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MathThink MOOC v4 – Part 5

Published December 21, 2013 Uncategorized 3 Comments

This post continues the previous four in this series.

In Part 5, I wrestle with grading, and once again expectations raise their troublesome head.

 Like all MOOCs coming out of Stanford, my Introduction to Mathematical Thinking course carries no college credit, nor does it lead to any kind of Stanford certificate. For the first three sessions, students whose aggregate grade was above a certain threshold did receive a Statement of Accomplishment, and if their grade was particularly high, their statement testified that the Accomplishment was “with Distinction”. Those terms were stipulated by Stanford, though it was I, as instructor, who issued them, not my university.

Though some courses on Coursera can, for a small fee, be taken in a fashion that provides a degree of certification that the individual named on the course completion certificate is indeed the person who took the course, that’s not the case for Stanford MOOCs. It is also very much inconsistent with my intent in offering the course, which is to offer a course where the entire focus is on learning, not credentialing.

To me, focusing entirely on learning, and not offering a credential, is particularly suited to a course that aims to provide general purpose thinking skills that can be used in many ways in different walks of life. There are so many ways in which mastery of mathematical thinking can be advantageous in other courses, it can help people get credentials in many subjects, to say nothing of the non-academic benefits it can yield in professional and personal life. In today’s world, mathematical thinking ought to be regarded as on a par with basic literacy, not something to emblaze on a certificate.

In fact, I structured the course to maximize those general benefits, by keeping the mathematical content at a very elementary level for the first six weeks, focusing instead on logical and analytic thinking and the process of bringing precision to issues that are initially vague or ambiguous.

My decision to focus on learning, not the awarding of credit, was also heavily motivated by the fact that, arguably much more than in any other discipline, misguided educational policy has turned mathematics from a creative human endeavor into a relentless and mind-deadening treadmill of test taking. Not in all countries, to be sure, but certainly in the two with which I am intimately familiar as a practitioner, the US and the UK, and many others whose educational systems I am also acquainted with.

The majority of my MOOC students would, I knew, have never encountered a course that focuses on (creative, original) mathematical thinking (as opposed to mastering and applying standard procedures – something that can often be done with almost no thought whatsoever, and indeed can be done far more efficiently these days by apps you can run on your smartphone). So why spoil their first taste of something different, something creative and rewarding, by testing them?

Of course, in a discipline that is about problem solving, each student’s work needs to be evaluated and the results transmitted to them, so they know how they are progressing. But I did not want there to be any more significance attached to those grades than that.

As a person who is fiercely competitive, and does not like to lose, I knew that many would seek to score the highest grades they could on each piece of evaluated work. My hope was that they could approach the course much the same way I and my cycling colleagues approach a race. During the event, no quarter is taken as we all fight to win. But the moment we have all crossed the finish line, the final result ceases to be important. (Okay, it can last until we hit the bar that evening and begin to exchange embellished personal stories of the event. But definitely no longer than that.)

Unfortunately, what works easily for amateur bike racing, does not seem to work for taking a math course. For all our ultra-light, carbon-fiber racing machines and skin-tight lycra, I and my two-wheeled buddies know we are not professional cyclists and our event is not the Tour de France. We do not approach our races with any expectations carried over from previous experience.

With a math course, unfortunately, people do come along with expectations. Though some students successfully managed to focus on the content and the learning thereof, and were not put off by continually getting grades down in the 30 – 40% range (results that I kept stressing were as good as could be expected for anyone who had not encountered this kind of mathematical activity before), many could not make that shift. For them, many years of high stakes testing had turned mathematics into fierce competition to “get an A”, and anything less was “failure.”

Calling my course “Mathematical Thinking” was not enough to counter those expectations, and there was a lot of forum obsessing with grades.

It is, to be sure, a difficult transition to make. (Courses like mine are often called (high school to university) “transition courses.”) To this day, I remember the trauma of going from being an ace at high school, procedural math to being totally lost in my first-year university courses. The main thing that kept me going was the recognition that all my classmates were having the same difficulties. Getting your work back with a 30% at the top is a lot easier to take when everyone sitting in the same room as you is having the same experience – a support mechanism often missing for students in an online course.

The expectations that color how people view course grade-points also affect how they perceive the course certificate. I assumed from the start that many people would attach personal value to the Statement of Accomplishment, even if it has no street value. (On occasion, it appears it does. See the story half way down this article.)

I definitely wanted to make the SoA as meaningful as possible for the person who earns it. My course is difficult, and anyone who completes it should feel proud of what they have done. Accordingly I set the threshold so that approximately 80% of students who completed the course received a SoA, and 20% of those SoAs were with Distinction.

Current platform limitations meant this was not ideal. (The Coursera platform is still under very active development.) Though the instructor was free to specify the algorithm whereby the final course grade was computed from the student grades the system had assigned for each individual piece of work, the only measure the Coursera platform provided on which to make the certificate decisions was that final grade. This meant that some students who did not complete the course, but who scored highly on what they did do, also got certificates. Some of them were not happy to do so.

My response to those was essentially, “Don’t bother to print off the statement then,” though contexed to make it clear I understood why it bothered them.

Still, after enduring for three iterations of the course what for me, given my goals, was a distraction of grades and certificates, at the end of the most recent session I decided to make the award of a SoA in future dependent purely on completing the course.

For students who sign up for my spring session, starting on February 1, SoAs will be awarded for time spent and effort, not level of performance. The grade points awarded for each individual piece of work will cease to have even minimal significance outside the course, not even by way of the SoA.

Students in the spring session will have a choice of two versions of the course. The Basic Course will last for eight weeks, and completion leads to a Statement of Accomplishment. The Extended Course will continue for a further two weeks, devoted to a process I am calling Test Flight, with completion resulting in the award of a Statement of Accomplishment with Distinction.

The Basic Course is designed to develop mathematically-based analytical thinking skills having wide applicability. The additional learning provided by the Extended Course is focused on applying those skills to mathematics itself, in particular building on the earlier analysis of mathematical proof to establish some basic properties of whole and real numbers.

The Statements of Accomplishment will be awarded on essentially a “Pass/Fail” basis, and the certificate will not state a grade.

Both courses will use grades points purely as a metric of progress, not a record of achievement, nor as a criterion for awarding a SoA.

Not quite. I will use grade points in each individual piece of submitted work to determine what constitutes “completion” of the course. To guarantee a SoA, a student will have to submit at least five of the course’s eight, machine-graded Problem Sets,  earning at least 5% for those five Sets. That will determine a lower bound for SoAs.

Again, it means that any student who earns a higher overall grade will get a SoA, even if they complete fewer than five Problem Sets, but as long as everyone knows that system, I have no problem with that. Some commentators say that the lack of a guarantee that the person with a SoA really earned it is a weakness of MOOCs, but that’s true only if you view the goal of education as being evaluation and certification — a problem I have with American education in general.

When Coursera develops a more fine-grained framework for determining the awarding of certificates, I will probably modify the process I use, but frankly I do not regard this as a big issue.

Anyone who “cheats” in my course simply cheats themselves, by not getting the benefit of actually learning. I think education will be a lot better if we separate certification from education. I would have no problem with a third party organization coming along and offering an accreditation service for my course. For sure, an equivalent check will already happen if one of my students uses a SoA from my course to secure a job interview at a large company. The first thing the company is likely to do is ask the applicant to take a short test. As long as that test involves mathematical thinking – and any reasonably well designed test surely will – it will become immediately clear if that individual really did benefit from my course.

I am in the education business, not credentialing. If someone simply wants a SoA for my course, the quickest way to get one is to find an image of a certificate on the Web using Google image search, and use graphics processing software to insert their own name. In doing so, they will be demonstrating useful skill with digital media, of course, but I don’t believe that process will provide them with good mathematical thinking skills.

If you really want mathematical thinking, you need to actually work through my course, using the grades to measure your progress, and stay for the full ten weeks, completing the Test Flight process at the end.

So what’s Test Flight? Tune in next time. Meanwhile, here is a clue.

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MathThink MOOC v4 – Part 4

Published December 19, 2013 Uncategorized 3 Comments

In Part 4, I describe how I try to avoid the educational danger of a well taught class.

This post continues the previous three in this series.

We’re still on the problems caused by the expectations students bring to a course – problems I hope can be alleviated (to some extent) by that new introductory video I talked about in my last post.

Unfortunately, working against any positive effect my pep talk may have is the fact that my MOOC looks like a traditional, instruction-based course.

So too does the regular classroom instantiation of the course, whenever I give it. It’s not.

Neither version of the course is instruction-based. When the twenty-five or so students in a physical class are glowering at me from a few feet away, because I have not shown them in advance how to solve the problems on the assignment, I can patiently explain that their expectations have led them astray, and I can coach them into seeing my lectures for what they are: motivational examples. It’s not about learning how to solve a particular kind of problem, I can tell them. It’s developing the ability to set about solving a completely novel problem. And doing that inevitably involves a lot of failure. For solving novel problems frequently boils down to acquiring the ability to respond positively and constructively to failure.

That kind of direct feedback is not possible in a MOOC. There’s the difference.

Yes, I do use those lectures to provide some initial coverage of some standard material (as well as to do some sample worked solutions). But it’s not pretty. It’s WYSIWOSG – “What You See Is What Our Students Get.” When I record one of those lectures, I imagine a student has come into my office asking for help, and we sit down together at the desk and work through whatever is giving them grief. It is a simulation of two people working together, rather than one giving instruction.

