MOQR, Anyone? Learning by Evaluating

Many colleges and universities have a mathematics or quantitative reasoning requirement that ensures that no student graduates without completing at least one sufficiently mathematical course.

Recognizing that taking a regular first-year mathematics course—designed for students majoring in mathematics, science, or engineering—to satisfy a QR requirement is not educationally optimal (and sometimes a distraction for the instructor and the TAs who have to deal with students who are neither motivated nor well prepared for the full rigors and pace of a mathematics course), many institutions offer special QR courses.

I’ve always enjoyed giving such courses, since they offer the freedom to cover a wide swathe of mathematics—often new or topical parts of mathematics. Admittedly they do so at a much more shallow depth than in other courses, but a depth that was always a challenge for most students who signed up.

Having been one of the pioneers of so-called “transition courses” for incoming mathematics majors back in the 1970s, and having given such courses many times in the intervening years, I never doubted that a lot of the material was well suited to the student in search of meeting a QR requirement. The problem with classifying a transition course as a QR option is that the goal of preparing an incoming student for the rigors of college algebra and real analysis is at odds with the intent of a QR requirement. So I never did that.

Enter MOOCs. A lot of the stuff that is written about these relatively new entrants to the higher education landscape is unsubstantiated hype and breathless (if not fearful) speculation. The plain fact is that right now no one really knows what MOOCs will end up looking like, what part or parts of the population they will eventually serve, or exactly how and where they will fit in with the rest of higher education. Like most others I know who are experimenting with this new medium, I am treating it very much as just that: an experiment.

The first version of my MOOC Introduction to Mathematical Thinking, offered in the fall of 2012, was essentially the first three-quarters of my regular transition course, modified to make initial entry much easier, delivered as a MOOC. Since then, as I have experimented with different aspects of online education, I have been slowly modifying it to function as a QR-course, since improved quantitative reasoning is surely a natural (and laudable) goal for online courses with global reach—that “free education for the world” goal is still the main MOOC-motivator for me.

I am certainly not viewing my MOOC as an online course to satisfy a college QR requirement. That may happen, but, as I noted above, no one has any real idea what role(s) MOOCs will end up fulfilling. Remember, in just twelve months, the Stanford MOOC startup Udacity, which initiated all the media hype, went from “teach the entire world for free” to “offer corporate training for a fee.” (For my (upbeat) commentary on this rapid progression, see my article in the Huffington Post.)

Rather, I am taking advantage of the fact that free, no-credential MOOCs currently provide a superb vehicle to experiment with ideas both for classroom teaching and for online education. Those of us at the teaching end not only learn what the medium can offer, we also discover ways to improve our classroom teaching; while those who register as students get a totally free learning opportunity. (Roughly three-quarters of them already have a college degree, but MOOC enrollees also include thousands of first-time higher education students from parts of the world that offer limited or no higher education opportunities.)

The biggest challenge facing anyone who wants to offer a MOOC in higher mathematics is how to handle the fact that many of the students will never receive expert feedback on their work. This is particularly acute when it comes to learning how to prove things. That’s already a difficult challenge in a regular class, as made clear in this great blog post by “mathbabe” Cathy O’Neil. In a MOOC, my current view is it would be unethical to try. The last thing the world needs are (more) people who think they know what a proof is, but have never put that knowledge to the test.

But when you think about it, the idea behind QR is not that people become mathematicians who can prove things, rather that they have a base level of quantitative literacy that is necessary to live a fulfilled, rewarding life and be a productive member of society. Being able to prove something mathematically is a specialist skill. The important general ability in today’s world is to have a good understanding of the nature of the various kinds of arguments, the special nature of mathematical argument and its role among them, and an ability to judge the soundness and limitations of any particular argument.

In the case of mathematical argument, acquiring that “consumer’s understanding” surely involves having some experience in trying to construct very simple mathematical arguments, but far more what is required is being able to evaluate mathematical arguments.

And that can be handled in a MOOC. Just present students with various mathematical arguments, some correct, others not, and machine-check if, and how well, they can determine their validity.

Well, that leading modifier “just” in that last sentence was perhaps too cavalier. There clearly is more to it than that. As always, the devil is in the details. But once you make the shift from viewing the course (or the proofs part of the course) as being about constructing proofs to being about understanding and evaluating proofs, then what previously seemed hopeless suddenly becomes rife with possibilities.

I started to make this shift with the last session of my MOOC this fall, and though there were significant teething troubles, I saw enough to be encouraged to try it again—with modifications—to an even greater extent next year.

Of course, many QR courses focus on appreciation of mathematics, spiced up with enough “doing math” content to make the course defensibly eligible for QR fulfillment. What I think is far less common—and certainly new to me—is using the evaluation of proofs as a major learning vehicle.

What makes this possible is that the Coursera platform on which my MOOC runs has developed a peer review module to support peer grading of student papers and exams.

The first times I offered my MOOC, I used peer evaluation to grade a Final Exam. Though the process worked tolerably well for grading student mathematics exams—a lot better than I initially feared—to my eyes it still fell well short of providing the meaningful grade and expert feedback a professional mathematician would give. On the other hand, the benefit to the students that came from seeing, and trying to evaluate, the proof attempts of other students, and to provide feedback, was significant—both in terms of their gaining much deeper insight into the concepts and issues involved, and in bolstering their confidence.

When the course runs again in a few week's time, the Final Exam will be gone, replaced by a new course culmination activity I am calling Test Flight.

How will it go? I have no idea. That’s what makes it so interesting. Based on my previous experiments, I think the main challenges will be largely those of implementation. In particular, years of educational high-stakes testing robs many students of the one ingredient essential to real learning: being willing to take risks and to fail. As young children we have it. Schools typically drive it out of us. Those of us lucky enough to end up at graduate school reacquire it—we have to.

I believe MOOCs, which offer community interaction through the semi-anonymity of the Internet, offer real potential to provide others with a similar opportunity to re-learn the power of failure. Test Flight will show if this belief is sufficiently grounded, or a hopelessly idealistic dream! (Test flights do sometimes crash and burn.)

The more people learn to view failure as an essential constituent of good learning, the better life will become for all. As a world society, we need to relearn that innate childhood willingness to try and to fail. A society that does not celebrate the many individual and local failures that are an inevitable consequence of trying something new, is one destined to fail globally in the long term.

For those interested, I’ll be describing Test Flight, and reporting on my progress (including the inevitable failures), in my blog MOOCtalk.org as the experiment continues. (The next session starts on February 3.)

The Educational Power of Elementary Arithmetic

The trouble with writing about, or quoting, Liping Ma, is that everyone interprets her words through their own frame, influenced by their own experiences and beliefs.

“Well, yes, but isn’t that true for anyone reading anything?” you may ask. True enough. But in Ma’s case, readers often arrive at diametrically opposed readings. Both sides in the US Math Wars quote from her in support of their positions.

