Tuesday, December 4, 2012

The Darwinization of Higher Education

Stanford president John Hennessy has described the current changes in higher education initiated by technological innovations as an approaching tsunami. His remark was prompted largely by the emergence and rapid growth of MOOCs (massively open online courses), first from Stanford itself, joined soon afterwards by MIT and Harvard.

Are MOOCs going to initiate, or be part of, an educational tsunami? I think it’s too early to say. But in true mathematical fashion, I’m going to pursue the hypothesis that this
is the case and examine what is happening. In doing so, I’ll draw on the insight into MOOCs I gained from giving my own a few weeks ago, which I wrote about in last month’s column, and have been blogging about regularly at MOOCtalk.org.

Hennessy's observation was widely interpreted as being about the structure and business of higher education, and that may indeed be what he had in mind. He does, after all, have the responsibility of ensuring the survival and continuing prosperity of one of the world's leading universities. (A task that, as someone who receives a Stanford paycheck every month, I wish him every success in fulfilling.)

But when you look a bit more deeply at the way MOOCs are developing, you see that the real tsunami is going to be a lot bigger than that. It's not just higher education that will feel the onslaught of the floodwaters, but global society as a whole.

Forget all those MOOC images of streaming videos of canned lectures, coupled with multiple-choice quizzes. Those are just part of the technology platform. In of themselves, they are not revolutionizing higher education. We have, after all, had distance education in one form or another for over half a century, and online education since the Internet began in earnest over twenty-five years ago. But that familiar landscape corresponds only to the last two letters in MOOC ("online course"). The source of the tsunami lies in those first two letters, which stand for "massively open."

Right now, the most popular MOOCs draw student enrollments of about 50,000 to 100,000. In this it’s not unreasonable to expect those numbers to increase by at least a factor of 10, once people realize what is at stake.

True, those numbers don't tell the whole story. In particular, roughly 90% of the students who sign up do not complete the course. But that leaves many thousands who do finish, many of them with near perfect scores. And when that tenfold increase kicks in, it will be tens of thousands that complete. Paradoxically, it's the high rate of dropouts that will generate the tsunami (if there is one).

A good analogy is Google. Before Stanford graduate students Sergei Brin and Larry Page came up with their search algorithm, finding information (on the Web or elsewhere) was a time-consuming, and often hit-or-miss affair. At heart, what makes Google work is the efficient way it discards almost every possible answer to your query. Occasionally, in so doing, it may throw away the one item you really should see. But, given the way the algorithm works, that happens very, very rarely. As a result, Google gives you answers that are good enough for your purposes, most of the time.

The ability to sift through a massive amount of data means that there is no need for precise identification in search; with enough data, "good enough" really is good enough. In information terms, it's survival of the fittest; the process has no respect for the individual, but overall is extremely effective.

Now, the same university that gave you Google has launched the truly massive, open online courses. (Earlier MOOCs were not really massive. Indeed, the really massive ones, with millions of students, are probably a year or two away - yes, it could be that short a timeframe.)

Right now, the media focus on MOOCs has been on their potential to provide (aspects of) Ivy League education for free on a global scale. But an educational system does more than provide education. It also identifies talent - talent which it in part helps to develop. That makes a MOOC the equivalent of Google, where it is not the right information you want to find but the right people.

And the world definitely wants to find the right people. Last year, the World Economic Forum and the Boston Consulting Group issued a report describing the scale of the increasing need for talented individuals in today's world, and the numbers are staggering. For instance, the report states, "The United States ... will need to add more than 25 million workers to its talent base by 2030 to sustain economic growth, while Western Europe will need more than 45 million." The educational systems of these countries are not coming anywhere close to meeting those needs.

At the level of the individual student, MOOCs are, quite frankly, not that great, and not at all as good as a traditional university education. This is reflected (in part) in those huge dropout rates and the low level of performance of the majority that stick it out. But in every MOOC, a relatively small percentage of students manage to make the course work to their advantage, and do well. And when that initial letter M refers not to tens of thousands but to "millions," those successes become a lot of talented individuals.

One crucial talent in particular that successful MOOC students possess is being highly self-motivated and persistent. Right now, innate talent, self-motivation, and persistence are not enough to guarantee an individual success, if she or he does not live in the right part of the word or have access to the right resources. But with MOOCs, anyone with access to a broadband connection gets an entry ticket. The playing field may still not be level, but it's suddenly a whole lot more level than before. Level enough, in fact. And as with Google search, in education, "level enough" is level enough.

Make no mistake about it, MOOC education is survival of the fittest. Every student is just one insignificant datapoint while the course is running. Do well, do poorly, struggle, drop out - no one notices. But when the MOOC algorithm calculates the final ranking, the relatively few who score near the top become very, very visible. Globally, talent recruiting is a $130BN industry (Forbes.com, 2.12.12). It's "Google search for people" in action.

For those of us in education, MOOC education requires a major adjustment in attitude. Most of us go into the profession because we care about the individual. We love to interact with our students. Moreover, universities have all kinds of structures in place to catch and help struggling students. But in a MOOC, all of that goes out the window.

Doubtless, some current higher educational institutions will step in and provide support for MOOC students who need it. But what they won't be able to do is make education a local affair, where it is enough to do better than most of your fellow students at University X, or even in country Y. The fight to hire top talent will be global. And for American students from even moderately affluent backgrounds, a lot of their competition will have far more to gain from doing well, with all the added motivation that will bring.

