*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 am a student taking your class, and also a K-12 math teacher who is looking for ways to integrate higher-level mathematical thinking into my classroom. I am so happy that you, as a professional mathematician and professor, talk and think about K-12 math education!

I too was jarred by the transition from high school to university mathematics–I’m hoping to augment my teaching with some extra mathematical depth that more closely resembles “real” mathematical thinking. It’s often tricky to balance, and sometimes my ideas fall flat, but I think this course is helping me open up to a broader mathematical education for my students.

This past week, my algebra students were learning about inductive and deductive reasoning, and talking about how to formulate conjectures. We looked at an example: adding all the odd numbers from 1 to n, and looked for a pattern. We described it together verbally, translated it into an algebraic expression, and drew a picture of nested square numbers to give them an example of geometric proof. They enjoyed themselves very much, and I felt pretty good. Then one student raised her hand.

“What happens when you add even numbers?”

My first instinct was to stall, quickly figure out the answer for myself, then draw out the pattern, prompt them to fill it in, help them derive the algebra, etc etc. But I tried to take a cue from both your class, and Sugata Mitra’s excellent lectures on child-driven education:

“Great question. I have no idea, and I’m going to get some coffee. Let me know if you find out.”

And lo and behold, they had it all worked out and then some when I came back! I may not be anywhere near a good teacher, but at least I am improving when it comes to not getting in the way of a mathematical education. :)

Thank you for the excellent class and for your thoughts on math teaching.

I think we are all shaken when we go from the high school to the university. I certainly was — and I was fortunate to be largely self-taught in my final two years at high school, mostly for logistic reasons.

Isn’t it great when a class solves a problem on its own! I find the hard part is stepping back and giving them the time and the freedom to work on it on their own. In my MOOC, Paul and I do that all the time, and it is what I asked my Community TAs to do. There are actually very few occasions where the right answer does not emerge sooner or later. But it’s much harder to do in a regular, physical class.

I am a student in the course mentioned above. It is my first MOOC but I consider myself a lifelong learner and a ‘survivor’ of secondary education so I have been thinking about these issues. One thing that has occurred to me is that the lessons learned as MOOCs mature will end up being applied to K-12. For example, if a MOOC can successfully engage several thousand students having very diverse backgrounds, learning styles and paces, then a k-12 teacher should be able to engage and support a class of 20-30 students with diverse backgrounds, learning styles and paces. I am not saying that MOOCs have completely figured that out yet, but they will need to do that to be successful in the long term.

Agreed. In fact, Stanford has made it clear that we are in the MOOC business for just two reasons: (1) to carry out research into the medium in particular and learning in general, and (2) to improve the on-campus learning experience for our regular students. I’m not part of the group that made that call, but I totally endorse it. I am sure that part of what it will take is to incorporate what we have learned about video games, which have taken engagement on a mass level to a whole new level. (Video game learning is the focus of my other blog, profkeithdevlin.org.) I am very slowly introducing elements of video game design into my MOOC.

Professor, your comments could not have been more accurate vis-a-vis my situation. To read them was to provide huge reassurance and revitalize my self-confidence. I am 70 and my math education was at a top Ivy League school fifty years ago. While not a math major I always was at the top. Your course has been humbling but your comments and admonitions have been right on. I am taking the course solely for my own intellectual advancement; no academic or professional rewards. My wife thinks I am crazy as I found myself frustrated in not “getting everything right”. I’m still a frustated “Type A” but I am beginning to perceive some of the elegance and beauty in the material. That’s the reward. That’s what satisfies. And I am sure that is why you put yourself through this wringer of the MOOC (what a ghastly acronym!). I’m in for the duration and offer you my whole-hearted support for your work. Wish we could get together and have a beer and talk some philosophy. Regards. Richard Lydecker.

I know exactly what it is to be a “frustrated Type A”! :-) And that, as you observe, is why I am throwing myself into MOOCs. :-) The one thing that Stanford gave me that I never found anywhere else is the freedom to fail. Whenever I wanted to try something new and risky back in Europe or elsewhere in the US, there was never any shortage of colleagues who told me it was bound to fail (and who would be lining up to gloat when it did – as most risky things do). But here, the response is at least “Wow, that sounds interesting,” and often “How can I help?” And if you fail, nobody thinks twice about it. It is a psychologically very supportive environment. I’ve been in this environment for 25 years now. I am sure it has rubbed off on my approach to teaching. Fear of failing is debilitating. Get past that, and pretty well anything becomes possible.

