MOOCtalk

The disruption to daily life caused by the global COVID-19 pandemic has disrupted many aspects of our daily lives. Education, in particular, has been hit hard, with systemic education for all age groups having to go online for at least a few months, maybe longer.

I’m no stranger to the online education domain. Back in 2012, I launched the first ever mathematics Massively Open Online Course (MOOC), part of the wave of such courses coming out of Stanford University: Introduction to Mathematical Thinking. I ran that ten-week course for several years, offering two sessions a year on the Coursera platform, itself a spin-off company from Stanford. Participants back then numbered in the tens of thousands, spread all across the Globe.

It was heady stuff, both for the students and the instructional team. (I recruited a group of experienced math instructors to help me as course tutors, as we answered the many questions students inevitably have when trying to learn how to think mathematically. It’s a skill rarely taught in the school system, where the focus is mastering a set of time-tested procedures for solving certain kinds of problems. Going from that to mathematical thinking is, for most of us, a very difficult transition.)

Over those early MOOC years, more and more such courses were offered, and with more choices available, the enrollments for any one dropped steadily. In order to survive as a company, Coursera eventually had to restructure the way courses were delivered, taking them away from the original model of a regular university course — Stanford’s original goal was to deliver normally-exclusive Stanford courses available for free to the world — and packaging them so that can be used in a part-time fashion by students whose lives were already full of regular commitments.

When the numbers of students online at any one time dropped to a trickle, I and my fellow tutors eventually decided it was not a good use of our time, and we pulled out to pursue other interests.

But now we have Coronavirus, and people suddenly find themselves with lots of time on their hands, together with a need (and desire) to interact socially with their fellow humans. So I decided to jump back in to my MOOC, and work through it week-by-week along with the students. An email call to the original group of Course Mentors met immediately with eleven positive responses, leaving seven I have yet to hear from. This is an impressive array of talented, experienced math educators.

The plan is, we try to operate as much as possible as we did in the early days, creating and sustaining a vibrant online community. For that is what it used to be, and what we all missed when the realities of commercial survival for Coursera morphed the course into something else. (To be sure, that new version of the course is still of value, as students in each session who have never interacted with me personally online often tell me in emails, and comment on the course website. But to my mind, and that of my team, it is not the experience it used to be.)

Starting on March 30, then, we are going to try to recreate that original experience; both to meet educational goals and provide an opportunity for people around the world, having a shared interest in learning how to think mathematically, to come together socially in a shared online space.

I should stress that the actual course I and my team will  be involved with over the coming ten weeks is the one already on the Coursera platform. We are not changing any of the content. So students already working their way through the course can switch in to this special session if they want, past students can re-enter and experience the course with the original instructor and his team, and new students can come in and taste it for the first time. The involvement of me and my team is the only change. But as I explain below, that is likely to be a significant change.

I created the MOOCtalk blog in May 2012 when I began planning and developing my original course, using my posts as a public “laboratory notebook.” Looking back on it now, I see how I got some things wrong, but on the whole most things right. In particular, my post of September 21, 2012, just four days into the new course, seems highly relevant to the value today that such an experience might provide. That, at least, is my hope.

Here is a (lightly) abridged copy of that post, where I have omitted parts that do not seem relevant to today.

Liftoff: MOOC planning – Part 7

Published September 21, 2012

It’s been three weeks since I last posted to this blog. The reason for the delay is I was swamped getting everything ready for the launch of my course four days ago, on Monday of this week. As of first thing this morning there are 57,592 students enrolled in the class.

The course was featured in an article on MOOCs in USA Today. It was a good article, but like every other news report I’ve seen on MOOCs, the focus was on the video lectures. Those certainly take a fair amount of time on the part of the instructor (me, in this case), and are perhaps the most visible feature of a MOOC, just as the classroom lecture is the most visible part of many on-campus courses.

For some subjects, lectures, either in-person or on a computer screen, may be a major part of a course. But for conceptual mathematics, which is what my course is about, they are one of the least important features.

Learning to think mathematically is like learning to swim, to ride a bicycle, to ski, to play golf, or to play a musical instrument. You can probably get some idea by having someone explain it to you, but you won’t learn how to do it that way. The key words in that last clause are “learn” and “do”. There is really only one way to learn how to do something, and that is by doing it. Or, to put it more bluntly, the only way to achieve mastery is by repeated failure. You keep trying until you get it. The one thing that can help is having someone who already has mastery look at your attempts and give you constructive feedback.

In fact, failing in attempting to do something new isn’t really failure at all in the sense the word is usually used. Rather, a failed attempt is a step towards eventual success. Edison put it well when asked how he felt about his many failures to make a light bulb. He replied, “I have not failed. I’ve just found 10,000 ways that don’t work.”

After just one week of my course, I’ve seen a lot of learning going on, but it wasn’t in the lectures. Even if I’d been able to see each student watching the lecture, I would not have seen much learning going on, if any.  Rather, the learning I saw was on the discussion forums, primarily the ones focused on the assignments I gave out after each lecture. As I explained to the students, the course assignments and the associated forum discussions are the heart of the course.

