The saga starts in 2012, when the New York Times published an opinion piece by Mr. Hacker, with the attention grabbing headline Is Algebra Necessary? As the headline writer surely intended, the article generated a large number of online comments, tweets, and discussions in various blogs. Since Hacker clearly has a valuable connection to the nation's premier national newspaper, it is then a pity he pitched his article the way he did. For, although Hacker says that he did not write the headline himself, his article was indeed promoting the removal of algebra as a required course in K-12 education (as does his new book). Not only did that suggestion alienate accomplished scientists and engineers and a great many teachers--groups you'd want on your side if your goal is to change math education--it distracted attention from what was a very powerful argument for introducing the teaching of algebra into our schools, something I and many other mathematicians would enthusiastically support.

Yes, you read that right:

Taking two examples from his article, Hacker lamented that there was no good reason to require K-12 students to "master polynomial functions and parametric equations" or to "force them to grasp vectorial angles and discontinuous functions." His use of these examples as illustrations highlights how narrow and off base is his understanding of mathematics.

He is simply wrong about his first example. There is good reason to study both polynomial functions and equations (parametric or otherwise), provided it is done properly. He would, however, be absolutely correct if you were to take his phrases to refer to mastering certain procedures for manipulating symbolic expressions (as he clearly does), which is what that valuable educational goal has largely morphed into in our classrooms and textbooks.

As for his second example, I had to google the term "vectorial angle." It turns out to be an uncommon (and unfamiliar to me) name for measuring an angle as you go around a circle, so I doubt any child needs to be forced to "grasp" that, even a child brought up in an age when most clocks are digital. And "discontinuous functions" have nothing to do with algebra; they are topics in the subject known as Real Analysis, which typically is studied only by university level math majors.

There is always a danger in setting oneself up as an advocate for change in a discipline one does not know. Hacker is not a mathematician. He is a retired college professor of political science, who has taught some courses in mathematics to non-majors. As a result of his experiences, he has arrived at some conclusions about K-12 mathematics education that in many ways are close to my own, and which I have written about extensively in my Devlin's Angle and my personal blog profkeithdevlin.org.

[In many ways, my position was articulated far more eloquently and passionately than I ever managed by math teacher Paul Lockhart in his essay

As I say, Hacker and I have very similar views about the abysmal state of much of today's K-12 mathematics education, and the negative effect it has on generations of school students who, as a result, graduate with a poor understanding of, and often great antipathy towards, mathematics. But, given the importance of mathematics in today's world, we absolutely should not abandon the obligatory teaching of algebra, as he advocates; rather, we should teach it right. Unfortunately, since Hacker plainly does not understand what algebra is, or more generally what mathematical thinking is, he instead proposes we throw away the healthy but neglected baby along with the depressing pool of lukewarm, dirty bathwater it currently hides in.

First codified by the Persian mathematician al-Khwarizmi in his book

First, algebra provides methods for handling entire classes of numbers, rather than specific numbers (which is what arithmetic does). (Those

Second, algebra provides a way to find numerical answers not by computing, which is often very difficult, but by reasoning logically to hone in on the answer, using whatever information is available. Thus, whereas in arithmetic you work forwards, starting with numbers and computing with them to arrive at an answer, in algebra you work backwards, starting by postulating an answer and reasoning logically to figure out what it is. True, this powerful application of human logical reasoning capacity frequently gets boiled down to mastering various symbolic procedures to "Solve for

When al-Khwarizmi wrote his book (the world's first algebra textbook), apart from the familiar ten symbols for numerals, there was not a single symbol anywhere. No formulas or symbolic equations to be seen. Al-Khwarizmi was showing the traders and engineers of the 9th Century how to solve the numerical problems they faced in their daily lives. The focus was on

Where Hacker goes wrong is confusing algebra with a specific implementation of algebra introduced by François Viète in 16th Century France. More accurately referred to as "symbolic algebra," it provides a set of formulaic procedures for carrying out algebraic reasoning in a largely mechanical fashion. It's very efficient, which is why it rapidly gained broad acceptance and widespread use. But the symbolic implementation is a procedural mental aid that only makes sense when learned and practiced in the context of real problems. If the symbolic method gets separated from the real world domains it was developed to handle, it ends up seeming like a meaningless and pointless game. That is what has happened with school algebra, as it has become codified in today's textbooks. [Some of us actually like that game, and it has proven time and time again to be valuable to society for some of us to play it. But that is a separate issue.]

As it happens, the separation of symbolic algebra from applications is no longer of much consequence in mass K-12 education. For there is a second, much more recent implementation of al-Khwarizmi's method of algebra--one of immediate use to everyone in today's world: the electronic spreadsheet. A perfectly correct description of Microsoft Excel is that it is a computer implementation of algebra, just as Viète's symbolic algebra was an earlier, paper-and-pencil implementation.

[Check back on my summary of what algebra is. The columns in a spreadsheet allow you to reason with entire classes of numbers, and the spreadsheet's macros are the algebraic formulas. Because computers are highly efficient at performing many arithmetical calculations in essentially the same fraction of a second, there is no need to work backwards as with symbolic algebra; you can solve problems in a different manner, by working forwards on entire columns at once. There are also computer systems that can carry out algebraic symbol manipulations as well; they are called Computer Algebra Systems.]

As I say, it is a pity that, because he is so far removed from mathematics as it is actually practiced in today's world, Hacker misses the large target that I am pretty certain he is trying to hit--a target that deserves to be hit. Namely, the degree to which the mathematics taught in many of the nation's schools has drifted away from the real thing used every day by large numbers of people, to the point where much of what is taught is not only of little use, but can do real harm. Kids who are put off math in school will find their life choices significantly narrowed.

How far is Hacker off base? Two examples jumped out of the page at me in a recent interview he gave for the Chronicle of Higher Education.

In arguing for teaching coding in schools, Hacker says, "

Second example: As part of his argument that learning mathematics is irrelevant in much of today's world, Hacker says, "

The pity is, Mr Hacker is right on target with his analysis of much that is wrong with what goes on in school math classes (through no fault of the teachers, I should add, since the majority have to teach what is mandated, and tested ad nauseam), and he is fortunate to have access to a large megaphone--the NYT--to make his analysis known. Unfortunately, his narrow, and in many cases out-of-date perception of what mathematics is, together with his many misunderstandings of the nature of subjects such as algebra and the importance of, say, parametric representations, mislead those who have similar misunderstandings, and alienates those of us in the math biz who would otherwise be lining up alongside him. For these reasons, I give his essays an A for observation, C for background knowledge, and an F for drawing the wrong conclusions.

No, make that a D/F for his conclusions. His arguments did yield correct conclusions, he just did not realize they did, and claimed the opposite.

I would give similar grades to Hacker's new book. In reading an advance copy to write my review, I annotated 20 pages (out of a total of 200) where he makes significant errors due to a lack of knowledge of, or a misunderstanding of, mathematics. That's an error rate of 10%; way too high for significant errors. You can read my review in the March 1 issue of Devlin's Angle.

MEMO TO SELF: Don't write essays or books on revolutionizing political science education.

FURTHER READING: For more background on the early development of algebra and its present day nature, see my earlier article What is Algebra?, published in 2011.]]>

The IMO, held this year in Chiang Mai, Thailand in July, complete with opening and closing ceremonies, takes place on two successive days, on each of which the competitors sit a grueling four-and-a-half hour examination.

Each exam comprises three questions, so the expectation is that it will take about an hour and a half to solve each problem - though many competitors are unable to answer more than one or two of the total six.

If you are having trouble coming to grips with the idea of spending nine hours struggling to solve really difficult math problems, under strict exam conditions,

Directed by British film-maker Morgan Matthews,

Clearly, with the best young mathematical minds in a nation needing ninety minutes to solve a problem (if they solve it at all), IMO questions are not the kinds of math problems you find in a typical high school math textbook. Some of them require knowledge of advanced math, but there are always a few that, on the face of it, look fairly simple. At least, you don't need to have completed an advanced math class to understand what the question says. Here is one from the recent 2015 Olympiad:

Determine all triples (a,b,c) of positive whole numbers such that each of the numbers ab - c, bc - a, ca - b is a (whole-number) power of 2.