It’s not scripted, I record “as live,” and I leave any mistakes there on the page. The technology is kept deliberately at the low end. (I do sometimes ramp up the speed of my handwriting at the editing stage to match my voice, but interestingly many students tell me they had not noticed that until they read or heard me point it out.)

Because it’s not about the content, you see. It’s all about the thinking. Those lectures are really about the how, not the what.

For anyone who wants to see a polished presentation of the what, the basic factual content, beautifully laid out, there is a short, ultra cheap, self-published (optional) textbook (more accurately, a course companion book) I wrote to accompany the course. Or the student can reconstruct their own polished account based on my lectures, basing them either on brief notes they take during the lecture or by printing off the screen after I have finished the page.

It’s important to see my lectures for what they are: examples of mathematical thinking. They are certainly not provided so the student can learn to replicate me. Each student has to develop mathematical thinking for themself, and it doesn’t have to be identical to mine in all respects.

Of course, the nature of mathematics places significant limitations on what constitutes mathematical thinking. There is an intrinsic limit to how far they can deviate from me or any other mathematician. But in many cases, the range can be much broader than most people realize – certainly so wide that the idea of a “model solution set” makes no sense.

If I were to provide a model solution to a given problem, as students frequently ask for, I am depriving them of one more well constructed novel problem on which to develop their ability to solve new problems. I will have shown them my way of solving it, and for ever after they will be that bit more likely to approach any similar problem my way, instead of developing their own ability.

It’s the same when I record a video of myself solving an assignment problem after I have given the students time to attempt it themselves. (I call those “tutorial videos.”) The actual solution I produce is not important – there are usually other ways to solve the problem, or at least other ways to express the solution.

Notice that I said I solve problems after I have asked the students to try them. I don’t assign problems that are slight variants of ones I have demonstrated. I give them an opportunity to try – and fail – to solve the problems I assign, before I demonstrate one of the possible solutions – again as an example, not “the solution.”

Of course, the whole thing is planned out. It could not be otherwise. But the planning is at a high level, well above the details. A casual viewer would not see my performance as a “well taught lesson.”

At least, I hope they would not, since “well taught lessons” rarely lead to good results, as has been demonstrated by a number of research studies, starting with Alan Schoenfeld’s much-cited 1988 paper When Good Teaching Leads to Bad Results: The Disasters of “Well-Taught” Mathematics Courses.

For students who seek the comfort of a well-presented coverage of the core content, there is my book. Though cheap, it is not free, and that too is deliberate. It is not a part of the course. It’s something that is there for anyone who wants it. There is a (token) cost for getting it. It is a carefully constructed, stand-alone artifact, of value in its own right. Take it or leave it. Consulting books when required is a valuable part of mathematical thinking. But my book is not part of the course; it is not really a course textbook as such, rather it is optional, supplemental reading. A course companion, or a reference, if you will.

The course is about thinking. About doing. About action. In my lectures I am demonstrating what it is like to do mathematics – to generate it within one’s own mind. Reading it is something else.

The videos are an attempt to take the student inside my mind, in real time, to get a sense of what it means to think mathematically. Again, let me stress, not to replicate me. Just to provide an example.

It should be obvious that having pre-prepared pages or slides in the videos, either hand-written or typed, with or without staged reveals, will not work. That approach might help a student learn how to read and comprehend mathematics, but not how to produce it.

That people can in fact acquire the ability to think mathematically from such limited exposure to another person doing so is, in many ways, remarkable. But it’s hardly unusual; countless people have learned this way over thousands of years.

The ingredients of mathematical thinking are already present in the human brain. (See my 2000 book The Math Gene, an account of the evolutionary development of human mathematical capacity.) It just takes a series of suitable triggers to bring those ingredients together. My role as instructor is to make those triggers available. But the student has to pull them and to have a reasonably good aim when doing so.

This is the only way I know how to “teach” mathematics (i.e., mathematical thinking). It’s how I learned.

It’s possible mathematical thinking may not be achievable for everyone, even with a personal tutor. But making the attempt is achievable.

An absolutely key ingredient to any level of success, however, is accepting that failure is part of the process, the focus of Part 2 in this series.

That’s why, having observed how three MathThink MOOC classes have responded to the three versions of the course I have offered, I have decided to make awarding of the course Statement of Accomplishment in future dependent entirely on course completion, not level of performance. I’ll pick up that theme next time.

Step by step I am moving away from features common in traditional, classroom courses. I am also constantly asking myself what exactly is a course, and why do we so often package learning into courses. There too, student expectations play a significant role. Clearly, this blog  is not close to ending. I, for one, am curious to know where it will go next. (Seriously. Like my MathThink course and MOOCs in general, this blog is a work in progress. A living document. Think of it as reflective lab notes.)

MathThink MOOC v4 – Part 3

Published December 17, 2013 Free online courses 3 Comments
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In Part 3, I describe some aspects and origins of the basic course pedagogy, and how they relate to student expectations.

This post continues the previous two in this series.

Expectations. So far I’ve talked about two expectations many students bring to my MOOC that cause problems:

(1) a perception that learning is a cycle of

instruction –> worked examples –> student exercises

(a process that’s better described as training, not learning), and

(2) a belief that failure is something to be avoided (rather than the essential part of learning that it is).

A third problematic expectation many students bring is based on the assumption that mathematics is a body of knowledge to be absorbed, rather than a way of thinking that has to be learned/acquired/developed. That belief is what can lead to the erroneous, and educationally debilitating, perception of mathematics that what makes it hard to learn is the sheer number of different rules and tricks that have to be learned, as described in the article about Jo Boaler’s work I cited in Part 1 of this series.

The view of mathematics as a large collection of procedures can get you quite a way, which explains the huge success of Khan Academy, which shows you all those rules – thousands of them! But it won’t get you to the stage of thinking like a mathematician. Mastering an array of procedures is fine if you are (1) willing to invest the time to keep learning new tricks and (2) prepared to end up working for someone who can do the latter (i.e., think mathematically). Because, increasingly, in the western world, it is that latter that is the valuable commodity. (I wrote about this back in 2008.) My use of the term “mathematical thinking”, rather than just “mathematics”, to title my course was designed to highlight the distinction, but many students nevertheless come to my MOOC expecting a mathematics course (in the sense they have come to understand that term), and are disappointed to discover that it is nothing of the kind. (Some have even asked why I don’t make it more like Khan Academy, a bizarre request which leaves me wondering why they don’t just enter the KA URL in their browser rather than navigate to my MOOC.)

Based on the kinds of issues I’ve been discussing regarding mathematical thinking, in designing my MOOC (and the classroom course that came earlier), I drew on a number of established pedagogies. Most notably among them is Inquiry-Based Learning. For a general background on this powerful and effective learning method, check out this 21-minute video.

Do please watch this video. The focus of much of the video is producing professional mathematicians, and that reflects a common use of the IBL method in mathematics majors classes. In my course, however, with its focus on general mathematical thinking skills for use in many life situations, I don’t ask the students to act as those in a regular IBL class – that would be impossibly hard to pull off in a MOOC in any case. But I believe the general learning principles apply (perhaps even more so), and some of the comments in the video from people who pursued careers in industry address that aspect.

Another pedagogic strategy I adopt is one that has been used in mathematic education since the time of the ancients, which I usually refer to as the Mr Miyagi Method, after the Japanese martial arts expert in the hit 1984 movie The Karate Kid. Having promised to teach karate to the young American Daniel Larusso, Mr Myagi makes his young student paint a fence, wax a floor, and polish several cars. Only with great reluctance does Daniel acquiesce, but in due course he discovers the value of all that effort, as you see from this brief clip.

As I say, this form of teaching has been used in mathematics for centuries. The reason is that in many cases it is impossible to appreciate how mathematics can be applied in a particular situation until enough of the relevant mathematics has been learned. So you design small, self-contained exercises to develop the individual component abilities. Mathematics textbooks have been doing this since they were written on clay tablets five thousand years ago. It’s what most people experience as “mathematics education.”

An attractive alternative is project-based learning. (Again, please do watch this short video.) Unfortunately, whereas PBL is fine for a regular course, in a MOOC that is designed to be of value both to students working on their own, with few if any additional resources, and to students who just participate in a part of the course, it is not an option. That leaves the Miyagi Method as the only game in town.

Even is a regular classroom, and for sure in a MOOC, I would however strongly recommend not adopting Mr Miyagi’s method of delivery. It would surely have been better (as an educational strategy, though not as a movie scene) if he had first explained to Daniel what those chores had to do with learning karate. If a student has to ask, “Why am I learning this?”, the teaching has failed. Why not tell the student from the start?

But remember, times change, and skills and abilities that were valuable in one era sometimes become far less significant, as we are reminded by another Hollywood blockbuster character, Indiana Jones. So you’d better be sure that when you tell a student why a particular topic is important, the reason you give is plausible. (Note: In today’s world, no one balances checkbooks any more – heavens, most people no longer have a checkbook – and no householder uses geometry to figure out how much carpet to order for a room.)

Turning the failing-as-part-of-learning meme on my own journey of learning how to design and give a MOOC, I think that so far I have definitely failed to make sufficiently clear to my MOOC students (1) the basic goals of the course, (2) the approach I am taking to try to achieve those goals, and (3) how those goals lead to adopting the methods I have just outlined above.

To be sure, I laid everything out in detail in the guide-notes I posted on the course website, and in some of my earlier posts to this blog, that I link to from the course site. The problem was, many students never read everything on the site; indeed, some appear not to have read any of the site information.

Now, you might say, they had an obligation to do so. It’s their education, after all, not mine. But MOOCs are about taking learning to a much wider audience than is reached by traditional higher education, and if a MOOC instructor does not manage to connect to that audience, then that is a failure of mission.

