Instructors: Prof. Greg Restall, Dr Jen Davoren
Quote:
 
(Part 1) This is an introduction to formal logic and how it is applied in computer science, electronic engineering, linguistics and philosophy. You will learn propositional logic—its language, interpretations and proofs, and apply it to solve problems in a wide range of disciplines…..you will learn how to use the core tools in logic: the idea of a formal language, which gives us a way to talk about logical structure; and we’ll introduce and explain the central logical concepts such as consistency and validity; models; and proofs.
 
(Part 2) This subject follows from Logic: Language and Information 1, to cover core techniques in first order predicate logic: the idea of formal languages with quantifiers, which gives us a way to talk about more logical structure than in propositional logic; and we will cover the central logical concepts such as consistency and validity; models; and proofs in predicate logic….We will also explore how these techniques connect with issues in linguistics, computer science, electronic engineering, mathematics, and philosophy.

(Addendum: as of January 2017, these courses are still undergoing conversion to Coursera’s new platform software. Stay tuned)

I loved this course; along with the Solar System course, it became a focus of my winter/spring MOOC schedule.

I’ve taken a couple of other logic MOOCs: Mathematical Thinking, and the Stanford logic course intended for computer programmers. But this was the first logic course that included modules on linguistics and philosophy. I was very happy. Not that it was easy – in fact, the Peer Assessment assignments made me wonder if I had any idea at all of what was going on, and there were some moments when I was ready to hang myself from the nearest Proof Tree – but I loved every head-banging moment.

Part 1 included introduction to the basics of propositional logic and proof trees in two core modules, then offered a choice of four additional application areas, at least two of which were required. Of course, I headed for the more humanities-aimed subjects, and fell in love with implicatures and the maxims of Grice’s Cooperation Principle – and recognized how politicians and other liars depend on them – as well as the “fuzzy logic” and infinitely-valued logic covered in the Philosophy module. I did take a quick stab at the Programming module, but it became clear that it required more time than I was able to devote. I hope someday I can go back and pick up that and the Digital Systems module (I have fond memories of logic gates from my days of hanging out with… oh, never mind).

Part 2 expanded to first-order predicate logic, with some indication of what’s involved in higher-order logics. Here three core modules were required, along with three of five application areas: mathematics was added. I was very happy to see that. It gave me a chance to do more work on the concepts from Mathematical Thinking – bounds, convergence, that sort of thing. Then in Philosophy, I had a great time figuring out how to negate “The present King of France is bald”, and the Linguistics module turned pronouns into fine art.

Each lecture set in both core and applications was followed by a set of ungraded practice homework which in most cases included a variety of problems; unlimited attempts were allowed, so plenty of practice was available. Detailed notes in PDF form were provided for all the material. Each core and application module also included a final exam and/or a peer assessment assignment. I found some of the the Part 1 peer assessment assignments to be extremely difficult; in fact, I was pretty discouraged at the close of the class, since I’d felt I’d had a pretty good grasp on things and then came face-to-face with the realization that maybe I understood a portion and need to broaden my conceptual grasp. I received extremely generously scores from my peers on the Part 1 assignments, by the way, as well as some great feedback. It’s important to know what you don’t know, so it was a worthwhile, if humbling, experience. The peer assessment assignment for the second course turned out to be too far over my head to even make a credible attempt, but at least it was optional, as it went beyond the material covered in class. The material for Part 2 was pretty humbling to begin with, particularly the philosophy module covering vagueness. Loving the material may be necessary, but is definitely not sufficient, for understanding.

The core material was covered by both instructors, but in the applications, they each took over what were presumably their personal specialties: Greg did Philosophy and Linguistics, and Jen handled computer applications and mathematics. I came to be extremely fond of both of them along the way. Now, I rarely mention instructors’ personal styles in these comments on MOOCs – first, it’s a good way to get myself into hot water, and second, it’s generally irrelevant to the educational experience. But in this case, although the courses were strictly business, each instructor brought a certain aura that added greatly to my enjoyment of the course – Greg with his vest and tie, a slightly Eleventh Doctor air about him, and his tendency to talk with his eyes shut, giving him an air of… bookish shyness? – the one exception being when he discovered in the middle of a proof tree that he’d actually worked out the example differently from what he’d started to explain. His laughter and slight fluster was completely charming; he should make little flubs more often. Then there’s his penchant for fitting wombats into examples.

Jen Davoren brought a completely different, but equally appealing, personna. I came to adore her constantly changing hair colors (seriously, check out her faculty page), and her ever-interesting kaliedoscope of logic t-shirts. Not to mention her asides about Alan Turing – I’d just seen The Imitation Game – and Angela Merkel (did you know Merkel has a doctorate in quantum chemistry and was an active research scientist before she was Chancellor of Germany? I didn’t, because we Americans learn as little as possible about world leaders other than our own, unless scandal is involved, in which case they’re all over the supermarket tabloids). Since I didn’t take the Digital Systems or Prolog modules she taught, I was delighted to find her teaching the Mathematics module, which began with the comment: “Admittedly, it is one of the most abstract branches of mathematics. This abstractness, together with a foundational role of logic, may at least partially explain why logicians tend to be the eccentrics within mathematics departments.” The most interesting mathematicians I’ve met through MOOCs have all been a little eccentric in one way or another – including Jen.

I’m not sure if my fondness for the instructors, limited as their involvement beyond the videos was, followed from my fondness for the material, or vice-versa (or maybe I just loved the font used for slide headers – I believe it’s one of the Barbedors. Yes, I gathered enough samples to Identifont it, what, there’s something wrong with that?), but in any case, I loved this course and highly recommend it – for anyone who wants to understand the basics of formal logic.

Now, that’s an important caveat. You have to want it. Because every logic course I’ve ever taken at the introductory level (that’s three now) eventually turns into ten minutes of:

And ‘Ga’ is true, so that’s why ‘Fa or Ga’ is true. Not everything has property ‘G’ is true. Why not? Because of the ‘b’ instance ‘not Gb’. So, not everything has property ‘G’. Not everything has property ‘F’. Not because of the ‘b’ instance, but because of the ‘a’ instance. And indeed, not everything does have property ‘F’. because ‘a’ doesn’t. Finally ‘not everything is F or everything is G’ is true, because both of the disjuncts here are false. ‘everything is F’ is false, ‘everything is G’ is false, so ‘not everything is F or everything is G’, indeed that formula is true.