Hello Mr. Devlin,

I am taking your course for the second time and loving it – learning as much more this time as last. I tutor elementary and middle school students and listen intently each time you speak of k-12 education. I agree with most of your opinions, but was confused about the comment about “the last four years” when you spoke of politics. What exactly is different in the Obama administration that concerns you? My feeling is that the future of education is much safer in the hands of the democrates than the republicans. Please expand on this comment. My ears and mind are open.

Again, thank you so much for all your work in the areas of MOOCS. I plan to take a grand total of 18 of them this year.

teresa

The problem is not of Obama’s making, though some of us believe that things would be very different had he been white, so in that sense there is a causal link. Rather, since his election, the Republican Party has simply refused to cooperate in governing the country, and have brought it, and with it the world’s economy, to the brink of disaster.

Glad you are enjoying the course. Taking 18 online course in one year is full-time education. Good luck to you!

“And textbooks grew from being a valuable educational support to the present-day mega-profit industry that has effectively killed US K-12 education.”

You should write an entire blog post on this. I’m curious to know more.

Good suggestion. Thanks. I’ll do just that at some stage.

Hi Dr. Devlin,

Great post – thanks for sharing your perspective as someone who is leading a course. That’s very interesting. I liked what you said the benefits of MOOCS: “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.”

This is a really important point. A similar benefit of MOOCs is that you can try so many different subjects without any risk other than your time. I think we are going to see a lot more people becoming interested in new subjects they otherwise never would have been exposed to because there of this lack of risk in trying it out. Unlike at university or school, you can drop out if you don’t like it or if you no longer have time to study and you don’t suffer academically or financially. This is often seen as a bad thing: the high drop-out factor of MOOCs (something I see you wrote about for the Huffington Post), but if you look at it from this perspective it can be a benefit.

It’s really quite a wonderful and exciting time for lifelong learners!

Also a wonderful and exciting time for those of us in the uni edu biz as well – of course, we are professional lifelong learners!

Hi Prof. Devlin,

I want to thank you first of all for offerering this course, but most of all for the thought and effort you’re putting into the general concept of a MOOC. I think it’s a very interesting concept and I hope it is further developped and becomes (even) more popular. (I admit to not understanding how it works financially, but I assume it must, if so many big universities are investing in it… perhaps you can explain this?)

I am currently enrolled in the MathThink course. It’s interesting to think about the wide variety of people who might be enrolled in the course. I certainly do not seem to be your target audience. I have a bachelor in mathematics from several years ago (but not TOO many years ago ;)) and I took the course mainly because I don’t really mathematics very often in my career, and wanted to refresh a little and prevent my brain from getting too rusty. Also, I have never taken a MOOC and wanted to start with a topic I was relatively familar with, to see what it was like.

Regarding the high school to university transition: for me, the biggest change was not the change in the level of mathematics. It was the change from being at the top of the class to being somewhere between average and below average. That was a difficult and frustrating (and humbling!) transition!

Dr. Devlin,

I am a student in your class and have spent a good portion of my life as a student and then professor (finance), with additional experience in investment consulting (technology support) for the past 25 years.

I will be retiring at the end of April. Your course has made me think and reflect not only on mathematics but also on other areas of my life. Regarding mathematics, there are troubling gaps in my knowledge. I need to transcend thinking primarily in a procedural way.

To me, the course is both engaging and difficult. Most of the course material is new to me. At times my brain feels like mush; hopefully it is in the process of re-wiring itself and growing.

MOOCS will have a transformational impact on education and the way that knowledge emerges from interactions among many individual participants. You are all to be congratulated for nurturing MOOCS, experimenting with various approaches, and encouraging all of us to persevere.

Thank you.

I bought your book through Amazon and I’m reading it. I’m an economist and I’m aware just how important your work is. Unfortunately, having a full-time job, I haven’t been able to keep up with the quizzes. But this is a long term commitment to me into relearning my mathematical thinking.