As a result, one change I am making with the new version of the course in February is that one of the first things the students will encounter is a video of me explaining the course pedagogy.

[From the very first offering of the course, I posted video discussions between me and my then course TAs, in which we discussed the course design, but those discussions really only made sense after a student had spent some time in the course. So from the second run onwards I cut them into short segments that were released on the site throughout the course. I suspect those discussions were perceived more as “Charlie Rose type” television conversations, rather than providing key information about how to take the course. (In the second of the two discussions, I even asked a professional television and radio host I know to moderate the discussion.) In any case, they did not have the effect I hope will be achieved by a face-to-face explanation by me, as the instructor, of the course goals and structure, given before the course starts. You can view those two earlier videos at: Team Discussion (8mins), What’s New in Number Two (10min 45sec).]

Will my new introductory video solve the problem? I don’t know. For sure, MOOC students do watch (almost) all the videos. Indeed, if there is a problem, it is that some seem to perceive the videos as the most important component of the course, a perception the news media seem to share. Why is that a problem? Because video instruction (i.e., direct instruction) in fact-based, science disciplines does not work. Indeed, instructional videos do actual harm by re-enforcing any prior-held false beliefs, as Derek Muller explains in this video. (Yup, putting math and science education out as a MOOC is hard!)

My guess is that my new introductory video will have an effect, but it will be limited, and many will still be left feeling confused as the course moves ahead. Unfortunately, since the only tools we have at our disposal in a MOOC are video, text, and social media, I don’t see what more I can do, so my gut feeling at this stage is that with the new video I will have gone as far as the medium allows. Nothing works for everyone. All we can do is design for a feasible maximum.

I’ll say (yet) more on this theme of recognizing, anticipating, and dealing with student expectations in my next post. Based on giving three successive versions of my MOOC now, I think the student expectations issue is much more significant in a MOOC than in a regular class. The reason is that in a MOOC, because you have no direct contact with the students, you have very limited ability to counter or correct or allow for those expectations. Your only real strategy is to identify them, and pre-emptively try to lessen their impact on the student.

great-expectations-posterTRAILER (LOOKS GOOD)

MathThink MOOC v4 – Part 2

Published December 15, 2013 Free online courses 6 Comments
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In Part 2, I reveal that I share with Steve Jobs, J K Rowling,  Sebastian Thrun, Thomas Edison, and a successful Finnish video-game studio head, a strong belief in the power of failure.

This post continues the one posted two days ago about the expectations students being to my MOOC.

One of the problematic expectations many students bring to my course is that I will show them how to solve certain kinds of problems, work through a couple of examples, and then ask them to solve one or two similar ones. When I don’t do that, some of them complain, in some cases loudly and repeatedly.

There are several reasons why I do not simply continue to serve up the pureed (instructional) diet they are familiar with, and instead offer them some raw meat to chew on.

Most importantly, the course is not about mastering yet more, specific procedures; rather the goal is to acquire a new way of thinking that can be used whenever a novel situation is encountered. Tautologically, that cannot be “taught.” It has to be learned. The role of the “instructor” is not to instruct, but to offer guidance and feedback – the latter being feasible in a MOOC by virtue of most beginners having broadly similar reactions and making essentially the same mistakes.

To progress in the course, the student has to grow accustomed to the way professional mathematicians (to say nothing of engineers, business leaders, athletes, and the like) make progress: learn by failing. That’s the raw meat I serve up: failure.

Not global failure that debilitates and marks an end to an endeavor; rather repeated local failures that lead to eventual success. (Though the distinction is really one of our attitude toward a failure – I’ll come back to this in a moment.)

Most of us find it difficult making the adjustment to regarding failing as an integral part of learning, in large part because our school system misguidedly penalizes (all) failures and rewards (every little) success.

Yet, it is only when we fail that we actually learn something. The more we fail, the better we learn; the more often we fail, the faster we learn. A person who tries to avoid failure will neither learn nor succeed. If you take a math test and score more than 75%, then you are taking a test that is too easy for you, and hence does not challenge you to learn. A score of 75% or more says you did not need to take the test! You were not pushing the frontiers of your current abilities.

I should add that I am not talking about tests and exams designed to determine what you have learned, rather those that are an integral part of the learning process – which in my case, giving a course that offers no credential, means all the “graded” work.

In my course, the numbers the system throws out after a machine-graded Problem Set, or the mark assigned by peer evaluation, are merely indicators of progress. A grade between 30% and 60% is very solid; above 60% means you are not yet at the threshold where significant (for you) learning will take place, while a score below 30% tells you either that you need to put more time and effort into mastering the material, or slow down, perhaps working through the remainder of the course at your own pace then trying again the next time it is offered. (Another great advantage of a free MOOC.)

What is important is not whether you fail, but what you do as a result. As I was working on this post, I came across an excellent illustration in an article in FastCompany about the Finnish video game studio Supercell. Though the young company has only two titles in the market – Clash of Clans and Hay Day – it grossed $100 million in 2012 and $179 million in the first quarter of 2013 alone.

Supercell’s developers work in autonomous groups of five to seven people. Each cell comes up with its own game ideas.  If the team likes it, the rest of the employees get to play. If they like it, the game gets tested in Canada’s iTunes App store. If it’s a hit there it will be deemed ready for global release.

This approach has killed off several games. But here is the kicker: each dead project is celebrated. Employees crack open champagne to toast their failure. “We really want to celebrate maybe not the failure itself but the learning that comes out of the failure,” says Ilkka Paananen, the company’s 34-year-old CEO.

It’s not just in the PISA scores where Finland shows the world it knows a thing or two about learning; you can find it manifested in the App Store download figures as well!

(And let’s not forget that another Finnish game studio, Rovio, produced over a dozen failed games before they hit the global App Store jackpot with Angry Birds.)

Where I live, in Silicon Valley, one of the oft-repeated mantras is, “Fail fast, fail often.” The folks who say that do pretty well in the App Store too. In fact, some of them own the App Store!

One of my main goals in giving my MOOC is helping people get comfortable with failing. You simply cannot be a good mathematical thinker if you are not prepared to fail – frequently and repeatedly. Failing is what professional mathematicians do maybe 99% of the time. Responding appropriately to failure is a key part of mathematical thinking.

And not just mathematical thinking. It’s definitely true of engineering as well. Remember Thomas Edison, who on being asked how he motivated himself to continue his efforts to build an electric light bulb when a thousand attempts had failed, replied (paraphrase), “They were not failures, I just found a thousand ways it won’t work.”

The metaphor I use regularly in my MOOC is learning to ride a bike. If you think about it, you don’t learn to ride a bike; you learn how not to fall off a bike. And you do that by repeatedly falling off until your body figures out how to avoid falling.

Incidentally, the fact that you really did not learn to ride a bike by learning how to is indicated by the fact that almost no one can correctly answer the question, What direction do you turn the handlebars in order for the bike to turn to the right? Your conscious mind, the one that would have been involved if you had learned how to ride a bike, says you twist the handlebars to the right in order to turn the bike to the right. But, if you are able to ride a bike, your body knows better. You turn the handlebars to the left in order to make the bike turn to the right. Your body figured that out when it learned how not to fall down.

Don’t believe me? Go out and try. Make a conscious attempt to turn right by twisting the handlebars to the right. Most likely, your body will prevent you carrying through. But if you manage to over-ride your body’s instinct, you will promptly fall off. So please, do this on grass, not the hard pavement.

Not surprisingly, six weeks in a MOOC is woefully little to adjust to the professionals’ view of failure. The ones who breezed through my course, unfazed by seeing the system return a grade of 30% on a Problem Set, were in most cases, I suspect (and in a fair number of cases that suspicion was confirmed), professional engineers, business people, or others with a fair bit of post-high-school education under their belts. Those for whom the course was one of their first ventures into collegiate education, often had a hard time of it. (Not a few gave up and dropped the course, sometimes leaving an angry, departing post on the class forum page.)

It’s not called a “transition course” for nothing.

I’ll continue this theme of dealing with student expectations in my next post.

Meanwhile, I’ll leave you with three more examples about the power of failing in the learning process.

The first is Steve Jobs’ 2005 commencement address at Stanford.

The second is J. K. Rowling’s 2008 commencement address at Harvard.

Finally, and very close to home, is Sebastian Thrun’s recent business pivot of his MOOC delivery company Udacity, which I discussed in a commentary in the Huffington Post. Though I would agree with the many commentators that his initial attempt had “failed,” where the tone of many was dismissive, I saw just another instance of someone on the pathway to (for him, yet another) success. It’s all about how you view failure and what you do next.

I’ll continue the theme of dealing with student expectations in my next post.

MathThink MOOC v4 – Part 1

Published December 14, 2013 Free online courses 5 Comments
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In Part 1 of a series, I focus on the distinction between high school math and university-level mathematics, suggesting they are effectively different subjects that are best learned in different ways.

One of the biggest obstacles in giving an online course on mathematical thinking, which my MOOC is, is coping with the expectations students bring to the course – expectations based in large part on their previous experience of mathematics classes. To be sure, prior expectations are often an issue for regular, physical classes. But there the students have an opportunity to interact directly with the instructor on a regular basis. They also have the benefit of a co-present support group of others taking the same class.

But in a massive open online class, apart from locally configured support groups and text-based discussions on the MOOC platform discussion forum, each student is pretty much on her or his own.

The situation is particularly bad for a course like mine, designed to help students transition from high school mathematics to university-level mathematics. For one thing, the two are so different as to be in many ways completely distinct subjects.