Yes, some organizations will make money from MOOCs, though it is unlikely to be the Ivy League course providers whose stellar faculty and exclusive brands make their courses so attractive. They cannot afford to lose their exclusivity. But new sources of revenue for some colleges and universities who can adapt to the arrival of MOOCs, and the possible death of those that cannot, is just a market adjustment. If we are going to witness a tsunami, it is likely to be the true globalization of higher education and talent search.

POSTSCRIPT ADDED DECEMBER 5By chance, a day after this column appeared, Coursera sent out a mass email informing current and former students of their new talent placement service. (I have no connection to Coursera other than using their platform for my MOOC, and no knowledge of their business plans.) See my recent post at mooctalk.org for more details, where I also give a brief history of this column. (Yes, it has a curious past.)

Monday, November 12, 2012

MOOC Lessons

Planning and giving a university-level MOOC (massively open online course), as I did this fall, requires a complete re-evaluation of what it means to “teach” at that level. (Or any level, come to that, but university teaching is all I have first-hand experience of. My knowledge of K-12 teaching is limited to some familiarity with current theories of learning and some widely – but not universally – accepted principles of good pedagogy. Lots of theory, but no practice.)

I’ve always felt that the focus in university mathematics education should be on student-led learning, not teacher determined instruction. The key ability the student needs to develop is being able to take a novel problem and figure out a solution. That is, after all, what professional mathematicians do! As far as I can tell, I share this model of university mathematics education with the vast majority of my colleagues in the professoriate.

That’s not to say we – at least some of us – don’t reflect on what we do in the classroom, nor that we don’t attend courses, workshops, webinars, and presentations on educational technique. But my sense is that those of us that do this end up spending far less time providing “well crafted instruction”, and putting more of our effort into creating an environment in which our students can learn for themselves, and stimulating and encouraging them to do so.

In adopting this approach we capitalize on a hugely important factor you find at university but typically not at school: we are professionals who love our subject with a passion and have devoted our lives to its pursuit. When we stand in front of a class and write on a blackboard (mathematicians still prefer a blackboard to a whiteboard), we are not giving instruction so much as providing an example of how a pro thinks.

For, at heart, contact with the pros is what university education is about. For the vast majority of students, university is the first time in their lives they come shoulder-to-shoulder with the disciplinary experts. Those disciplinary pros do not have the pedagogic content knowledge required of a good K-12 teacher – at least nothing like to the same degree – but that is compensated by something that I think is far more important at that more advanced stage of a student’s development: learning by up-close observation of, and interaction with, a domain expert.

For sure, you will find university professors who have a different overall philosophy than the one I just sketched, but as I noted already, I think most of my colleagues have a similar view to mine.

Certainly, the celebrated physicist Richard Feynman, in the Preface to his 1963 book Six Easy Pieces, wrote:

The best teaching can be done only when there is a direct individual relationship between a student and a good teacher – a situation in which the student discusses the ideas, thinks about the things, and talks about the things. It’s impossible to learn very much by sitting in a lecture, or even by simply doing problems that are assigned.
A key element of operating in the fashion I am advocating (as is Feynman) is, then, the person-to-person interaction that takes place between a student and a professor (admittedly, often limited to the few short weeks of a university term). When a professor tries to port a course to a MOOC, however, that personal interaction goes out the window.

The primary issue is not the second O in MOOC – “online”, with the professor and students in different physical locations. That may be a significant factor, but just how significant is not yet known. (With the availability of rich social media, I think only empirical research will tell us the answer. Intuition is no longer a reliable guide to the importance of physical co-presence, if indeed it ever was.)

Rather, the key factor is that initial M – “massively”. In an online class with twenty-five students, the professor may be able to interact regularly with each student. But when there are 65,000 students, scattered around the Information Superhighway, there can be no meaningful interaction. The flow is asynchronous and entirely one way, from the professor to all those students.

That means the student becomes totally responsible for his or her learning. There can be one-on-one interaction, but it has to be student-to-student, perhaps within small study groups.

The task of the professor is then to design a course that can succeed as a result of student-student and small-student-group interactions.

As I was planning, and even more so when I was giving, my first MOOC, I felt very much like the conductor of a 65,000-player orchestra. I got to choose the pieces the orchestra will perform, I controlled when to start each piece and when to stop, and to some extent I dictated the tempo. I observed and occasionally commented on the overall group’s performance. I sometimes gave hints and advice. But each one of those 65,000 members of the orchestra did the actual playing. In principle, by pulling together, they should have been able to complete the current piece tolerably well, if I had suddenly been taken ill and had to put down the baton.

In fact, for the first time in my career, I was able to conduct a class the way I’d always wanted to: as an experienced guide who helps the committed learner in a minimal way, only when absolutely necessary.

That approach to university “teaching” can be done in a traditional class setting, but it takes an unusual individual and an even more unusual environment in which to do it. R L Moore is the most famous example of a mathematician who “taught” that way. (It’s so unusual, I need to put quotes around the key verb.) (See my MAA columns from May 1999 and June 1999.)

I tried the Moore Method, as it is called, a few times in my career, but it never worked well. I have enormous respect for my colleagues who have made it work – and some have. But, faced with teaching a MOOC (better make that “teaching”), I had to rely on one (but by no means all) major element of the Moore Method: the students would have to figure things out for themselves.

Moreover, I was of necessity relieved of the factor that has always led to my abandonment (or severe weakening) of the Moore method whenever I tried it: students who can’t handle the approach drop out.

In a physical class of maybe twenty-five students, I always felt a responsibility to do the best I could for each one. Particularly problematic were the ones who had gotten to university by virtue of “good teaching,” who could jump through all the templated hoops that were placed before them, but were floored when presented with a totally novel problem. After all, it was not their fault they were disadvantaged by “good teaching.” I felt it was my job to rescue them as best I could.)