~~ Transcript for Part 2, Lecture 3.4, “Trees for Predicate Logic”, which is why I didn’t use the transcripts at all for this course

And that sort of thing can drive you crazy. Unless you really want it. But if you want it, this is a great place to get it.

Heart Stats? I Hardly Even Know Them MOOC

Course: I Heart Stats
School: Notre Dame (via edX, free)
Instructors: Dan Myers
Quote:

Statistics can be confusing and opaque. Symbols, Greek letters, very large and very small numbers, and how to interpret all of this can leave to feeling cold and disengaged—even fearful and resentful.
But in the modern information age, having a healthy relationship with statistics can make life a whole lot easier…. The purpose of this course, then is to help you develop a functional, satisfying, and useful life-long relationship with statistics….
What you’ll learn:

• Select appropriate statistical tests for data according to the levels of measurement
• Perform basic calculations to determine statistical significance
• Use standard methods of representation to summarize data
• Interpret and assess the credibility of basic statistics

I began to realize, midway through Genetics & Evolution, that I needed a much better understanding of statistics. Think of all the jokes you’ve heard: there’s no better way to obfuscate an issue than to come up with a statistic that sounds impressive, even though, when examined more closely, it doesn’t hold up, because few people know how to examine stats more closely.

So I took this course. Two MOOC-friends of mine took this course at the same time I did; they both loved it. For that matter, it seems a thousand people Liked the Facebook page. Me, not so much, but I say it every time I do one of these: every course I hate, someone else loves. In this case, a lot of people.

So what was my problem? Mostly, I just really, really hate stats.

In addition to arachnophobia (which actually contributed to my dropping the Animal Behavior course this week, believe it or not), I’ve discovered I have sigmaphobia: a fear of summation signs. Those are the Greek capital-Sigmas, the things that Greek restaurants use as capital E’s even though they’re really S’s. There’s a great line in Sylvia Plath’s The Bell Jar about protagonist Esther Greenwood’s visceral reaction to reading German: “…each time I picked up a German dictionary or a German book, the very sight of those dense, black, barbed-wire letters made my mind shut like a clam.” That’s how I am with summation signs. It’s just a personal quirk, and it’s necessary that I keep calm and carry on, but it makes things like stats extra-special difficult.

I hated all the calculating – and this course was about 85% calculating. Sure, adding a column of numbers, calculating an average, squaring each number in the column, summing the squares, etc etc, isn’t that hard – that’s what Excel is made for. It’s just incredibly tedious and I hate it. But… I do need to understand stats better, and I had to start somewhere.

I did learn a few things beyond calculating. I even recognized a few things I’d seen in other courses: “Oh, so that’s what he was talking about when he said he was using a stricter standard of significance testing… oh, that thing we did back then, that must’ve been a chi square.…” Just yesterday, I was watching someone talk about his research and he mentioned within and between factors, ANOVA and t-tests; I was so excited – I know what that is! I’d have to listen way more carefully to it to say I understood his research design, but at least I recognize the tests he’s using on his data, and that’s more than I could’ve said a month ago.

As for more comprehensive understanding, Dan made reference several times to more advanced courses for theory, and I’m going to need that. There was some attention to concepts in this class: discussion of data types and the requirements of the various stats, and a very good “Hypotheses Testing Q&A” with Dan and Sara early on (my favorite thing in the course), and the first half of the final was relatively conceptual: Here’s the situation, what stat should you use. But one eight-week intro course isn’t anywhere near enough, at least for a mathematical idiot like me.

Despite my antipathy towards the subject itself, the course did a lot of things very right. The calculations for each of the statistical methods – Standard Deviation, Chi Square, T-test, ANOVA, Regression, Correlation – were demonstrated three times by three different people in different formats, with PDFs of detailed Notes available as well, so if one route of explanation didn’t appeal, there were alternatives (my personal preference was for Sara’s runthroughs). Plenty of practice problems were available, both as practice and in graded form. The course even offers “I Heart Stats” t-shirts for sale, just like the one Sara wears in the videos. How many courses offer that? I’ve said before MOOCs should offer swag. I would’ve bought a bunch of course-specific t-shirts or coffee mugs or tote bags. Just… well, not for this course.

They did their job. And I did mine, even when it hurt. I’m kind of proud of that, that I kept going with a course I hated, since I’ve been dropping courses all over the place at the first sign of “This isn’t anywhere near as interesting as I thought it would be.” I finished. Granted, I kept an eye on the progress meter, and I stopped as soon as I’d done enough of the final for a pass (ok, not all that proud of myself), because I really didn’t want to calculate another σβ.

One of the minor quibbles I had was partly of my own doing, I think. I seem to recall being asked if I’d be willing to participate in a research study during the course. I love MOOCs so I always agree to this stuff – it’s possible it was another course, in fact, not this one at all. But throughout, little questionnaires kept coming up. “Checking In”: “Which emotions best describe how you’re feeling about the course?” with a list of maybe a dozen emotion words (anger, contentment, hope, isolation, shame). Then there was SAM, the Self-Assessment Manikin. It seemed like overkill to have two kinds of mood-assessment, but maybe one was the study I’d signed up for, or maybe they were thinking of people who weren’t fluent in English, or maybe they wanted to compare results between the two different representations (which is an interesting idea, by the way). Thing is, these things cropped up so often early on, I started to get really pissed off whenever I saw them… so I kept entering things like “angry” and “anxious” and “confused” because that’s how I was feeling, even though I was doing fine with the course material. Which is also kind of an interesting result. The instructors are sociologists, after all.

I also took great exception to one of the examples used, with self-described fake data showing differences between IQ scores for different races, with a reference to Murray’s book The Bell Curve, which to me was very controversial back in the day. I was pretty upset about this. It seemed, at best, insensitive to start flinging around fake data showing white people have higher IQ scores than black or Hispanic people. I was relieved when we went back to things like evaluation of popcorn brands, and relating hair products to gender or exercise to work productivity. I had some discomfort with how some results were phrased as well; I wish someone had said, at some point, “Correlation is not causation.” Because that’s where the fun starts. Maybe I just don’t understand the concept.