School mathematics tends to be almost exclusively procedural, mastering established methods to solve artificially constructed problems designed to be amenable to such an approach. The student who best masters all the techniques in the syllabus and becomes skillful in pattern-matching problems to solution methods, does well. (I know that first hand; it’s how I got to university to study mathematics!)

In contrast, university mathematics is about learning how to deal with a novel situation of a kind you have not encountered before. (If no one else has encountered it, we call it mathematics research.) Though it certainly can involve pattern matching and the application of established, standard procedures, it usually does so only as components of a novel solution you develop to deal with that particular situation. Moreover, at university level, the problems are typically of a “prove that this is true (or false)” variety, rather than “solve this equation” or “compute the value of that formula.”

What is more, while a school math problem typically has a right answer, university mathematics generally involves much more than mere correctness. Indeed, there may not be a unique “right answer.”

Not only is the subject matter different, so too is the pedagogy. Almost all students’ experience of mathematics learning in school is teacher instruction. The teacher describes a method, does a few worked examples, and then asks the students to do a few similar ones. Rinse and repeat.

It’s a very efficient way to cover a lot of ground when the goal is pattern matching and procedure application. It works for school mathematics. Unfortunately, it does not prepare the graduates for the other kind of mathematics. (It also leaves them without ever having a satisfactory answer to their question “What is this good for?”, a question that leaves anyone versed in mathematics astounded. “What is it not good for?” is a more interesting question. It does not have a simple answer, by the way. It’s a very nuanced question.)

It’s like teaching someone the elements of bricklaying, carpentry, plumbing, and electrical wiring, and then asking them to go out and design and build a house. You need all of those skills to build a house, but on their own they are not enough. Not even close.

In deciding, almost two years ago now (before the New York Times had heard of MOOCs) to develop a MOOC to help people learn the other kind of mathematics, what I call mathematical thinking, I knew I was taking on a big challenge. I’d found it hard to teach that kind of course in a physical classroom with just 25, carefully selected students at elite colleges and universities.

On the other hand, most people go through their entire mathematics education without ever encountering what I and my colleagues would call “real mathematics,” and many of them eventually find they need to be able to handle novel situations that involve – or may involve – or could productively be made to involve – mathematical thinking. So I felt there was a need to have a resource publicly available to help them acquire this valuable ability.

The huge dropout rates in MOOCs did not really bother me. For a mathematical thinking course, it’s possible to gain value from dropping into the course for just a few days – and to keep coming back at future times if required. The focus was not on credentialing, it was developing a valuable mental ability – a powerful way of thinking that our ancestors have developed over three thousand years.

That way of thinking can be utilized profitably in many other courses that do yield a certified credential, so students could approach the course as a low-stress, no-risk way of preparing for subsequent learning.

The course is structured as course for those students who seek an encapsulated experience, and in many ways that yields the greatest benefits, in large part because of the interactions with other students working on the same stuff. But the majority of students who have taken it the three times I have offered it have just taken a part of the course.

Each time I gave the course, I changed it, based on what I had learned. When it launches again in February, it will be different again. This time, in some fairly significant ways. In the coming days, I’ll describe those changes and why I made them.

First out of the gate, I’ll describe what exactly were the problems caused by those expectations many students brought to the course, and  how did I try to deal with them. Also, what am I changing in the coming version of the course to try to help more people make what is a very difficult transition: from being taught (i.e., instructed) to being able to learn. The reward for making that one transition is huge. It opens up all of mathematics, and in the process makes it much, much easier.

The traditional, instructional way of teaching procedural mathematics frequently leaves students with the impression (dramatically documented by my Stanford colleague Jo Boaler) that mathematics consists of a large number of rules to be learned. But at the risk of sounding like those weird web advertisements (you know, the ones with a drawing or photo of a strange looking person) promising to teach that “one great trick” that will change your life, let me leave you by telling you the one great trick that all mathematicians learn:

You just have to master, once, a particular way of thinking, and you no longer need all those rules.

That’s what my course focuses on. Stay tuned.

The MOOC Express – Less Hype, More Hope

Published December 8, 2013 Free online courses Closed
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A real-time chronicle of a seasoned professor just about to launch the fourth edition of his massively open online course.

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Last week, I headed off to Arlington, Texas, to participate in a large, international conference on MOOC education, part of the Gates Foundation funded MOOC Research Initiative (MRI). While the founders of the big, massively-funded American MOOC (“MFAM”) platforms Coursera, edX, Udacity, and Novo Ed capture most of the media’s attention, this conference was led by the small band of far less well known Canadian online-education pioneers who actually developed the MOOC concept some years earlier, in particular George Siemens and Stephen Downes who organized and ran the first MOOC in 2008, and David Cornier who forever has to live with having coined the name “MOOC”.

(There were so many Canadians in Arlington, they brought their own weather with them, as you can see from the photograph. The conference ended with participants having to change flights and book additional nights in the conference hotel, as a severe ice storm hit the area. With a return flight that happened to lie within a brief lull in the storm, I managed to get out on time, though only after a slip-sliding taxi ride at a snail’s pace to get to DFW Airport. Others had it a lot worse.)

While teams of engineers in Silicon Valley and Cambridge (MA) are building out MOOC platforms that provide huge opportunities for massive scale-up, the two hundred or so researchers and educators who came together in Texas represent the vanguard of the educational revolution that is underway. If you wanted to know what MOOC learning might look at in a few years time, you would have better spent your time at the University of Texas in Arlington last week rather than in Silicon Valley. In his October announcement of the conference, organizer Siemens described it as “the greatest MOOC conference in the history of MOOCs,” a bit of satirical hype that is almost certainly true.

You need both, of course, the technology to reach millions of people and the appropriate, quality pedagogy. For example, the movie industry required a series of major advances in motion picture technology, but fancy film cameras alone are not what gave us Hollywood. As the technology advanced, so too did the art and craft of motion picture writing, acting and directing. Similarly with MOOCs. The focus in Arlington was on the educational equivalent of the latter, human expertise factors.

To pursue the movie analogy a bit further (and with that great spoof movie poster as a visual aid, how could I resist?), the MOOC action that gets reported in the Chronicle of Higher Education and the breathless (but hopelessly off-base and over-hyped) prose of Thomas Friedman in the New York Times is the equivalent of taking an early movie camera into a theater and recording what happens on the stage.

If you want to see the future of MOOCs, you need to hang around with the instructors in the lesser known, small universities and community colleges who, for many years, have been experimenting with online learning. Most of the leaders of that loosely knit band could be found in a large, cavernous room in Arlington for two days last week – along with key figures from such new-MOOC, Ivy League players as Stanford, MIT, and Georgia Tech.

What was the take-home message? In my case, and I suspect everyone else’s, confirmation that we really don’t know where the MOOC train is taking us. The problem is not an absence of good ideas or useful leads; rather the opposite. Don’t expect a “Conference Proceedings” volume any day soon. The best summary you will find is probably the conference Twitter stream (hashtag #mri13).

The fact is, there isn’t even a clear definition of what a MOOC is. The common classifications of c-MOOC (for the original, Canadian, connectionist animal) and x-MOOC (for the later, scalability-focused, Stanford version) don’t help much, since many (most?) of the MOOCs being developed now have elements of both. Commenting on the lack of a generally accepted definition in his welcoming remarks, conference organizer George Siemens opined that it may be that no agreed definition will emerge, and that the term “MOOC” will be similar to “Web 2.0”, remaining an undefined term that “reflect[s] a mess of concepts that represent foundational changes that we don’t yet understand, can barely articulate, but that will substantially impact the system.”

Actually, I should modify my remark above that Stanford is a MOOC player. That’s not really accurate. Some Stanford scholars (including me) are developing and offering MOOCs, but the main focus at my university is research into learning in a digital age, including online learning. There is a lot of such research, but MOOCs are just one part of it. And those MOOC platform providers you keep reading about, Udacity, Coursera, and Novo Ed? They are all private companies that span out of Stanford. They are now (and  were from their creation) Silicon Valley entities, outside the university. Stanford’s original, in-house MOOC platform, eventually called Class2Go, was recently folded into edX, to be run by MIT and Harvard as an open source platform.

As the MOOC research train goes forward, don’t expect the big, Ivy League universities to exclusively dominate the leading edge of the research. The reason so many participants in the Arlington conference came from far less wealthy, and often smaller institutions is that they have been developing and using online education for years to cater to their geographically-diverse, economically-diverse, and education-preparedness-diverse student bodies. The connections made or strengthened at the Arlington conference are likely to result in many collaborations between institutions both big and small, from the wealthy to the impoverished. The Digital World is like that. When it comes to expertise, today’s academic world is just as flat as the economic one – as Siemens clearly realized in drawing up his conference invitation list.

Expect to see MOOC pioneer Stephen Downes leading a lot of the action, as the head of a new, $19 million, 5-year R&D initiative for the Canadian National Research Council. Keep your eyes too on Siemens himself, who at the conference announced his move from Athabacsa University in Canada to be a professor in, and executive director of, the  University of Texas at Arlington’s new Learning Innovations & Networked Knowledge (LINK) Research Lab. (So new, it doesn’t have a website yet!) Track conference keynoter Jim Groom, who whiled away the time holed up at snowed-in DWF Airport to draft this summary of the gathering.  And don’t overlook the University of Prince Edward Island, where Dave Cormier and Bonnie Stewart brave the weather of the far North East. There are a whole host more. Check the list of recipients of MRI grants for other names to follow, and remember that this is just the tip of, dare I say it under the circumstances, the iceberg.