With a volunteer student body of tens of thousands, on the other hand, you can’t avoid losing a few thousand, and you can afford to do so. There will still be many thousands of students who remain. Indeed, the “end of course evaluation” is bound to be overall positive, because the ones who don’t like, or cannot cope with, your approach simply drop out along the way.

In short, a MOOC is very much a survival-of-the-fittest affair.

At this early stage, MOOCs are being developed and offered very much in an experimental mode. But if, and when, they become an accepted part of the global educational landscape, then it’s not just higher education that will change, but society, as the international playing field gets truly leveled, with the most talented and ambitious people from everywhere in the world competing on merit alone.

Facing that possible future, maybe we need to ask ourselves if we do the best for our own students here in the US by being “too helpful.” And if the answer is “no” at university level, maybe it should be “no” in the high school as well.

For further discussion of my MOOC, see my blog MOOCtalk.org.

Monday, October 1, 2012

The Fibonacci dedication in Pisa

The commemorative tablet to Leonardo Fibonacci in Pisa.
If you find yourself with time to spare in Pisa, after you have completed the obligatory (and well worthwhile) visits to the famous Leaning Tower and the splendid buildings around it, you should stroll over to Lungarno Mediceo, the road that follows the northern bank of the Arno, and look for building number 30. 

The State Archive in Pisa at 
Lungarno Mediceo 30.
Formerly the Palazzo Lanfranchu and later Palazzo Toscanelli, this historic building dates back in its present form dates back to the nineteenth century, and  was at one time (1821-22) the residence of the famous British writer Lord Byron. 

Today, it is the home of the Archivio di Stato (State Archives) of Pisa. The stone plaque shown in the photo is one of two memorials in Pisa to one of their most famous residents, Leonardo “Fibonacci,” the writer of the book Liber Abbaci, generally credited with introducing Hindu-Arabic arithmetic to western Europe. (The other memorial is a statue in the Camposanto.) 

In 1241, the Comune of Pisa decreed that an amount of money should be given annually to Leonardo for his service to the city. The text of the proclamation was reproduced on the stone tablet the city placed there on 16 June, 1867 to honor their great ancestor. The medieval text follows an introductory declaration written in 1865.

The inscription is written in a very formal nineteenth century form of Latin, which translates literally as:

The Rulers and People of Pisa in the year 1865 after ignoring old crushing falsehoods and where the will of the Elders was to study what was better known and proven about Leonardo Fibonacci ordered the city archives to file a copy of the decree by the same Most Eminent Republic of Pisa that one monument equal to so great a man survive.
[The 1241 decree] 
In consideration of the honor brought to the city and its citizens and their betterment by the teaching and zealous cooperation of that discreet and wise man, Master Leonardo Bigolli, as well as by his regular patriotic efforts in civic and patriotic affairs, the Pisan Commune and its Officials in certain right and conscious of our prerogative to make recompense for work that he performed in heeding and consolidating the efforts and affairs already mentioned confer upon this same Leonardo so meritorious of our love and appreciation an annual salary or reward from the Commune of 20 free denarii and the usual accompaniments. This we affirm with the present statement.  

The above translation was carried out for me by medieval scholar and Leonardo translator Barnabus Hughes. For further details of the life and works of Leonardo, see my 2011 book The Man of Numbers, or come along to the talk I’ll be giving at the MAA’s History of Mathematics group meeting (HOMSIGMAA) at the Joint Mathematics Meetings in San Diego on January 9, in the evening.

Saturday, September 1, 2012

What is mathematical thinking?

What is mathematical thinking, is it the same as doing mathematics, if it is not, is it important, and if it is different from doing math and important, then why is it important? The answers are, in order, (1) I’ll tell you, (2) no, (3) yes, and (4) I’ll give you an example that concerns the safety of the nation.

If you had any difficulty following that first paragraph (only two sentences, each of pretty average length), then you are not a good mathematical thinker. If you had absolutely no difficulty understanding the paragraph, then either you are already a good mathematical thinker or you could acquire that ability pretty quickly. (In the former case, you most likely pictured a decision tree in your mind. Doing that kind of thing automatically is part of what it means to be a mathematical thinker.)

Okay, I had my tongue firmly in my cheek when I wrote those opening paragraphs, but there is such a thing as mathematical thinking, it can be developed, and it is not the same as doing mathematics.*

In my last column, I talked about my decision to self-publish a really cheap textbook to accompany my upcoming MOOC (massively open online course) on Mathematical Thinking. At the time of writing this column, just shy of 40,000 students have registered – and there are over two more weeks before the class starts.

As a result of sending out a number of tweets, chronicling my experiences in developing my MOOC in a blog MOOCtalk.org, and posting some videos about the upcoming course on YouTube, I’ve already received a fair number of emails asking for details about the course. (At one point, so many so I had to temporarily shut off comments on MOOCtalk.org, lest WordPress closed me down under the assumption that with so much traffic it must be a porn site.)

In this column, I’ll answer one question that came up a number of times: What is mathematical thinking? In fact, I’ll do more, I’ll answer the four questions I opened with.

To people whose experience of mathematics does not extend far, if at all, beyond the high school math class, I think it’s actually close to impossible for them to really grasp what mathematical thinking is. I used to try to convey the distinction with an analogy. “K-12 mathematics is like a series of courses in digging trenches, pouring concrete, bricklaying, carpentry, plumbing, electrical wiring, roofing, and glazing,” I would say. And then, after a brief pause, I would continue,  “Mathematical thinking is the equivalent of architecting. You need all of those individual house-building skills to build a house. But putting those skills together and making use of them requires a higher-order form of thinking. You need someone who can design the building and oversee its construction.”

