Posts Tagged 'education design'



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.

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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.

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 …

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.

 

Coming up for air (and spouting off)

A real-time chronicle of a seasoned professor who has just completed giving his first massively open online course.

Almost a month has passed since I last posted to this blog. Keeping my MOOC running took up so much time that, once it was over, I was faced with a huge backlog of other tasks to complete. Taking a good look at the mass of data from the course is just one of several post-MOOC activities that will have to wait until the New Year. So readers looking for statistics, analyses, and conclusions about my MOOC will, I am afraid, have to wait a little bit longer. Like most others giving these early MOOCs, we are doing so on the top of our existing duties; the time involved has yet to be figured into university workloads.

One issue that came up recently was when I put on my “NPR Math Guy” hat and talked with Weekend Edition host Scott Simon about my MOOC experience.

In the interview, I remarked that MOOCs owed more to Facebook than to YouTube. This observation has been questioned by some people, who believe Kahn Academy’s use of YouTube was the major inspiration. In making this comment, they are echoing the statement made by former Stanford Computer Science professor Sebastian Thrun when he announced the formation of Udacity.

In fact, I made my comment to Scott with my own MOOC (and many like it) in mind. Though I have noted in earlier posts to this blog how I studied Sal Khan’s approach in designing my own, having now completed my first MOOC, I am now even more convinced than previously that the eventual (we hope) success of MOOCs will be a consequence of Facebook (or social media in general) rather than of Internet video streaming.

The reason why I felt sure this would be the case is that (in most disciplines) the key to real learning has always been bi-directional human-human interaction (even better in some cases, multi-directional, multi-person interaction), not unidirectional instruction.

What got the entire discussion about MOOCs off in the wrong direction – and with it the public perception of what they are – is the circumstance of their birth, or more accurately, of their hugely accelerated growth when a couple of American Ivy League universities (one of them mine) got in on the act.

But it’s important to note that the first major-league MOOCs all came out of Stanford’s Computer Science Department, as did the two spinoff MOOC platforms, Udacity and Coursera. When MIT teamed up with Harvard to launch their edX platform a few months later, it too came from their Computer Science Department.

And there’s the rub. Computer Science is an atypical case when it comes to online learning. Although many aspects of computer science involve qualitative judgments and conceptual reasoning, the core parts of the subject are highly procedural, and lend themselves to instruction-based learning and to machine evaluation and grading. (“Is that piece of code correct?” Let the computer run it and see if it performs as intended.)

Instructional courses that teach students how to carry out various procedures, which can be assessed to a large degree by automatic grading (often multiple choice questions) are the low hanging fruit for online education. But what about the Humanities, the Arts, and much of Science, where instruction is only a small part of the learning process, and a decidedly unimportant part at that, and where machine assessment of student work is at best a goal in the far distant future, if indeed it is achievable at all?

In the case of my MOOC, “Introduction to Mathematical Thinking,” the focus was the creative/analytic mathematical thinking process and the notion of proof. But you can’t learn how to think a certain way or how prove something by being told or shown how to do it any more than you can learn how to ride a bike by being told or shown. You have to try for yourself, and keep trying, and falling, until it finally clicks. Moreover, apart from some very special, and atypical, simple cases, neither thinking nor proofs can be machine graded. Proofs are more like essays than calculations. Indeed, one of the things I told my students in my MOOC was that a good proof is a story, that explains why something is the case.

For the vast majority of students, discussion with (and getting feedback from) professors, TAs, and other students struggling to acquire problem solving ability and master abstract concepts and proofs, is an essential part of learning. For those purposes, the online version does not find its inspiration in Khan Academy as it did for Thrun, but in Facebook, which showed how social interaction could live on the Internet.

When the online version of Thrun’s Stanford AI class attracted 160,000 students, he did not start a potential revolution in global higher education, but two revolutions, only the first of which he was directly involved in. The first one is relatively easy to recognize and understand, especially for Americans, who for the most part have never experienced anything other than instruction-based education.

For courses where the goal is for the student to achieve mastery of a set of procedures (which is true of many courses in computer science and in mathematics), MOOCs almost certainly will change the face of higher education. Existing institutions that provide little more than basic, how-to instruction have a great deal to fear from MOOCs. They will have to adapt (and there is a clear way to do so) or go out of business.

If I want to learn about AI, I would prefer to do so from an expert such as Sebastian Thrun. (In fact, when I have time, I plan on taking his Udacity course on the subject!) So too will most students. Why pay money to attend a local college and be taught by a (hopefully competent) instructor of less stature when you can learn from Thrun for free?

True, Computer Science courses are not just about mastery of procedures. There is a lot to be learned from the emphases and nuances provided by a true expert, and that’s why, finances aside, I would choose Thrun’s course. But at the end of the day, it’s the procedural mastery that is the main goal. And that’s why that first collection of Computer Science MOOCs has created the popular public image of the MOOC student as someone watching canned instructional videos (generally of short duration and broken up by quizzes), typing in answers to questions to be evaluated by the system.