And never, ever forget to read the excellent (and highly knowledgable) ed tech commentator Audrey Watters, who was invited to attend the Arlington conference but had a scheduling conflict. New readers interested in MOOCs, start here.

Recent headlines may have given the impression that MOOCs were a short-lived bubble, and the experiment has largely failed. Nothing could be further from the truth. For one thing, even the much reported business pivot of Udacity was a familiar event in Silicon Valley, as I argued in a commentary in the Huffington Post.

As Siemens put it in his conference welcome, “the ‘failure’ of MOOCs is a failure of hype and an antiquated notion of learning,  not a failure of open online learning.”

Evaluation rubrics: the good, the bad, and the ugly

Published August 13, 2013 Free online courses 6 Comments
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A real-time chronicle of a seasoned professor just about to launch the third edition of his massively open online course.

With the third session of my MOOC Introduction to Mathematical Thinking starting on September 2, I am busy putting the final touches to the course materials. As I did when I offered the second session earlier this year, I have made some changes to the way the course is structured. The underlying content remains the same, however – indeed at heart it has not changed since I first began teaching a high school to university “transition” course back in the late 1970s, when I was a young university lecturer just starting out on my career.

With the primary focus on helping students develop an new way of thinking, the course was always very light on “content” but high on internal reflection. A typical assignment question might require four or five minutes to write out the answer; but getting to the point where that is possible might take the student several hours of thought, sometimes days. Students who approach the course thinking it is an introductory course on logic – some of whom likely will, as they have in the past,  post on the course forum that they cannot understand why I am proceeding so slowly and making such heavy weather of the material – will, if they don’t walk away in disgust, eventually (by about week four) realize they are completely lost. Habituated to courses that rush through a pile of material that required mostly procedural mastery, they find it challenging, and in many cases impossible, to slow down and adopt the questioning, reflective approach this course requires.

My course uses elementary linguistics and formal logic as a vehicle to help develop new thinking skills that are essential for university mathematics majors, very valuable for STEM majors, and of considerable value for anyone who wants to lead a more rewarding life. But it is definitely not a course in linguistics or logic. It is about thinking.

Starting with an analysis of certain features of ordinary language, as I do, provides a starting point that is accessible to everyone – though because the language I examine is English, students for whom that is a second language are at a disadvantage. That is unavoidable. (A Spanish language version, embedded in Hispanic culture, is currently under development. I hope other deep translations follow.)

And formal logic is so simple and structured, and so accessible to a beginner, that it too is well suited to an introductory level course on analytic, and in particular mathematical, thinking.

Why my course videos are longer than most

The imperative of a student devoting substantial periods of time engaged in sustained contemplation of the course material has led to me making two decisions that go against the current grain in MOOCs. First, the pace is slow. I speak far more slowly than I normally do, and I repeat each point at least once, and often more so. Second, I do not break my “lectures” into the now-almost-obligatory no-longer-than-seven-and-ideally-under-three-minutes snippets. For the course’s second running, I did split the later hour or more long videos into half-hour sections, but that was to make it easier for students without fast broadband access, who have to download the videos overnight to watch them.

Of course, students can speed up or slow down the videos, they can watch them as many times as they want, and they can stop and start them to suit their schedules. But then they are in control and make those decisions based on their own progress and understanding. My course does not come pre-digested. It is slow cooking, not fast food.

Learning by evaluation

The main difference returning students will notice in the new session is the much greater emphasis on developing evaluation skills. Fairy early in the course, students will be presented with purported mathematical proofs that they have to evaluate according to a grading rubric.

At first these will be fairly short arguments, designed by me to illustrate various key features of proofs, and often incorporating common mistakes beginners make. Later on, the complexity increases. For those students who elect to take the final exam (and thereby become eligible to earn a Distinction grade for the course), evaluation will culminate in grading three randomly assigned, anonymized exam submissions from fellow students, followed by grading their own submission.

Peer evaluation is essential in MOOCs that involve work that cannot be machine graded, definitely the category into which my Mathematical Thinking course falls. The method I use for the Final Exam is called Calibrated Peer Review. It has a long history and proven acceptable results. (I describe it in some detail on my MOOC course website – accessible to anyone who signs up for the course.) So adopting peer evaluation for my course was unavoidable.

The first time I offered the course, I delayed peer evaluation until the final couple of weeks, when it was restricted to the final exam. Though things went better than I had feared, there were problems. The main issues, which came as no surprise, were, first, that many students felt very uneasy grading the work of others, second, many of them did not do a good job, and third, the rubric (which I had taken off another university’s Internet shelf) did not work at all well.

On the other hand, many students posted forum comments saying they found they enjoyed that part of the course, and learned more in those final two weeks than in the entire earlier part of the course.

I had in fact expected this would be the case, and had told the class early on that many of them would have that reaction. In particular, evaluating the work of fellow students is a very powerful, known way to learn new material. Nevertheless, it came as a great relief when this actually transpired.

As a result of my experience in the first session, when I gave the course a second time this spring, I increased the number of assignment exercises that required students to evaluate purported proofs. I also altered the rubric to make it better suited to what I see as the main points in the course.

The outcome, as far as I could ascertain from reading the comments student posted on the course discussion forum, was that it went much better. But it was still far from perfect. The two main issues were the rubric itself and how to use it.

Designing a rubric

Designing a good rubric is not at all easy for any course, and I think particularly challenging for a course on more advanced parts of mathematics. Qualitative grading of mathematical arguments, like grading essays or works of art, is a holistic skill that takes years to acquire to a degree it can be used to evaluate performance with some degree of reliability. A beginner attempting evaluation needs guidance, most typically provided by an evaluation rubric. The idea is to replace the holistic application of a lifetime’s acquisition of tacit domain knowledge with a number of categories that the evaluator should look for.

The more fine-grained the rubric, the easier it will be for the novice evaluator, but the more onerous the grading task becomes. The rubric I started with for my course had six factors, which I felt was about right – enough to make the task doable for the student yet not too many to turn it into a dull chore. I have retained that number. But, based on the experiences of students using the rubric, I changed several categories the first time I repeated the course and I have changed one category for the upcoming third session.

In each of the six categories in the rubric, the student must chose between three levels, which I name Novice, Apprentice, and Practitioner. I chose the names to emphasize that we are using evaluation as a way to learn, and the focus is to measure progress along a path of development, not assign summative performance judgments of “poor”, “okay”, and “good”.

The intention in having just three levels is to force a student evaluator to make a decision about the work being assessed. But this can be particularly difficult for a beginner who is, of course, lacking in confidence in their ability to do that. To counter that, in this third session, when the student enters the numerical value that course software will use to track progress, the numerical equivalents to those three categories are not 0, 1, 2, but 0, 2, and 4. The student can enter 1 or 3 as a “middle value” if they are undecided as to which category to assign.

Using the rubric

Even with “middling” grades available for the rubric items, most students will find the evaluation process difficult and very time consuming. A rubric simply breaks a single evaluation task into a number of smaller evaluation tasks, six in my case. In so doing, it guides the student as to what things to look for, but the student still has to make qualitative judgments within each of the categories.

To help them make these judgments, the last time I gave the course, I provided them with tutorial videos that take them through the grading process. I record myself grading the same sample arguments that they have just attempted to evaluate, verbalizing my thinking process as I go, explaining why I make the calls I do. They are not the most riveting of videos, and they can be a bit long (ten minutes for some assignment questions). But I don’t know of any other way of conveying something of the expertise I have built up over a lifetime. It is essentially a modern implementation of the age-old apprentice system of acquiring tacit knowledge by working alongside the expert.

Unfortunately, as an expert, I make calls based on important distinctions that for me jump from the student’s page, but are not even remotely apparent to a beginner. The result last time was, for some questions, considerable frustration on the part of the students.

To try to mitigate this problem (I don’t think it can be eliminated), I changed some aspects of the way the rubric is formulated and described, and decided to introduce the entire evaluation notion much earlier in the course. The result is that evaluation is now a very central component of the course. Indeed, evaluating mathematical arguments now plays a role equal to constructing them.

If it goes well – and based on my previous experience with this course, I think it will go better than last time – I will almost certainly adopt a similar approach if and when I give the course in a traditional classroom setting once again. (A heavy travel schedule associated with running a research lab means I have not taught a regular undergraduate class for several years now, though an attractive offer to spend a term at Princeton early next year will give me a much welcomed opportunity to spend some time in the classroom once again.)

Evaluating to learn, not to grade

One feature of a MOOC – or at least a MOOC like mine that does not offer college credit – is that the focus is on learning, not acquiring a credential. Thus, grading can be used entirely for formative purposes, as a guide to progress, not to provide a summative measure of achievement. As an instructor, I find the separation of the teaching and the grading extremely freeing. For one thing, with the assignment of grades out of the picture, the relationship between teacher and student is changed significantly. Also, it means numerical grades can be used as useful indicators of progress. A grade of 35% can be given for a piece of work annotated as “good” (i.e., good for someone taking an introductory course for the first time). The number indicates how much improvement would be required to take the student to the level of an expert practitioner.

To be sure, students who encounter this use of grades for the first time find it takes some getting used to. They are so habituated to the (nonsensical but widespread) notion that anything less than an A is a “failure” that they can be very discouraged when their work earns them a “mere” 35%. But in order to function as a school-to-university transition course, it has to help them adjust to a world where 35% if often a respectable passing grade.

(A student who regularly scores in the 90% range in advanced undergraduate mathematics courses can likely jump straight into a Ph.D. program – and some have done just that. 35% really can be a good result for a beginner.)