It is a great analogy. I felt sure it would convey the essence of mathematical thinking. But many conversations and email exchanges over the years eventually convinced me it was not working. Saying A is to B as C is to D works fine when the recipient has good understanding of A, B, and C and some understanding of D. But if they have not even a clue about D, or even worse, if they believe that D actually is C, then the analogy simply does not work. It’s one of those analogies that is brilliant if you are sufficiently familiar with all four components, but hopeless as a way to explain one in terms of the other three.

Once I realized that, I set out to find a better way to describe it. It took me most of a whole book to do it. Not the ultra-cheap textbook I mentioned above. That has a different purpose. Rather, my book on using video games in mathematics education.

Below, in about 850 words, is the nub of what I say in that book in about 75 pages. (Yes, that’s quite a compression ratio. Clearly, it’s lossy compression!) After the quote, I’ll give you a specific example of mathematical thinking from my own past involvement in national security research. (Don’t worry, my part was not classified. You can read it without me having to kill you.)

BEGIN QUOTE [pp.59–61]:

[Mathematical thinking is more than being able to do arithmetic or solve algebra problems. In fact, it is possible to think like a mathematician and do fairly poorly when it comes to balancing your checkbook. Mathematical thinking is a whole way of looking at things, of stripping them down to their numerical, structural, or logical essentials, and of analyzing the underlying patterns. Moreover, it involves adopting the identity of a mathematical thinker.]

[For instance] like most people, when I am doing something routine, I rarely reflect on my actions. But if I’m do ing mathematics and I step back for a moment and think about it, I see myself [not just as someone who can do math, but] as a mathematician.

“Well, duh!” I hear you saying. “You are a mathematician.” By which I assume you mean that I have credentials in the field and am paid to do math. But I have a similar feeling when I am riding my bicycle. I’m a fairly serious cyclist. I wear skintight Lycra clothing and ride a $4,000, ultralight, carbon fiber, racing-type bike with drop handlebars, skinny tires, and a saddle that resembles a razor blade. I try to ride for at least an hour at a time four or five days a week, and on weekends I often take part in organized events in which I ride virtually nonstop for 100 miles or more. Yet I’m not a professional cyclist, and I would have trouble keeping up with the Tour de France racers even during their early morning warm-up while they are riding along with a newspaper in one hand and a latte in the other. […] Being a bike rider is part of who I am. When I am out on my bike, I feel like a cyclist. And you know, I’d be willing to bet that the feeling I have for the activity is not very different from [the professional bike racers].

It’s very different for me when it comes to, say, tennis. […] I don’t have the proper gear, and I have never played enough to become even competent. When I do pick up a (borrowed) racket and play, as I do from time to time, it always feels like I’m just dabbling. I never feel like a tennis player. I feel like an outsider who is just sticking his toe in the tennis waters. I do not know what it feels like to be a real tennis player. As a consequence of these two very different mental attitudes, I have become a pretty good cyclist, as average-Joe cyclists go, but I am terrible at tennis. The same is true for anyone and pretty much any human activity. Unless you get inside the activity and identify with it, you are not going to be good at it. If you want to be good at activity X, you have to start to see yourself as an X-er  – to act like an X-er.

A large part of becoming an X-er is joining a community of other X-ers. This often involves joining up with other X-ers, but it does not need to. It’s more an attitude of mind than anything else, though most of us find that it’s a lot easier when we team up with others. The centuries-old method of learning a craft or trade by a process of apprenticeship was based on this idea. [The video games scholar James Paul Gee, in his book What Video Games Have to Teach Us About Learning and Literacy, p. 18] uses the term semiotic domain to refer to the culture and way of thinking that goes with a particular practice – a term that reflects the important role that language or symbols plays in these “communities of practice,” to use another popular term from the social science literature. […]

In Gee’s terms, learning to X competently means becoming part of the semiotic domain associated with X. Moreover, if you don’t become part of that semiotic domain you won’t achieve competency in X. Notice that I’m not talking here about becoming an expert, and neither is Gee. In some domains, it may be that few people are born with the natural talent to become world class. Rather, the point we are both making is that a crucial part of becoming competent at some activity is to enter the semiotic domain of that activity. This is why we have schools and universities, and this is why distance education will never replace spending a period of months or years in a social community of experts and other learners. Schools and universities are environments in which people can learn to become X-ers for various X activities – and a large part of that is learning to think and act like an X-er and to see yourself as an X-er. They are only secondarily places where you can learn the facts of X-ing; the part you can also acquire online or learn from a book. […]

The social aspect of learning that goes with entering a semiotic domain is often overlooked when educational issues are discussed, particularly when dis cussed by policy makers rather than professional teachers. Yet it is a huge factor. […]


In my blog MOOCtalk.org, I will explain what persuaded me to try to prove that the pessimism I expressed in the above passage about someone becoming an X-er through a remote experience like a MOOC might be misplaced, at least in part. But my focus here is describing mathematical thinking.

In many cases, the real value of being a mathematical thinker, both to the individual and to society, lies in the things the individual does automatically, without conscious thought or effort. The things they take for granted – because they have become part of who they are. This was driven home to me dramatically in the years immediately following 9/11, when I was one of many mathematicians, scientists, and engineers working on national security issues, in my case looking for ways to improve defense intelligence analysis.