But this kind of course occupies the space in the overall educational landscape that McDonalds does in the restaurant business. (As someone who makes regular use of fast food restaurants, this is most emphatically not intended as a denigratory observation. But seeing utility and value in fast food does not mean I confuse a Big Mac with quality nutrition.)

Things are very, very different in the Humanities, Arts, and most of Science (and some parts of Computer Science), including all of mathematics beyond basic skills mastery – something that many people erroneously think is an essential prerequisite for learning how to do math, all evidence from people who really do learn how to do math to the contrary.

[Ask the expert. We don’t master the basic skills; we don’t need them because, early on in our mathematic learning, we acquired one – yes, just one – fundamental ability: mathematical thinking. That’s why the one or two kids in the class who seem to find math easy seem so different. In general, they don’t find math easy, but they are doing something very different from everyone else. Not because they are born with a “math gene”. Rather, instead of wasting their time mastering basic skills, they spent that time learning how to think a certain way. It’s just a matter of how you devote your learning time. It doesn’t help matters that some people managed to become qualified math teachers and professors seemingly without figuring out that far more efficient path, and hence add their own voice to those who keep calling for “more emphasis on basic skills” as being an essential prerequisite to mathematical power.]

But I digress. To get back to my point, while the popular image of a MOOC centers on lecture-videos and multiple-choice quizzes, what Humanities, Arts, and Science MOOCs (including mine) are about is community building and social interaction. For the instructor (and the very word “instructor” is hopelessly off target in this context), the goal in such a course is to create a learning community.  To create an online experience in which thousands of self-motivated individuals from around the world can come together for a predetermined period of intense, human–human interaction, focused on a clearly stated common goal.

We know that this can be done at scale, without the requirement that the participants are physically co-located or even that they know one another. NASA used this approach to put a man on the moon. MMOs (massively multiplayer online games – from which acronym MOOCs got their name) showed that the system works when the shared goal is success in a fantasy game world.

Whether the same approach works for higher education remains an open question. And, for those of us in higher education, what a question! A question that, in my case at least, has proved irresistible.

This, then, is the second MOOC revolution. The social MOOC. It’s outcome is far less evident than the first.

The evidence I have gathered from my first attempt at one of these second kinds of MOOC is encouraging, or at least, I find it so. But there is a long way to go to make my course work in a fashion that even begins to approach what can be achieved in a traditional classroom.

I’ll pursue these thoughts in future posts to this blog — and in future versions of my Mathematical Thinking MOOC, of which I hope to offer two variants in 2013.

Meanwhile, let me direct you to a recent article that speaks to some of the issues I raised above. It is from my legendary colleague in Stanford’s Graduate School of Education, Larry Cuban, where he expresses his skepticism that MOOCs will prove to be an acceptable replacement for much of higher education.

To be continued …

Answering the 64,000-Students Questions

A real-time chronicle of a seasoned professor who has just completed giving his first massively open online course.

With the “instructional” part of the course finished and the remaining students working on the Final Exam (it will be peer graded next week), at last I can sit back and take a short breather. The next step will be to debrief and reflect with my two course assistants (both PhD students in the Stanford Graduate School of Education) and decide where to ride the MOOC beast next.

For sure I’ll offer another version of this course next year, with changes based on the huge amounts of data you get with a global online class of 64,000 students. Despite the enormous effort in designing, preparing, and running such a massive enterprise, there are three very good reasons to pursue this.

First, and this I believe is one of the main reasons why Stanford is supporting the development of MOOCs (I am not part of the central, policy-making administration), designing, running, and analyzing the learning outcomes of MOOCs is a tremendous research opportunity that will almost certainly result in new understandings of how people learn, and as a result very likely will enable the university to improve the learning experience of our regular on-campus students. After just five weeks, my two graduate assistants have enough data to write several dissertations, in addition to the one they need to get their doctorates.

Second, there is a huge, overall, feel-good factor for those of us involved, knowing that we can help to provide life-changing opportunities for people around the world who would otherwise have no access to quality higher education. Is what they get as good as being at Stanford? I very much doubt it, though the scientist in me says we should keep an open mind into the eventual outcomes of what is at present a very novel phenomenon. But if you compare a Stanford MOOC with the alternative of nothing at all, then already you have an excellent reason to continue.