One final point about peer evaluation is an issue I encountered last time that surprised me, though perhaps it should not have, given everything I know about a lot of high school mathematics instruction. Many students approached grading the work of others as a punitive process of looking to deduct points. Some went so far as to complain (sometimes angrily) on the discussion forums about my video-streamed grading as being far too lenient.

In fact, one or two even held the view that if a mathematical argument was not logically correct, the only possible grade to give was 0. This particular perspective worried me on two counts.

Firstly, it assumes a degree of logical infallibility that no living mathematician possesses. I doubt there is a single published mathematical proof of more than a few paragraphs that does not include some minor logical slips, and hence is technically incorrect. (Most of the geometric proofs in Euclid’s Elements would score 0 if logical correctness were the sole metric!)

Second, my course is not a mathematics course, it is about mathematical thinking, and has the clearly stated aim of looking at the many different aspects of mathematical arguments required to make them “good.” Logical correctness is just one item on that six-point rubric. As a result, at most 4 of the possible 24 points available can be deducted in an argument is logically incorrect. (Actually, 8 can be deducted, as the final category is “Overall assessment”, designed to encourage precisely what the phrase suggest.)

To be sure, if my course were a mathematics course, I would assign greater weight to logical correctness. As it is, all six categories carry equal weight. But that is deliberate. Most of my students’ entire mathematical education has been in a world where “getting the right answer” is the holy grail. One other objective of transition courses is to break them of that debilitating default assumption.

Finally, and remember, this is for posterity, so be honest. How do you feel?

I’ve written elsewhere that I think MOOCs as such will not be the cause of a revolution in higher education. Rather they are just part of what is more like to be an evolution, though a major one to be sure. From the point of view of an instructor, though, they are providing us with a wonderful domain to re-examine all of our assumptions about how to teach and how students learn. As you can surely tell, I continue to have a blast in the MOOCasphere.

To be continued …

The MOOC will soon die. Long live the MOOR

Published June 3, 2013 Free online courses 16 Comments
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A real-time chronicle of a seasoned professor who just completed giving his second massively open online course.

The second running of my MOOC (massive open online course) Introduction to Mathematical Thinking ended recently. The basic stats were:

Total enrollment: 27,930

Number still active during final week of lectures: ca 4,000

Total submitting exam: 870

Number of students receiving a Statement of Accomplishment: 1,950

Number of students awarded a SoA with Distinction: 390

From my perspective, it went better than the first time, but this remains very much a research project, and will do for many more iterations. It is a research project with at least as many “Can we?” questions as “How do we?”

From the start, I took the viewpoint that, given the novelty of the MOOC platform, we need to examine the purpose, structure, and use of all the familiar educational elements: “lecture,” “quiz,” “assignment,” “discussion,” “grading,” “evaluation,” etc. All bets are off. Some changes to the way we use these elements might be minor, but on the other hand, some could be significant.

For instance, my course is not offered for any form of college credit. The goal is purely learning. This could be learning solely for its own sake, and many of my students approached it as such. On the other hand, as a course is basic analytic thinking and problem solving, with an emphasis on mathematical thinking in the second half of the course, it can clearly prepare a student to take (and hopefully do better in) future mathematics or STEM courses that do earn credit – and I have had students taking it with that goal in mind.

Separating learning from evaluation of what has been learned is enormously freeing, both to the instructor and to the student. In particular, evaluation of student work and the awarding of grades can be devoted purely to providing students with a useful (formative) indication of their progress, not a (summative) measure of their performance or ability.

To be sure, many of my students, conditioned by years of high stakes testing, have a hard time adjusting to the fact that a grade of 30% on a piece of work can be very respectable, indeed worth an A in many cases.

My typical response to students who lament their “low” grade is to say that their goal should be that a problem for which they struggle to get 30% in week 2 should be solvable for 80% or more by week 5 (say). And for problems they struggle with in week 8 (the final week of curriculum in my course), they should be able to do them more successfully if they take the course again the next time it is offered – something else that is possible in the brave new world of MOOCs. (Many of the students in my second offering of the course had attempted the first one a few months earlier.)

Incidentally, I think I have to make a comment regarding my statement above that the MOOC platform is novel. A number of commentators have observed that “online education is not new,” and they are right. But they miss the point that even this first generation of MOOC platforms represents a significant phase shift, not only in terms of the aggregate functionality but also the social and cultural context in which today’s MOOCs are being offered.

Regarding the context, not only have many of us grown accustomed to much of our interpersonal interaction being mediated by the internet, the vast majority of people under twenty now interact far more using social media than in person.

We could, of course, spend (I would say “waste”) our time debating whether or not this transition from physical space to cyberspace is a good thing. Personally, however, I think it is more productive to take steps to make sure it is – or at least ends up – a good thing. That means we need to take good education online, and we need to do so for the same reason that it’s important to embed good learning into video games.

The fact is, we have created for the new and future generations a world in which social media and video games are prevalent and attractive – just as earlier generations created worlds of books and magazines, and later mass broadcast media (radio, films, television) which were equally as widespread and attractive in their times. The media of any age are the ones through which we must pass on our culture and our cumulative learning. (See my other blog profkeithdevlin.org for my argument regarding learning in video games.)

Incidentally, I see the points I am making here (and will be making in future posts) as very much in alignment with, and definitely guided by, the views Sir Ken Robinson has expressed in a series of provocative lectures, 1, 2, 3.

Sir Ken’s thoughts influenced me a lot in my thinking about MOOCs. To be sure, there is much in the current version of my MOOC that looks very familiar. That is partly because of my academic’s professional caution, which tells me to proceed in small steps, starting from what I myself am familiar with; but in part also because the more significant changes I am presently introducing are the novel uses I am making (or trying to make) of familiar educational elements.

The design of my course was also heavily influenced by the expectation (more accurately a recognition, given how fast MOOCs are developing) that no single MOOC should see itself as the primary educational resource for a particular learning topic. Rather, those of us currently engaged in developing and offering MOOCs are, surely, creating resources that will be part of a vast smorgasbord from which people will pick and choose what they want or need at any particular time.

Given the way names get assigned and used, we may find we are stuck with the name MOOC (massive open online course), but a better term would be MOOR, for massive open online resource.

For basic, instructional learning, which makes up the bulk of K-12 mathematics teaching (wrongly in my view, but the US will only recognize that when virtually none of our home educated students are able to land the best jobs, which is about a generation away), that transition from course to resource has already taken place. YouTube is littered with short, instructional videos that teach people how to carry out certain procedures.

[By the way, I used the term “mathematical thinking” to describe my course, to distinguish it from the far more prevalent instructional math course that focuses on procedures. Students who did not recognize the distinction in the first three weeks, and approached the material accordingly, dropped out in droves in week four when they suddenly found themselves totally lost.]

By professional standards, many of the instructional video resources you can find on the Web (not just in mathematics but other subjects as well) are not very good, but that does not prevent them being very effective. As a professional mathematician and mathematics educator, I cringe when I watch a Khan Academy video, but millions find them of personal value. Analogously, in a domain where I am not an expert, bicycle mechanics, I watch Web videos to learn how to repair or tune my (high end) bicycles, and to assemble and disassemble my travel bike (a fairly complex process that literally has potential life and death consequences for me), and they serve my need, though I suspect a good bike mechanic would find much to critique in them. In both cases, mathematics and bicycle mechanics, some sites will iterate and improve, and in time they will dominate.

That last point, by the way, is another where many commentators miss the point. Something else that digital technologies and the Web make possible is rapid iteration guided by huge amounts of user feedback data – data obtained with great ease in almost real time.

In the days when products took a long time, and often considerable money, to plan and create, careful planning was essential. Today, we can proceed by a cycle of rapid prototypes. To be sure, it would be (in my view) unwise and unethical to proceed that way if a MOOC were being offered for payment or for some form of college credit, but for a cost-free, non-credit MOOC, learning on a platform that is itself under development, where the course designer is learning how to do it, can be in many ways a better learning experience than taking a polished product that has stood the test of time.

You don’t believe me? Consider this. Textbooks have been in regular use for over two thousand years, and millions of dollars have been poured into their development and production. Yet, take a look at practically any college textbook and ask yourself is you could, or would like to, learn from that source. In a system where the base level is the current college textbook and the bog-standard course built on it, the bar you have to reach with a MOOC to call it an improvement on the status quo is low indeed.

Again, Khan Academy provides the most dramatic illustration. Compared with what you will find in a good math classroom with a well trained teacher, it’s not good. But it’s a lot better than what is available to millions of students. More to the point, I know for a fact that Sal Khan is working on iterating from the starting point that caught Bill Gates’ attention, and has been for some time. Will he succeed? It hardly matters. (Well, I guess it does to Sal and his employees!) Someone will. (At least for a while, until someone else comes along and innovates a crucial step further.)

This, as I see it, is what, in general terms, is going on with MOOCs right now. We are experimenting. Needless to say – at least, it should be needless but there are worrying developments to the contrary – it would be unwise for any individual, any educational institution, or any educational district to make MOOCs (as courses) an important component of university education at this very early stage in their development. (And foolish to the point of criminality to take them into the K-12 system, but that’s a whole separate can of worms.)

Experimentation and rapid prototyping are fine in their place, but only when we all have more experience with them and have hard evidence of their efficacy (assuming they have such), should we start to think about giving them any critical significance in an educational system which (when executed properly) has served humankind well for several hundred years. Anyone who claims otherwise is probably trying to sell you something.