Keith, what you’re doing is inspiring. (I haven’t even looked at any MOOC lectures yet — I got to your blog via a Facebook recommendation from Kai von Fintel). Just wanted to share that when I started teaching mathematics for linguists (because half of us starting out in linguistics at MIT in the early 60’s were former math majors and half weren’t and suffered from considerable math insecurity and anxiety), I realized that in both high school and college all my math courses had combined the goal of teaching with the goal of “sorting” — there was always as strong element of “how good are you at this?”. And I simultaneously realized that in college I had been able to take lots of courses in music (Russian musical history, a seminar in Bach, an ear-opening course in contemporary music) with no presumption at all that we were supposed to be budding musicians or in any sense “good at it”. Music wasn’t a “specialization” at our small liberal arts college, and we were all studying it just because we wanted to enrich our personal lives. So I applied that philosophy in my math course (which I taught starting in graduate school, and continued even for several years after retirement), recognizing that none of my students were expecting to “go on in mathematics”. That made it tremendous fun to teach! I tried to make all the obligatory exercises ones you could get right if you were just paying attention, with occasional more challenging or adventurous ones thrown in as optional extras. The students liked it, and some of them got over their math anxiety, and I think a number of them managed to see that math can be beautiful. (We even lost a good department secretary who took the course, enjoyed it, and then declared “I CAN do math! I’m going to business school!”) Your course sounds orders of magnitude more exciting! I’m so happy to hear about it, and plan to watch some of it as soon as I can!

Barbara, Thanks for the comments. Of all people, you will be amused if you look at the first few lectures in my MOOC — the introductory examples are all taken from natural language. It is such an “obvious” place to start. Typically, mathematics instructors start with examples from naive physics (not interesting to many) or everyday finance (most of us find this boring, and largely irrelevant in today’s society), but we all use language all the time, and who isn’t intrigued by its many quirks? The problem if you want to use the course to grade progress to rank students is that the questions are usually ambiguous and often don’t have definitive answers, and they can be much harder for students for whom English (in my case) is a second language. (I wrote that sentence for other readers!) But when you separate learning from certification, and allow the student to re-run lectures as often as they want, and to even take the entire course again, the whole game changes.

I took the first iteration of the course, but dropped out of the second — for time, mostly. (I am taking an EdX course in Biology and one in complex systems.)

Your point that anyone who gets a grade of 90 or more probably should not be taking the course is well taken. I was taking the course mostly to see what a well-run MOOC is like. I was first exposed to formal logic in high school when I read Whitehead and Russell’s Principia, which had just come out in paperback to *56. My teachers were of no help here — I was told to sit in the back of class in Geometry so the other students could get a chance. (I invented a geometry during class — won a science fair.)

I would say that I succeeded in maths despite my education. I wish that I had a professor like you (or Gilbert Strang, etc.) when I was studying maths. Now, with the internet, I can, and other students can.

One problem with maths education as I see it is that the curriculum for high school and especially college maths has been designed by people like me, most of whom assume that the easy stuff is ‘intuitively obvious’. Personally, I have a hard time working on maths in a social context after decades of doing the work alone. One other problem is that K-8 maths is usually taught by people who either do not understand maths or hate it. (That was my experience in the 1950s and early 1960s, and I see no evidence things have changed.)

I think that MOOCs are here to stay, in some form. I would also like to see more choices, so that people with different learning styles can find something that works.

I am a very amateur flute player. One thing that has happened over the last 50 years or so is that the standards of performance has risen. Students in high school learn pieces that used to be held until graduate school, for example. What is shocking is that the same thing has not happened in maths, science, or even writing. Of course, the reason that standards have risen for instrumental performance is that only those who are motivated continue beyond the early stages of learning an instrument. The vast majority of people who start an instrument in grade 4 drop out by grade 6.

With maths, unmotivated students cannot drop the subject, so a way must be found to make it possible for them to learn anyway. The other side of the coin is that motivated students are often held back because the educational system wants to move people in lock-step. (When I was in high school, ‘smart’ students were tracked, but that did not help much, and I don’t think it is done as much any more.)

I think that the best thing that can come out of education reform is a diversification of teaching methods and ways to deliver the experience. For that to happen, the old idea of what education is simply has to change.

Hi Professor, you mentioned that you were self-taught in the last two years of high school. I wanted to know from you, what will be the best approach for those who are also trying to teach themselves a subject? Especially after you mentioned that textbooks are killing the educational system. So what other resources do you recommend for those who are trying to learn by themselves? And do you think that reading textbooks alone will be useless?

Thanks!