See how far you get in an hour and a half. Or spend the same amount of time watching the movie.

The story follows Nathan Ellis (played by Butterfield), a British student (the movie was made by BBC Films) as he goes through the grueling process of preparing for and taking the test to qualify for team pre-selection in the British National Mathematical Competition, going off to a training camp in Taiwan, where the final team of six is selected in a mock IMO competition, and then heading to Cambridge, England, for the international competition itself.

Both the mathematics and the mathematics competitions are handled well. You won't learn any math in the film, but you do get a voyeuristic look at the world of competitive math problem solving.

On first viewing, I felt that the romantic thread between the Asa Butterfield character (Nathan Ellis) and the young female Chinese math whiz he meets at the training camp, played by Jo Yang, was a crass Hollywood device to create a movie with mainstream audience appeal. ("Not many people like math, but millions like a good love story." There's no sex in the film, by the way. The two lead characters do spend a night together, but all you see is them reading a math book together in bed as they prepare for the competition the next morning.)

But then I watched the original BBC documentary that

I still found the film's final ending formulaically disappointing, but no more so than many other box office successes. Other than that, it's definitely worth watching.

Watch the official trailer for

See Morgan Matthews' BBC documentary on which the movie is based here.

Interestingly enough, an American documentary film maker, George Csicery, followed the US IMO team at the same time that Morgan was filming the UK team. Csicery's film

Finally, read the recent Huffington Post article by Daniel Lightwing, the British math whiz on which the Asa Butterfield character was based. (The movie was originally released in the UK last fall under the title

$1.36 billion is, after all, a lot of money. In fact, that 2014 figure is an all-time record. But it's not an anomaly. The figure for 2013 was $1.2 billion, just a little lower. So however you look at it, a lot of money is now regularly going into edtech.

In fact, the bulk of those investment dollars go into higher education. The K-12 market has long sales cycles and is dominated by a small number of entrenched incumbents, which make most of the big-player investors unwilling to put funds into edtech for schools. As a result, the total invested in K-12 educational technology in 2014 was relatively tiny: a mere $642 million, a figure that was already up 32 percent from the previous year. Moreover, just 10 investments accounted for over half the funding.

Those K-12 figures come from a report published by New Schools Venture Fund, one of the larger players in the K-12 investment world.

The situation may be starting to change a bit. In 2014, a few of Silicon Valley's top-tier venture investors dipped their financial toes into the K-12 market for the first time in over a decade, putting funds into companies such as Remind, Edmodo, BrightBytes, and Clever.

That those companies had started to gain a sufficiently strong foothold in the K-12 market to attract substantial investment indicated that they were clearly providing products that sufficiently many schools and school districts, in particular, were finding useful. But how many those new products were for instructional use in classes, and do teachers show any sign of wanting to use them? So far, the signs are not encouraging.

As a mathematician, I'll take math teachers as my example. Mathematics is, after all, an obligatory discipline throughout K-12 education, and any use of mathematics post-school is likely to involve technology use. What is more, mathematics education offers considerable potential to be enhanced by new technology. So it should provide as good a canary as possible for any imminent revolution (or

The National Council of Teachers of Mathematics (NCTM), the dominant US professional organization for math teachers, has the use of technology in classrooms as a main pathway to improving learning. The NCTM's Principles to Actions says, on page 5: "An excellent mathematics program integrates the use of mathematical tools and technology as essential resources to help students learn and make sense of mathematical ideas, reason mathematically, and communicate their mathematical thinking."

So one way to find out what the vanguard of K-12 mathematics teachers are doing in their classrooms -- and are planning to do -- is to look at the list of presentations given at the huge annual NCTM meeting. How many of those presentations are about, or at least make reference to, technology?

Ihor Charischak, president of the NCTM-affiliated Council for Technology in Math Education, has done just that. He released his findings in a recent blogpost.

According to Charischak, at the NCTM Annual Meeting to be held in Boston, MA, next April, there will be 733 sessions. He combed through them and identified just 97 that highlight technology in some form. At 13.2 percent, not only is that low, it indicates a continuing drop in interest in educational technology. At last year's NCTM Meeting in New Orleans, 21 percent of the sessions were technology-oriented, a year earlier, in 2013 in Denver, 28 percent of the sessions had a technology theme, and the year before, in Philadelphia, there were 38 percent tech sessions, an all-time record.

Not only is there relatively little evidence of teacher interest in incorporating any kind of technology in the classroom, but the trend is clearly down. Moreover, what technology interest Charischak could identify was hardly in new technologies: It was predominantly the use of handheld calculators and Computer Algebra Systems (like

What these data show is that, to date, practically all that much-hyped edtech funding has had virtually no direct impact on what goes on in the K-12 math classroom. Overall, K-12 math teachers are not incorporating new technology in their teaching.

True, the more progressive ones do incorporate the occasional technology product into their lessons, to help their student master difficult concepts. But, with little freedom and no institutional purchasing power, they overwhelmingly go for supplementary apps they can download for free (or are cheap enough to pay for themselves) on the Internet.

That indicates massive untapped potential for classroom edtech, but free educational apps do not offer investors an attractive financial opportunity. Instead, they focus overwhelmingly on the organizational tools bought by the people in education that do have purchasing power: the learning management systems, data collection and analysis packages, and student performance dashboards, bought by system administrators.

Those technology products can make K-12 education more efficient and perhaps more cost effective. But student learning takes place in the classroom, with a teacher. If we truly want to make dramatic improvements in school education, we have to find a way to redirect the bulk of edtech funding into the development of tools that teachers can use in their classes. That means making the creation of curriculum products by for-profits a much more attractive proposition for an investor. It's hard to see how that could be achieved without some form of government incentives.

For sure, quality learning tools have been produced (the math learning tools from the non-profit MIND Research Institute are a great example), and will continue to be made, even if we leave the current system as it is. But progress will be painfully slow. Genuinely revolutionizing K-12 education

How badly do we want 21st-century-relevant, first-class education for the nation's children?]]>

Before you read any further, you might want to try to solve it.

This very problem came up on my Twitter feed some while ago. (I follow a number of sources that generate this kind of thing.)

If you are well trained in Pre-Common-Core school math, you probably jumped right in and came up with a number, most likely 12. That's the answer I would surely have come up with when I was at school, having been well trained to perform like a mathematical circus animal.

But today, as a professional mathematician who uses mathematics to try to solve real life problems, my reaction was very different, and so was my answer: 3,840. The kind of thinking I brought to the problem was exactly what the Common Core is supposed to promote: sensible, reflective thinking that can be of real use.

(But note my use of that word "supposed" in that last sentence. Though rejection of the Common Core mathematics standards would leave US children unable to compete with the rest of the world in science and technology, there is much to worry about when it comes to some early attempts to implement the standards.)

My first reaction to the key ring problem was this:

What a nice little problem to form the basis of a class discussion. Sure, it is a bit contrived. How often has anyone needed to know how many different arrangements there are?" But that aside, it has a number of features that give it great pedagogic potential:

- it is simple to state
- it is easy to visualize
- a teacher could get the class to each bring in a key ring and some keys, and begin the investigation experimentally, first with one key, then two, etc.
- the question evidently has multiple solutions, depending on how you interpret what is being asked, making it a super exercise in mathematical modeling of a real-life situation
- those different solutions are in some cases orders of magnitude different
- the simplicity means that students should be able to formulate their own models and put forward clear explanations of why they chose the particular model they did.

Just what the (eight) Common Core Mathematical Practice standards ask for!

The Tweeter who posed the question posted a subsequent tweet giving the answer 12. That individual presumably interpreted the question as (in math-problem-speak): "How many ways are there to order five points on a non-oriented circle having no distinguished point?" If that is the question, the answer is indeed 12.

But that was not the question as posed, which was about keys on a key ring. If you stop yourself from switching off your common sense and going into math-class-mode, looking for the fastest way to put numbers into a formula and get a number out, and instead approach the problem for what it is, a straightforward question about a real-world situation, you will find yourself on a very different trajectory. A trajectory of

Here is how I tackled the problem, approaching it as I would any mathematical problem I have been hired to solve.

