2013-06-18T23:02:16-04:00 Dr. Keith Devlin http://www.huffingtonpost.co.uk/author/index.php?author=dr-keith-devlin Copyright 2008, HuffingtonPost.com, Inc. HuffingtonPost Blogger Feed for Dr. Keith Devlin Good old fashioned elbow grease. tag:www.huffingtonpost.com,2013:/theblog//3.3420851 2013-06-11T17:05:15-04:00 2013-06-11T17:05:22-04:00

Dr. Keith Devlin http://www.huffingtonpost.com/dr-keith-devlin/
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 Innumeracy, the US, in particular, fares poorly in that crucial twenty-first century life skill.

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 we should give up basic Constitutional freedoms to reduce drunk driving. Or cell phone use or texting while driving. Or smoking. Or obesity. All far greater killers than a terrorist attack.

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 far more likely to die in your bathtub than in a terrorist attack, you can live your entire life free of any fear of terrorism - and fear of taking baths, since that risk, while greater than a terrorist attack, is still tiny.

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?]]> tag:www.huffingtonpost.com,2013:/theblog//3.2946591 2013-03-27T14:58:18-04:00 2013-05-27T05:12:01-04:00

Dr. Keith Devlin http://www.huffingtonpost.com/dr-keith-devlin/ Stanford MOOC on "mathematical thinking," now into its fourth week.

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]]> tag:www.huffingtonpost.com,2013:/theblog//3.2785808 2013-03-02T14:11:55-05:00 2013-05-02T05:12:02-04:00

Dr. Keith Devlin http://www.huffingtonpost.com/dr-keith-devlin/
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 admitted, 79 percent graduate in four years and 96 percent within six. But that's comparing apples with oranges. Anyone who decides to take a MOOC simply logs onto the website and signs up, thereby becoming one of the statistics. So a fair comparison would be to take the number of students who apply to Stanford. That figure is around 35,000, by chance about the number of students I expect will sign on for my course. So considerably more students who sign up for my free online course will graduate than will occur with students who "sign up" (i.e., apply) to Stanford, which graduates about 1,700 students a year.

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 is becoming clear, is that evaluating MOOCs in terms of traditional higher education will prove to be about as useful -- and just as misleading -- as the early twentieth century pundits who thought of the first automobiles as "horseless carriages." (Or, for that matter, the 1980s commentators who saw the Internet as a useful research tool for academics, but nothing more.)

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.]]> tag:www.huffingtonpost.com,2012:/theblog//3.1850674 2012-09-02T16:06:03-04:00 2012-11-02T05:12:01-04:00

Dr. Keith Devlin http://www.huffingtonpost.com/dr-keith-devlin/
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.]]> tag:www.huffingtonpost.com,2012:/theblog//3.1829050 2012-08-27T15:37:03-04:00 2012-10-27T05:12:03-04:00

Dr. Keith Devlin http://www.huffingtonpost.com/dr-keith-devlin/
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.]]> tag:www.huffingtonpost.com,2012:/theblog//3.1374255 2012-03-26T10:38:28-04:00 2012-05-26T05:12:01-04:00

Dr. Keith Devlin http://www.huffingtonpost.com/dr-keith-devlin/ Touch, starring Kiefer Sutherland, has as one of its central characters a mathematically gifted, autistic, 11-year-old child Jake, played by David Mazouz. How accurate is the portrayal of mathematics in the show? Based on the first episode, the answer is, "Not very." (The caveat is, it doesn't really matter.)

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 Life by the Numbers, that was first broadcast in 1998. He got the math right then. He should have done. I helped write the script.)

It would have been easy to get the math right in Touch. On the other hand, taken literally, the portrayal of the application of mathematics to the world by the young Jake is so way-over-the-top fictitious that inaccuracy in specific details does not adversely affect the storyline.

In this respect, Touch is very different from the previous television series in which one of the main characters was a mathematician, the CBS crime series NUMB3RS, on which I was an adviser for the first three seasons. There, the intention was always to portray mathematics accurately, and the producers went to great lengths to ensure that outcome.

In contrast, it's as a metaphor for the role of numbers and mathematics in today's world that Touch comes into its own, and does so brilliantly.

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 Mathematics: The Science of Patterns. (Don't worry, Fox lawyers; it wasn't.)