Third, and this is something that anyone in education will acknowledge makes up for our earning a much lower salary than our (often less formally qualified) friends in the business and financial worlds, there is the pleasure of hearing first-hand from some of our more satisfied customers. The following is one of many appreciative emails and forum posts I have received as my course came to and end:

Mr. Devlin and all members of the Introduction To Mathematical Thinking team, I just wanted to say Thank You for everything that you have done to share your knowledge and giving your time and great effort to help others learn. I imagine that this is not an easy project to lead and sustain on a continuous basis. However, you have done a wonderful job in relaying your message. Through your efforts, you have helped many people in the process; especially me. Until this class, I hated math. I hated the idea of learning math or thinking in mathematically analogous methods that are applicable to real world situations. I just didn’t get it. I’m still a little confused about why I am able to comprehend your lessons as effectively as I am (which is saying a lot considering how much I hated math) when I have not been able to do so in the past. Now, I find myself looking forward to your classes everyday! I look forward to using what I have learned from the last video lectures or assignments and using those lessons in situations I did not think possible. And now, I love math! Your instruction has helped me to think more logically and to draw more concise conclusions with issues that I am trying to handle. This is indeed a skill. This is also a skill that you can build upon throughout your lifetime if one chooses to do so. Though I may not be at the level of learning that I should be at, I have learned more in the past three weeks than I have learned throughout my life; and I will continue to learn. I am very serious about this statement. So, thank you All. Thank you, Mr. Devlin. Great Job and Cheers!

Nice!

To be sure, there were trolls on the course discussion forum, for whom nothing we did was right. But one of the benefits of having tens of thousand of students is that within at most an hour of a flame post appearing, tens of others jumped on the offending individual, and within a short while all that was left was a “This comment has been deleted” notice. As the course wore on, the trolls simply dropped away.

Though there was the one individual who, in week four, posted a comment that he hated my teaching style and was learning nothing. Given that this was a free course that no one was under any compulsion to take, and for which no official credential was awarded, one wonders why this person stuck around for so long!

That example provided no more than an amusing anecdote to tell when I start to give talks on “What’s it like to teach 64,000 students?” (Invitations are already coming in.) But there is a somewhat closely related issue that I find far more significant.

Like almost all current MOOCs, there was no real credentialing in my course, so the focus was entirely on learning for its own sake. (As a lifelong math professor, used to teaching classes where many of the students were there because they needed to fulfill a mathematics requirement, having a class of students who were there purely voluntarily added appeal to my giving a MOOC.) To be sure, there were in-lecture quizzes, machine-graded assignments, and a peer evaluated final exam, but the only people who had access to any student’s results were myself, my two course assistants, and the student. Moreover, there was no official certification to back up a good result (the course offered two levels, Completion and Completion with Distinction), and turn it into a form of credential.

Yet many students had an ongoing obsession with their grades, and indeed pleaded with me from time to time to re-grade their work. (Clearly not possible in a 64,000 student MOOC. Besides, I never saw their work. How could I?) As a competitive person myself, I can appreciate the desire to do well. But with literally nothing at stake, I was at first surprised by the degree to which it bothered some of them. When I figured out what was probably going on, I found something that bothered me.

Unlike most MOOCs, mine, being at first-year university level, can be taken by high school students. Indeed, since my primary target audience comprised students entering or about to enter university to study mathematics or a math-related subject, I expected to get high school seniors, and designed my course as much as possible to accommodate them.

I’m guessing that the majority of students who were obsessed with grades were still at high school – indeed, most likely a US high school. That grade obsession I observed is, I suspect, simply a learned behavior that reflects the way our K-12 system turns the learning of a fascinating subject – one of humankind’s most amazing, creative, intellectual achievements – into a seemingly endless sequence of bite-sized pieces that are fed to the student in a mandated hamster-wheel.

No wonder they could not relax and enjoy learning for its own sake. Any natural curiosity and desire to learn – something all humans are born with – had been driven out of them by the very institution that is supposed to encourage and develop that trait. In its place was mere grade hunting.

Do I know this for a fact? No. That’s why I used those hedging words “guess” and “suspect”. But something has to explain that grade obsession in my course, and it certainly brought to mind Paul Lockhart’s wonderful essay A Mathematician’s Lament, which I had the privilege to bring to a wider audience some years ago.

But now I digress. Time to wrap up and check the dashboard on the course website see how many students have submitted the Final Exam so far.

Though this post has dropped the title “MOOC Planning”, I am going to keep posting here, as the project goes forward. Stay tuned.

To be continued …

It’s About Time (in Part): MOOC Planning – Part 10

 A real-time chronicle of a seasoned professor embarking on his first massively open online course.

Well, lectures have ended and the course has now switched gears. For those still left in the course (17% of the final enrollment total of 64,045), the next two weeks are focused on trying to make sense of everything they have learned, and working on the final exam — which in the case of my course involves peer evaluation.

Calibrated Peer Review is not new. A study of its use in the high school system by Sadler and Good, published in 2006, has become compulsory reading for those of us planning and giving MOOCs that cover material that cannot be machine graded. [If you want to see how I am using it, just enroll in the class and read the description of the “Peer Review system”. There is no obligation to do anything more than browse around the site! No one will know you are not simply a dog that can use a computer.]