That happened with the book that brought her to most people’s attention, Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States, first published in 1999. And I fear the same will occur with her recent article "A Critique of the Structure of U.S. Elementary School Mathematics," published in the November issue of the American Mathematical Society Notices.

Still, if I stopped and worried about readers completely misreading or misinterpreting things I write, Devlin’s Angle would likely appear maybe once or twice a year at most. So you can be sure I am about to press ahead and refer to her recent article regardless.

My reason for doing so is that I am largely in agreement with what I believe she is saying. Her thesis (i.e., what I understand her thesis to be) is what lay behind the design of my MOOC and my recently released video game. (More on both later.)

Broadly speaking, I think most of the furor about K-12 mathematics curricula that seems to bedevil every western country except Finland is totally misplaced. It is misplaced for the simple, radical (except in Finland) reason that curriculum doesn’t really matter.  What matter are teachers. (That last sentence is, by the way, the much sought after “Finnish secret” to good education.) To put it simply:

BAD CURRICULUM + GOOD OR WELL-TRAINED TEACHERS = GOOD EDUCATION

GOOD CURRICULUM + POOR OR POORLY-TRAINED TEACHERS = POOR EDUCATION

I am very familiar with the Finnish education system. The Stanford H-STAR institute I co-founded and direct has been collaborating with Finnish education researchers for over a decade, we host education scholars from Finland regularly, I travel to Finland several times a year to work with colleagues there, I am on the Advisory Board of CICERO Learning, one of their leading educational research organizations, I’ve spoken with members of the Finnish government whose focus is education, and I’ve sat in on classes in Finnish schools. So I know from firsthand experience in the western country that has got it right that teachers are everything and curriculum is at most (if you let it be) a distracting side-issue.

The only people for whom curriculum really matters are politicians and the politically motivated (who can make political capital out of curriculum) and publishers (who make a lot of financial capital out of it).

But I digress: Finland merely serves to provide an existence proof that providing good mathematics education in a free, open, western society is possible and has nothing to do with curriculum. Let’s get back to Liping Ma’s recent Notices article. For she provides a recipe for how to do it right in the curriculum-obsessed, teacher-denigrating US.

Behind Ma’s suggestion, as well as behind my MOOC and my video game (both of which I have invested a lot of effort and resources into) is the simple (but so often overlooked) observation that, at its heart, mathematics is not a body of facts or procedures but a way of thinking. Once a person has learned to think that way, it becomes possible to learn and use pretty well any mathematics you need or want to know about, when you need or want it.

In principle, many areas of mathematics can be used to master that way of thinking, but some areas are better suited to the task, since their learning curve is much more forgiving to the human brain.

For my MOOC, which is aimed at beginning mathematics students at college or university, or high school students about to become such, I take formalizing the use of language and the basic rules of logical reasoning (in everyday life) as the subject matter, but the focus is as described in the last two words of the course’s title: Introduction to Mathematical Thinking.

Apart from the final two weeks of the course, where we look at elementary number theory and beginning real analysis, there is really no mathematics in my course in the usual sense of the word. We use everyday reasoning and communication as the vehicle to develop mathematical thinking.

[SAMPLE PROBLEM: Take the famous (alleged) Abraham Lincoln quote, “You can fool all of the people some of the time and some of the people all of the time, but you cannot fool all the people all the time.” What is the simplest and clearest positive expression you can find that states the negation of that statement? Of course, you first have to decide what “clearest”, “simplest”, and “positive” mean.]

Ma’s focus in her article is beginning school mathematics. She contrasts the approach used in China until 2001 with that of the USA. The former concentrated on “school arithmetic” whereas, since the 1960s, the US has adopted various instantiations of a “strands” approach. (As Ma points out, since 2001, China has been moving towards a strands approach. By my read of her words, she thinks that is not a wise move.)

As instantiated in the NCTM’s 2001 Standards document, elementary school mathematics should cover ten separate strands: number and operations, problem solving, algebra, reasoning and proof, geometry, communication, measurement, connections, data analysis and probability, and representation.

In principle, I find it hard to argue against any of these—provided they are viewed as different facets of a single whole.

The trouble is, as soon as you provide a list, it is almost inevitable that the first system administrator whose desk it lands on will turn it into a tick-the-boxes spreadsheet, and in turn the textbook publishers will then produce massive (hence expensive) textbooks with (at least) ten chapters, one for each column of the spreadsheet. The result is the justifiably maligned “Mile wide, inch deep” US elementary school curriculum.

It’s not that the idea is wrong in principle. The problem lies in the implementation. It’s a long path from a highly knowledgeable group of educators drawing up a curriculum to what finds its way into the classroom—often to be implemented by teachers woefully unprepared (through no fault of their own) for the task, answerable to administrators who serve political leaders, and forced to use textbooks that reinforce the separation into strands rather than present them as variations on a single whole.

Ma’s suggestion is to go back to using arithmetic as the primary focus, as was the case in Western Europe and the United States in the years of yore and China until the turn of the Millennium, and use that to develop all of the mathematical thinking skills the child will require, both for later study and for life in the twenty-first century. I think she has a point. A good point.

She is certainly not talking about drill-based mastery of the classical Hindu-Arabic algorithms for adding, subtracting, multiplying, and dividing, nor is she suggesting that the goal should be for small human beings to spend hours forcing their analogically powerful, pattern-recognizing brains to become poor imitations of a ten-dollar calculator. What was important about arithmetic in past eras is not necessarily relevant today. Arithmetic can be used to trade chickens or build spacecraft.

No, if you read what she says, and you absolutely should, she is talking about the rich, powerful structure of the two basic number systems, the whole numbers and the rational numbers.

Will that study of elementary arithmetic involve lots of practice for the students? Of course it will. A child’s life is full of practice. We are adaptive creatures, not cognitive sponges. But the goal—the motivation for and purpose of that practice—is developing arithmetic thinking, and moreover doing so in a manner that provides the foundation for, and the beginning of, the more general mathematical thinking so important in today’s world, and hence so empowering for today’s citizens.

The whole numbers and the rational numbers are perfectly adequate for achieving that goal. You will find pretty well every core feature of mathematics in those two systems. Moreover, they provide an entry point that everyone is familiar with, teacher, administrator, and beginning elementary school student alike.

In particular, a well trained teacher can build the necessary thinking skills and the mathematical sophistication —and cover whatever strands are in current favor—without having to bring in any other mathematical structure.

When you adopt the strands approach (pick your favorite flavor), it’s very easy to skip over school arithmetic as a low-level skill set to be “covered” as quickly as possible in order to move on to the “real stuff” of mathematics. But Ma is absolutely right in arguing that this is to overlook the rich potential still offered today by what are arguably (I would so argue) the most important mathematical structures ever developed: the whole and the rational numbers and their associated elementary arithmetics.