A final remark. I’m not saying that massive open online courses will go away. Indeed, I plan to continue offering mine – as a course – and I expect and hope many students will continue to take it as a complete course. I also expect that higher education institutions will increasingly incorporate MOOCs into their overall offerings, possibly for credit. (Stanford Online High School already offers a for-certificate course built around my MOOC.) So my use of the word “die” in the title involved a bit of poetic license. 

But I believe my title is correct in its overall message. We already know from the research we’ve done at Stanford that only a minority of people enroll for a MOOC with the intention of taking it through to completion. (Though that “minority” can comprise several thousand students!) Most MOOC students already approach it as a resource, not a course! With an open online educational entity, it is the entire community of users that ultimately determines what it primarily is and how it fits in the overall educational landscape. According to the evidence, they already have, thereby giving us a new (and more accurate) MOOC mantra: resources, not courses. (Even when they are courses and when some people take them as such.)

In the coming posts to this blog, I’ll report on the changes I made in the second version of my MOOC, reflect on how things turned out, and speculate about the changes I am thinking of making in version 3, which is scheduled to start in September. First topic up will be peer evaluation – something that I regard as key to the success of a MOOC on mathematical thinking.

Those of us in education are fortunate to be living in a time where there is so much potential for change. The last time anything happened on this scale in the world of education was the invention of the printing press in the Fifteenth Century. As you can probably tell, I am having a blast.

To be continued …

Overcoming the legacy of prior education

Published March 20, 2013 Free online courses 25 Comments
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A real-time chronicle of a seasoned professor who is giving his second massively open online course.

We’re now into the third week of the course. The numbers are down on the first edition, almost certainly because the six months that have passed have seen the appearance of hundreds of other MOOCs students have to choose from. But the numbers are still huge. As of today:

Total registration: 27,014

Active students last week: 9,608

Total number of streaming views of lectures: 120,925

Total number of lecture downloads: 35,888

Number of unique videos watched: 87,155

Number of students submitting homework assignments: 5,552

Based on what we (my TA, Paul, and I) learned when I gave the course the first time last fall, I made some changes this time round. Paul and I discussed those changes in a video-recorded discussion we had with media host Angie Coiro just before edition 2 launched, that I referred to in my last blog.

Although the overall numbers are down by about 60%, the profile of the class activity is very similar. The most obvious one, the huge drop in numbers from the total number of enrollments to the number who are still active in week three, has been discussed ad infinitum, often being referred to as “a big problem with MOOCs.” As I observed in a recent blog in the Huffington Post, I don’t think there is a problem at all. The drop off is just a feature of what is a very new form of human experience. Old metrics are simply not appropriate, “retention rate” being one such. (Unless you pay attention to the base for the retention computation, in which case MOOC “retention” is not that different from retention in traditional college education.)

Some of the early research into MOOC participants that has been carried out by my colleagues at Stanford (including studies of my first MOOC) has already demonstrated what we suspected about why so many drop out of MOOCs: many people who register for a MOOC never have any intention of completing the course, or even getting beyond sampling one or two lectures and perhaps attempting one or two of the assignments. Some are motivated by pure curiosity into this new phenomenon, others just want to get a flavor of a particular discipline or topic, and doubtless others have different reasons.

For example, one reason some students enroll that I had not anticipated, reflects the fact that a MOOC offers a large number of eyeballs to be accessed. A very  small number of students enrolled for my course in order to advertise products. (At least, that was one reason they enrolled; they may also have wanted to learn how to think mathematically!) In the long run, this may or may not turn out to be a positive thing. Certainly, the products advertised in the discussion forums for my course (at least the ones I saw) were all education related and free. (Moreover, I also included my own course-related textbook in my short list of suggested – but not required – resources.)

Still, the very wide reach of MOOCs means we are likely to see new kinds of activities emerge, some of them purely commercial. The example I cite above, though right now a very isolated one, may be a sign of big things to come – which is why I mention it. There is, after all, a familiar pattern. The Internet, on which MOOCs live, began as a military and educational network, but now it is a major economic platform. And textbooks grew from being a valuable educational support to the present-day mega-profit industry that has effectively killed US K-12 education.

Talking of which (and this brings me to my main focus in this post), the death – or at least the dearth – of good K-12 mathematics education becomes clear when you look through the forum posts in a MOOC such as mine, which assumes only high school knowledge of mathematics.

To be sure, generalizing is always dangerous, particularly so when based on comments in an online forum, which always attracts people with something to complain about. (Case in point: See my Twitter feed when it comes to banks, United Airlines, and bigoted politicians.) But with that caveat in mind, some themes become clear.

First, many forum posters  seem to view education as something done to them, by other people who are in control. This is completely wrong, and is the opposite of what you will find in a good university (and a very small number of excellent K-12 schools).  “To learn” is an active verb. The focus should be creating an environment where the student can learn, wants to learn, and can obtain the support required to do so. There is no other way, and anyone who claims to do anything more than help you to learn is trying to extract money from you.

Second, there is a common view of education as being primarily about getting grades on tests – generally by the most efficient means (which usually means by-passing real learning). In education, tests are metrics to help the student and the instructor gauge progress. That does not prevent tests being used to assess achievement and provide credentials, but that is something you do after an educational experience is completed. Their use within the learning process is different, and everyone involved in education – students, instructors, parents, bureaucrats, and politicians – needs to be aware of the distinction.

Even worse, is the belief that a test grade of less than 90% is an indication of failure, often compounded by the hopeless misconception that activities like mathematics depend mostly on innate talent, rather than the hours of effort that those of us in the business know is the key. (Check out Carol Dweck’s Mindset research or read Malcolm Gladwell’s book Blink. Better still, read both.)

This is compounded by the expectation that a grade of 90% is possible within just a few days of meeting something new. For example, here is one (slightly edited) forum post from a student in my class:

Right now I want to quit this class. I don’t understand ANY of it. Hell I don’t understand anything regarding to math except basic equations and those barely. When asked to give a theorem on why something (let’s say a right angle) is that way my answer always was “it is because it is”). So now I don’t know what to do. I got 14 out of 40 … 14, and the perfectionist in me is saying might as well give up … you gave it a shot … there is no way to catch up now. The person in me who wants to learn is saying to keep trying you never know what will happen. And the pessimist in me says it doesn’t matter – I dumb and will always be dumb and by continuing I am just showing how dumb I am.

In this case, I looked at other posts from this student and as far as I can tell (this is hard when done remotely over the Internet) she is smart and shows every indication she can do fine in mathematics. In which case, I take her comment as an indication of the total, dismal failure of the education system she has hitherto been subjected to. No first-line education system should ever produce a graduate who feels like that.

Certainly, in learning something new and challenging, getting over 30% in the first test, less than a week after meeting it for the first time, is good. In fact, if you are in a course where you get much more than that so quickly, you are clearly in the wrong course – unless you signed up in order to fine-tune something you had already learned. Learning is a long, hard process that involves repeated “failure”. And (to repeat a point I made earlier) anyone who says otherwise is trying to extract money from you.

Turning to the third theme emerging on the course forums, there is a perception that the most efficient way to learn is to break everything down into the smallest possible morsels. While an important component of learning – if the breaking down is done by, and not for, the student – it is just the first part of a two-part process. The second part, which is by far the most important, and is in fact where the actual learning takes place, is putting it back together into a coherent whole. Textbooks and YouTube videos can provide morselized edubits (I just made that word up), and they do so by the bucketload. What they cannot do, is deliver real learning.

Suitably designed, I see no reason why MOOCs cannot be made to provide good learning, at least up to sophomore college level in many, if not most, disciplines. But a key to doing that is to leverage the power, not of machines, but of people. For fairly well understood evolutionary reasons, human learning is a social activity. We learn best from and with other people. That is how we are built!

Part of the benefit from learning in a social context is that it can offer the learner not just feedback, but also the – at a fundamental level, more important – human support that people need to succeed in education. You can find both of these in a MOOC. Within a short time of the student above posting her feelings, another student responded with this:

Hi. Don’t be discouraged. This course will give you the opportunity to think in a different way. I took the course last year and struggled with most of it. I am taking the course again as I find the subject of mathematical thinking fascinating. My scores this time round are better than the last time which indicates that given enough time even the most mathematically challenged can improve! Only have one caveat for you. If you don’t enjoy the struggle in trying to comprehend and feel that it is not worth the effort then maybe this course is not for you.

With that comment we can see one huge benefit of MOOCs. (At least, all the time they are free.) You can take them as many times as you need or want.

The one essential ingredient in order to take advantage of the huge opportunity MOOCs offer, is knowing how to learn. That should be the main ability graduates of the K-12 system get from their education. Unfortunately, with the current US (and elsewhere) system built around “being taught” and “being tested,” only a few students emerge with that crucial ability, and the ones who do usually say it is in spite of their school education.

The problem, by the way, is not the teachers. Certainly, most of the ones I meet agree with me, and are very clear as to what the problem is: a system that simply does not give them the freedom and support that is necessary for them to really help students learn. (See Jo Boaler’s excellent, well researched book What’s Math Got To Do With It? for a distressing account of how the current, overly micro-regulated system fails our students in the case of mathematics.)

Okay, that’s enough ranting for one post. Let me finish with a couple of examples where MOOCs are already working well. One student in my MOOC posted the following comment:

I have taken this course on a whim to get myself back in gear to return to school in the fall. I always despised the math classes that I was forced to attend in high school and early college. I was frustrated with the endless formulas and cookie cutter style problem solving. If you can solve one you can solve them all so being forced to endlessly solve these equations and proofs over and over seemed to be a futile act of nonsense.