First, I reflected on what the problem is about. Well, it's about keys and keyrings. I know about those. In a drawer at home, I have several key rings I have acquired over the years. So many that I suspect that, from that one source alone, a future historian might be able to retrace many places I have visited and institutions I have interacted with.

Those keyrings all have one feature in common, namely a topologically circular ring, and the majority share a second feature, namely to the ring is attached some kind of object, often a badge or shield, sometimes a ball-shaped object, an occasional pen or flashlight, etc.

I'll add that some of those badge-like attachments have two distinct faces and are rigidly attached to the ring, some have two distinct faces but the entire badge can be rotated on an axle, and some are symmetrical so that the entire keyring looks the same from both sides. ("What difference can these features make?" a student might ask. If they don't, a teacher should ask them.)

I also have an even larger collection of keys. (Why do we keep old keys, by the way? I have no idea what the majority of them lock, and for sure they no longer fit anything I still have access to.) A very small number of those keys are symmetrical: when I turn them over, they look identical. (Again, is this significant to how we model the problem?) But the vast majority do not have such symmetry.

In my case, when I modeled the keyring probem, I based my model on my nostalgia-based collection of real keyrings and my irrational collection of old keys. In other words, I did what I think any sane, sensible person would do

I then had to decide what constitutes an "arrangement". Again, I interpreted this word (I almost said "key word", but decided that would be too tacky, though puns are their own reword) in the way I thought most natural. The problem asked about different

In the end, I came out with 120 x 32, namely 3,840. A far cry from the "expected", pre-Common-Core classroom answer of 12. Which of us is right?

Neither. There is no right answer for the problem as posed. It's a modeling task -- using mathematics to analyze something we encounter in the world. As such it's a super example of how mathematics is used in the world on a daily basis. (What people like me get paid to do.)

The fact is, when used wisely, "simple" word problems provide great exercises in the kind of mathematical thinking that the world needs. The kind of thinking we should want our kids to master: thinking that provides reliable answers to real questions.

As a nation, we should stop the current suicidal cry to turn back the clock to a form of math teaching that did no one any good and which those of us who became professional mathematicians first had to unlearn, and focus on making good, practical, sensible use of centuries old teaching tools (such as word problems) to produce a generation well equipped for life in the Twenty-First Century.

NOTE: Much of this article is abridged from a much longer post, which first appeared in my Mathematical Association of America "Devlin's Angle" column in May 2011.]]>

The Common Core State Standards (CCSS) for mathematics continue to generate a lot of online debate. As a mathematician and math educator, I usually follow the latest missives that do the rounds on social and traditional media, and in almost all cases the post or story has little to do with the CCSS, but rather methods for standardized testing. That was definitely the case with comedian Louis CK's recent series of tweets on the topic, as he himself later acknowledged in subsequent posts and TV interviews.

Ironically, confusion of the CCSS with methods for testing mathematical proficiency is a prime example of the kind of imprecise thinking that the CCSS intend to rectify. (I will henceforth focus entirely on the mathematics standards. The CCSS also cover English language arts and literacy.)

Since politicians and policy makers are generally not immune to popular opinion, and many such may not be well-versed in mathematics education, a public debate that is so widely off target has the potential to do dangerous harm to the nation's future.

The CCSS were created to ensure that all students who graduate from an American high school do so with the skills and knowledge necessary to succeed in college, career, and life in the Twenty-First Century, regardless of where they live.

The reference to the current century is significant. The last 50 years (particularly the last twenty) have seen such dramatic changes in the way human beings live in the developed world that skills important for many centuries have rapidly become largely irrelevant, their place being taken by new skills that the grandparents -- and in some cases the parents -- of today's students never required.

For a good explanation of why some CCSS-oriented math homework problems can, as a result, have parents baffled, see this excellent blog post by a practicing teacher.

At their heart, the CCSS comprise a set of eight basic principles of "Mathematical Practice":

MP1. Make sense of problems and persevere in solving them.

MP2. Reason abstractly and quantitatively.

MP3. Construct viable arguments and critique the reasoning of others.

MP4. Model with mathematics.

MP5. Use appropriate tools strategically.

MP6. Attend to precision.

MP7. Look for and make use of structure.

MP8. Look for and express regularity in repeated reasoning.

(The CCSS document provides a paragraph that elaborates each principle.)

If we do not continue to evolve our education system with those goals in mind, we will fall drastically behind in the Flat World of the New Millennium. (See here and here.)

Though I played no role in drafting he CCSS, every single one of those principles have formed the heart of my mathematics teaching for a career of over 40 years. They are the very heart of what I (and others) call

Anyone who opposes the CCSS needs to say which of these eight guiding principles they believe should not be followed, and why.

Beyond the guiding principles, we start to get into implementation issues. (All of the opinions I have seen expressed about the "Common Core" have focused on implementation.)

The CCSS document takes those general principles and breaks them down into sets of specific items that should be mastered at each grade level. These items (there are many) provide learning goals that teachers should seek to achieve within a specific school year.

For example, one of the standards for Grade 5 algebra says

This breakdown of the overall learning goals into specific items is done to provide the kind of nationwide uniformity required in an equitable society that provides equal opportunities to all students. It provides classroom teachers with a list of specific topics to cover during the course of a school year.

The order in which to address those specific goals, and the methods used, are left to the individual teachers, who can obtain or design lesson plans suited to their particular students.

In particular, the CCSS are not a curriculum; they do not prescribe any particular method or approach. Teachers may choose to cover items individually or in connected groups. Because, at the end of the day,

A mathematically knowledgeable teacher given the goal of producing mathematically able students well-equipped for Twenty-First Century life and career(s), without the need to meet detailed systemic metrics (a position I have the good fortune to be in but the vast majority of teachers do not) would need look no further than the eight guiding principles.

The guiding principles do such a good job of articulating the requirements of good mathematical thinking, that pretty well everything a student really needs to master in terms of mathematical thinking (not just for graduation but for life!) can be developed by following the eight principles

The reason why many people find that hard to believe is that they were only ever exposed to the algorithmic-skills math instruction developed for earlier times -- a form of teaching that is hopelessly inadequate for life in today's world. (A fact ably demonstrated every time a parent -- taught in a previous era of rules and procedures -- is unable to help their child with what is inevitably a very simple mathematical homework problem of the kind relevant to today's world.)

So why are there good mathematic teachers who I know (and know of) and respect, who express opposition to the CCSS?

Like all seasoned math teachers, I propose leaving that final question for you, the reader, to answer.

But I will give some hints that may help you find a solution. (Guiding Principles MP1, MP3, and MP6 in particular should help here.)

By my reading of what they write, those teachers are opposed not to the CC principles themselves, rather to what they suspect (likely with good reason) will be required of them in the classroom when the Standards become codified into contractual obligations.

The CCSS were created by highly qualified and credentialed experts in both mathematics and mathematics education. They are intended to be operationalized in the classroom by highly qualified and credentialed mathematics teachers. In between the two, are a whole host of politicians, system administrators, and educational materials providers, the vast majority of whom are not expert in either mathematics or mathematics learning. Where do you think the weak link is likely to be, and how can that weak link be addressed? Is there more than one weak link? Can you suggest solutions?

By way of comparison, do the same problems arise in, say, medicine or aviation, where there is also a need for domain experts to work with and through politicians, administrators, and suppliers? If not, why not? (See that teacher's blog I referred to earlier.)]]>

We mathematicians are one of many social groups who tend to make a big thing of coffee. The famous, late Hungarian mathematician Paul Erdos coined the oft-repeated slogan that "a mathematician is a machine for turning coffee into theorems." While only a mathematician's spouse might take that as more than metaphorical, it is the case that mathematicians do tend to make a big deal about coffee. In my experience, the only human grouping that makes a bigger deal are the inhabitants of Melbourne, Australia.

Oddly, the Italians don't have a "coffee thing," but that is surely because no native-born Italian ever experienced anything other than excellent coffee.