But overall, this was a detailed and effective basic introduction to a topic that befuddles a lot of us; if you want to know how to calculate ANOVA or standard deviations, strap on your calculator, crank up your spreadsheet, and go for it. Who knows – you might end up hearting stats.

Logical MOOC

logic 210Course: Introduction to Logic
School: Stanford via Coursera (free)
Instructors: Michael Genesereth, Eric Kao
Quote:
 Logic is one of the oldest intellectual disciplines in human history. It dates back to the times of Aristotle; it has been studied through the centuries; and it is still a subject of active investigation today.
     This course is a basic introduction to Logic. It shows how to formalize information in form of logical sentences. It shows how to reason systematically with this information to produce all logical conclusions and only logical conclusions. And it examines logic technology and its applications – in mathematics, science, engineering, business, law, and so forth.
     The course differs from other introductory courses in Logic in two important ways. First of all, it teaches a novel theory of logic that improves accessibility while preserving rigor. Second, the material is laced with interactive demonstrations and exercises that suggest the many practical applications of the field.

[Addendum: This course has been converted to the new Coursera platform; content may have changed, and the experience is likely to be different]

I keep looking for some area of math that isn’t such a nightmare for me. Logic seemed a, mmmm-hm, logical choice: no numbers, and I greatly enjoyed the logic portion of Intro to Mathematical Thinking – so this should work out well, I should be able to do this!

Not so much, no.

The first three weeks went fine. Lecture videos introduced material, quizzes were very manageable, and the supplementary logic puzzles were fun. I was feeling pretty cocky: I got this.

Then week 4 hit.

Caution: steep drop-off. Mendelson. Fitch. Perfectly innocent-sounding common-sense sentences like “If by assuming φ we can infer ψ, then we can infer the sentence φ implies ψ” turned into vicious monsters laughing at me in the dark in nightmares. That’s what happens when you take a course given to Stanford computer science majors.

The exercises were designed to provide instant feedback. This is the first time I’ve seen a MOOC instructor up-front admit that there’s no way to prevent “cheating,” so why not take pedagogic advantage of what’s possible:

Finally, a few words about the online problems. First of all, you can submit your answers to any problem as often as you like, and the system will take your highest grade. Second, all problems provide immediate feedback. When you check a checkbox or make a selection from a menu or compete a proof, the system will tell you immediately whether the answer is correct, even BEFORE you submit your answers for grading. The upshot is that, for some problems, there is no reason not to get a perfect score.
         Yes, we realize that it is possible to “game” the system by dumbly trying all answers until you get the right one and then submitting that answer. However, this is already possible. There is nothing stopping you from signing up twice, getting the right answers from one account and using them in your other account. Besides, for many problems, mostly proofs, finding a correct answer is a challenge, and you will have to work hard to get that coveted green checkmark. And we do not reveal proofs until after the hard deadline.

This wasn’t as much of a giveaway as it seems. In fact, I flunked the course (correction: To my surprise, I didn’t flunk, at least not in terms of obtaining a Certificate of Achievement with a score of 78%) because so many of the proofs eluded me. And, to be frank, the discussion boards were so loaded with “hints” that most of those, I might’ve been able to eke out as well. But call me crazy; I have a conscience, so I used the hints, sure, but if I couldn’t come up with something on my own, I didn’t submit the solution for points. I don’t see any particular advantage to “passing” these courses (other than ego, and I have little ego when it comes to math), so I’d rather have an accurate record of what I was able to do, should I try again some time. The instant feedback on the checkbox and multiple choice questions, on the other hand, was a great idea. Yes, I probably got a higher “score” than I would have otherwise, but it helped to clarify some points. I’m a little surprised they didn’t have ungraded homework questions with instant feedback, and then more traditional quiz questions, but I don’t teach MOOCs, I just take them.

Staff was very active on the boards – the TA was even available on Sunday evenings, time I tend to use a lot, and he was very helpful. Mike (funny, how some professors are so clearly Professors, and others are so clearly Mike) also did a lot of board-prowling, and while that has tremendous value in and of itself, I’m afraid I wasn’t able to understand his explanations any better than the original material being questioned.

However, there was Rachel.

If the proofs were monsters lurking in the dark, Rachel was the angel from heaven sent to show the way through the forest. A fellow student with some background in math and logic, she had a knack for explanation, and the patience to deal with troubled souls lined up for miles. I personally would have never gone past week 4 had it not been for Rachel. Many others felt the same way.

Plenty of supplementary material was available in the course. We had access to proof editors for each system, so we could work out simple examples or the samples in the lectures, to get the hang of how they worked. The entire syllabus was available in a single document, complete with formatted examples and diagrams, which are so important in a course like this. A set of extra resources was available for each week – tips on proofs, background and advanced materials. Several “logic puzzles” were provided for group discussion, each demonstrating some topic from the lectures; I’m afraid I was too mired in proofs to work on most of them, but I enjoyed the few I did.

Staff seemed quite restrained, somehow, but that may be because the humor was dry and ironic. The tip sheet on handling the Fitch proof editor was titled “Be-Fitched.” Michael picked up on my Stevie Smith reference when I signed a post, “Not Waving, But Drowning.” And, best of all: Box logic. I was probably the only student stupid enough to think this was an actual thing, like the Fitch editor or Relational Logic, so feverishly took notes and made diagrams. Then I “got” it. Coming right after Week 4, I needed a smile.

I was impressed with the instructor involvement, with their responsiveness to questions and problems, and with the resources provided; it’s not their fault I can’t think. And I will admit I’m not as well-prepared in mathematics as the typical Stanford student (how’s that for dry, ironic humor; I’m probably not as well-prepared in mathematics as the lawn at Stanford). I hope they follow through with the “boot camp” idea. There has to be some middle ground between week 3 and 4, between “I got this” and “Huh?” I’ll be taking the upcoming logic courses from University of Melbourne shortly, maybe that will help fill in the gaps, or maybe I just need to try again.

Some day. When I can handle the nightmares again.

The Amazing Technicolor CalcuMOOC

Course: Calculus: Single-Variable
School: University of Pennsylvania via Coursera (free)
Instructor: Robert Ghrist
Quote:
 
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.