For what is often not realized is that there is absolutely nothing elementary about elementary arithmetic.

Incidentally, for my video game, Wuzzit Trouble, I took whole number arithmetic and built a game around it. If you play it through, finding optimal solutions to all 75 puzzles, you will find that you have to make use of increasingly sophisticated arithmetical reasoning. (Integer partitions, Diophantine equations, algorithmic thinking, and optimization.)

I doubt Ma had video game instantiations of her proposal in mind, but when I first read her article, almost exactly when my game was released in the App Store (the Android version came a few weeks later) that’s exactly what I saw.

Other games my colleagues and I have designed but not yet built are based on different parts of mathematics. We started with one built around elementary arithmetic because arithmetic provides all the richness you need to develop mathematical thinking, and we wanted our first game to demonstrate the potential of game-based learning in thinking-focused mathematical education (as opposed to the more common basic-skills focus of most mathematics-educational games). In starting with an arithmetic-based game, we were (at the time unknowingly) endorsing the very point Ma was to make in her article.

Math Ed? Sometimes It Takes a Team

In last month’s column, I reflected on how modern technology enables one person—in my case an academic—to launch enterprises with (potential) global reach without (i) money and (ii) giving up his day job. That is true, but technology does not replace expertise and its feeder, experience.

In the case of my MOOC, now well into its third offering, I’ve been teaching transition courses on mathematical thinking since the late 1970s, and am able to draw on a lot of experience as to the difficulties most students have with what for most of them is a completely new side to mathematics.

Right now, as we get into elementary, discrete number theory, the class (the 9,000 of 53,000 registrants still active) is struggling to distinguish between division—a binary operation on rationals that yields a rational number for a given pair of integers or rationals—and divisibility—a relation between pairs of integers that is either true or false for any given pair of integers. Unused to distinguishing between different number systems, they suddenly find themselves lost with what they felt they knew well, namely elementary arithmetic.

Anyone who has taught a transition course will be familiar with this problematic rite of passage. I suspect I am not alone in having vivid memories of when I myself went through it, even though it was many decades ago!

As a result of all those years teaching this kind of material, I pretty well know what to expect in terms of student difficulties and responses, so can focus my attention on figuring out how to make it work in a MOOC. I know how to filter and interpret the comments on the discussion forum, having watched up close many generations of students go through it. As a result, doing it in a MOOC format with a class spread across the globe is a fascinating experiment, when it could so easily have been a disaster.

My one fear is that, because the course pedagogy is based on Inquiry-Based Learning, it may be more successful with experienced professionals (of whom I have many in the class), rather than the course’s original target audience of recent high school graduates. In particular, I suspect it is the latter who constantly request that I show them how to solve a problem before expecting them to do so. If all students have been exposed to is instructional teaching, and they have never experienced having to solve a novel problem—to figure it out for themselves—it is probably unrealistic to expect them to make that leap in a Web-based course. But maybe it can be made to work. Time will tell.

The other startup I wrote about was my video game company. That is a very different experience, since almost everything about this is new to me. Sure, I’ve been studying and writing about video game learning for many years, and have been playing video games for the same length of time. But designing and producing a video game, and founding a company to do it, are all new. Although we describe InnerTube Games as “Dr. Keith Devlin’s video game company,” and most of the reviews of our first release referred to Wuzzit Trouble as “Keith Devlin’s mathematics video game,” that was like referring to The Rolling Stones as “Mick Jagger’s rock group.” Sure he was out in front, but it was the entire band that gave us all those great performances.

In reality, I brought just three new things to our video game design. The first is our strong focus on mathematical thinking (the topic of my MOOC) rather than the mastery of symbolic skills (which is what 99% of current math ed video games provide). The second is that the game should embed at least one piece of deep, conceptual mathematics. (Not because I wanted the players to learn that particular piece of mathematics. Rather that its presence would ensure a genuine mathematical experience.) The third is the design principle that the video game should be thought of as an instrument on which you “play math,” analogous to the piano, an instrument on which you play music.

In fact, I was not alone among the company co-founders in favoring the mathematical thinking approach. One of us, Pamela, is a former middle-school mathematics teacher and an award winning producer of educational television shows, and she too was not interested in producing the 500^th animated-flash-card, skills-mastery app. (Nothing wrong with that approach, by the way. It’s just that the skills-mastery sector is already well served, and we wanted to go instead for something that is woefully under-served.) I may know a fair amount about mathematics and education, and I use technology, but that does not mean I'm an expert in the use of various media in education. But Pamela is.

And this is what this month’s column is really about: the need for an experienced and talented team to undertake anything as challenging as designing and creating a good educational learning app. Though I use my own case as an example, the message I want to get across is that if, like me, you think it is worthwhile adding learning apps and video games to the arsenal of media that can be used to provide good mathematics learning, then you need to realize that one smart person with a good idea is not going to be anything like enough. We need to work in teams with people who bring different expertise.

I’ve written extensively in my blog profkeithdevlin.org about the problems that must be overcome to build good learning apps. In fact, because of the history behind my company, we set our bar even higher. We decided to create video games that had all the features of good commercial games developed for entertainment. Games like Angry Birds or Cut the Rope, to name two of my favorites. Okay, we knew that, with a mathematics-based game, we are unlikely to achieve the dizzying download figures of those industry-leading titles. But they provided excellent exemplars in game structure, game mechanics, graphics, sounds, game characters, etc. In the end, it all comes down to engagement, whether the goal is entertainment and making money or providing good learning.

In other words, we saw (and see) ourselves not as an “educational video game company” but as a “video game company.” But one that creates video games  built around important mathematical concepts. (In the case of Wuzzit Trouble, those concepts are integer arithmetic, integer partitions, and Diophantine equations.)

Going after that goal requires many different talents. I’ve already mentioned Pamela, our Chief Learning Officer. I met her, together with my other two co-founders, when I worked with them for several years on an educational video game project at a large commercial studio. That project never led to a released product, but it provided all four of us with the opportunity to learn a great deal about the various crucial components of good video game design that embeds good learning. Enough to realize, first, that we all needed one another, and second that we could work well together. (Don’t underestimate that last condition.)

By working alongside video game legend John Romero, I learned a lot about what it takes to create a game that players will want to play. Not enough to do so myself. But enough to be able to work with a good game developer to inject good mathematics into such a game. That’s Anthony, the guy on our team who takes a mathematical concept and turns it into a compelling game activity. (The guy who can give me three good reasons why my “really cool idea” really won’t work in a game!) Pamela, Anthony, and I work closely together to produce fun game activities that embed solid mathematical learning, each bringing different perspectives. Take any one of us out of the picture, and the resulting game would not come close to getting those great release reviews we did.