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, Touch is not a math-based TV series. It's about human connectivity.

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."

Touch stretches the power of mathematics to understand and to accurately predict beyond the physical world, to the lives of individual people, and math doesn't do that. But that's how big-theme fiction works, by taking sweeping ideas and shrinking them down to a personal level. (Think of all those famous Russian novels!)

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 Touch's pilot episode. And if the series lives up to the promise of the pilot, I probably will.]]> tag:www.huffingtonpost.com,2012:/theblog//3.1371977 2012-03-23T11:04:42-04:00 2012-05-23T05:12:01-04:00

Dr. Keith Devlin http://www.huffingtonpost.com/dr-keith-devlin/ answer Steve Jobs reportedly gave to Barack Obama in February of last year, when the president asked him if it was possible for Apple to bring back the manufacture of some of its products to the United States.

Repetitive tasks such as high-tech assembly-line manufacturing, airline reservations, and customer support are not the only things that can be outsourced in the flat world of the twenty-first century. So too can many less routine tasks that require a university education in science, technology, engineering and mathematics (STEM).

In particular, procedural mathematics (solving differential equations, optimizing systems of inequalities, etc.) can be outsourced. In fact, many mathematical tasks are already routinely "outsourced" -- to machines. Admittedly, a person often has to do some mathematics to put the problem into a form where an existing software package can solve it, and sometimes a new program has to be written, which also requires human mathematical ability. But those human parts too can be outsourced, at electron speed along an ethernet cable or a wireless link. With a few keystrokes, a designer or a CTO in New York or San Francisco can send a mathematical problem to India at 5:00 PM and by 9:00 AM the next morning the solution is back, ready to be used.

In fact, this is happening now, with companies such as Infosys, Tata Consultancy, Cognizant, HCL, Wipro, and iGate Patni. For example, iGate Patni is a Silicon Valley headquartered, Indian IT outsourcing company with over 26,000 well-educated employees who perform such tasks as writing smartphone apps, handling complex financial matters, and optimizing business logistics processes.

The outsourcing of mathematics and mathematics-dependent STEM activities is only going to increase. It's a question of sheer numbers. In China, with a population of 1.3 billion, and India, population 1.1 billion, there is enormous pressure on children (both parental and self-motivational) to secure a good education leading to a secure future, and that will inevitably produce more and more highly able mathematicians, scientists, and engineers. The US, with a total population of 300 million, less than a third of each of those two giants, cannot possibly compete -- even if we were to completely overhaul our STEM education.

It's a salutary thought that, for someone like me, with bachelor and doctoral degrees in mathematics, what were once highly marketable skills that on graduation presented me with a wide choice of possible careers, are now available elsewhere, far more cheaply and in abundance. (In my case, I eventually opted for university research, but only after investigating careers with IBM and BP.) Every mathematical skill, procedure, or technique I learned over six years at university is now essentially obsolete from a US market perspective.

If we cannot compete, then we need to play a different game. Fortunately, that other game is one we already do well at: originality and innovation. Nowhere is the US lead in those areas more apparent than in those major outsource destinations.

For instance, Phaneesh Murthy, the CEO of iGate Patni, quoted in Fast Company last September, lamented the difficulty he has finding truly innovative thinkers in India, noting that "The U.S. education system is much more geared to innovation and practical application. It's really good from high school onward." Summarizing his views, the Fast Company article concluded that for the US, "To compete long term, we need more brainstorming, not memorization; more individuality, not standardization."

(This is why I am not unduly worried for my own future. I learned two things at university far more valuable than a bunch of techniques: I learned to think a certain way -- as a mathematician -- and I learned how to master new techniques quickly whenever I need them.)

For many years, we have grown accustomed to the fact that advancement in a technology-driven society required a workforce that has mathematical skills. But if you look more closely, those skills fall into two categories.

The first category comprises people who, given a mathematical problem (i.e., a problem already formulated in mathematical terms), can find its mathematical solution.

The second category comprises people who can take a new problem, say in manufacturing, identify and describe key features of the problem mathematically, and use that mathematical description to analyze the problem in a precise fashion, picking up whatever mathematical techniques are required along the way.