At the other end of the spectrum from Italians are the Americans and the Brits, who live under the illusion that the lower ranks of society drink instant coffee, the middle class and 7-Eleven stores everywhere make it in a filter system, and the more sophisticated make it in a device called a "percolator."

The drawback of the Italian approach (Melbournians imported good coffee expertise with waves of Italian immigrants) is it requires a large, chrome-plated device the size of a small car, which operates on the little known physical law that the volume of coffee that ends up in the cup is inversely proportional to the size of the machine that produces it, a problem that can be easily disguised by (i) using a very small cup, or (ii) topping it up with highly frothed milk.

For the rest of us, including all mathematicians outside of Italy and Melbourne, there was only one viable method: the French Press. Coupled with a small coffee grinder and good, freshly roasted coffee beans, this simple device could be relied upon to produce an excellent brew. Grind the beans just before you tip them into the jar, add water just off the boil, stir the scum lightly so the grounds sink, wait half a minute or so, then gently push down the sieve to press the grounds to the bottom. All the while, enjoying the aroma of those freshly ground beans.

Then, a few years ago, the "Keurig system" came along. Want a cup of coffee? Take a small plastic pod from the box, pop it into the elegant looking machine on the kitchen counter, hit a button, and voila: a fresh cup of coffee. Discard the pod when you are done.

No hassle. And no mess -- coffee comes from a grinder with a high electrostatic charge that flings fine coffee powder everywhere, and the thick black mess left in the bottom of the press uses every trick to avoid being disposed of. I tried a pod system once, and was hooked. Sure, all those preparatory aromas are then just a memory, as is the whole routine. But, although the coffee does not have the same quality of taste or smell, it is good enough. Definitely better than filters or percolators. And the sheer convenience and lack of mess more than compensates for the losses.

Or so I kept telling myself. I suppressed my sense of loss and a touch of guilt. The nostalgia and guilt got the better of me recently when I was preparing to head across country for half a year, taking leave from Stanford to take up the offer of a Visiting Professorship at Princeton. Rather than buy a second Keurig for a mere half year, I would use the occasion to rekindle my earlier obsession with the whole routine of making freshly ground coffee in a press.

I dug out the grinder and the press from the back of the kitchen cupboard, bought a fresh pack of Peet's Italian Roast coffee (a Bay Area wonder), and packed them along with my road bike, safe inside the industrial-strength plastic shell of the airline-baggage-handlers-proof bike box.

For my first few days in Princeton, I reveled in re-running my old routine, timing the grinding just right, basking in the aromas, gazing at the hot black liquid as it poured majestically from the press, and bringing the cup to my lips for that first sensual sip. This was the way the coffee experience was supposed to be. How could I have ever given in to the crass temptations of convenience, in the process adding yet more plastic litter to the environment with every discarded pod? I was a child of the Sixties. I knew better. I had discovered my former self.

This morning, nine days after arriving in Princeton, I drove four miles down US 1 to Macy's and bought a small, single-cup Keurig coffee machine and a supply of coffee pods. I just could not stand all that hassle and all that mess. As with buying your first Macintosh, once you get used to the ease and simplicity, there is no going back to anything else.

My brief nostalgia trip into "handmade" coffee was every bit as enjoyable as my occasional dives into Unix. Both put you in direct touch with the "real thing" -- computing in the case of Unix, coffee making with a grinder-press combo -- and as a result are far more satisfying. Your amygdala loves you. But it doesn't take long before your more rational neo-cortex starts to ask, "Why are you putting up with all this hassle? There's a simpler way."

Make no mistake, pod coffee ain't great. But, unlike filters and percolators, it's good enough. And good enough always wins. (Pod) coffee anyone?]]>

With every page, I found my mind's eye conjuring up a fictional image of the book's author, writing by candlelight in the depths of the Siberian winter like Omar Sharif's Doctor Zhivago in the David Lean movie adaptation of Pasternak's famous novel.

The reality is somewhat different from that mental image, of course. But only in what are largely irrelevant details. Though Frenkel was born in 1960s Russia, then part of the Soviet Union, since 1989 he has lived in the United States, and the book was written in Berkeley, California, where the howling wolves of the Russian night are replaced by the mountain lions and coyotes that roam the Oakland hills, and penetrating Bay Area fogs chill the body in place of the icy winds sweeping across the snowy Steppes.

But make no mistake about it: History, culture, tradition, and national character are not cast aside in the course of a ten-hour airplane ride from Moscow to the USA.

What is more, I fear your loss is far greater than the one I suffer by being unable to read Dostoyevsky in the original Russian, though Frenkel can mitigate your loss to some extent by providing you with a "translation" of his mathematics from the complex, symbolic formulas of his many research papers to the accessible prose and simple line drawings you will find in his book.

Equally, if you are a mathematician and do not recognize the sheer romance, the love, and the poetry in Frenkel's words, if you see the mathematical techniques but not the love - and I know mathematicians who almost certainly fit this description - then you too are destined to live a life of one fewer dimension than those of us who experience both on a daily basis. As the famous German mathematician Karl Weierstrass said, "A mathematician who is not also something of a poet will never be a perfect mathematician."

An easy way to describe the structure of Frenkel's book is to say it comprises two themes. One theme is a "popular science" description of an important area of modern mathematics - the Langlands Program, a sort of Grand Unified Theory of mathematics that weaves together large parts of algebra, geometry, number theory, analysis, and quantum theory, the entire tapestry explained at great length in the book.

The other theme is an autobiographic account of Frenkel's own life, first growing up as a mathematically gifted child of Jewish heritage in the then notoriously anti-Semitic Soviet Union, overcoming all the obstacles put in his way in order to secure a mathematics education, and then moving to the United States to lead the jet-setting life of a world famous mathematician.

But to view it that way, as two storylines, would be to miss what it is really about, much like Woody Allen's famous (satirical) three-word summary of

Frenkel desperately wants you to "get" this mathematics that he has fallen in love with; to understand, at least to some extent, what mathematics is and why he has devoted his life to it. If he does not succeed - and the criminally poor, culturally impoverished nature of much Western mathematics education means that for many readers he will not - it will not be for want of trying.

He writes (p.7): "My dream is that all of us will be able to see, appreciate, and marvel at the magic beauty and exquisite harmony of these ideas, formulas, and equations, for this will give so much more meaning to our love for this world and for each other."

He may dream this - being a dreamer is another familiar trait of the characters in Russian novels - but he knows that for many readers his task is hopeless, and says so early on, with an edge of both despair and frustration: "Mathematical knowledge is unlike any other knowledge," he writes in his Preface. "While our perception of the physical world can always be distorted, our perception of mathematical truths can't be. They are objective, persistent, necessary truths. A mathematical formula or theorem means the same thing to anyone anywhere - no matter what gender, religion, or skin color ... There is nothing in this world that is so deep and exquisite and yet so readily available to all. That such a reservoir of knowledge really exists is nearly unbelievable. It's too precious to be given away to the 'initiated few.' It belongs to all of us."

Yet for all that mathematics belongs to all Humankind, the teenage Frenkel himself was denied access to the Soviet Union's flagship institute of learning, Moscow State University, despite having aced all the entrance exams. "The doors were slammed shut in front of me. I was an outcast," he laments - but only briefly, continuing, "But I didn't give up. I would sneak into the University to attend lectures and seminars. I would read math books on my own, sometimes late at night. And in the end I was able to hack the system. They didn't let me in through the front door; I flew in through a window. When you are in love, who can stop you?"

Who indeed? When Frenkel's student papers were smuggled out to the west, they found their way to Harvard University, where the Mathematics Department persuaded the President to invite the twenty-one-year old prodigy to come over as a Visiting Professor. Putting him on a fast track to a Ph.D., Harvard provided a launch pad for what has since been a spectacular career of ground breaking, international research, based first at Harvard, then more recently at the University of California at Berkeley.

Along the way, this now Russian-American wrote a screenplay,

Intrigued? I hope what little I have told you about Frenkel - the man, the life, and the mathematics - persuades you to get his book and read it. Not because I am his agent; I am not. I have only met Frenkel once, a few weeks ago, at a dinner party in Berkeley. Rather, because I share his love for mathematics, and I too have poured effort into trying to convey the reasons for my love to my fellow humans. And because he writes so well.