Meh MOOC

Course: Pre-Calculus
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.

Pressure!

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.

Mathematical MOOCing

If I ever try to take four concurrent math classes again, I need to remember: hitting myself over the head with a hammer every hour for 12 weeks would be less painful. This is why I put the Pushcart read on hiatus?!? (I’ll be getting back to it soon, I promise) The degree of suffering math engenders in me is well-documented in these pages, but I can’t seem to leave it alone now that free online classes are available.

These four – two of which I’ve taken before, and two of which have a couple of weeks yet to go – are all (for me) good-to-great MOOCs, but all in their own unique ways; they have different goals and atmospheres, so likely will appeal to different students for different reasons.

Calculus (Calc1, Calc2 from Ohio State via Coursera):

I took this course a year ago; it was my first MOOC, in fact, and I praised it at length at the time. I decided to repeat it to pick up some additional details. That’s the great thing about MOOCs: if you don’t quite get everything the first time around, you can take them more than once, without fighting with a Registrar and the Bursar’s Office, and without stigma; lots of people take classes more than once, and some are even designed that way.

And guess what: it only got better! While Jim Fowler’s terrific videos (they’re available to anyone via YouTube, by the way; they not only include detailed theory and examples, they’re also cute as he explains things in a variety of ways using everything from high-tech gizmos to paper cutouts) were the backbone of the course, Jenny George was now the lead instructor; her patience and talent for clear, step-by-step explanation is extraordinary, and she bravely made herself very accessible via email for questions that couldn’t be answered on the forums – not to mention, I was thrilled to (finally!) see a woman teaching math on Coursera! The division of labor worked out well, since it offers different approaches that will likely provide something for everyone. There’s one more week to go (I had a bit of a hissy fit over integration by parts the other day, in fact), but at this point I’m pretty sure I’ll get through it, and I’ve definitely strengthened my grasp of calculus – at least, as much as I can strengthen my grip of anything mathematical. One of the benefits of taking a course a second time is that what seemed so difficult before, now seems a lot easier (integration by parts notwithstanding).

I was so encouraged by the first few weeks, in fact, I decided to take a look at Jim’s Calculus 2 class – “Sequences and Series” – which was intentionally timed to coincide with Calc1. I didn’t expect to get through it all, but I figured I could get some idea of what was going on so it would look familiar when I took it “for real” later. Turns out, things went better than I’d hoped, though I left some things for my second pass. The “homework” was particularly well designed, similar to Calc1’s MOOCulus exercises but, in my opinion, even better, in that links to pertinent resources appear alongside the question. I ended up doing the full spread of six to nine homework problems per chapter nearly daily once I caught on (the questions change each time the test is reloaded), which reduced the “hey, I don’t remember this, we had this six weeks ago” confusion that usually sets in when confronted by a final exam. And, to my surprise – I like Taylor series! Who knew? Jim has a third class (multivariable calc, sounds scary) on the rotation, and I’m hoping to be up for it next time it rolls around.

Effective Thinking Through Mathematics (Utexas/Austin via edX):

I’ve admitted developing something of an attitude towards edX, but this was my favorite edX course so far. What’s interesting is that it may have been the least planned; go figure. If nothing else, it confirmed once again that putting together a MOOC, even if you’ve got a great concept and the best of intentions, isn’t easy, and it isn’t just a matter of taping what happens in the classroom.

True to the name of the course, this wasn’t, strictly speaking, a math course; the idea was to demonstrate a series of effective thinking techniques through working on mathematical problems. Some of those techniques – like “make mistakes” and “ask questions” – were familiar, and others – “understand simple things deeply” – were new. I found it a relatively easy course, partly because there was only a minimum of testing and evaluation, but mostly because I found a used copy of the non-required textbook the course is based on, and worked through it last Fall so I’d be prepared.

The topics were terrific: infinity, the fourth dimension, number theory, topology, fourth-dimensional geometry, etc. The instructional concept seemed like a great idea: instead of lectures, professor Michael Starbird sat with one or two students – not heavy-duty math students, either; these were relative neophytes – and ran through a series of concepts with them, using a more-or-less Socratic approach. He’d present a puzzle, or a question, and let the students fumble with and streamline ideas until a solution was reached.

It may be the best way to present material when you’ve got a one-to-two teacher/student ratio, or in a physical classroom, but I’m not sure it quite translated to MOOCland. The problem, for me, was: this can get a little frustrating when you’ve got different questions, ideas, and solutions than the participants. I was bored when they were struggling with something I understood, and never found out the flaws in some of my own ideas. A strong self-motivation is part and parcel of the MOOC process, but here, it was nearly impossible NOT to sit back and let the students in the videos do all the work, since after the first few minutes, our paths of reasoning typically diverged in some important aspect. Still, I’m intrigued by the concept, and I wonder if there isn’t a way to make it work better – perhaps via a “choose-your-own-adventure” based approach (which might be overly complicated for a MOOC), or by requiring more participation at the initial stages of a problem before the “answers” are released (which would require a better communication system than what they laughingly refer to as discussion boards). But I have to remember, again, that this isn’t primarily a math course; the idea was to show the five elements of effective thinking in action.

The logistical organization seemed to disintegrate a bit in later weeks (we’re in the last week now), with weekly lessons released later and later; I got the impression they were taped week to week, which seems pretty risky to me. In one case, an introduction segment wasn’t released at all so things started in the middle, leaving me (and some others in the class) perplexed – how did the featured participants know those particular features were salient? Am I really so stupid that it’s obvious and I don’t see it when they do? Turns out, the initial part of the discussion hadn’t been included in the material released. This was fixed within a couple of days (days?!?), but for me, already highly insecure about math, it created unwelcome tension. But it’s one of my primary tenets of MOOCs: technical sh*t happens; deal with it and move on.

I still think Prof. Starbird has a great idea here, but there might be a better way to translate it to work better within the inherent limitations and advantages of the MOOC format. Still, I’d recommend it to anyone who wants a gentle, low-pressure introduction to a variety of heavy-duty mathematical concepts. If a new crop of topics were introduced, I’d take it again myself.