My brief was to look at ways that reasoning and decision making are influenced by the context in which the data arises. Which information do you regard as more significant? How do you weight, and then combine, information coming from different sources. I’d looked at questions like this in pre-9/11 work – indeed that was the research that brought me from the UK to Stanford in 1987, and by the time the Twin Towers came down, I had written two research books and a number of papers on the topic. But that research focused on highly constrained domains, where the complexity was limited. The challenge faced in defense intelligence work is far greater – the complexity is huge.

I did not have any great expectations of success, but I started anyway, proceeding in the way any professional mathematician would. I could give you a list of some of the things I did, but that would be misleading, since I did not follow a checklist, I just started to think about the problem in a manner that has long become natural to me. I thought about it for many hours each day, often while superficially occupied with other life activities. I was not aware of making any progress.

Six months into the project, I flew to D.C. to give a progress report to the program directors. As I fired up my PowerPoint projection and copies of my printed interim report were passed around the crowded meeting room, I was sure the group would stop me half way through and ask me (hopefully politely) to get on the next plane back to San Francisco and not waste any more of their time (or taxpayers’ dollars).

In the event, I never got beyond the first content slide. But not because I was thrown out. Rather, the rest of the session was spent discussing what appeared on that one slide. I never got close to what I thought was my “best” work. As my immediate research report told me afterwards, beaming, “That one slide justified having you on the project.”

So what had I done? Nothing really – from my perspective. My task was to find a way of analyzing how context influences data analysis and reasoning in highly complex domains involving military, political, and social contexts. The task seemed impossibly daunting (and still does). Nevertheless, I took the oh-so-obvious (to me) first step. “I need to write down as precise a mathematical definition as possible of what a context is,” I said to myself. It took me a couple of days mulling it over in the back of my mind while doing other things, then maybe an hour or so of drafting some preliminary definitions on paper. The result was a simple statement that easily fitted onto a single PowerPoint slide in a 28pt font. I can’t say I was totally satisfied with it, and would have been unable to defend it as “the right definition.”
But it was the best I could do, and it did at least give me a firm base on which to start to develop some rudimentary mathematical ideas. (Think Euclid writing down definitions and axioms for what had hitherto been intuition-based geometry.)

The fairly large group of really smart academics, defense contractors, and senior DoD personnel in that meeting room spent the entire hour of my allotted time discussing that one definition. Not because they were trying to decide if that was the “right” definition, or the best one to work with. In fact, what the discussion brought out was that all the different experts had a different conception of what a context is, and how it can best be taken account of – a recipe for disaster in collaborative research if ever there was.

What I had given them was, first, I asked the question “What is a context?” Since each person in the room besides me had a good working concept of context – different ones, as I just noted – they never thought to write down a formal definition. It was not part of what they did. And second, by presenting them with a formal definition, I gave them a common reference point from which they could compare and contrast their own notions. There we had the beginnings of disaster avoidance, and hence a step towards possible progress in the collaboration.

As a mathematician, I had done nothing special, nothing unusual. It was an obvious first step when someone versed in mathematical thinking approaches a new problem. Identify the key parameters and formulate formal definitions of them. But it was not at all an obvious thing for anyone else on the project. They each had their own “obvious things.” Some of them seemed really clever to me. Others seemed superficially very similar to mine, but on closer inspection they set about things in importantly different ways.

“Your work is not classified, so you are free to publish your results, if you wish,” the program director told me later, “but we’d prefer it if you did not make specific reference to this particular project.” “Don’t worry,” I replied, “I have not done anything that would be accepted for publication in a mathematics journal.” Which is absolutely the case. I had not done any mathematics in the familiar sense. I had not even taken some mathematical procedure and applied it. Rather, what I had done was think about a complex (and hugely important) problem in the way any experienced mathematician would.

I’ve had a number of similar experiences over the years, and though they appear on the surface to be widely different (from analyzing children’s fairy stories to looking at communication breakdown in the workplace to trying to predict the endings of movies like Memento to trying to make sense of the modern battlefield), at their (mathematical) heart they all have the same general pattern.

That then, is mathematical thinking. How do you teach it? Well, you can’t teach it; in fact there is very little anyone can teach anyone. People have to learn things for themselves; the best a “teacher” can do is help them to learn.

The most efficient domain to learn mathematical thinking is, perhaps not surprisingly (though it’s not such a slam-dunk as you might think) mathematics itself. Particularly well suited parts of mathematics for this purpose are algebra, formal logic, basic set theory, elementary number theory, and beginning real analysis. These are the topics I have chosen for my MOOC. Other topics could serve the same purpose, but would require more background knowledge on the part of the student. But it’s not about the topic. It’s the thinking required that is important.

*One of the features of mathematical thinking that often causes beginners immense difficulty is the logical precision required in mathematical writing, frequently leading to sentence constructions that read awkwardly compared to everyday text and take considerable effort to parse. (The standard definition of continuity is an excellent example, but mathematical writing is rife with instances.) The opening paragraph is a parody of such writing. This comment was added a day after initial publication, when a letter from a reader indicated that he missed the fact that the opening was a parody, and complained that he found it difficult to read. That difficulty was, of course, the whole point of the opening, but that point is lost if readers don't recognize what is going on. So I added this remark.

Wednesday, August 1, 2012

The future of textbook publishing is us

In my May column, I announced my intention to give a free online course (a MOOC) this coming fall, and asked for assistance from the mathematical community.

That course, a school-to-university transition course, titled Introduction to Mathematical Thinking, is going ahead, with the first lecture on September 17. There is a brief description of the course, together with a short promotional video, on the Coursera website.