Hitherto, our mathematics education process has focused primarily on producing people of the first variety. As it turned out, some of those people always turned out to be good at the second kind of activities as well, and as a nation we did very well. But in today's world, and the more so tomorrow's, with a growing supply of type 1 mathematical people in other countries -- a supply that will soon outnumber our own by an order of magnitude -- our only viable strategy is to focus on the second kind of ability.

In other words, the only mathematical niche for the US -- and, luckily for us, it is a crucial niche in today's world economy -- is at the innovation end. Fortunately, innovation is an area where we still lead the world, in large part because our political system allows for and rewards innovation.

Traditionally, a mathematician had to acquire mastery of a wide range of mathematical techniques, and be able to work alone for long periods, deeply focused on a specific mathematical problem. Doubtless there will continue to be native-born Americans who are attracted to that activity, and our education system should support them. We definitely need such individuals. But our future lies elsewhere, in producing people who fall into my second category: what I propose to call the innovative mathematical thinkers.

This new breed of individuals (actually, it's not new, it's just that no one has shone a spotlight on them before) will need to have, above all else, a good conceptual understanding of mathematics, its power and scope -- when and how it can be applied -- and its limitations. They will also have to have a solid mastery of a few very basic mathematical skills, but they do not have to be stellar. A far more important requirement is that they can work well in teams, often cross-disciplinary teams, they can see things in new ways, they can quickly come up to speed on a new technique that seems to be required, and they are very good at adapting old methods to new situations.

Arguably the worst way to educate such individuals is to force them through a traditional mathematics curriculum, with students working alone through a linear sequence of discrete mathematical topics. To produce the twenty-first century, innovative mathematical thinker, you need project-based, group learning in which teams of students are presented with realistic problems that require mathematical and other kinds of thinking for their solution.

Of course, you still need a curriculum, in the sense of a list of topics that students need to master at some point or other. But it should be a short list, and should not be used as a list to proceed through topic by topic, as is current practice in the US. There needs to be a shift in STEM education from (topic-based) instruction (hashtags #traditional and #back-to-basics) to guided-discovery and project-based learning (#reform, #inquiry-based-learning). The primary focus needs to be not on what people know, but on how they think.

This is the promised follow-on to my previous blog on 60 Minutes and Khan Academy. An earlier, somewhat different, and longer, version of this article first appeared in my "Devlin's Angle" column in MAA Online, in July 2010.]]> tag:www.huffingtonpost.com,2012:/theblog//3.1345925 2012-03-20T11:56:31-04:00 2012-05-20T05:12:01-04:00

Dr. Keith Devlin http://www.huffingtonpost.com/dr-keith-devlin/ 60 Minutes segment on Khan Academy recently, opened with former hedge fund manager turned world-educator Salman Khan riding home on a bicycle, evoking (I suspect deliberately on the part of 60 Minutes) one of America's most cherished images: the lone stranger who rides into town and fixes what needs to be fixed.

Whereas John Wayne or Clint Eastwood would dismount and walk into the saloon with guns at the ready, Sal Khan got off his bike and walked into his tiny home office to record a math lesson on his computer, to upload onto YouTube. But the message was the same: The outsider who rides into town to save us is a part of our mythology.

Like most mythologies, when you analyze it you find many reflections of ourselves and the society we have built. In particular, it captures that "can do" attitude that attracted me, like many before me and since, to emigrate here. But it does so in a way that is totally unrealistic. There are no such lone heroes, and real life's problems are never so simple that one individual can fix them.

The real West was not won by people like Clint Eastwood's "Man With No Name," and it is equally naïve to think that one person at a computer terminal can "fix" mathematics education. But we Americans are suckers for the myth, even extending it to our election of a president, on whose decidedly human shoulders all sorts of unrealistic expectations are regularly placed.

For me, the most wildly inaccurate "lone outsider" statements were made by former Google CEO Eric Schmidt, who repeated Silicon Valley's own favorite creation myth, that the major changes arise from the activities of mavericks outside the system. This is just another variant on that same romantic story.