Introduction to Mathematical Thinking (Stanford via Coursera):

“I have lost my faith, Josephus.” That’s the opening line in a favorite quasi-literary-historical mystery of mine (no, not that one), and it’s the refrain that started running through my head around the dreaded Week 6 of Stanford’s Intro to Mathematical Thinking (my second time through) when I again failed to grasp induction. The premises of the class – of the general philosophy of mathematical pedagogy Prof. Keith Devlin advocates – are that 1) we are all capable of learning basic math; there is no such thing as a “math person” and 2) we learn more from our failures than our successes, so struggle is to be cultivated; if the class is easy, you need a harder class. Compelling ideas, especially to someone (like me) who equates math with failure. But: though I’m the Fox Mulder of the math world – I want to believe – this is Kool-Aid I just can’t drink. I have yet to find a limit to my capacity for learning-less failure, and at some point, you need to be able to come up with a correct answer. It may be it’s simply too late for me, for a variety of reasons. I still haven’t had the fortitude to look at my scores; the only score I need to know is the one I gave my own final as part of the self-grading requirement: 14 out of 40. Not as bad as last time, when I couldn’t even attempt about half the questions, but a pretty poor showing still.

That said –while I do pretty horribly in this class, it’s been extremely helpful with other classes. Both Calculus sessions were far more understandable because of it, both from the general caveat to take the time I need to figure something out, to the advantage of having seen some things (like the definition of limits, and summation signs) often enough that I could pay attention to what was actually happening when those things were used. It’s the most instructive class I’ve ever sucked at.

Perhaps the greatest benefits of the first run of this class (which I talked about at length last year, calling it “ModPo for Math”) was that, between repeats, I tried to get better prepared for this second pass through a variety of avenues. Any course that can get me to do a couple of hours of math a day, when I’m not even enrolled in a math class, is a big deal, whether I do well in it or not. And just doing that much work every day has to have some effect. But I doubt I’ll take it again in the Fall; it just takes too much out of me.

By the way, I lied about that line from the novel. It’s actually in the past perfect tense: “I had lost my faith, Josephus.” The novel is the story of reclamation of faith, in various things, through various means, by each of the three characters. This explains – it must explain – why, even having lost my faith, I’ve signed up for How to Learn Math: For Students – a course based on the premise: “Everyone can learn math well. There is no such thing as a ‘math person’….evidence shows the value of students working on challenging work and even making mistakes….” Sound familiar? No wonder – Dr. Jo Boaler, the instructor, is a Stanford colleague of Dr. Devlin.

Turns out it’s not all that easy to lose one’s faith. I guess I’ll have to keep trying.

To be continued…

My Continuing Mathematical Misadventures

Addendum: This course has been converted to the new Coursera platform; the experience and effectiveness will be very different

 

I really tried to hit the ground running with this one: “Introduction to Mathematical Thinking, taught by Dr. Keith Devlin at Stanford. I didn’t just show up for the game; I worked out beforehand at Khan. Before the start of the race, I backed up to get some momentum by watching a series of prior public lectures by the instructor, a total of about nine hours of very cool stuff (shhh, nobody tell the Right that the word “algebra” is derived from Arabic, or they’ll outlaw it) and grubbed up a used copy of the not-required textbook to play around with. And once things started, I hammered in a variety of pitons (that’s a mixture of three metaphors, shall I go for four?), to hold me securely when things got rough.

And yes, I fell off the cliff anyway. But you know, while some cliffs were mistakes in the first place, there are some cliffs that are worth falling off, even worth trying again. This cliff, painful as it was, was very worth it.

True to the course title, the emphasis throughout was on reasoning rather than computation, on understanding and constructing proofs for mathematical statements rather than solving problems. That at least gave me a shot. Still, a certain amount of mathiness was necessary, and my threshold for mathiness is extremely low.

I had no reason to take this course, other than mathematical masochism; it’s intended primarily, though not exclusively, for 17-year-olds about to start a university degree in a STEM field. It’s hard to tell from message boards where anyone can be anything, but I don’t think I met any 17-year-olds in the class; maybe they didn’t need the message boards, maybe they just cranked out the work in a couple of hours and went partying. Maybe they were very mature or just didn’t seem like my idea of 17-year-olds. I met a lot of STEM people, elhi math teachers, and, yes, a few from my natural peer group, middle-aged mathtastrophes trying for a second (or third, or seventh) chance: splendid pitons, all. It’s amazing the affection you can feel for someone with whom your only conversation has been along the lines of, “there is an even number x greater than 2 , which is not a prime, so the contradictive assumption made in the beginning is False so the problem statement is correct, is that right?” Especially when s/he helps you figure it out. Then there were the Community TAs, fellow students who “got” it and helped us all. Most Coursera courses have CTAs, but in this one, they were exceptionally effective.

I felt good the first couple of weeks, when the focus was language. Language? I can do language; I did pretty well, all things considered, though I’d prefer to never hear about melanoma again. Then I faded a bit for a couple of weeks, until week 6, when the Induction Bomb hit and I just dissolved into a puddle of goo. I picked up some of the last two weeks, but I never really recovered. The good news is that I may have gleaned enough background to build back up for next time. Yes, next time: it’s not just ok to take this course more than once, it’s encouraged, even expected. So it’s less like “oh, gee, I flunked out” and more like, “End semester 1, semester 2 starts next Spring.”

In spite of myself, I genuinely came away from this with a lot. I didn’t even realize it, until, near the end of the class, I went through some old lecture questions out of curiosity, as a review; I was amazed at how easy some things were that weeks before had me dripping tears on my keyboard (that’s not a metaphor; seriously, I worried about shorting something out at one point). Seems grasp lags a couple of weeks behind exposure, but while I was perpetually in a state of WTF, I did in fact learn something.

So what did I learn:

I learned enough about implication, conjunction, disjunction, negation, and sets to be willing to go back and try Intro to Mathematical Philosophy when it runs again [addendum: it appears this course is no longer offered on the Coursera website]. I wiped out of that this summer in W1, stomped to death by Achilles’ Tortoise. I understand logic a lot better now. And – wow – I kinda sorta vaguely get converging sequences. Maybe I can get to W2, where, IIRC, Hilbert’s Hotel awaits (and I kinda, sorta vaguely get that, too). Lesson: there’s a benefit to coming at things from different angles.