I have started a blog, MOOCtalk.org, to chronicle my experiences working with this new format and to provide a platform for feedback and discussion once the course gets going.

I also wrote a short textbook to accompany the course, Introduction to Mathematical Thinking. Though the textbook is not required for the course, some of my Stanford colleagues who gave the first generation of “Ivy League MOOCs” – just a few months ago, so fast has this new movement taken off – told me that many students want an old-fashioned physical book. On the other hand, all the transition textbooks I am familiar with are fairly pricey, which would put them well beyond many of the students who are likely to enroll. Moreover, none of them are designed to accompany a MOOC. So I decided to write one.

My two main criteria were: it had to be short (no more than 100 pages) and cheap (less than $10 in the US). The only option was to self-publish in print-on-demand format with Amazon’s CreateSpace service. I did not quite hit my page-limit; when I include the front material, the tally comes out at 102 pages, but that’s close enough. But I figured I’d cover my costs if I set the retail price at $9.99, just below my target.

The procedure is so ridiculously straightforward, I can see no reason why anyone should ever publish another textbook a different way, given the huge expense of textbooks. We authors have been typesetting our own manuscripts ever since Don Knuth first released TeX in 1978, and all that is required to produce a book with CreateSpace is to generate a PDF file that fits the page-size you select.

CreateSpace does not provide TeX support (by way of a style file), but they do provide sample pages for authors submitting manuscripts in Word, and I just played around with the page parameters in LaTeX until the output matched their samples for both odd and even numbered pages, which I checked by printing out copies of both, putting my output on top of theirs, and holding the two up to the light. (Low tech, but effective.)

In my case, I decided to produce my book in the standard 6 in x 9 in format, and the key LaTeX parameters I came up with are (for the record)
\oddsidemargin 1 in \evensidemargin .55 in
\marginparwidth .75in \marginparsep 7pt
\topmargin -.5in \headheight 12pt
\headsep .25in \footheight 12pt \footskip .35in
\textheight 7.5in \textwidth 4.95 in

When I submitted my final PDF file, CreateSpace’s automated checking system flagged the manuscript as possibly not being correctly formatted, but I pressed forward, since the next stage is that one of their employees examines the manuscript, and indeed that individual accepted it, confirming my suspicion that I was probably off by a millimeter or two, something that could upset an automated checking system but is close enough to pass a human eyeball test.

The point is, the whole process is so well designed, there is no reason why anyone who can use LaTeX should do anything other than self-publish from now on. With a very small number of exceptions, no one who writes a university-level textbook does so to make money. Our goal is to get material in front of students as quickly and cheaply as possible. If there were a way to do so that can save the students money, I am sure we would all want to do so, the more so given the way textbook costs have skyrocketed in recent years. With modern print-on-demand technology, we now can do just that.

You don’t need to know anything about publishing to do this. CreateSpace does for book publishing what TurboTax did for filing your tax return, and it does it in much the same way, by taking you through the entire process in a simple, step-by-step fashion, including cover design, securing an ISBN code, and selecting marketing channels.

For sure, the finished product is not quite as good as would be achieved with the professional expertise of a good publishing house. But to my mind, for a textbook, it’s close enough, especially when the resulting book can be sold for as little as a tenth the price a publisher would charge.

The one thing I paid someone else to do was copy-editing. I have written enough books to value highly the services of an experienced copy editor. (You might also want to pay an indexer. I did my own, but I have done so for several of my previous books.)

Of course, even with good copy editing, occasional errors creep through. Not long after my book was on the market, I was looking through one of my author’s copies (you have to buy them, but at an even lower price than the retail mark), and spotted a couple of small typos. A few minutes editing the LaTeX file, followed by a quick upload of the replacement PDF, and the correction was made, ready for the next person to buy a copy.

Returning to the MOOC now, let me re-iterate the request I made in my May column. I am giving my MOOC in the early fall to coincide with the many transition courses offered at colleges and universities across the US, in the hope that instructors of such courses will incorporate my MOOC in their courses in some way. My reason for this is that I think the only way to make a transition course MOOC work is to have enough participants who either are already familiar with the material (such as instructors) or else have direct access to such expertise (e.g., their students in a transition course). I see no other way for students struggling to understand the material to get the help, advice, and feedback they will need to progress. Social media provides various platforms for students to interact, to ask questions of one another and to comment on others’ work. But there has to be a mechanism for mathematical truth to find its way into the discussions!

So the key to making something like this work is, I think, to build up a Wikipedia- like community of instructors who, for five weeks each year, will make available their expertise to the thousands of students around the world who are taking advantage of a MOOC to obtain an education they would otherwise not have access to.

The benefit to the students in the transition classes given by MOOC-participating instructors is that their learning will assuredly be enhanced by acting as tutors for the students who are not so privileged. Both because teaching others is a powerful way to learn – as most of us discover when we become TAs at graduate school – and because those students will surely feel much more incentivized to understand by playing such a feel-good role.

Stay tuned to my MOOCtalk blog for updates on the project. And if you are an instructor giving a transition course this fall, please consider getting involved.

Sunday, July 1, 2012

“Can’t we all get along?”

Unless you follow several mathematics education blogs or subscribe to certain Twitter feeds, you may well have missed the recent revival of the US Math Wars. The protagonists in the latest salvo are not the traditional foes, Mathematically Correct versus NCTM and MAA, but two groups who are making use of new technology in education, with Khan Academy squaring up against a number of web-savvy, younger mathematics and science educators who believe the US can and should do a lot better (and differently) in math ed than we do.