As is true for all the great Russian novels, you will find in Frenkel's tale that one person's individual story of love and overcoming adversity provides both a penetrating lens on society and a revealing mirror into the human mind.

To be sure, many readers may have to skim through some of the mathematical parts. If that applies to you, then as a result you will miss a lot. But you will still come away the richer.

And what of that "formula of love" that the

"Yes, but is it really a formula of love?" you will surely ask. "Can there ever be such a thing?"

Frenkel has an answer - the answer. "Every formula we create is a formula of love."

FOOTNOTE: This review was originally written in response to a request from a large daily print newspaper when the book first appeared last year. Unusually for me, after a long delay, they decided not to publish it. My suspicion is they did so because they felt the "romantic prose" I adopted did not fit well with their image of mathematics and how it should be portrayed. If so (and I do not know for certain), I think that is a great pity, and may reflect why many people are turned off math. Mathematicians

But as Hack Education's Audrey Watters commented, those conclusions and predictions miss the point that, from the start, Udacity was a very different player in the online education space from the other initiatives coming out of Stanford and MIT.

Udacity's focus, and business model, were always directed at (and around) computer science education, a very special, atypical case. Those of us that also got into the MOOC act in the early days, and many who have entered the space since then, came in with very different goals and expectations - and very definitely without a business model.

Thrun created Udacity as a for profit company, funded by Silicon Valley venture capitalists, and is now following a well-trodden path that has led many Valley startups to become successful companies with global reach. From a business perspective, it does not matter that the original plan or product was flawed. In any startup, it almost always is. It's what you do next that matters.

Venture capitalists, like many successful technology entrepreneurs, have a habit of badly misunderstanding education, but it is equally easy to misunderstand what they do. Which is basically to bet on people. As VC Peter Levine, a partner with Andreessen Horowitz and a Udacity board member, says of the $15 million investment in Udacity he led in October 2012, "If it hadn't been for Sebastian, we wouldn't have done this investment." (Ref: first link above.)

In making big bets on individuals, today's VCs are following in the footsteps of people like DARPA program manager Bob Taylor who, in the 1960s, poured millions of (taxpayers') dollars into Doug Engelbart's "crazy project" to develop a radical new way to operate computers using things called "mice" that allowed a user to move objects on a screen that looked like a desktop and to open files called "documents" that were stored on another computer. In the case of Thrun (whom I have met but do not know), my guess is it will turn out that Levine made a good bet - from a business, but not an educational, perspective.

The fact is, Silicon Valley has yet to come to terms with education. It's a massive, global market, but a notoriously complex one that is tricky to negotiate. A lot of what goes on in good (

The hard part is getting any new product into that market. The immediately apparent resistance to change in education (at all levels) is for sure driven in large part by inertia. But not everyone in education is a Luddite. Many of us look for and strive to engender change from within, but being familiar with what it takes to provide good learning, we know that change cannot be wrought with simplistic quick-fixes. We have a reasonably good sense of what can scale and what probably cannot. We also know that simple numerical metrics are poor guides to progress.

As someone who has now taught three iterates of a Stanford MOOC, and who talks regularly with other MOOC instructors, I will tell you that they are alive and well, and getting better all the time as we learn what they really are and how to give them. They are not "regular university courses online" and they won't replace universities. They may well, however, reach a stage where they disrupt higher education, and if so, institutions that don't adapt to a changing landscape are indeed likely to go out of business.

Education is about individual learners. You want to understand where MOOCs are going, don't start with the 50,000 who sign up for a MOOC and never get beyond the first week (if indeed they log on at all). Look at what (the many thousands of)

Here is one. The following two emails (name redacted) from a student who had just completed my recent Mathematical Thinking MOOC, were forwarded to me by the Coursera learning support staff who work with students of need, and who had interacted with the student throughout the course. [Referring to myself as "KD" in all my course postings was a design decision I made.]

MESSAGE 1.

As my family was not rich enough to put me to University, I am very happy for this chance to get educated like this. Because of my educational background, I am now having difficulty getting a proper job. But thanks to the [certificate] with distinction, it will (I am hoping) serve as a gate of my breakthrough to get a better job. I will put this certificate on my resume/CV.

Give my greatest sincerity and love to KD, and tell him that he has given me a great confidence with which to overcome my mental boundaries.

MESSAGE 2.

Okay, that's just one person's story. That's the part that makes me feel good. But here is where scaling comes in. With millions of students worldwide having taken MOOCs, there are already thousands of stories like this. I have hundreds of emails and MOOC forum posts from students who tell me my MOOC has changed (or helped change) their lives, often in dramatic ways. If MOOCs continue to develop and grow - and get better - soon there may be tens of thousands of such transformations, then hundreds of thousands, and eventually millions.

And there you have tomorrow's talent supply. Those huge dropout rates that were once regarded as a big problem turn out to have been our first glimpse of an amazing global filter for people with commitment, persistence and ability.

MOOCs do not and, I believe, cannot replace a good university education. But they can, and in some cases already have, provide a pathway to such education for millions of people around the world who, for various reasons, do not at present have any access. Scale that across the entire world, and you have disruption.

Come on Silicon Valley, you are superb at developing products to find a place in the market. But for the most part, that marketplace is (like Silicon Valley itself) a legacy of the Cold War. Transforming lives on a global scale is one of the biggest potential new markets there is, and every year it renews. Short term profits are probably non-existent, and mid-term probably meager at best, but the downstream payoff is huge. Particularly when you factor in the associated long term returns that come from being a facilitator of those enabling life transformations.

Can I

Quality global education is, potentially, a huge market. But it's not sitting there, waiting for us to simply tap into with a good product, a smart CEO, and a large marketing budget. We have to create that market as we go. If we want to live up to, and surpass, the successes that came from our Cold War initiatives, I'd say that may be our next grand challenge.]]>

The more publicized trainwreck, though arguably the lesser significant of the two, was the partnership between San Jose State University and for-profit MOOC provider Udacity, initiated last January in a blaze of publicity by Udacity co-founder Sebastian Thrun and California governor Jerry Brown. The public-private agreement called for Udacity to support three remedial classes developed and run by professors at San Jose State.

The potential prize was big: The course fee was a mere $150 per student -- covered by foundation grants for the initial trial -- a fraction of the cost of a regular course. But when the results came in, the euphoria quickly evaporated. The passing rates were 29 percent, 44 percent and 51 percent, respectively, much lower than hoped for. As a result, the university and Udacity have announced that no further such courses would be offered until they had analyzed what went wrong.

To be fair, the project was hugely ambitious. Many of the students had already failed a remedial class or a college entrance exam, and it is not clear they would have done significantly better in a traditional class. It would be just as unwise to conclude that the initiative cannot be made to work as it was to set unrealistically high expectations for what was very much an experiment. The two parties have declared -- rightly -- that the next step is to examine the data (of which MOOCs, by virtue of being online, generate copious amounts, albeit much of it having hitherto unproved value), modify the course structure and try again. That's how progress is made.

The other railcrash, also in California, was the announcement that Senate Bill 520, a controversial piece of legislation seeking to incorporate for-credit, partially-outsourced online education in all three of the state's higher education institutions -- the California Community Colleges, California State University, and the University of California -- has now been put on hold until at least 2014.

Though SB520 was not focused on MOOCs per se, the MOOC explosion had spurred legislators to take a pro-active role to overcome what they felt was too slow a pace by the state's three higher education systems in embracing new delivery technologies to reduce the costs to students. Both Udacity and Coursera were involved in formulating the bill, which also had some high profile support from business leaders. Opposition came from faculty unions as well as the UC and CSU systems. (The CCC system did not publicly express an official opinion.)

As with the SJSU-Udacity project, however, it would be unwise to view the possible-death of SB520 as anything more than the end of phase one of what will be an ongoing process. The idea behind the bill was to tie funds ($16.9M for CCC, $10M for CSU, $10M for UC) to "increas[ing] the number of courses available to matriculated undergraduates through the use of technology, specifically those courses that have the highest demand, fill quickly and are prerequisites for many different degrees."