Faris, I have nothing against textbooks per se. My gripe is with the big, mass-produced textbook industry, which puts market share and profits ahead of good learning. So my advice would be to look around for well written textbooks that are appropriate for your background and suit your learning style. In my final two years at high school, I used first-year university textbooks, which worked for me. (By and large, university textbooks are way better than K-12 textbooks.) Good luck.

You’ve given me quite a bit to think about. I’m enjoying your class and consider myself so fortunate to be one of your students. I am a 58 year old female and I graduated high school after eleventh grade at the age of 16. I loved math and wanted to teach it. I entered Russell Sage College going on 17 and with a math background through what NY called Intermediate Algebra and Trigonometry. Looking back I realize that I was ill prepared to be in math-major calculus classes with math majors that had a better foundation. Perhaps they were further along the continuum of learning to think mathematically. Turns out I did fine. But, at the end of my freshman year they offered some math/combo degrees and I switched to math economics. I had to take various math classes and economics classes and a few others like statistics and BASIC programming, AND I took my secondary ed classes as electives. So, I became certified to teach math grades 7-12. I also had the experience of having to take Diff Equations in a one-month january term (soph year) due to the school reworking the calculus series. Point is, by switching to math/economics, I avoided having to take Analysis and Physics. Perhaps if I had learned to think like a mathematician back then, I would not have been quite so frightened to try those classes. And, I know a lot of my fears were performance anxiety.

In the end, I ended up going to graduate school for economics and worked at Texaco doing computer liaison work. After leaving industry to become a mom, I have taught high school math as a substitute, and I tutor.

Math economics did not mean we took math-economics classes. When I got to graduate school to study economics, the calculus was intimidating, even to me with a math background. When I wrote to my former economics professor and told him they should make calculus a requirement for an economics major (not math/economics) because of all the calculus in economics, he wrote back saying very few of their students pursued graduate work. That was not the answer I wanted to hear.

Perhaps with MOOCs I can attempt some of these higher level classes that I managed to avoid.

This is my third MOOC class and I must say that I appreciate your presence in the forums. It is clear to me that you are committed to making yourself available to us.

I learn a lot from browsing the forums. not about the underlying content, but about how different people react to it and interpret it, and new ways to make it accessible. The latest exchanges about the tennis match problems was something of an eye-opener to me. I’ve been looking at the world through mathematical eyes for so long, I simply did not see many of the readings others had. It isn’t a question of a right or wrong interpretation, or even a better or worse one, they were simply different readings, each one valid in its own way. So on this occasion, the class has helped me refine what turned out to be a couple of excellent questions – but when I formulated them, I had no idea they were going to turn out to be so effective.

Another great benefit of giving MOOCs, is we can use the feedback from collective intelligence to improve that we do. Truly, the “wisdom of the crowd” at work.

Thanks for commenting.

Hi Professor Devlin!

I’m currently a student at Georgetown University in D.C. and also a student of yours in Coursera.

My High School math experience was abysmal, and when I took Calculus during my Freshman at GU, I lacked the necessary skills to succeed in the course. I’ve only begun your Intro class, but I’m excited with what I ve seen so far.

I think there’s defnitely has got to be some change in the education system. I wish I knew what that was because I’d save my nephews and niece a whole heap of trouble.

Dr. Devlin, that last point about what you learn in the forums is interesting. It’s common for teacher — myself included — to point out how much they learn from their students. Do you think that’s more true or true in a different way in a MOOC space?

I saw another instructor in a stats course make the point that the heterogeneity of the massive number of students turned out to be a big advantage, because the least prepared students, far from holding everyone back, would ask question that provoked discussion that cut out a lot of presumption built in to the lessons and forced people to get at the heart of the concepts.

Hi Keith,

First of all, thanks for putting so much hard work into this course. Being a maths teacher (with a maths PhD) I signed up for your course not with the intention of completing any coursework, but in order to check out the material before recommending next year’s course to some of my very talented high school students.

One of the reasons I find the MOOC developments so exciting is that I used to live (and teach) in East Africa, where there are so many talented students who never had the opportunity of good schooling or good university courses. To have courses like yours available to everyone, is just… amazing.

While I’m commenting, I just want to thank you for something else as well – when I started my undergraduate studies in engineering, I found your book (The Joy of Sets) in the library, and it was one of those things that made me love mathematics, and eventually switch to pure maths.