And without Randy, there would not even be a game to get reviewed! Video games are, after all, a business. (At some point, we will have to bring in revenue to continue!) The only way to create and distribute quality games is to create a company. And yes, that company has to create and market a product—something that’s notoriously difficult. (Google “why video game companies fail.”) Randy (also a former teacher) was the overall production manager of the project we all worked on together, having already spent many years in the educational technology world. He’s the one who keeps everything moving.

Like it or not, the world around us is changing rapidly, and with so many things pulling on our students’ time, it’s no longer adequate to sit back on our institutional reputations and expect students to come to us and switch off the other things in their lives while they take our courses.

One case: I cannot see MOOCs replacing physical classes with real professors, but they sure are already changing the balance. And you don’t have to spend long in a MOOC to see the similarities with MMOs (massively multiplayer online games).

We math professoriate long ago recognized we needed to acquire the skills to prepare documents using word processing packages and LaTeX, and to prepare Keynote or PowerPoint slides. Now we are having to learn the rudiments of learning management systems (LMSs), video editing, the creation of applets, and the use of online learning platforms.

Creating video games is perhaps more unusual, since it requires so many different kinds of expertise, and I am only doing that because a particular professional history brought me into contact with the gaming industry. But plenty of mathematical types have created engaging math learning apps, and some of them are really very good.

Technology not only makes all of these developments possible, it makes it imperative that, as a community, we get involved. But in the end, it’s people, not the technology, that make it happen. And to be successful, those people may have to work in collaborative teams. 

Two Startups in One Week

Last week turned out to be far more hectic than most, with the simultaneous launch of two startups I have been involved in for the past few years.

When I went into the life of academic mathematics some 42 years ago, I could never have imagined ever writing such a sentence. Nor, for that matter, would I have had the faintest idea what a “startup” was. It’s a measure of how much society has changed since 1971, when I transitioned from being a “graduate student” to a “postdoc,” that today everyone knows what a startup is, and many of my doctorate-bearing academic colleagues have, as a sideline to their academic work, started up labs, centers, or companies. What was once exceptional is now commonplace.

Massive changes in technology have made it, while not exactly easy, at least possible for anyone in academia to become an “edupreneur,” to use (just once, I promise) one of the more egregious recent manufactured words. This means that, when our academic work leads to a good idea or a product we think could be useful to many of our fellow humans, we don’t have to sit back and hope that one day someone will come along and turn it into something people can access or use. We can make it available to them ourselves.

MOOCs are one of the most recent examples. If any of us in the teaching business finds we have developed a course that students seemed to have benefited from and we are proud of, we can (at least to some extent) bottle it and make it available to a much wider audience.

Of course, we have had versions of that ability since the invention of the printing press. Today, millions of people, academics and non-academics alike, use those printing press descendants, websites and blogs, to achieve a much wider audience for their written word.

A somewhat smaller (but growing) number have used platforms such as YouTube and Vimeo to make video-recordings of their lectures widely available.

To some extent, MOOCs can be viewed as an extension of both of those Internet media developments. A MOOC sets out to achieve the very ambitious goal of bottling an entire college course and making it available to the entire world—or at least, that part of the world with broadband access.

The launch this past weekend of the third iteration of my constantly-evolving MOOC on Mathematical Thinking was one of the two startups that gobbled up massive amounts of my time over the past few weeks. Even though, having given essentially the same course twice before, the bulk of the preparatory work was done, implementing the changes I wanted to make and re-setting all the item release dates/times and the various student submission deadlines was still a huge undertaking. For with a MOOC, pretty well everything for the entire course needs to be safely deposited on (in my case, with my MOOC on Coursera) Amazon’s servers before the first of my 41,000 registered students logged on over the weekend.

When you think about it, the very fact that a single academic can do something like this, is pretty remarkable. What makes it possible is that all the components are readily available. To go into the MOOC business, all you need is a laptop, a word processor (and LaTeX, if you are giving a math course), possibly a slide package such as PowerPoint, some kind of video recording device (I use a standard, $900 consumer camcorder, others use a digital writing tablet), a small microphone (possibly the one already built in to your laptop), and a cheap consumer video editing package (I use Premiere Elements, which comes in at around $90). Assuming you already have the laptop and a standard office software package, you can set up in the MOOC business for about $1500.

Sure, it helps if your college or university gives you access to the open source MOOC platform edX, or is willing to enter an agreement with, say, the MOOC platform provider Coursera. But if not, there are options such as YouTube, websites, Wikis, and blogs, all freely available.

My second startup was supposed to launch at least a month before my MOOC, but a major hacking event at Apple’s Developer Site delayed their release of the first (free) mathematical thinking mobile game designed by my small educational software company, InnerTube Games. Both launches falling in the same week is not something I’d want to do again!

Why form a company to create and distribute mathematics education video games that incorporate some of the findings and insights I’d developed over several years of research? The brutal answer is, I had no other viable option. Though several years of research had convinced me that it was possible to design and build “instruments” on which you can “play” parts of mathematics, in the same way a musical instrument such as a piano can be used to play music (in both cases by passing the need for static symbolic representations on a page, which are known to be a huge barrier to learning for many people), I simply was not successful in convincing funders it was a viable approach.

Clearly then, I had to build at least one such instrument. More precisely, I had to team up with a small number of friends who brought the necessary expertise I did not have. Again, a few years ago, it would have been impossible for an academic to found and build a small company and create and launch a product in my spare time. But today, anyone can.

Sure, even more so than with MOOCs, to form and operate an educational software company, you need to work with other people—three in my case. (That, at least, has been my experience.) But the key point is, the technology and the resources infrastructure make it possible. You don’t have to give up your day job as an academic to do it! And just as a MOOC provider (or a YouTube, website, blogging platform combo) takes care of the distribution of your course, so too the Web (in my case, in the form of Apple’s App Store) can make your creation available to the world. At no cost.

We are not talking about enterprises designed with the purpose of making money here—I am essentially in the same game as the writing of academic works or textbooks, and in my case less so, since my books cost money but my MOOC and my game are free. Rather we are making use of a global infrastructure to make our work widely accessible. If that infrastructure involves for-profit MOOC platforms or software companies, so be it.

The fact is, it has never been as easy as it is today for each one of us to take an idea or something we have created and make it available to a wide audience. Sure, for both my examples, I have left a lot unsaid, focusing on one particular aspect. (Take a glance at my video game website to see who else was involved in that particular enterprise and the experience they brought to the project. That was a team effort if ever there was!) But the key fact is, it is now possible!

For more about my MOOC, and MOOCs in general, see my blog MOOCtalk.org. For my findings and thoughts on mathematics education, see many of the posts on my other blog profkeithdevlin.org together with some of the articles and videos linked to on the InnerTube Games website.