No one, to my knowledge, thinks that goal is anything but laudable. Governor Brown killed (or at least stunned) the bill -- but not the goal -- by imposing a line-item veto on his own earmarks. Significantly, however, he did not take away the funds. Rather, he left it to the three systems to decide how to use the money in order to assist matriculated students complete degrees at a lower cost. And technology will play a major role in those initiatives.

What both episodes tell us is that, while there may be (I would say there almost certainly are) ways we can use technology to reduce the student costs -- and perhaps the waiting lines to get into courses -- that currently bedevil higher education, last year's naïve predictions of an imminent revolution are being replaced by a more sane attitude, including a recognition that the current higher education faculty have valuable expertise that cannot be ignored or overridden roughshod.

Teaching and learning are complex processes that require considerable expertise to understand well. In particular, education has a significant feature unfamiliar to most legislators and business leaders (as well as some prominent business-leaders-turned-philanthropists), who tend to view it as a process that takes a raw material -- incoming students -- and produces graduates who emerge at the other end with knowledge and skills that society finds of value. (Those outcomes need not be employment skills -- their value is to society, and that can manifest in many different ways.)

But the production-line analogy has a major limitation. If a manufacturer finds the raw materials are inferior, she or he looks for other suppliers (or else uses the threat thereof to force the suppliers to up their game). But in education, you have to work with the supply you get -- and still produce a quality output. Indeed, that is the whole point of education.

In this connection, it is worth noting that the MOOC explosion came out of two of the most prestigious private universities in the world, Stanford and MIT, where the incoming raw material is preselected to be of the highest quality on the planet! The MOOC platforms that form the basis for the for-profit online education companies Coursera, Udacity, and Novo Ed, and the open-source edX, all came from a Stanford research project (called Class2Go) to develop a range of tools to support flipped classrooms for its own, highly-capable, on-campus students. So it was highly unlikely that those platforms would work immediately with less well-prepared students.

The feature of that platform that initially excited Sebastian Thrun, Daphne Koller, Andre Ng and others (myself included) was not using it as a tool to reduce the cost of remedial courses at colleges and universities, rather the possibility of making quality higher education available to the entire world, for free.

That goal has already been achieved -- very effectively for some courses in some disciplines, with other cases being very much works in progress with uncertain outcomes. (The former tend to be introductory-level courses that focus on the assimilation of knowledge and the acquisition of procedural skills, testable by machine-graded quizzes, the latter usually involve higher-order thinking, qualitative judgements and interpersonal activity, requiring human expertise to evaluate.)

But as I have noted elsewhere, it is very much a Darwinian, "survival of the fittest" kind of education, that leaves many by the wayside. In particular, success in a MOOC requires that the student has already learned how to learn -- something that for many students is the principle outcome of a college education.

To my mind, those students around the world whose lives have already been changed by MOOCs (by having access to higher education that would otherwise be unavailable to them) provide reason enough to be pleased with what we have already achieved with this new educational structure, and justifies its continuation.

But what are the benefits for all the rest -- the vast majority of students who, for various reasons, are not able to benefit significantly from taking a MOOC? Indeed, can the technologies on which MOOCs are built offer

My own view is that there are indeed benefits to be had, but figuring out exactly what they are and how to achieve them is going to take time. In adopting this perspective, I am very much in line with my own university, Stanford, as can be seen in this short video summarizing the current (significant) efforts the university is making in the development, use, and study of educational technologies.

Note that when MOOCs are mentioned in the video, they come up as a final afterthought. In fact, Stanford has already gotten out of the MOOC business, a mere 18 months after they first exploded onto the world scene. The original Class2Go platform has been spun out to the three for-profit companies mentioned above, and Class2Go itself has been folded into MIT-Harvard's edX, to be developed as an open source system.

(To be sure, many Stanford faculty are giving MOOCs on one of those platforms, but they are not offered for Stanford credit and do not lead to a Stanford diploma. Those of us who offer them do so unremunerated, in addition to our normal university duties -- in many cases viewing them as experiments in teaching and learning.)

As indicated in the video, what we are doing is stripping higher education down to its smallest components and seeing how technology can be used to enhance or make more efficient each part, before reassembling them into what may be a new higher education landscape.

Unfortunately for those who write about education, that process is likely to be slow and punctuated by many false starts, with progress for the most part being made in small increments. More evolution than revolution. Just like most other research, in fact.

There will, I am sure, be disruption in higher education. On a local basis, some colleges and universities will go out of business and others (having a different structure) may spring into existence. But, unlike the rapid and dramatic way technology upturned the music industry and journalism, I think that, apart from some changes (such as flipped classrooms), it is only in retrospect that we will recognize the disruption of the entire higher education system.

Whatever your view of NSA leaker Edward Snowden, the issue his disclosure has raised is one that any genuinely democratic society needs to have ongoing public debate about, particularly the US, which since its birth has seen itself as a global beacon of democracy.

Unfortunately, given the nature of national security, which involves the balancing of risks, for that debate to amount to more than mere opinions, one of several basic cognitive abilities the participants need to have is a good, general sense of quantity. Yet as mathematician and author John Allen Paulos observed in his 1988 bestselling book

With our education system focusing largely on drilling students to carry out mathematical procedures that can now be done effortlessly on cheap devices we carry in our pockets, the hugely important ability known as quantitative literacy goes undeveloped.

The price we are now paying as a nation for this educational neglect is driven home dramatically in the Pew Research Center's June 10 report, headlined Majority Views NSA Phone Tracking as Acceptable Anti-terror Tactic.

As a naturalized American, I have an immigrant's reverence for those words of our National Anthem, "Land of the free, home of the brave." For many of my fellow citizens born here, I fear these are just words they learned to recite in elementary school. For the fact that 56% of Americans declare that they would give away fundamental freedoms to reduce the risk of terrorist attack indicates that we may become the "land of the enslaved, home of the scared." (Imagine another J Edgar Hoover, but with today's information infrastructure at his disposal.)

If there really were a terrorist threat, this willingness to trade-off our Constitutional rights might be understandable - though surely nothing to be proud of. But the plain fact is, US citizens living on US soil do not face a terrorist threat of any significance. We simply don't. The false belief that we do is where our nation's lack of basic quantitative literacy comes home to roost. Bigtime, with potentially major consequences.

Yes, those images of the Twin Towers falling were, and remain, vivid to civilized people the world over. In my case, they resulted in my changing my career, and directing my mathematical research skills to a series of projects for the Department of Defense - so I take the threat seriously. But I have never made the mistake of thinking it is a major threat for which we should give up the basic freedoms the Founding Fathers fought for and enshrined in our Constitution. I worked on intelligence analysis because my skills are of particular relevance there and I was in a position to do my bit. But I could have far great impact on protecting my fellow Americans had my expertise been in, say, medicine or public policy.

The fact is, it does not require my level of mathematical training to recognize that we do not face a credible terrorist threat. All it needs to put those terrifying images into proper context is basic quantitative literacy; in particular, the ability to assess and balance risks. It's not rocket science. Heavens, it's not even shopping sense!

Take the ten year period starting with the 9/11 attack through to 2011. That includes the one and only significant loss of life in the US due to terrorism, when over 3,000 people died, so in terms of risk assessment it is an anomalous spike, but le's go with it. In that same period, roughly 360,000 Americans lost their lives in traffic accidents. Over 100 deaths for every one terrorism death. (In fact, the greatest loss of life due to 9/11 was not in the Twin Towers but on the US roads, caused by an increase in traffic deaths during the months when few people were flying!)

About 10,000 of the 30,000 annual traffic deaths in the US are caused by drunk drivers. Yet despite the fact that every one of us is over thirty times more likely to be killed by a drunk driver than to be a victim in a terrorist attack, no one has argued that

True, there is an extensive research literature analyzing why humans give irrationally inflated significance to dramatic forms of death such as airplane crashes or terrorist attacks. But those misperceptions are easily countered with just a modicum of quantitative literacy. We're not talking mathematics, which most of us (I definitely include myself) find difficult. No fancy calculation is required. Just a general sense of number - a reliable sense of risk - which anyone can acquire with just a little coaching.