As I was working on my course, Coursera was still frantically building out their platform to support peer evaluation. There was a lot of just-in-time construction. It’s been a long time since I’ve had to go behind a user-friendly interface and dig into the underlying code to do something on a computer, and the programming languages have all changed since I last did that.

One thing I had to learn was one of the ways networked computers keep time. I now know that at the time of writing these words, 7:00AM Pacific Daylight Time on October 22, 2012,  exactly 1,350,914,400 seconds have elapsed since the first second of January 1st, 1970, Eastern Standard Time. That was the start of Unix Time.

I needed to learn to work in Unix Time in order to set the various opening times and completion deadlines for the exam process. I expect that by the time the next instructor puts together a MOOC, she or he will be greeted by a nice, friendly Coursera interface with pulldown menus and boxes to tick — which probably will come as a great relief to any humanities professors reading this, who don’t have any programming in their background.

[By coincidence, Unix was the last programming language I had any proficiency in, but I did not need to know Unix to use Unix Time. I just used an online converter. Unix was developed in 1969 at AT&T Bell Laboratories in New Jersey. Hence the 1970 EST baseline.]

In fact, time conversion issues in general turned out to be a  continuing, major headache in a course with students all over the world. One thing we will not do again is have 12:00PM Stanford Time, aka Coursera Time (i.e., PDT), as any of the course deadlines. It might seem a nice clean stopping point, and there are all those memories of Gary Cooper’s deadline in the classic Western movie High Noon, but many students missed the deadline for the first submitted assignment because they thought 12:00PM meant midnight, which in some parts of the world made them a whole day late.

The arbitrary illogicality of the AM/PM distinction is not apparent to those of us who grew up with it. But my course TA and I are now very aware of the problems it can lead to! In future, we’ll stick to unambiguous times that stay away from noon and midnight. But even then, with local computer systems usually working on local time, to say nothing of the different Summer and Winter Times, which change on different dates around the world, timing events in MOOCs is going to remain a problematic issue, just as it is for international travelers and professionals who collaborate globally over Skype and other conferencing services. (When I used the Unix Time conversion app, I had to remember that Unix thinks New Jersey is currently just two hours ahead of California, not the three hours United Airlines uses when it flies me there. Confusing, isn’t it?)

The reason why times are an issue in my course is that it is a course. At first glance, it may look little different from Khan Academy, where there are no time issues at all. But Khan Academy is really just an educational resource. (At least, that’s the part most people are familiar with and use, namely the video library that started it all. People use it as a video version of a textbook — or more precisely a video equivalent to that good old standby Cliffs Notes, which got many of us through an exam in an obligatory subject we were not particularly interested in.)

In contrast, in my case, as I’ve discussed earlier in this blog series (in particular, Part 6), my goal was to take a standard university course (one I’ve given many times over the years, at different universities, including Stanford) and make it available to anyone in the world, for free. To the degree I could make it happen, they would get the same learning experience.

That meant that the main goal would be to build a (short-lived) learning community. The video-recorded lectures and tutorials were simply tools to make that happen and to orchestrate events. Real learning takes place when students work on assignments on their own, when they repeatedly fail to solve a problem, and when they interact (with the professor and with one another) — not when they watch a lecture or read a book.

To achieve that goal, the MOOC would, as I stated in Part 6, involve admissions, lectures, peer interaction, professor interaction, problem-solving, assignments, exams, deadlines, and certification. To use the mnemonic I coined early on in this series, the basic design principle is WYSIWOSG: What You See Is What Our Students Get.

As we go forward, I intend to iterate on the course design, based on the data we collect from the students (and 64,000 students very definitely puts us into the Big Data realm). But my basic principle will remain that of offering a course, not the provision of a video library. And the reason for that should be obvious to anyone who has been following this blog series, as well as some of the posts on my other blogs Devlin’s Angle and profkeithdevlin.org. The focus is not on acquiring facts or mastering basic skills, but on learning to think a certain way (in my case, like a professional mathematician). And that requires both a lot of effort and (for most of us) a lot of interaction with others trying to achieve the same goal.

Our ancestors in the 11th Century started to develop what to this day remains the best way we know to achieve this at scale: the university, where people become members of a learning community in which learning takes place in a hothouse atmosphere that involves periods of intense interaction as deadlines loom, sustained by the rapidly formed social bonds that emerge as a result of that same pressure.

While I will likely experiment with variants of this model that allow for participation by students who have demanding, full-time jobs, I doubt I will abandon that basic model. It has lasted for a thousand years for a good reason. It works.

To be continued …


I'm Dr. Keith Devlin, a mathematician at Stanford University. I gave my first free, open, online math course in fall 2012, and have been offering it twice a year since then. This blog chronicles my experiences as they happen.

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