And for another (dramatic) example of how one person with a good idea can quickly reach a global audience, see Derek Muller’s superb STEM education resource Veritasium. 

“Will this (mathematics) be of any use?”

Readers who follow me on Twitter will have noticed many tweets on the recent revelations about illegal NSA surveillance. Here is why I think that none of us in mathematics and mathematics education can ignore that debate.

There’s a popular conception that mathematicians are unworldly, and that mathematics is, at its heart, walled off from the real world, its pursuit a form of escapism that takes the pursuer into a realm of pure, abstract thoughts.

Certainly, that’s a general sense of mathematics that I held for many years. Yes, like all my fellow mathematicians, I always knew that mathematics – all of it – arose, directly or indirectly, from real world problems, and that any branch of mathematics having any discipline-internal significance almost always turns out to have real-world applications. But neither of those was why I did mathematics. For most of my life as a mathematician, I simply did not care about the history or application of what I was doing. It was all about the chase – the search for new knowledge in a beautiful domain.

Early on in my career, when more politically active colleagues urged me to boycott conferences and workshops funded by NATO (a big issue back in the 1970s), or to avoid applying for research funds from commercial or military sources, I essentially turned a deaf ear to what they were saying, and got on with the work that interested me.

As a mathematician working in axiomatic set theory, with particular foci on the properties of sets of large infinite cardinality and on undecidability proofs, I felt fairly confident that nothing I did would ever find practical application, so for me the issue was purely one of where the money came from to support my research. I felt “clean,” and not under any moral pressure regarding potential unethical uses being made of my work.

True, I was aware that the famous early twentieth century mathematician G. H. Hardy had made the same claim about his work in number theory, yet in the mid-1970s his work found highly significant application in the design of secure cryptographic systems. But I felt that a similar outcome was unlikely in the case of infinitary set theory. (I am no longer quite as sure about that; I speculated about possible applications of Cantor’s set theory in my June column.)

I think we all have to address the morality-of-possible-applications question about our work as mathematicians at one time or another. Some, from Archimedes to Alan Turing, have actively engaged in military research; others try to avoid any direct contact with commercial or warfare-related activities.

The rise of math-based corporations such as Google that form a large and influential part of today’s global world, and the closely related growth of the modern, math-driven security state, as iconicized by the NSA, make it impossible to maintain any longer the fiction (for such it always was) that we can pursue mathematics as a pure activity, separate from applications, be they good or ill.

The uncomfortable fact is, we are in no different a situation than manufacturers of sporting guns who deny any agency when their product is used to kill people. (Yes, people pull the trigger, but as comedian Eddie Izzard pointed out, “the gun helps.”)

If we want to be able to maintain that our work will not play a role in someone’s death, torture, or incarceration – or in someone else achieving enormous wealth and power – our only option is to not go into mathematics in the first place. The subject is simply way too powerful as a force – for good or for evil.

Shortly after September 11, 2001, I was asked to join a research project funded by the U.S. intelligence service. For me, that was my crunch time. The work that led to that invitation was an outgrowth (described in my 1995 book Logic and Information) of my earlier research in mathematical logic and set theory. Like it or not, I was already in deep. To say no to that invitation would have been every bit a positive action as to say yes. Sitting on the fence was not a possibility. I was a mathematician. I’d already made the gun.

As the Google founders Larry Page and Sergei Brin eventually discovered, “Do no evil” is a wonderful grounding principle, but the power of mathematics renders it an impossible goal to achieve. The best we can do is try to make our voice heard, as many mathematicians and nuclear physicists did during the Cold War, who spoke publicly about the massive scale of the danger raised by nuclear weapons.

Finding out (as I have over the past few weeks) that the work I’d done over the past twelve years – for various branches of the U.S. government (intelligence and military) and for commercial enterprises (in my case, the video game industry) – was part of a body of research that had been subverted (as I see it) to create a massive global surveillance framework, I felt I could not remain silent.

Not because I felt that I, as an individual, did anything of significance. I worked on non-classified projects, and made no major breakthroughs. I was a very tiny cog in a very big machine. (If “they” are keeping an eye on me, they are definitely wasting our tax dollars!)

But I did take the money and I did do the work. I don’t regret doing so. The fact is, I’d made the crucial choice long before 2001; back in my youth when I decided to become a mathematician.

Those of us in mathematics education have always told our students that math is useful. In today’s world more than ever, we cannot at the same time pretend it is free of moral issues. Agnosticism is not an option (if it ever really was). To say or do nothing is inescapably a positive act, just as significant as saying or doing something.

We humans have created our mathematics, and used it to help shape our world. Now we have to live in it. Not only are we the ones who bear a large responsibility for that world, we are also, by our very expertise, the ones who (in many fundamental ways) understand it best. (It often seems that only the mathematically sophisticated really appreciate that an American is more likely to die in his or her bathtub than from a terrorist attack, and that more people died on the roads due to increased traffic during the time after 9/11 when all flights were grounded than did in the Twin Towers attack.)

So, to return to the question implicit in my title, “What is mathematics used for?” Douglas Adams provided the answer: “Life, the universe, everything.” With such reach and power comes responsibility. 

FOOTNOTE: For a more personal take on the above issues, see the interview I did on June 21 on Shecky Riemann’s Math Tango blog.

“It Only Takes About 42 Minutes To Learn Algebra With Video Games”

When tech folk dabble in education (and tech writers cover it), the excess of hype is sometimes matched only by their breathtaking lack of knowledge about education. Even so, the above headline to the July 1 post by Forbes contributor Jordan Shapiro must rank as one of the most stupid and ignorant statements in human history.

It would be somewhat less ludicrous, though still open to debate, if the headline had said “learn some algebra.” But “algebra”? All of it?

Almost certainly, Shapiro himself did not write the headline—writers rarely do. In fact, the article itself is fine. I have no problem with what Shapiro wrote. But the fact that the ludicrous headline had not been changed 24 hours later indicates that Forbes’ editors feel happy with it. Sigh.

What the article itself reports is that, on average, students who played a particular video game (DragonBox, of which more later) completed a sufficient part of it in 42 minutes. Since the game itself is based on algebraic principles, they could, therefore, be said to have engaged in algebraic thinking. (I would be inclined to say just that, though with any kind of machine learning—and human teaching if the instructor is not paying close attention—one should always be on the lookout for an instance of Benny’s Rules.)

Whether such performance in a video game justifies saying that the students learned some (!) algebra in 42 minutes depends on what metric you use to determine what learning has taken place.

Of course, if you define algebra to be (or to include) symbolic manipulation, then successful completion of any video game is not going to count as “doing algebra.” That is why I used the term “algebraic thinking” a couple of paragraphs back. (See my previous blog post What is Algebra? for a discussion of the distinction.) But is that the appropriate measure? What do we want K-12 students to learn under the title “algebra”?