The truth is, the vast majority of technology companies in the Valley emerged from either Cold War DARPA funded research, from decidedly corporate AT&T Bell Laboratories and Xerox PARC, or from federally funded research at Stanford University and SRI. Hewlett-Packard, Shockley, Fairchild, Intel, Cisco, Sun, etc. all came from one or more of those sources. (Though Apple does not fit that mold -- it is one of the few examples that match Schmidt's description -- its breakthrough Macintosh came from research at SRI and Xerox PARC.) Schmidt's own Google was a result of a federally funded program carried out by two graduate students in Stanford's Computer Science Department. It doesn't get more "in the system" than that!

True, in many of those cases, it was one or more young researchers who ignored the advice of their seniors that a particular research path was unlikely to succeed and pressed ahead anyway, but young people have always been at the cutting edge of "the system."

Though I found the 60 Minutes coverage of Khan Academy disappointingly superficial, and in places plain inaccurate, it took some time before I realized what it was that left me bothered.

It was not the claim that there was something (potentially) revolutionary going on. I think there is, though no one in the program pointed to what it is. Nor was it the fact that there was no mention of the debate going on in the educational world as to whether Khan Academy is, in the metaphor of Sergio Leone's famous Spaghetti Western, the Good, the Bad, or the Ugly.

No, what bothered me was the program's unspoken implication that the many thousands of American mathematics teachers did not know what they were doing, and that they, or perhaps the kids in our schools, needed "saving." (Some probably do, but some is not the whole system.) It was not Sal Khan himself who gave that impression -- what he said, very clearly, as he has on many occasions, was quite the opposite. Rather it was the way the program was structured and narrated.

People both for and against KA (for the record, I am a significantly qualified "for") tend to portray the issues involved as black or white. But like most things in life, they are many shades of grey.

What is without doubt, however, is that millions of people around the world have found Khan's videos valuable aids to help them to pass a crucial math test. Some of them (almost certainly a minority) have, in the process, learned some mathematics -- meaning that, faced with a real world problem whose solution requires the use of math, they will, as a result of watching Khan's videos, be able to use math to solve the problem. (That, of course, is the ultimate goal of mathematics education.)

I would myself have been such a person. Had KA been around when I was at school, I would have loved it. So would Bill Gates, whose public statements and financial support have enabled Khan to build on the initial success of his home-made videos. So, I suspect, would most scientists and engineers. So would many teachers who are critical of KA's pedagogy. (Former Google Education Fellow Dan Meyer, for example.) Almost certainly, the younger Sal himself would have been able to teach himself math using his videos.

For those of us who find ourselves with the ability to learn math, we will do so with whatever tools we can find, and most of us do just fine. For people like us, Sal Khan's videos are a great resource.

Unfortunately, we are a minority, and a school teacher has the responsibility of teaching all children. And there's the rub. For the majority who find mathematics extremely difficult, instructional videos have known problems, and we currently know of no approach that comes close to regular group interactions with a good, inspiring, human teacher. Changing the way a human mind works, which is what teaching amounts to, is a difficult task. Moreover, it involves emotional, psychological, and social factors. It would be impossibly hard, were it nor for the fact that teachers are themselves emotional, psychological, and social creatures, at heart very much like their students. The key is for the teacher and the student to establish human contact.

In short, teaching is complex and hard, and math teaching particularly so. A lone stranger riding into town on a bicycle is not going to provide the "answer," not even if he has a broadband connection to the Internet and Bill Gates behind him. Sal Khan is not the Clint Eastwood of Math Ed. No one is. No one can be. Such a person is a myth.

By focusing on the great American lone-stranger myth, 60 Minutes missed what I think are the causes of the relatively poor performance in mathematics of America's children and a number of practical ways we might turn things around. KA -- and there is now more to KA than just the videos -- or something like it is likely to be part of the answer. Unfortunately, the US has yet to agree on the question. What exactly do we mean by "mathematics education"?

Those are far less romantic stories. But they are real.

To be continued ...
]]> tag:www.huffingtonpost.com,2012:/theblog//3.1338163 2012-03-12T17:15:07-04:00 2012-05-12T05:12:01-04:00

Dr. Keith Devlin http://www.huffingtonpost.com/dr-keith-devlin/
It sounds like a simple enough question. But a recent group discussion on LinkedIn generated over 60 contributions when I last checked. People seem to have very clear preferences as to what is "right." Unfortunately, those preferences differ.

In expressions such as "5 - 3" there is general agreement: You say "Five minus three." In this case, the symbol "-" denotes the binary arithmetic operation of subtraction, expressed as "minus", and the expression "5 - 3" means "subtract 3 from 5," or "5 minus 3."