When you realize that, as an American, you are

We Americans are so attached to our personal freedoms that many argue fiercely to maintain the right to fill their closets with assault rifles (accidental deaths by guns are a significant risk in the US, much higher than terrorism) and rally against universal health care (which would drastically reduce our third-world-level infant mortality figures). (Count me in the freedom camp but not the other two.)

Yet 56% of us would ditch the Fourth Amendment, and perhaps the First as well, to reduce a risk that is already lower than we incur when we take a bath!

Are we really, as a nation, going to give up personal freedoms that are the envy of the world - a beacon to humanity - because of a collective numerical stupidity which could be eradicated in a single generation by a small change in K-12 education?]]>

Using some elementary parts of mathematics as a basis, the course sets out to develop the kind of creative, "out of the box" thinking that practically every forward-looking government report around the world tells us is going to be critical as we move through the 21st Century. The kind of creativity that education expert Sir Ken Robinson talks about in his virally famous talks. (For example, this one given at the RSA in 2010, and subsequently animated by them.)

Other than standard high school mathematics, the only real prerequisite for my course is knowing how to learn. That ability is, as Sir Ken and many others have observed, the one thing above all that schools should be developing in their students.

I see that ability in many of my MOOC students. But they are the adult students who have spent some years in the workforce. What I see in the students in high schools or those currently enrolled in a traditional college, is a total dependence on the "show me five similar examples and then ask me to do a sixth that is essentially the same" approach.

Put frankly, that is the educational method animal psychologists use to train Bonobo apes. More to the point, it is also the method that was developed (for children) in the early 19th Century, when countries around the world were introducing universal education. Its purpose was to prepare a workforce to fuel the post industrial revolution society. A key requirement was to train millions of people to think inside particular boxes. And that is what it did, very effectively.

Which was fine back then, but is woefully inappropriate in today's world. So much so, that the kids who are most likely to be the leaders in tomorrow's world are regularly diagnosed as having a problem (ADHD) and anaesthetized by Ritalin and other drugs in order to force them through an outdated, factory-production-line form of education that bores the most creative to distraction.

In my MOOC class, I have (I extrapolate) thousands of young people from around the world, who have enrolled because they want to acquire the kind of mathematically-grounded, creative thinking they know they will need, but whose school education has simply not prepared them to take ready advantage of the opportunity that MOOCs offer.

It's definitely not their fault. Indeed, I see the ones who have shown the initiative to sign up for the course -- which is purely about the learning, and offers no credential. But all they have ever experienced in their educational journey is examples-rich instruction, generally with an emphasis on working alone. When presented with a problem for which I have not shown them any examples, they have no idea how to proceed. They cannot follow my advice as to how to set about solving a novel problems (ask yourself exactly what the problem says, note down what you know that may be relevant, look at it from different angles, formulate a simpler version, discuss it with others working on the same problem, etc., a list well known to the older students in my class who get paid to do just that every day), because they have never been asked to do anything of this kind.

Given the stranglehold on U.S. public K-12 education held by various powerful groups with a vested interest in preserving the status quo, buttressed further by others who want to enter the same lucrative market, MOOCs offer a wonderful opportunity to overcome the damage schools do (often against the wishes of the teachers), and provide the workforce the nation now needs and will increasingly need in even greater numbers. But to achieve that, those of us developing these new courses need to resist the pressures - from many sides, including many of the students themselves -- to conform to existing educational models.

One feature MOOCs offer, that is phenomenally powerful educationally, is to separate credentialing from the learning process. When the marks a student receives on each assignment or test count towards the final grade on which a credential is awarded, as familiarly happens in K-12, the awarding of course grades can no longer be used as an effective way for a student (and an instructor) to gauge progress. The grade becomes more important than the learning. But in a MOOC, the two can (and should!) be kept separate.

This issue has been moot until now, since no institution has awarded a credential for a MOOC, but that is in the process of changing. Still, we can have the best of both worlds. Since a student can take a MOOC as many times as she or he wants, with the only cost being time (learning time!), the student can elect which iteration of the course to take for a credential.

Another powerful feature of MOOCs we can take advantage of, is that the traditional course structure of assignment release and submission times can be repurposed to create periods of intense activity, when thousands of social-media connected students around the world are all working on the same set of problems, and can form small working groups to collaborate and help, support, and encourage one another to achieve a difficult common goal.

There are other features of MOOCs too that can be leveraged to provide good learning. Even so, I doubt that a MOOC can ever provide the kind of first-rate learning you can get at a top ranked university like Stanford, which regularly turn out young people equipped for today's world. But I think that with some effort, we can scale enough of it so that MOOCs can make up for much of the damage resulting from putting 21st Century students through a 19th Century school system. And we can do it on a global scale.

Alternatively, perhaps driven purely by economic considerations, we can let MOOCs settle to become simply a Web version of the traditional educational system everyone is familiar -- and comfortable -- with.

Right now, my MOOC students fall into two camps, those who value the former (21st Century learning), the others crying for the latter (19th Century instruction). Most worrying, the split appears to be largely based on age -- with the ones who will most desperately need the former (tomorrow's generation) being the ones asking for the latter (yesterday's education).

For more details on my MOOC, see my blog MOOCtalk.org]]>

By the end of week three, that number will likely have dropped to 10,000 (it was 20,000 last time round), and by the end of the course a "mere" 5,000 (10,000 before), with maybe as few as 500 taking the optional final exam in order to earn a certificate with distinction (1,200 in 2012).

This seems to fit the attrition pattern that commentators have most typically described as "worrying" or "a problem," hinting that therein lies a seed of the MOOC's eventual demise. But is an 85 percent attrition rate really a problem? In fact, is it significantly different from traditional higher education?

For comparison, the equivalent figure for my own university, Stanford, is 95 percent. That's right, 95 percent; a higher attrition rate than my online course. That's not Stanford's published "graduation rate," of course. Of students

The (only) point I am trying to make with this comparison (which has numerical significance, but says nothing about quality of education or utility), is that applying the traditional metrics of higher education to MOOCs is entirely misleading. MOOCs are a very different kind of educational package, and they need different metrics -- metrics that we do not yet know how to construct.

Once thing that we have learned from the research done on the first twelve months of MOOCs is that, besides their students being typically much older than the traditional college population (median ages seem to be in the mid-thirties, but the spread is large), people sign up for a MOOC for very different reasons.

A great many never intend to complete the course. Rather, their goal is to sample, in order to get a general sense of a subject or topic. In other words, they come looking for education. Pure and simple.

For those students, the issue of certification never arises. And thereby goes another myth about MOOCs: that they are doomed by the lack of a reliable accreditation. (In fact, there are ways to provide reliable certification, and the different MOOC platforms either offer it already or have it in their plans to offer it in the near future. But that may not be the most significant feature of tomorrow's MOOC.)

MOOCs mean so many different things to so many different people, only time will tell which sections of society they most serve, and what they will ultimately offer. Those of us currently experimenting with their design are already able to take into account the massive amount of feedback data that any online activity yields, and as a result MOOCs are already starting to evolve. But a mere one year in, I don't think any of us could confidently predict where this will all lead.

What

With MOOCs, we have a very different entity in our midst. What they become will be determined not by Coursera or Udacity or Stanford or MIT, but by the millions of people around the world who, by the way they use them, will shape their future. That's a flat, global community at work. But don't blink, or you'll miss the action.]]>

The surprising thing is that it wasn't just racial, which, as the whole world knows, is rife in US history, but also class-based, something that in my experience is endemic in the UK but not common here.

One of the things that attracted me to Stanford (way back in 1987, when I first arrived) was that it is a meritocracy. Your origins do not matter; it's all about how smart you are, how hard you work, and how good your work and ideas are. I've lived in the US and worked at Stanford for so long, I've grown used to that state of affairs. It therefore came as something of a surprise to read the following description of me recently in an education blog: "a damaged East Yorkie boy desperate to seem part of the aristocracy of math and the cognitive elite."