[ASIDE: There is another definitional question as to the classification of DragonBox as a video game. Game developers have different views as to what constitutes a video game. Some would describe DagonBox as an entertaining, interactive, digital app, but would stop short at classifying it as a game.]

Before I go any further, I should give some disclaimers. First, as readers of my blog profkeithdevlin.org (or my book Mathematics Education for a New Era) will be aware, I am a strong proponent of the use of video games in mathematics education. In fact, I advocate an approach to the design of math ed video games that definitely includes DragonBox. I’ve met the developer, Jean-Baptiste Huynh, and one of the co-founders of his company WeWantToKnow, and I used their game as an example in a feature article on math ed video games I wrote for American Scientist in March of this year. I am about three-quarters of the way through the second, greatly expanded version of the game, DragonBox2. Among the designs for math ed video games that my own company, InnerTube Games, has been working on for several years, are a couple that have much in common with DragonBox. (We are due to release our first one, Wuzzit Trouble, this summer, but chose one based on arithmetic and number theory to be our initial release, with algebra-based games to come later.) So I am not a dispassionate outsider here.

For his Forbes article, Shapiro interviewed Jean-Baptiste Huynh, and everything the DragonBox designer says, I agree with 100%. Here is my take on the benefits of playing DragonBox (besides the fact that is it fun).

A student who plays through the new, greatly expanded version of the game will undoubtedly engage in a substantial amount of (contextualized) algebraic thinking focused on the solution of linear equations in one variable. The score they obtain in the game will provide a good measure of how well they have mastered that form of thinking (i.e., solving single-variable linear equations).

Does that mean the student can then sit down and ace a standard written algebra exam? Not at all. Even though the later stages of DragonBox and DragonBox2 involve on-screen manipulations of the standard symbolic representations of equations, the step from physically moving digital objects to manipulating symbolic expressions on a page is a much harder cognitive challenge than one might first think. The human mind simply finds it very difficult to reason in a purely abstract fashion. (In my book The Math Gene, published in 2000, I investigated the reasons for that difficulty.)

At issue is the notorious transfer problem, which, roughly speaking, is the difficulty humans face in taking something that has been learned in one context and applying it in another.

Huynh is of the opinion that it requires a human teacher to help the student take the difficult step from completion of his game to mastery of symbolic algebra, and I agree with him. I suspect that not everyone will be able to make the transition, no matter how good the teaching, but many will.

There is certainly a lot to be gained from mastery of symbolic algebra. First of all, learning at that level of abstraction is readily applicable to any specific domain. Second, being able to reason free of the complexities of any application domain is extremely powerful.

On the other hand algebra (or, more accurately, algebraic thinking) was successfully used in commerce for many hundreds of years before the modern, symbolic variety was introduced in the sixteenth century. So acquiring useful algebra skills is not totally dependent on mastery of symbolic algebra.

A major question is, will playing DragonBox increase the likelihood that a student will be able to master symbolic algebra, compared with a student who does not have that game experience? There is good reason to assume the answer is “Yes,” but that remains to be fully tested—something that can be done only now the game (and others like it) is out. (The analogous question remains to be answered for my own company’s forthcoming games.)

My reason for suspecting that playing video games like DragonBox is highly beneficial in learning symbolic mathematics—the kind that is tested in our school system—is perhaps best explained by an analogy from Hollywood. In the 1984 movie The Karate Kid (I can’t bring myself to watch the 2010 remake) and its sequel (KK2), martial arts instructor Mr Miyagi prepares his young pupil Daniel for Karate tournaments by getting him to polish a car, sand a floor, catch a fly with chopsticks, and paint a fence, all of which develop the reflexes and muscle memory required for key Karate moves, which Daniel uses to great effect later in the movies.

True, this is not sound educational theory, though many teachers (and most athletic coaches) adopt a similar approach. (This is a blog, remember, not a research journal.) But until we have something more concrete, the analogy works for me. Indeed, I am betting my company on it—as is Jean-Baptiste Huynh.

Will Cantor’s Paradise Ever Be of Practical Use?

We really have no way of knowing what early humans thought when they gazed up at the sky. Since everyday practical experience is, by definition, limited to a very small region of space and time, it requires considerable cognitive sophistication to conceive of something – say the night sky – “going on for ever,” let alone to ponder whether that means it is “infinite,” or indeed what “infinite” actually means.

What we do know is that the ancient Greeks made what may have been the first substantial attempt to analyze the notion of infinity, with Zeno of Elea (ca 490-430 BCE) of particular note for his discussion of a number of (seeming) paradoxes that arise from the assumption that space and time are (or are not) infinitely divisible.

Archimedes’ (ca 287-112 BCE) calculations of areas and volumes made implicit use of infinity, and from today’s perspective can be recognized as the forerunner of integral calculus.

Skillful formal – though by modern standards not rigorous – use of the infinitely large and the infinitely small was made by Isaac Newton and Gottfried Leibniz in their development of modern infinitesimal calculus in the seventeenth century, though it was not until the nineteenth century when Bernard Bolzano, Augustin-Louis Cauchy, and Karl Weierstrass finessed the lurking problems of infinity by means of the famous (and for many a first-year mathematics major, infamous) epsilon-delta definitions of limits and continuity.

But none of these developments was about infinity as an entity; the focus rather was on the unending nature of certain processes, starting with counting. It was Georg Cantor (1845 – 1918) who really tackled infinity head on. His proof that the set of real numbers cannot be put into one-one correspondence with the natural numbers, and hence is of a larger order of infinitude, led to a series of papers, published in a remarkable ten-year period between 1874 and 1884, that formed the basis for modern abstract set theory, including the development of a fully formed arithmetical theory of infinite numbers (or “cardinals”).

Reactions to Cantor’s revolutionary new ideas ranged from outraged condemnation to fulsome praise. Henri Poincaré called Cantor’s work a “grave disease” that threatened to infect mathematics, and Leopold Kronecker described Cantor as a “scientific charlatan” and a “corrupter of youth.” Ludwig Wittgenstein, writing long after Cantor's death, complained that mathematics had become “ridden through and through with the pernicious idioms of set theory,” a theory he dismissed as “utter nonsense,” “laughable,” and “wrong.”

At the other end of the spectrum, in 1904, in the UK the Royal Society awarded Cantor its highest award, the Sylvester Medal, and in Germany David Hilbert declared that “No one shall expel us from the Paradise that Cantor has created.”

Having devoted the early part of my professional career to work in (infinitary) set theory, starting with my Ph.D. in “large cardinal theory,” completed in 1971, and moving on to work on alternative universes of sets (a particularly hot topic after Paul Cohen’s introduction of the method of forcing in 1963), in the early 1980s my interests started to shift elsewhere, to questions about information, communication, and human reasoning.