It is when the expression "- 3" appears on its own that the fun begins. Traditionalists (the kind of people who rely on the Chicago Manual of Style and insist I should have put the colon and question mark inside the quotes in my opening question) will say it should be read as "negative three." (Your last math teacher probably said that too.) But in everyday situations, most people say "minus three." For example, I doubt you have ever heard the TV weather person say that the temperature will fall to "negative three (degrees)." No, she or he will have said "minus three."

My sense (and it is nothing more than that) is that almost all professional mathematicians will probably come down on the side of "minus three." The reason is that those of us in the math biz put the minus sign in front of numbers all the time, and those numbers may themselves be positive or negative. For example, we frequently find ourselves referring to numbers such as "- N" where N might turn out to be -3. Since the result in such a case is in fact a positive number, it seems totally wrong (to us) to refer to it using "negative," which we take as indicating the sign of a number. In other words, we view "negative" as an adjective, which tells us that the number is less than 0.

We generally view "-", on the other hand, not as a symbolic adjective but an operation. Usually it is a binary operation, but sometimes we think of it as a unary one. In such cases, it simply changes the sign (or reflects it in the origin on the number line, if you want to use geometric language). In other words, we do not view "-N" as indicating that the number is positive or negative, rather that it has the opposite sign to N. Quite simply, "minus" can function as a sign-changing, unary operator.

You might claim that we could use "negative" similarly, so that "negative negative three" means "three," but to me at least (and I know I am not alone) it sounds bizarre (in a way that "minus minus three" does not).

If the symbol "-" were only ever put in front of positive numbers, it would be fine to read it as "negative". In fact, it could be advantageous to do so, as it would tell us the sign of the number. Since many people outside of mathematics, science, and engineering may in fact never encounter a double-negative (except to wonder how it works out to be positive), that might help explain why the "negative three" camp is well occupied.

But for anyone dealing with numbers in a scientific context, for whom "-" is frequently applied to a negative (sic) quantity or to one whose sign is not known, rather than use "negative" on some occasions and "minus" on others, the latter is the default reading on all occasions.

That, I think, makes the case for professional scientists and mathematicians always using "minus three." But why do so many non scientists use the same terminology? My guess is that they either pick it up from their science teachers at school (but perhaps not their math teachers), or maybe from the TV weather person. TV weather forecasters may well not be trained scientists, but they do have to read scientific reports from professional meteorologists, and that could account for the domination of "minus."

So, if you want a ruling from a qualified mathematician, I'll give one: Always read "-N" as "minus N." Feel free to use my name to try to settle any dispute. But if you ask me to get personally involved, I'll give you an answer right out of Top Gun: "That's a negative."]]> tag:www.huffingtonpost.com,2012:/theblog//3.1315664 2012-03-05T18:00:06-05:00 2012-05-05T05:12:01-04:00

Dr. Keith Devlin http://www.huffingtonpost.com/dr-keith-devlin/
To the rest of the world, the Valley is a place of enormous commercial successes in the tech world: HP, Intel, Cisco, Apple, Sun, eBay, Yahoo!, Google, LinkedIn, Zynga, the list goes on. (Facebook started in Boston, but moved here to become big.) But to those of us who live there - and with a home in the center of Palo Alto and an office at Stanford, I live right in the heart of it - the dominant and recurring theme is failure.

Companies fail here all the time. The Valley draws its strength and amazing resilience from the fact that failure is not only accepted, it is expected and encouraged. One of the mantras in the Valley is this: "If four-out-of-five (the ratio varies depending who says it) of your attempts don't fail, you're not being bold enough." Failure is the fastest route to success there is.

My debut blog last week was viewed by many as an attack on the Valley, and on two of its stars, one old the other new: Sun co-founder Vinod Khosla and Khan Academy founder Salman Khan. But to anyone familiar with the Valley, my attention-grabbing headline was not an attack but a (mundane) observation of life in these parts.

Though several of the comments on my blog took me to task for my words, it's telling that one of the supposed targets, Sal Khan, whom I know, responded by noting that he, Khosla, and I basically agree, and Khosla, whom I have not met, replied by suggesting we might meet and talk. Very different reactions from many other respondents, and typical of the Valley! Incidentally, it seems to have slipped past many readers' notice that I was able to use Khosla's recent blog post as my jump-off point precisely because I follow his blog!