I learned that a writing technique might just be valuable in math as well. A few years ago, when I still had pretentions about being a “real” fiction writer, I read From Where You Dream, Robert Olen Butler’s writing process book that advised getting new writing (not revision) done first thing in the morning, before you encountered other language at all – just pee, switch on the coffee maker, and write, don’t look at the newspaper or CNN (good advice no matter what, IMHO), and let your new writing be your first linguistic engagement with the world following sleep. Back then I modified that and used it for a couple of scenes I was struggling with: I literally got into bed, covers over my head and cat next to my pillow (oh, Lucy; another raw, fresh ghost of you I find) and hovered between sleep and wake, imagining myself as the point-of-view character. That got a little rough sometimes, especially when the kid got beat up in the boys’ room at school, but I got the gritty dirt on the floor getting into his mouth, the hexagonal tiles, I got the pier scene right, too, the wet hands slipping, the different tones of voice, so it was worth it. I transferred that to math by accident with the last equation of the course – insomnia can be your friend – and kinda sorta ended up understanding what the epsilon and the a and the m and the n all had to do with each other (maybe). I wish I’d thought of using this technique earlier in the course. It requires a certain degree of memorization, but I seem to grasp things better when I can visualize, even “feel” myself pointing at and moving the symbols around in an immersed state, as opposed to staring at my computer screen, crying. Lesson: try this again. Try this for everything. Try things that work, in different areas when you need something that works.

I learned, for the 27,364,582nd time this year, that I am still my own worst enemy. I completely skipped a question because I saw a summation sign, and I panicked. In my defense, I was having a horrible week for a variety of reasons (that was the Friday LDM came to town, an interlude I badly needed). But when I saw the solution, I realized I actually had the skills do interpret and even answer the question, if I’d just put the effort into remembering what I knew about summation signs, maybe with a quick refresher (and what’s the good of having an internet if you’re just going to use it for hilarious South American men’s underwear commercials). Lesson: I need to get out of my own way.

I learned that I’m learning; yes, me, I’m learning more about math. Things most fourteen-year-olds know, maybe; things maybe I knew when I was fourteen (then again…). Ridiculously small things, like what an even number or a rational number really is. Some of what I learned was even more basic, grade school level, and had nothing to do with this course – all those elhi math teachers, talking about something called math talks and number sense, which I realize now I lack completely, but now that I understand what it is I lack, I can do something about it. Lesson: Progress, even very slow and minor progress, is progress.

I learned about a next step. One of the other ancillary benefits of the huge population of MOOCs and the chatter that goes on in the discussion forums is that you find out about other classes; it’s how I originally found out about this one, in fact, back when I was in MOOCulus last Spring. Some of us are signed up for Effective Thinking Through Mathematics starting in January through EdX (I’m really nervous about EdX, I’m in my first class there now and I’m not too comfy with the platform, though I didn’t understand Coursera at first, either, and now it’s like home). I’m getting a running start on it as well; I found a cheap used copy of the not-required textbook (for the first time in my life, I have more math books on my Read Next shelf than fiction). I was looking through it the other night. I came across the “genie” problem – you’ve probably seen variations of it, it’s a pretty standard logic puzzle. Nine stones, one’s a diamond and is heavier than the glass, but you don’t know which, and you can only use a balance scale twice; how do you figure out which is the diamond? I’ve never been good at these things. I tend to skip over them, leave them for the smart people to play with. But because it’s part of this upcoming course, I took a crack at it, and… I knew what to do almost immediately! It was the same thing as some proof we did somewhere in the last eight weeks where you show that a number is either even or odd, and you go from there. It’s a bit embarrassing to be so psyched (I was pacing in circles for fifteen minutes) over something a clever eight-year-old has learned how to do, but it’s better than not knowing, right? Lesson: Come at things from as many angles as possible, until something somewhere coalesces.

I’ve added Keith Devlin to my Pantheon of Mathematical Heroes (it’s a very small pantheon: Jim Fowler, Vi Hart, and now Keith. But I’m working on adding a few more). He’s kind of Mr. MOOC; before I started this course I came across his blog, MOOCtalk (there’s a lot more to this than just filming a few lectures lifted from the classroom – at least there is when you do it right), and he’s in considerable demand as a speaker on the topic. At one point, he posted: “When you are teaching a real class, you put effort into accelerating students who show promise and trying to rescue those that are struggling. Situations like this show up some of the things that are lost when you go to a MOOC…” It wasn’t that lost this time, at least not on me. I was rescued, very specifically. He didn’t even answer my question; he told me to figure it out and gave me a general sense of how, and then sent me to see the “What it Feels Like to be Bad at Math” by the Math with Bad Drawings guy (a great site I now follow). A world-class Stanford mathematician rescued me, the middle-aged mathtastrophe. So now, like Private Ryan, I need to earn this.

So the next time someone tells you there’s no way you can learn anything in a MOOC with 80,000 students, send them to me and I’ll tell them a few things. It does take a Teacher (and some day I’ll post my definition of that word; like Mel Gibson said in Man Without a Face, it’s not about a piece of paper, it’s about a moment of grace. Mel may have fallen off his own cliff, but at least he got that much right). And, of course, a Student has to show up, too. My Student doesn’t always show up for every MOOC I take, but she does when she finds a Teacher. Keith Devlin’s a Teacher. A great deal of care went into designing a MOOC that would approach the effectiveness of an in-person class, and it showed. I’ve said before that not all MOOCs work (“just like real college!”), but when they do – when I’m excited about a math class I really have no business taking – they’re pretty extraordinary

I wish I could’ve been a success story, but if I can get this much out of a course when I do so horribly (score-wise, at least), imagine what I can get out of it when I do well.

So goes another chapter in my lifelong love-hate relationship with math (it’s a source of some amusement that I actually have a “math” tag on this blog). Some day I have to figure out where this comes from, this insistence on banging myself over the head with a hammer for months on end. And make no mistake about it: math, even the conceptual math used here, is psychically painful for me, something that’s a little different from anxiety. Anxiety I take for granted.