For the benefits of those not familiar with the UK, I originate from Yorkshire (known to millions of US television viewers as "Herriott country," after the (late) Yorkshire veterinary surgeon who made the county famous through his bestselling books and television series). The phrase "East Yorkie boy" manages to combine three standard slurs into a single three-word phrase: the "East" refers to the fact that of the three Ridings (subunits) of Yorkshire, the East Riding has always been the most impoverished region; "Yorkie" has enough US equivalents for its strong racial overtones to be self-evident here; and "boy" - well, that will also be clear to US readers.

As to the content of the blogger's remark, (and I am not going to identify the blogger), he suggests that, coming from working class origins in the North of England, I really do not belong among the "cognitive elite" of Stanford.

A quick online check of the blogger indicated he is an American, not British, so this is our racism (speaking as an American citizen), not Britain's. He is also elite-university educated and the president of a US financial company, so presumably not without standing and influence. He also, clearly, is sufficiently familiar with the UK to tap into its social nuances. This is well-informed, sophisticated racism.

One incident in twenty-five years - a drop in a vast ocean, a mere comma in a library full of daily racial slurs. And to be honest, apart from the message it sends about society, I simply shrugged it off, aware of many, far more significant examples happening daily that result in real hurt. Nonetheless, it is a reminder that racial prejudice and discrimination are alive and well. With an African American president, we have come a long way in just a few decades. But as countless ongoing examples far more significant (and more damaging) than mine make clear, we have a long way further to go. Something particularly worth remembering in an election year.]]>

Apart from hearsay evidence from two disgraced former cycling teammates of Armstrong, the USADA bases its case (at least according to what they have said) on the blood and urine samples taken from the cyclist in 2009 and 2010, when he made a brief comeback to the sport after four years in retirement. In a June letter to Armstrong, subsequently made public, the USADA said those samples were "fully consistent with blood manipulation including EPO use and/or blood transfusions."

Though a recreational cyclist, my interest in this case is fairly minimal. It is that term "fully consistent with" that piqued my mathematician's interest. It is a very odd phrase to use in a situation like this, not least because it has absolutely no evidentiary force. It says nothing of any significance.

[Certainly, after two years deliberation, including testimony from former team-mates obtained under oath through a grand jury, the U.S. federal criminal investigation of the allegations made against him finally dropped the case early this year, saying there was no real evidence against him.]

Though the layperson typically thinks of mathematicians as being focused on numbers, that is actually not the case. That false view is a consequence of the mathematics taught in high school. Only at university are you likely to encounter the mathematics done by the professionals. High among our real areas of expertise are logical reasoning, rigorous proof, and the precise use of language.

Incidentally, I am not referring here to using language and reasoning precisely in esoteric discussions of arcane mathematical topics. Yes, we do that too. But we also apply our expertise in everyday, practical domains. (Homeland Security, to name one domain I myself have worked on.)

There are a number of terms we use to describe evidence. The strongest is "proof" (or "conclusive proof", but any mathematician will tell you the adjective is superfluous.) We might say that, "Evidence X proves that Y happened."

An alternative that might seem weaker, but in actuality is not, is that "Evidence X implies that Y happened."

Definitely weaker, is "Evidence X suggests (or indicates) that Y happened."

All of these have evidentiary power of differing degrees. And there are others.

At the other end of the spectrum, we can say, "Evidence X contradicts Y having happened." X proves Y did not occur.

Evidence collected to uncover wrong-doing, such as doping controls in sport, by virtue of their design, rarely (if at all) provide proof of innocence. At best, when a doping test does not come up positive, the most you can say is it did not yield proof. It does not rule out (i.e., does not contradict) doping, just as a negative result from a cancer screening does not mean you are cancer free, merely that the test did not detect any cancer.

So what does that USADA term "fully consistent with" mean? Well, first of all, let's drop the "fully"; it's superfluous. Consistency is a definitive term. Something is either consistent or not; no half measures. It's also a term mathematicians like myself are very familiar with -- again for real world uses as much if not more than within theoretical mathematics. It means "does not contradict". Nothing more, nothing less.

Given the availability of terms such as "proves," "indicates," "suggests," or more evocative terms such as "raises the distinct possibility that," why did the USADA decide to use the curious term "consistent with"? Since they surely spent a lot of time, and consulted with a number of lawyers, in drafting their letter, their choice of wording was clearly deliberate. Why choose a term that means "does not contradict"?

After all, I can say "Drinking milk as a child is (fully) consistent with using crack cocaine as an adult." Should we take that as evidence that milk producers are to blame for adult drug use? Of course not. But this example has exactly the same logical heart, and the same evidentiary force, as the USADA letter's "fully consistent with blood manipulation including EPO use and/or blood transfusions."

Why not say "suggest" or "indicates"? They fall well short of "proof", but they do carry some weight.

"Does not contradict" is, then, it appears, a key part of their case against Armstrong. In which case, I find it troubling. The USA should have far higher standards of proof than that.]]>

The first number we encounter, by way of Jake's disembodied voice (he does not speak, so we only hear him as a thought-track) is the golden ratio, approximately 1.618. Thematically, that's good, since that number does occur a lot in nature, often by way of its closely associated Fibonacci sequence. Which makes it all the more perplexing that, midway through the first episode, we have Danny Glover's character repeating a series of oft-recycled falsehoods about the Fibonacci sequence.

He begins by saying that it was discovered by the twelfth-century mathematician Fibonacci, which is not true. Fibonacci (who was in fact a thirteenth-century mathematician, and who was not given that nickname until the 19th century) simply included in a book he wrote, an ancient arithmetic problem that yields those numbers when you solve it. There is no evidence that he ever investigated the sequence. Besides, most of the sequence's interesting mathematical properties and its connections to the natural world were not discovered until many centuries later.

Though there are many fascinating examples of the occurrence of the Fibonacci sequence in the natural world, the three that Glover cites are all wrong: that the sequence can be found in the curve of a wave, in the spiral of a shell, and in the segments of a pineapple.

Almost certainly, the writer looked at one of many popular math books that are available, or consulted some of the even greater number of Fibonacci-related websites, where such false claims are repeated with uncritical regularity. (Ironically, Danny Glover was the host for the PBS math documentary series

It would have been easy to get the math right in

In this respect,

In contrast, it's as a metaphor for the role of numbers and mathematics in today's world that

As the first episode opens, we hear Jake say, "Patterns are hidden in plain site. You just have to know where to look." That line could have been taken right out of my 1996 book

To most people, mathematicians spend most of their time scribbling obscure looking symbols into notebooks or on blackboards -- or, in TV and movie portrayals, on windows and bathroom mirrors. But to the mathematician, those symbols are what is required to describe those hidden patterns in the world around us, in much the same way that the equally obscure looking symbols of musical notation capture the melodious patterns of music.

Later on in the first episode, we hear Danny Glover's character say much the same thing, but with a focus on human connectivity:

"The universe is made of precise ratios and patterns, a quantum entanglement of cause and effect where everyone reflects on each other... Your son sees everything -- the past, present, future -- he sees how it's all connected."

The fact is,

It is just possible that there is an underlying quantum-level connectivity, as Glover claims, which gives rise to human connections, but if so that is for scientists centuries hence to figure out.

But what is the case, today, is that mathematics lies beneath all of the transportation and communications technologies that really have created a world where many of us -- and very soon all of us -- will be (potentially) connected, and can affect each others lives, instantly.

Not only has mathematics given us that world, with radio and TV, jet aircraft, cell phones, the Internet, the social web, etc., it is mathematics that enables us to understand it. In that sense, Jake is a metaphorical representative of all mathematicians -- the ones who really do possess the power to see -- and to create -- those connections.

"You're telling me my son can predict the future," Sutherland replies incredulously when he hears Glover's words.

"No I'm telling you it's a roadmap," comes the reply.

That's not literally true. Mathematics provides roadmaps for the physical universe, but not for the social world. If you want a literal answer, Glover would have had to say "It's a contour map."

Whether the series creators and writers set out to create, by way of Jake, a metaphor for the role of mathematics in the modern (and to some extent the ancient) world of human connectivity, I have no idea. At some level, they must have. Regardless of their intent, however, if the first episode is anything to go by, they have done so superbly.

How well? I could base an entire college level math course just on