I summarized some of my own views on KA in an article in The Huffington Post on March 20 of this year, but this article focuses on another issue than the one I discussed there. Namely, the large numbers of fanatical supporters Khan has gathered, who respond to even the mildest and well meaning criticisms of anything Khan with venom, hatred, and personal attacks.

The first round of this new skirmish I was aware of (likely not the first to occur, since I check in only occasionally on developments in K-12 mathematics education, though now the excellent EdSurge provides me more useful news than I can possibly keep up with) was in February of this year when Mathalicious founder Karim Kai Ani posted a critique of KA pedagogy on his company blog. (Disclosure: I knew that posting was coming up, since Karim emailed me beforehand with questions about gamification and individually prescribed instruction, both of which he wanted to discuss in his article. I also voluntarily endorse his educational materials.)

The KA–Mathalicious deluge began within a few hours of the blog being published, with a lot of the venom unleashed on Y-Combinator’s Hacker News. From the nature of many of the comments, which were personal, and often visceral attacks on Ani rather than reasoned responses to what was a very well thought out and cogently presented argument, it’s a reasonable assumption that they were contributed by schoolkids. Which to my mind makes this phenomenon worth thinking about. For schoolchildren are what K-12 education is about. When a substantial number of our main customers speak, we should listen, even if the method by which the message is delivered leaves a lot to be desired.

Actually, I don’t know what percentage of the K-12 population is represented in those Hacker News (and other) blog posts, but what we do know is that Khan Academy has a large and devoted following. In some ways, I too am a fan of sorts, though less so than when I participated in a TV discussion with Sal back in 2010. For sure, I am dismayed by the huge number of mathematical errors and pedagogic shortcomings in his videos. (We’ve learned a lot about mathematics education in the past fifty years.) In the early days it was easy to overlook them – any new enterprise is buggy, particularly a one-person show like the early KA. But once the Silicon Valley millions started to roll in, and with those funds a staff, it would have been an easy Web exercise to crowdsource curriculum improvement/development to the many mathematics teachers and university mathematics education specialists who would surely have been eager to help.

My enthusiasm came not from the KA site’s contents, but from something else I saw: Sal Khan himself. Up close – and I have met him a few times – he comes across as a really nice, approachable guy with a great sense of humor. Well, so do a lot of people. But there is something else. Sal has the ability to project that identity and that personality over the Web, using just his voice and the trace of a digital pen on a tablet. In the Age of Social Media, that, it seemed to me, was powerful. And, at least to date, extremely rare.

Effective teaching is a human-to-human activity. Good teachers strive to achieve a connection with our students in the classroom. (I say “our students,” since I like to think I am a good teacher, albeit not at the K-12 level.) I’m told my voice conveys my enthusiasm for mathematics when I go on NPR as the Math Guy. If true, and some people seem to think so, then I can’t really take credit for it. I just speak into a microphone. No training, no rehearsing. Just me.

Sal Khan has that natural ability in spades. As he tells us (all Silicon Valley enterprises have a creation story, and in his case it’s true), Khan Academy began as his attempt to help his younger cousins with their math homework, by posting video tutorials on YouTube. He did not set out to build an education empire. People outside his family simply came across his videos on the Web and found them useful.

From a pedagogic perspective, his videos do not provide student learning, they deliver instruction – a distinction I discussed in my March Devlin’s Angle. It would be easy for me to critique them, and many have. (Read on.) But to me at least, providing learning is not where their real value lies.

Learning mathematics is hard. Very hard. It is easy to get discouraged and give up. Some of us, when we are learning math the first time, are lucky enough to have a parent, grandparent, uncle or aunt, older sibling, or family friend who can sit down alongside us and help us. I suspect that a great many of today’s professional mathematicians owe their eventual success in the subject to someone who mentored them in the early days.

But not everyone has such a person in their lives. At least, they did not until Sal Khan came along: friendly, non-threatening, patient, and a good explainer (actually not brilliant, but that might be all to the good, since a brilliant instructor could easily discourage a less-brilliant student). Above all, human. A regular guy. Just think about that for a moment. It’s a valuable weapon in the educational landscape.

Much of the attraction of KA came from its very amateurish nature. It really was just Sal Khan in his converted closet. Sure he did not really understand (as a mathematician does) a lot of the math he explains (though he knows an awful lot more than most people), nor was he trained in mathematics pedagogy. (Ditto for me, BTW – on both counts! – though I think I know more about both than Sal, which makes it possible for me to critique many of his lessons.) But that was a huge part of what made KA so popular. Millions of people around the world, young and old, whose experiences of math class was or had been awful, saw him as on their side. He was everyone’s friendly, helpful Uncle Sal. (I hope this does not come across as a hagiography. I have a lot of issues with KA. But I am trying to understand what leads to KA having such a passionately supportive fan base. I think there is something for us all to learn here.)

For millions of users of KA, what they find on the site is far better than anything they have had or are getting. For them, it’s not just a homework helper, it’s a lifeline. Their only lifeline.

Sure, they probably don’t really learn any mathematics. (I have yet to be convinced you can really learn mathematics over the Web, though that does not stop me wanting to try, as indicated by my other blog MOOCtalk.) The fact is, many people actually don’t want to learn mathematics, they want to pass a math exam. And if Sal Khan helps them do that – and millions say that he has done exactly that – then is it surprising they become KA fans? (And educational establishment enemies!)

Then Bill Gates comes along, and KA goes global. Expectations change. Now things have gotten more tricky. When a resource like KA becomes the primary vehicle by which millions of people acquire many or all of their mathematical skills, the stakes become dramatically higher than when it was a one-man homework-help service. Like it or not, ask for it or not, KA now has (in my view) an obligation to get things right. Doing so without destroying a major part of its appeal, is clearly going to be tricky.