I have yet to cry because I can’t understand a Gertrude Stein poem, and lord knows, I haven’t understood any of them (but I’m getting there, thanks, ModPo (the other day I called this course “ModPo for math” since they both found ways to transcend the innate limitations of a MOOC); I don’t feel like a waste of oxygen when I don’t see the point of a Pushcart story, and puzzling over TNY stories is practically a team sport in the blogosphere. But I’ll cry if I can’t figure out how old Ben is if he’s 3 times as old as Omar and 16 years ago he was 7 times as old as Omar (that’s an actual Khan question… when was the last time you asked a question like this? Say “Hey, Ma, 12 years ago I was 4 times older than Bob and now I’m 3 times older, how old are we?” and Ma is likely to slap you upside the head). Maybe because even ten-year-olds figure out how old Ben is, but there are MFAs and PhDs who throw up their hands at Gertrude Stein.

There must be some very deep Freudian stuff going on here, since I have an obsession with this particular mountain range called Math. When I’m not watching insane Rube Goldberg pop music videos (89 devices, 85 takes; if they can do that, I can do this), that is.

Next chapter, in January. p

What I learned from MOOCulus

This is an example of the way in which mathematics is a democratizing force: problems that at one time would have only been accessible to the geniuses on earth are now accessible to everyone. At one time in history, you would have had to have been the smartest person on earth to have calculated the area of some curved object. But now, armed with the Fundamental Theorem of Calculus, we can all take part in these area calculations.

Jim Fowler, MOOCulus, Calculus 1, Winter/Spring 2013

[ADDENDUM: This course has been converted to “self-paced”; two of the best elements – the discussion forums, and the MOOCulus exercises – are no longer available. I’m not enthusiastic about the new format]

I’ve commented before on my lifelong adversarial relationship with mathematics. But I can’t seem to leave it alone. And even The New Yorker knows writers should learn math. So yeah, I took another math course. Calculus, via Coursera. Fifteen weeks of (free) online math class. Sheesh. What am I, crazy? Yeah.

But this course was special.

How can you not have fun in a class where the teacher considers calculus to be a democratizing force? Where the windowsill features a bowl of coffee beans with a fork stuck in it, and there’s a shelf held up by a dozen Pellegrino bottles? Where lectures are sprinkled with references to LOTR and Douglas Adams (“Suddenly, A Whale…”) and video games (I’d never heard of Portal before, or the closing song “Still Alive,” and now I’m obsessed with it; it was the perfect theme song for the class)? Where videos feature little paper cutouts of a guy moving in front of a light to show his shadow lengthening at a rate related to his speed, drawings of gobbling and belching of functions (complete with music and sound effects), ninja sheep leaping over ladders, the integral monster conquered by donning a wizard hat?

“One does not simply walk into calculus”

Jim Fowler, MOOCulus, Calculus 1, Winter/Spring 2013

It’s hard not to become engaged in the process thanks to lead instructor Jim Fowler, who might be able to make a go of a supplementary career talking math with Jon Stewart and Steven Colbert the way Neil deGrasse Tyson talks astrophysics; who oozes enthusiasm for the subject and hangs out on the forums (as do the other team members) to answer questions and offer encouragement. Sure, I spent every Monday and Tuesday, when the new materials went up (sometimes Wednesday and even Thursday) tearing my hair out, cursing, and crying. But I couldn’t quit; I’d miss all the fun, and anyway, I couldn’t let these guys down.

I couldn’t let down Steve Gubkin, designer of the MOOCulus online exercises, who, when a student posted a comment on the forums about having the memory span of a goldfish, pointed him to a goldfish video with a Pete Seeger soundtrack. We really were never alone.

I couldn’t let down Bart Snapp who only got a little screen time early on but showed up regularly on the forums to address questions about the textbook he put together from existing open-source materials (and if you’ve priced calculus textbooks lately, you’ll know how much that’ll save you).

I couldn’t let down Tom Evans, who wrote the music that started each video lesson and who introduced me to the astonishingly beautiful mathematical music of Michael Blake during Pi Day festivities on the Forums.

And I sure couldn’t walk away from Jim, who somehow found time to drop by a Mathfic topic (new project: fiction related to math, similar to Updike’s “Problems“) I’d started on the Forums (of course I did, what else would you expect?) and recommend some particularly appropriate Jorge Luis Borges pieces. Who, even though he has a frightening array of academic degrees, never made us feel like Calculus – the Freshman Composition of the math department – was trivial. To the contrary: he radiated a thoroughly genuine excitement about topics he’s probably taught dozens, maybe hundreds of times before.

Lecture Topics:

“Morally, why is the product rule true?” 4.02
“Why Shouldn’t I Fall in Love with L’Hopital?” 7.03
“How Long Until the Grey Goo Destroys the Earth? 7.04

Jim Fowler, MOOCulus, Calculus 1, Winter/Spring 2013

I’ll admit, though: the first few weeks weren’t easy. I was a little worried for a while there.

See, this is what I was used to: I’d study a chapter (or, in this case, watch a video), take a test that measured how well I paid attention to the chapter/video, and the grade would scold or praise me. This is what I was expecting. This is what I’d trained for all my life. This is what education is all about: doing well on tests. Isn’t it?

But that’s not what this course was about. It wasn’t about being rewarded for diligence or punished for sloth. It was more of a system to get us to move from theory to application. To use what was in the video to reason out a way to solve the problem. The exercises were part of an overall learning experience, with a Hint button for when we got stuck so the MOOCulus system (written by Steve, based on the open-source code used by Khan Academy) could tell when we were struggling, when we’re ready for more difficulty, and when we’d mastered an application (though it’s not quite that sophisticated, at least not yet, but that’s the goal).

The weekly quizzes – which again, seemed unfamiliar at first – allowed virtually unlimited attempts, that pushed the concepts and applications to their limits, twisted things around so we’d recognize them from all angles. The idea of the exercises and tests wasn’t to judge me, to make sure I was paying attention in class, to see how smart or hard-working I was: the idea was to actually teach me something. Or, more accurately perhaps, to get me to learn.

“Mathematics isn’t just about isolated facts, it’s really about analogies between ideas… it’s like in literature where metaphor plays such an important role.”

Jim Fowler, MOOCulus, Calculus 1, Winter/Spring 2013

I discovered the underpinnings of this kind of thing about halfway through the course in an article by another Coursera teacher, who talks about what I now think of as “the bonobo method” in a HuffPo article. I wish I’d seen before the class started. I think I would’ve been a lot calmer had I realized I wasn’t supposed to look at the questions and know them all, the way I’d been doing it for a very long time.