I found myself temporarily back in the world of set theory and the arithmetic of infinite numbers recently, when I was approached by the organizers of the World Science Festival to moderate a panel discussion on the topic of infinity and a more in-depth follow-up the following day.

Both discussions raised the question as to whether study of infinity – in particular the hierarchy of larger infinities that Cantor bequeathed to us – would ever have any practical applications. As panelist Hugh Woodin remarked at one point in the discussion, it is a foolish mathematician who declares that a particular piece of mathematics will not find applications. For instance, G. H. Hardy’s famous statement (in his book A Mathematician’s Apology) that his work in number theory would never find practical application, proved to be spectacularly wrong less than a century later, when number theory became the foundation for internet security.

Hardy’s observation was based on his familiarity of the world he lived in, a world in which the World Wide Web was not even a dream. Today, we cannot know what the world of tomorrow will look like. On the other hand, whatever our children and grandchildren will take for granted, their world will surely be finite, which makes it unlikely that Cantor’s theory – and the almost a century of development in set theory since then – will have practical use.

Or does it? What about calculus? Infinitesimal (!) calculus not only has applications in the modern world, but much of the science, technology, medicine, and even financial structure the underpins our world depends on calculus for its very existence. Applications don’t get more real than that.

True, but the dependence on infinities you find in calculus is essentially asymptotic. What really drives calculus is the unending nature of certain processes on the natural numbers. Talk of “infinitely large” or “infinitely small” is little more than a manner of speaking. Indeed, the epsilon-delta definitions (which do not involve infinities or infinitesimals) are a way to formalize that manner of speaking, effectively eliminating any actual infinite or infinitesimal quantities.

In contrast, much of the work on infinity (more precisely, infinities) carried out in the second half of the twentieth century (when I was working in that area) focused on properties of sets that made their cardinalities super-infinities of different orders: inaccessible cardinals, Ramsey cardinals, measurable cardinals, compact cardinals, supercompact cardinals, Woodin cardinals, and so on. Infinities which dwarfed into invisibility the puny cardinality of the set of natural numbers. Indeed, each one in that sequence dwarfed all its predecessors into invisibility. How could that work find an application?

I’ll lay my cards on the table. I think the chances are that it won’t. But I don’t think it is impossible. Indeed, I began to suspect a possible application in the very domain I worked in after I left set theory.

[This may of course be nothing more than a reflection of having at my disposal a large hammer which made everything look a bit like a nail. But let’s press on.]

The post 9/11 world saw me involved in a series of Defense Department projects the first being improving intelligence analysis (and the others essentially variants of that).

In today’s information rich world, the major nations can be assumed to have access to all the information they need to predict (and hopefully thence prevent) the majority of terrorist attacks. The trouble is, the few data points which must be identified and connected together to determine the likelihood of a future attack are just a tiny few in an overwhelming ocean of data. Even in the era of cloud computing, identifying the key information is analogous to using the naked eye to find a handful of proverbial needles in a non-proverbial field of haystacks.

To all intents and purposes, the available data is infinite. The only hope is to impose some structure on the data that makes it possible for humans and computers to work together on it, narrowing down the focus to the regions more likely to be of significance. Though modern computing systems can sift through massive (finite) amounts of data in a relatively short time, they need to be programmed, and writing those programs (at least, some kinds of them) will require some structure on those large sets of data. A possible place to find the appropriate structure(s) is infinitary set theory. In other words, to develop the appropriate structures, assume the data is infinite. View the infinite as a theoretical simplification of the very large finite. (Economists sometimes make a similar simplifying assumption about economies.)

Do I think this is likely? Frankly, no. But then, neither could Hardy conceive of any practical application of his work in number theory. [Incidentally, like Hardy, I don’t think mathematics needs applications to justify itself. It’s just that the question of application is what this article is about!]

The discussion about large cardinals you will find in those panel discussions at the World Science Festival might seem impossibly abstract and far removed from the everyday world. Indeed, it is. But the questions being discussed all resulted from a process of rigorous, logical investigation that arose directly from late nineteenth century attempts to understand heat flow. History tells us that what begins in the real world, very often ends up being used in the real world.

Prediction is hard, particularly about the future.

Incidentally, how did I end up working on a project for the DoD? They asked me. I might not be the only person to speculate about a possible use of Cantor’s paradise. This is your taxpayer dollars at work.

The Mother of All NCTM Addresses

This month’s column is short, but I am asking you to set aside 51 minutes and 36 seconds to watch the embedded video. It is a recording of the Iris M. Carl Equity Address given on Friday April 19 at this year’s NCTM Annual Conference in Denver, Colorado. The title of the talk is “Keeping Our Eyes on the Prize” and the speaker is Uri Treisman, professor of mathematics and of public affairs, and director of the Charles A. Dana Center, at the University of Texas at Austin.

I was not able to be at NCTM, but on the recommendation of several colleagues, I watched the YouTube video. I simply cannot write a column on mathematics or mathematics education in the same month as Treisman’s immensely more important, profound—and powerfully articulated—words became part of mathematics education history. As a community, we now have our own “I have a dream” speech. Thank you, Uri.

PDFs of Treisman's presentation slides are available here.

Only in Silicon Valley

ADDED MAY 1: NOTE THAT THIS COLUMN WAS POSTED ON APRIL 1, "ALL FOOLS DAY" IN THE USA AND SEVERAL OTHER COUNTRIES.

One of the benefits of being at a university like Stanford is that we occasionally get the opportunity to see up close the emergence of an amazing mathematical talent—someone who may turn out to be the next Euler or Gauss.

Just over 18 months ago, Avril Wan was, to all appearances, just another bright fourteen-year-old living in Taiwan, where her father Yewful Wan runs a large shipping company and her Welsh-born mother Melanie Wan is a university mathematics professor (and a former student of Timothy Gowers in Cambridge).

Then, in September 2011, Stanford computer science professor Sebastian Thrun and Google researcher Peter Norvig offered what turned out to be the first of what is now a flood of Massively Open Online Courses (MOOCs), which make advanced university courses available to the entire world over the Internet. Ms. Wan enrolled for that first MOOC, in artificial intelligence, and was the only student to score a perfect 100% for the course.

When initial investigations made it clear that Ms. Wan’s performance was legitimate, Thrun moved quickly, and arranged for Stanford to offer her a place in Stanford’s famed Symbolic Systems Program (which has produced a whole string of graduates who have founded and led successful Silicon Valley companies, such as Reid Hoffman, who founded LinkedIn, and Marissa Meyer, an early employee of Google and the new—and controversial—CEO of Yahoo!).

By the time Wan arrived at Stanford, Thrun had left to form Udacity, a Silicon Valley start-up offering free online courses to the world, and the newly arrived student, who had just turned 15 (and was accompanied by her mother), was assigned to the educational care of another famous Stanford mathematics professor, Persi Diaconis, known for his ability to see familiar problems in novel ways.