Which brings me to the more recent skirmishes.

On June 18, two math professors, John Golden and David Coffey, posted to YouTube a parody of Mystery Science Theater 3000, which they called MTT2K, in which they watched and critiqued a KA video (an old one, as it happens) about multiplying and dividing negative numbers. To my mind, the target of their theatrical sarcasm was not KA, rather the reverence that many seem to have for KA, but many Khan fans (perhaps the reverent ones) seem to have reacted differently.

In any event, KA immediately took down the video, replacing it with two new videos, one on Multiplying Positive and Negative Numbers, the other on Dividing Positive and Negative Numbers. The two new videos appear to have been made in direct response to the MTT2K critique. A short while later, Khan released another new video, Why a Negative Times a Negative is a Positive, providing further elaboration.

Unfortunately, as John Jay High School (New York) physics teacher Frank Noschese noted, that third video was awfully similar to one produced some years earler by James Tanton. Things were escalating (or spiraling down, depending on your favored metaphor).

The pity was that critiques by knowledgeable teachers and pedagogy experts resulting in modifications to KA instructional materials is surely the way to take something that has value and make it even more valuable. But that tended to get lost in the MTT2K-parody sarcasm and the barrage of name calling that followed.

Incidentally, Noschese is a 2011

Presidential Award for Excellence in Math and Science Teaching awardee, and the author of an excellent science education blog Action-Reaction. He has commented and tweeted extensively (and to my mind constructively) on KA. I hope KA follows his blog and takes note of what he says.

Another experienced STEM teacher with excellent suggestions (for KA and for teaching in general)  is Dan Meyer, a former Google Education Fellow and now a PhD student in education at Stanford. He joined in the MTT2K exchange in his usually witty fashion in two blogposts, Bill Gates Just Put Out a Hit on John Golden and David Coffey, on June 20 and Sal Khan Comments On MTT2K In Chronicle of Higher Education, on June 28.

The latter was about an article in the Chronicle of Higher Education that day, summarizing the MTT2K affair. The Chronicle article was inspired in part by the article by Justin Reich in Education Week on June 22 about the “MTT2K Prize”, a cash prize to be awarded for the best video commentary on a Khan Academy video. (Actually it’s a great idea, if it can be done collaboratively with KA, absent any negativity.)

The final episode in this skirmish (at least so far, based on what I have seen) was an entry video to the MTT2K Prize competition on June 29, by Wired blogger Rhett Allain, an Associate Professor of Physics at Southeastern Louisiana University.

To my mind, Allain’s blog description is better than his videoed critique. Allain just does not come across on video as well as Khan does. Many teachers, and most academics, spend a lot of their time critiquing one another. We get used to it. It’s how we learn and improve. But when you put it on YouTube, it is viewed by millions of people totally unfamiliar with the process, and they can – and, in the cases I am citing, did – react negatively.

Which brings me back to my starting point. Education is hard, mathematics education particularly so. It takes a lot of different kinds of knowledge and expertise to get it right. The focal point is – and I think has to be – a single person, a teacher. Either in the flesh or over the Web. That teacher has to be able to connect to the students. Being a great classroom teacher does not make someone good at doing it on YouTube. In fact, most teachers come across pretty badly on video. But that does not matter if the people who are able to use the medium will listen to what they say.

Sal Khan’s strength is that he comes across extremely well on video (with or without his face on screen!). I just wish he would work (with real experts) on the content more.

But the real problem is not the stuff on the KA site. Flawed as it is, it is, as I noted earlier, a lot better than many people have, or ever had, access to. The fact that many of Khan’s fans describe him as “the best teacher ever” speaks volumes about the poor quality of the mathematics education that many receive. I’ve visited many math classrooms both in this country and around the world, and I’ve seen great math teaching. You won’t find it on KA. Instead, you will find something else, something unique and of value.

Sure, KA has lots of weaknesses and could be improved. That goes for any product. The real problem is that the US (and other nations) identify mathematics learning with instruction and passing procedural tests. In that world, KA meets a clear market need for instruction to help people pass procedural math tests.

In contrast, Ani, Noschese, Golden, Coffey, Meyer, Allain, and all the other KA critics in the educational world are interested in facilitating something quite different: real learning among their students.

Sal Khan says he is trying to move into the real, conceptual learning space as well, but so far I have not seen much that would qualify, and as I noted earlier, my own interest in trying out the MOOC format notwithstanding, I have yet to be convinced that it is possible over the Web.

Khan has something of real value to offer, most uniquely his ability to do well both locally and at a distance what many teachers find hard or impossible even in the classroom, namely connect with and inspire students. But he seems not to have deep conceptual understanding of mathematics or knowledge of the highly complex field of mathematics learning, and given his background it would be strange if he did. (For a good, reasoned series of articles pointing out the problems with KA, see the 2011 articles by Sylvia Martinez of Generation YES.)

On the other side, there are many teachers and education researchers who do have knowledge of mathematics and mathematical pedagogy, but are not able to connect well, at least on video, where they come across as cold or impersonal or condescending – video is a harsh medium.

If each party recognized that the others had something of value to offer, and if  we could get beyond the squabbles and the name-calling, we could produce something that benefits the people we all care about: the students.

Los Angeles police brutality victim Rodney King died recently. The words he famously uttered during the riots that followed his beating in 1992 seem equally pertinent to the current state of affairs regarding KA: “Can't we all get along?”