And part of the idea wasn’t just to teach me calculus; it was to teach the team members how to teach calculus. Jim Fowler laid it out pretty clearly in a presentation (in an OSU session aptly titled, “Steal My Idea,” an attitude that needs more legs) that makes it clear how cool math education can be, and how MOOCs can be extremely beneficial to students and teachers.

But does it work? Did I actually learn any calculus?

Damn right I did. When it came time for the final – which, by the way, was announced as more of an evaluation than a learning experience – I zipped through nearly all of it without a hitch, only needing some time for the material covered in the final week or two. Ok, I’ll admit, I wagged the last question; it was multiple choice, with three attempts, so I think it would’ve been pretty stupid not to wag it. And I still have trouble with minus signs and remembering the derivatives and antiderivatives of trig functions. But I can’t believe how much I actually, really, truly learned. A long time ago, when my brain was a lot younger, I took an in-person calculus course. I got an outrageously high score (103, I think, thanks to bonus questions). But the main thing I learned there was how to write really small to fit every possible equation I might need on the single 3×5 index card we were allowed to bring into the test with us. This way worked much, much better.

MOOCs have seen some bad press recently (not helped by the ironically tragic, or tragically ironic, total meltdown of Coursera’s own “Fundamentals of Online Learning” course in February). Some of it is just the usual fear of anything new, the fear of impersonal technology taking over and losing the human connection. Yet in this online math course with thousands of students (35,000 to start) I felt more engaged with the course, more connected to the staff and the students, than I have in pretty much any in-person course, including some in pretty touch-feely humanities subjects. I had a blast. In Calculus. And this week, at the end of the course, many of us have left notes on the forum about how sad we are that it’s come to a close.

This was a triumph
I’m making a note here:
HUGE SUCCESS.
It’s hard to overstate my satisfaction.

– “Still Alive” from Portal (yet another cool thing I learned in Calculus)

[re-ADDENDUM: This course has been converted to “self-paced”; this makes me sad. The lectures and exercises are (presumably) the same, but it’s a solo experience. I’m sure that works for some people, but it would eliminate much of what I found to be the fun of the course, as well as the support of other student and staff.]

Just call me Princess SohCahToa

If it seems like I’ve been commenting on fewer short stories recently, it’s because math takes a lot of my time.

I missed out on math in school. Not that I didn’t take math classes – and lots of them. But at some point things like completing the square started going by me, and I took it hard. Like when I wrote a suicide note as the answer to the freebie extra-credit question on my Advanced Algebra II midterm: “Why is math fun?” It was the 60s, teachers still laughed at stuff like that back then, especially when my score on the test was a B (I, a caucasian of mixed northern-European heritage, understood the Asian F decades before Glee picked up on it). No one even bothered to ask why I, a freshman, was taking this class, typically in that era populated by juniors.

I gave up at that point and never took another high school math course, though I still struggled in chemistry and physics with the same issues: I don’t know my multiplication tables, and I don’t have a sense of things like inverses and halving a fraction by doubling the denominator. I can do that stuff, but I have to muscle it. You know how some people can’t figure out complete sentences or commas or quotation marks or spelling? That’s me with math. I like to think it gives me a sense of empathy for those who aren’t as strong in some areas as I am. But the truth is, it’s a pain in the ass.

In college (I was a non-traditional student, doing part-time night school then finally, in my 30s, a couple of full-time years to finish up, and even a little grad school) I tried again. Twice. Algebra killed me. Factoring was a nightmare. Trigonometry baffled me. I had to wrestle every problem to the ground. I got through a condensed calculus class with a final score of 105 by spending two to four hours a day working problems, then working them again until I found a way to do them. I have no idea how calculus works, or what it is, though I vaguely remember something about speed at a single point in time.

See, that’s the problem. Concepts like that interest me. If speed is meters per second, you can’t have a speed without two points in time – except you can, with calculus. How cool is that? Don’t ask me to understand it, but it’s amazing. Like infinity. Hey, even Italo Calvino likes infinity.

(Did Italo Calvino add on his fingers? Could he balance his checkbook – did he know whether you add or subtract the 42 cents you forgot to record when you wrote that check for $37.42? Or did he have to go back and recalculate the whole month, the way I do?)

When that wretched imp Zin discovered Khan Academy a few months ago, I tried again: I still want to prove I’m good enough, I’m smart enough, and dog-gone it, people like me. All my friends in high school understood algebra; most went on to Trig, Analytical Geometry, and Calculus without me. When I worked as a computer programmer and then systems analyst (an era I try to repress), my friends played with ideas for random number generators and fractal calculators and I sat quietly with my hands folded and my eyes downcast. I’m a little less distressed now that I hang out online with writers and artists, but I still come across the occasional writer who can calc, and it reminds me of my failings.

I’m also hoping by exercising different parts of my brain, I’ll develop complementary smart muscles. I told my Zoetrope buddy Marko about a voice teacher who encouraged me to work out the lower regions of my voice as a way of adding tone and support to the high registers. Maybe the same thing works with words and numbers.

So I started the (free) Khan course. Slowly. With arithmetic. 273 times 13. One half divided by one fourth. Numbers on the screen the size of my thumb. I finally got a calculator. I had to. The 8s and 6s still elude me, forty+ years later. Nine times six might be 32 or 34 or 36. Four times eight might be 48 or 46 unless I think about it, in which case it might be 32 or 36 or 24. Something about those two numbers, it’s like there’s a broken wire in my brain. I’m ok with the nines and sevens threes and most of the fours, and of course the fives, but if there’s a six or an eight in a problem, I’ll get it wrong. I have lots of “Persistence” badges. You get those when you do 20 problems without getting 5 in a row right.

I’ve progressed – slowly. I’m an Artisan Algebraist, in fact, a Journeyman Statistician and now an Apprentice Trigonometer, bless my aching cosecant and Princess SohCahToa. And if you think that didn’t cost some blood, toil, tears, and sweat, think again. Not to mention a sheaf of notes and my calculator.

It’s no easier now than it was back when I was 13, or 22, or 30. But I think I finally understand how to Complete the Square. (addendum: nope; I forgot again)