In late spring of 2012, there was a buzz across the Palo Alto campus when it seemed that, under minimal guidance from Diaconis, the young Ms. Wan had solved the notorious P = NP problem, but Ron Graham of the University of California at San Diego quickly found an error, pointing out that she had implicitly assumed the existence of a complete, two-valued measure on the power set of the natural numbers—a question first raised by the famous (Second World) Wartime mathematician Stan Ulam.

Meanwhile, Ms. Wan’s mathematics blog had started to attract attention back in her home country, making her somewhat of a Taiwan celebrity. In particular, motivational videos she had posted on YouTube to encourage young Taiwanese girls to study mathematics, eventually came to the attention of News Corporation’s Rupert Murdoch, who pledged $5M to make her videos available throughout the developing world.

But then, online tech journalist Dan Gillmor posted an article pointing out that Murdoch’s funding was contingent on the distribution being restricted to streaming to tablets supplied by his own, for-profit company Amplify. If so, that would surely have killed the deal, since Ms. Wan recognizes the value of free educational resources to the development of the less affluent countries of the world.

At that point, events started to unfold at the kind of breakneck speed that only happens in Silicon Valley. Ms. Wan, still just 15 years old, remember, and technically without even a high school diploma, found herself inside the Palo Alto offices of the famed venture capital company Greylock Partners, which was willing to commit $100M to fund the establishment of a global, free, online mathematics education platform, tentatively called “Wan World.”

With Greylock having been early stage funders of some of the most successful start-up companies in recent years, most of which required several years before anyone had the faintest idea how they would make money, that interest was all it took to unleash the floodgates. Within a few days, Ms. Wan (or rather, the group of advisers her father quickly assembled to cope with the interest) had been approached by Apple, Google, and Facebook, each of which wanted to develop the platform on which Wan World would run, and by McGraw Hill, Pearson Education, and Amazon, who wanted to own the content.

Meanwhile, despite all this frenzy, Ms. Wan herself seems remarkably unfazed by the sudden changes in her life. Speaking to an unusually full room in a recent meeting of Stanford’s Education’s Digital Future lecture/discussion series (which is where I first met her), she concluded her presentation by admitting to her fellow students, “Like you, right now, I just want to graduate.”  

Can we make constructive use of machine-graded, multiple-choice questions in university mathematics education?

A good metaphor for the current state of MOOC education is provided by this historical video. But when you look at those images, please remember what those events led to. Unless you are able to keep that history in mind, you should not at this stage get into the MOOC business. For there be only dragons.

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

Once I had made the decision to try to take (elements of) my 35-year-old mathematics transition course into the then emerging MOOC format—less than a year ago!—I was immediately brought face-to-face with the necessity of making use of two educational devices I had loathed (and never used) throughout my entire career in higher education:

1. machine-graded pop quizzes
2. machine-graded multiple-choice questions

For MAA readers, I don’t think I need to explain my dislike for either of these über-simplistic devices, which can surely be justified in a regular classroom only in terms of making life easier for the instructor.

Simply putting a class online does not require the use of either device, of course. Technologies such as video conferencing and screen sharing can make learning at a distance almost as good as traditional classroom learning, and in some circumstances can make it better in some respects. But making a class available to tens of thousands of students online changes everything. With such large numbers, the “class” dynamics change dramatically. But it’s not all for the worse.

The first thing to realize is that a MOOC is in many ways like radio or TV. Though both of those familiar features of modern life are referred to as “mass media,” they are in fact highly individual. The newsreader on radio or TV is not addressing a large audience; she or he is talking to millions of single individuals. The secret to being good on the radio or TV is to forget the millions and think of just one (generic) person. After all, the listener or viewer is not in a room with millions of other people; in fact, if the broadcast is successful, that listener or viewer is cognitively in a room with just the presenter. The really successful radio and TV newsreaders and presenters are the ones who can do that really well. They create that sense that they are talking just to You.

In my own case, I already knew that from many years of occasional media work, but I think all MOOC instructors come to that realization very quickly. When your voice, with or without your face, is in someone’s living room, there is a direct human connection that in important ways is far more intimate than is possible in a lecture hall filled with anything more than a handful of students.

Once you realize this feature of the MOOC medium, the underlying pedagogic model is obvious. It’s one-on-one teaching/learning—something that in the traditional academy is (of necessity) reserved only for doctoral students.

At which point, the appropriate use of both pop quizzes and multiple-choice questions starts to look feasible. (They ought to; doctoral advisers use both extensively, and to great positive effect, though they do not refer to them as such, and there is no machine-grading!)

Of course, in a MOOC it remains the case that the student cannot communicate directly with the professor, nor can the professor see and comment on an individual student’s work. That means two further techniques have to be used as well:

3. peer tutoring
4. peer evaluation 

In the first version of my MOOC, last September, I built the course around the doctoral-student education model, deliberately setting out to create the experience of a student sitting alongside me at my desk. (There is a low resolution example here.)

But as a result of a career-long dislike of the first two and a deep suspicion of the fourth, I used all but the third of those auxiliary devices reluctantly and as little as possible. (The one I did embrace, peer tutoring, did not work well the way I set it up. See below for details of Attempt Two.)

Because of my caution, I think I avoided a fate reminiscent of NASA’s first attempts to launch a rocket into space. But that was a first, exploratory experience, and I wanted to live to try again. This time around, based on what I learned, I am going to use all four much more aggressively, but in ways I think might work.

I’ll be describing how I’ll be using them in a series of posts to my blog MOOCtalk.org. For a brief—and decidedly limited—foretaste, check out this video excerpt of a conversation my MOOC TA Paul Franz and I had recently with radio and TV personality Angie Coiro, host of the syndicated radio and television interview show In Deep.

The goal of Version 2 of the course is not to reach the Moon. Chances are high that we’ll crash and burn. The goal is to at least get off the ground before we do, and, if we are lucky, maybe even reach the upper atmosphere. For sure, there will still be a long way to go.

If you want to live dangerously and be part of this huge experiment, and if you have a Ph.D. (or pending Ph.D.) in mathematics and several years of college teaching behind you, I am still looking for well qualified volunteers to act as “Community TAs” for the course, to answer students' questions on the course discussion forums. So far I have 14 volunteers, comprising 5 college professors, 3 Ph.D. students, 3 individuals currently working in the software industry, a K-12 education consultant, a research laboratory scientist, and a stock analyst. If you want to volunteer, and have the requisite experience, please drop me an email at devlin@stanford.edu. (There is no payment for doing this—that includes me!) But being part of a large and truly global community, who come together for several weeks for the sole purpose of learning how to think mathematically (the course carries no college credit), is truly a wonderful experience.