So what is my part in all of this? Well, first of all, I have to admit I am uncomfortable with the title “instructor,” since that does not really reflect my role, but it’s the name society generally uses. “Course designer, conductor (as for an orchestra), and exemplar” would be a much better reflection of what I have been doing. Once the course was designed, the lectures recorded, and all the ancillary materials prepared, my task was to set the agenda, provide motivation and context for the various topics, and give examples of mathematical thinking.

The rest is up to the students. It has to be. (At least, I don’t know of any other way to learn how to think mathematically.) To be sure, in a physical class, the instructor (and or the TAs) can interact with the students, and (if it occurs) that can be a huge factor. But that simply helps the students learn by repeated failure, it does not eliminate the need for that learning-by-trying-and-failing process. Let’s face it, if you are not failing at something, you have already learned it, and should move on to the next step or topic. (With understanding, once you get it, you don’t need to practice!)

In a MOOC, that regular contact with the instructor and or the TAs is missing, of course. That means the students have to rely on one another for feedback. This is where the Coursera platform delivers. Here are some recent stats from my course website:

Though I’d like to see a lot more students posting to the forums, with almost 120,000 views (after just one lecture and one course assignment!), it’s clear that that is where a lot of the action is.

As I surmised in an early blog-post, I don’t think it was the widespread availability of video technology and sites like YouTube that set the scene for MOOCs. To my mind, Facebook opened the floodgates, by making digitally-mediated social networking a mainstream human activity. (I’d better add Skype, since there are already several Skype-based study groups for my course. And of course, students who live close together can do it the old-fashioned way, by getting together in person to work through the assignments.)

One feature of the course that did not surprise me was the sense of feeling lost some students reported (and I’m sure many more felt), in some cases maybe being accompanied by panic. For most students, not only does my course present a side of mathematics they have never seen before (the world of the professional mathematicians), on top of that, none of the strategies they were taught to succeed in high-school math work any more.

Because the focus of the course is on mathematical thinking, I can’t provide the students with a list of rules to follow, templates to recognize, or procedures to follow. The whole point is to help them develop the ability to solve novel problems for which no rules are known.

Of course, at this stage, the problems I give them are ones that have been solved long ago, and which have been shown to provide good learning material. But to the student, they are new, and that’s what matters in terms of learning. Unless, of course, they look for the solution on the Web, which defeats the whole purpose. But in a voluntary course where the focus is on process, not “getting answers,” and which provides no college credential, I hope that does not occur. In fact, one of the things that attracted me to free MOOCs was that the students would enroll because they wanted to learn, not because they were forced to learn or simply in need of a diploma. (We mathematicians get a lot of students like that! But we get paid to teach those classes. So far, no one is paying MOOC faculty for their efforts.)

Most US students have a particularly hard time with this “there are no templates” approach, because of the way mathematics is typically taught in American schools.  Instead of helping students to learn mathematics by figuring it out for themselves, teachers frequently begin by providing instruction and following it up with examples. Michael Pershan has a nice summary of this on YouTube. (His initial focus is on Khan Academy, but Khan is simply providing a service that is molded on, and fits into, the US system. The crucial issue Pershan’s video addresses is the system.)

The pros and cons of the two approaches, instruction based or guided discovery, remains a topic of debate in this country, but in the case of my course, there can be no debate. The goal is to develop the ability to encounter a novel problem and eventually be able to figure it out. Providing instruction in such a course would be like giving a golf cart to someone who wants to walk to lose weight! It might get them to their destination with less effort, but it would defeat the real goal.

Having thought at length about how to structure this first version of the course, and played around with some approaches, I ended up, as I thought I probably would, going minimal.  Virtually no instruction, and what little there is presented as examples of mathematical thinking in action, not by way of a carefully planned lesson. I was pretty sure I’d do that, because that’s how I’ve always conducted classes where the goal is student learning (as opposed to passing a standardized test).

There are a number of studies pointing out the dangers of over-planned lessons, one of the most famous and influential being Alan Schoenfeld’s 1988 paper in Educational Psychologist (Vol 23(2), 1988), When Good Teaching Leads to Bad Results: The Disasters of “Well Taught” Mathematics Courses. Still, as I said, I did play around with alternatives, since I was worried how students would fare without having regular access to the instructor and the TAs. I may have to re-visit those other approaches, if things go worse this time than I fear.

But this time round, what the student gets is as close a simulation as I can produce of sitting next to me as I work through the material. The result is not perfect. It’s not meant to be. There are minor errors in there. It’s meant to provide an example of how a professional mathematician sets about things. Definitely not intended as something to be perceived as an entry in an instruction manual.

After those work sessions were video-recorded, they were edited, of course, but only to cut out pauses while I thought, and to speed up the handwriting in places. I found that on a screen, watching the handwriting in real time looked painfully slow, and rapidly became irritating, particularly in places where I had to write out an entire sentence. So I took a leaf out of Vi Hart‘s wonderful repertoire. The speed ramping ended up being the only place that modern digital technology actually impinged on the lecture. Everywhere else it merely provided a medium. The approach would be familiar to Euclid if he were somehow to come back and take (or give) the class.

END of abridged September 12, 2012 post

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.

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.)