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MathThink MOOC v4 – Part 3

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

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

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

A real-time chronicle of a seasoned professor just about to launch the fourth edition of his massively open online course.


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

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

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 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

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.


I'm Dr. Keith Devlin, a mathematician at Stanford University. In fall 2012, I gave my first free, open, online math course. I repeated it in spring 2013, then in fall 2013, and in February I am giving it a fourth time, each time with changes. This blog chronicles my experiences as they happen.

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