Heading into week three three of this class, my mind has been completely changed. I not only enjoy this more logic based math, but have, in the course of some personal reading and problem solving, discovered i have a knack for it. I have found the challenge of solving more and more difficult problems from a few books i have purchased much more gratifying and interesting than any other area of previous study.

I would like you know that I now plan to switch majors to mathematics. I would like to thank you and your team for an eye-opening experience.

Oh, all right, I admit that included more ranting about US K-12 education. But, heavens, it is bad, and it is likely to remain so all the time that real, knowledgable educators are not part of the conversation, with all the important decision being made by people whose primary interests are profits or political career advancement. (BTW, I have nothing against the profit motive. Heavens, I have two for profit companies of my own and am talking with colleagues about launching a third. But financial ROI is not the same as educational ROI – and again, anyone claiming otherwise, as one head of a major textbook publisher did not long ago, is motivated by the former. I do have something against many politicians, but then I am an American citizen, so after what we have experienced in the past four years, I would.*)

Here’s the other example, this one sent to me in an email, rather than posted on the course discussion forum.

I am enrolled in your course “Introduction to Mathematical Thinking.” It is incredible. You have alleviated my fears that my college professors will have the same attitude towards mathematics that my high school teachers do. Mathematics is beautiful and certainly emotional. I am surrounded at school by people who believe mathematics is systematic. Through all of the videos you have posted so far and your archived NPR clips, I am now confident that mathematics is the direction I want to pursue. I am excitedly awaiting next week’s lectures. 

With tears in my eyes and more gratitude than I know how to express,

It’s that kind of feedback that makes teaching one of the most rewarding professions in the world. It’s why people become teachers. If society would just get off teachers’ backs and let them get on with what they were trained to do, what they know how to do,  and what they want to do, we’d all be a lot better off. (Check out Finland.)

To be continued …

*ADDED LATER IN RESPONSE TO A QUERY FROM AN OVERSEAS READER: The problem is the complete refusal of the Republican Party to cooperate with a now twice-elected President of the US, in governing the country as they are all elected and paid from public funds to do, choosing instead to drive the country, and with it most of the world, to the brink of financial and thence  social disaster.

 

MOOCs are So Back to the Future

Published March 1, 2013 Free online courses 2 Comments

 A real-time chronicle of a seasoned professor who is about to give his second massively open online course.

With the second edition of my Stanford MOOC Introduction to Mathematical Thinking starting this weekend on Coursera, I have once again been wrestling with the question of the degree to which good, effective mathematics learning can be achieved at scale, over the Internet.

I describe some of my reflections in my latest post in my monthly Devlin’s Angle column for the MAA — a column aimed primarily at college mathematics faculty, which makes up most of the MAA’s membership.

When I started to plan the first iteration of the course last spring, my main goal was to be able to walk away alive in order to try again. I stayed as close as I could to the way I had taught such a course in a traditional classroom setting, since I knew how to make that work (in the traditional classroom). This time round, armed with what I learned from that first attempt, I am making a number of changes.

My course TA (last time and this) is Paul Franz, a doctoral student in Stanford’s Graduate School of Education, and the two of us went into the Stanford TV studio recently to talk about the new course with broadcaster Angie Coiro, currently host of the syndicated radio and television interview show In Deep. You can find the first clip (10 minutes) from that hour-long interview here.  I’ll release further clips in future posts to this blog.

One change I’ve made to the course is to stretch it from seven weeks (five weeks of lectures followed by two weeks of examination work) to ten weeks (eight plus two). As we observe in that video clip, that change was a direct result of the information we collected from giving the course the first time.

Students in a MOOC exhibit a very different — and far more varied — profile from the traditional university cohort. Not just in age and backgrounds, but also in their reasons for enrolling in a MOOC. For instance, many people enroll in a MOOC with no intention of completing the course. They simply want to get a sense of the topic or subject.

But there is another group that wants to complete the course,  and come in prepared to work very hard to do so. They want the course to be as close as possible to a regular university course — essentially the classroom course I have been giving off and on at a number of elite colleges and universities since the late 1970s, most recently at Stanford — not a watered down version. But as the course went on, a substantial number of them submitted forum posts and emailed me to say that the pressures of their professional lives occasionally made it impossible to keep up. With instructional videos being released three times a week, on Monday’s Wednesdays, and Fridays, if a business trip caused them to miss a couple of days, they were never able to recover, and eventually had to drop out.

So I have reorganized the course so it runs slightly longer, but with instructional videos coming out only twice a week (Mondays and Wednesdays). That still maintains the pressure that is a major component of my course (and primarily, I see it as a course, for reasons I have articulated in several earlier posts in this blog), but provides what I hope is sufficient flexibility for busy people to cope.

The adoption of a different schedule is almost certainly the most obvious change I have made. But that one is purely logistic. Far more significant, and to me (and my education graduate student TA) more interesting, are the pedagogic changes I have implemented.

As the first course progressed, I gradually came to realize that the underlying pedagogical model I had adopted enabled me to make much more extensive and aggressive use of a number of educational devices I had used only minimally the first time round, namely:

  1. machine-graded, multiple-choice pop quizzes
  2. machine-graded, multiple-choice (substantive) problem sets
  3. student evaluation/grading of work.

I have been strongly opposed to the first two (as are most of my colleagues, and for good reason) for my entire career in university education, and had never seen the need for the third (though I am familiar with the research that shows the beneficial effects on student learning of being asked to evaluate and grade the work of their peers). In a MOOC, where there are thousands of students, all three seem unavoidable. And so I used them all. But I did so as little as possible.

This next time round, all three play a much more prevalent role. And they do so because of that recognition that my underlying pedagogic model eliminated many of the objections and hestitations I had to those devices.

What is that pedagogic model? One-on-one teaching/learning, the kind of learning experience that in the traditional academy is reserved only for doctoral students. For inescapable personnel reasons — sheer numbers — it is not possible to provide one-on-one learning experiences for undergraduates or masters students at a traditional university.

But surely, isn’t it even more problematic in an online course with tens of thousands of students? Strange though it may seem, the answer is no.

The reason is that a MOOC is, in many ways, like radio or TV — and not just because MOOCs make use of video-recorded lectures. Of far more educational significance, though TV and radio are both referred to as “mass media,” they are in fact highly individual. The newsreader on radio or TV is not addressing a large audience; she or he is talking to millions of single individuals.
The secret to being good on the radio or TV is to forget the millions and think of just one (generic) person. After all, the listener or viewer is not in a room with millions of other people; in fact, if the broadcast is successful, that listener or viewer is cognitively in a room with just the presenter. The really successful radio and TV newsreaders and presenters are the ones who can do that really well. They create that sense that they are talking just to You. And the same is true for a MOOC.
When your voice, with or without your face, is in someone’s living room, especially if on a regular basis, there is a direct human connection that in important ways is far more intimate than is possible in a lecture hall filled with anything more than a handful of students.
Once you realize this feature of the MOOC medium, the one-on-one pedagogic model is obvious. I used it extensively in the first version of the course, going to considerable lengths to create a sense of the student sitting next to me at my desk as we worked through the material. (You can see a low resolution example here. The Coursera videos on their site are larger and of much higher resolution.)
Although the entire course was planned meticulously in advance — for such a complex system, with so many moving parts, it has to be — when I sat down to record an instructional session, everything was recorded as live, without notes. I simply plonked a piece of paper down on my desk, beneath an overhead camcorder, and began talking to an imaginary (single!) individual student sitting alongside me.
The only editing done to what was captured by the camcorder was to speed up some of the handwriting to match my voice. All my false starts and my inevitable writing and speaking errors, together with my moments of indecision, made it into the video that was released. For the fact is, the focus of the course, namely mathematical thinking, is an error-prone, messy, human activity,  that often proceeds at a pedestrian pace, punctuated with uncertainties, and that is exactly what I wanted to convey.
To make the result as realistic as possible, I made a point of not thinking in advance about the topic or problem I would discuss — and certainly never “rehearsed.” After all, none of those would have been possible if the student had simply knocked on my office door and said, “Professor Devlin, do you have a moment to explain something to me?”
True, this does not make for riveting, slick television. It’s not meant to. It’s WYSIWOSG teaching: “what you see is what our students get.”
In other words, MOOCs enable us to go back to the oldest, and to this day by far the most effective method of education the world has ever known: the one-on-one apprenticeship system. Once you realize that, many things become possible that can’t be effectively used in traditional undergraduate education.
That includes the use of pop quizzes and multiple-choice questions! For when you think about it for a moment, PhD advisers use both all the time. True, they don’t think of them in those terms, and they don’t implement them with machine grading, but they do use both techniques.
Of course, a MOOC still does not allow the apprentice to talk back to the instructor — though in reality most apprentice learners feel pretty timid next to the expert, and rarely do that. Moreover, the instructor is not able to view and comment on the learner’s work. This is where, in a MOOC, crowd sourcing (something not available in a traditional apprenticeship situation) must be brought to bear. Now we are into something new!
I’ll continue this thread in future posts.
Since most of the instructional materials in a MOOC have to be created and assembled before the course begins, once it gets underway you are pretty well locked into the approach you start with, and any changes have to wait until you give the course again. So this second version of the course is my first opportunity to implement the changes I would have liked to make last time, and to make adjustments based on looking at the course data after the first edition finished.
I’ll be blogging about those changes in real time as the course proceeds. Will they make things better, and if so how? You will know soon after I do.
To be continued …
I'm Dr. Keith Devlin, a mathematician at Stanford University. I gave my first free, open, online math course in fall 2012, and have been offering it twice a year since then. This blog chronicles my experiences as they happen.

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