That's how different the culture is here. And because it is almost impossible to move a culture, that goes a long way to explaining why attempts to replicate Silicon Valley elsewhere have all failed, some more so than others.

In most parts of the world, people focus on other people, not their ideas, and the result is often polarization, personal attacks, and conflict. Just take a look at the current GOP primaries. Choosing a president or candidate for president should be about policies and ideas, but that has not been the focus at all. Walk into any coffee shop in Palo Alto, however, and you'll hear mostly discussion about ideas. Who espouses those ideas is of little relevance, except, to some extent, at funding time.

So is it the case that, as my previous headline claimed, "Silicon Valley executives keep getting [something] wrong about education"? Yes, I believe they do. (My post would have had little point otherwise!) What they keep getting wrong is that they do not involve experienced education experts in the loop - at least not often enough.

I think that should change, and in the same "I am a fan and a believer" spirit as my Huff Post blog, I recently "berated" my academic colleagues about the same thing in my monthly blog for the (academically oriented) Mathematical Association of America.

Education presents technology companies with a particularly hard challenge, in that the success of its products depends on changing the way a young human brain works - something centuries of teachers' experiences have shown to be extremely hard to achieve, particularly in mathematics (my area and the focus of my previous blog).

Cognitive scientists and education researchers have learned a lot about math ed over the past half century, but as I noted in my MAA blog, a lot of what has been learned has yet to cross the academia-industry divide. Both sides need to do more to change that.

Not that I expect the Valley techs to stop trying just because the education experts think something won't work. After all, when two young Stanford grad students told their supervisor (a friend of mine) they wanted to try to build a better search engine, he told them that the problem was hopeless, and that Alta Vista was probably as good as could be achieved. Fortunately, Sergey Brin and Larry Page went ahead anyway, and now the world has Google.

Vinod Khosla's suggestion that we should be "trying hundreds of new ways of doing things" in education is very much in the spirit of the Valley. It's what it does well. Most will fail, as they always do. (The latest casualty in the educational software space, Airy Labs, occurred just this past week, but I'd be amazed if the folks involved don't just bounce right back.) But if there is an educational equivalent to Google to be found, chances are high it will be found in Silicon Valley.

The point of taking note of all the things that education research has found to be difficult is not to give up in despair. Rather, it's to better know what we are up against, and thereby increase the chances of success. And, in the final analysis, success is what Silicon Valley strives for.

Oh, and for the record, I have enormous admiration for risk taking, Silicon Valley executives. I'd have to. Like everyone else in these parts, I have my own startup company!
]]> tag:www.huffingtonpost.com,2012:/theblog//3.1301297 2012-02-25T15:45:35-05:00 2012-04-26T05:12:01-04:00

Dr. Keith Devlin http://www.huffingtonpost.com/dr-keith-devlin/
The always interesting and provocative reflections of the legendary Silicon Valley investor (and Sun Microsystems co-founder) Vinod Khosla provided the latest example of this when, in his Feb. 19 blog-post in TechCrunch, he wrote:

"Education 2.0 [...] we have not experimented enough with [...] out-of-the-box approaches but have instead tried to force-fit [...] traditional (often broken) ideas into the 'computerized' model."

Which might sound fine if this statement were not preceded by his explicit mention of Khan Academy as one of the new experiments. For KA is precisely a traditional approach transported onto the Web, namely one-to-one instruction, sitting side-by-side with the teacher. Is KA valuable? Sure it is? But "all" Sal Khan has done is take the traditional textbook instruction and put it up on YouTube.

Those quotes around "all" just then are important. It proved to be a significant leap forward, in large part because Khan is a good instructor -- he explains well in a highly non-threatening, "I am your friend" way. That's not an easy thing to achieve when the entire information channel consists of his voice and a screen-trace of what his hand writes on a tablet screen.

But what resources like Khan Academy provide is instruction, not teaching/learning. Anyone who has been lucky enough to experience good teaching will know the difference, but it's a sad fact of American life that most people's mathematics schooling consisted entirely of instruction and exercise sheets. They simply do not know what teaching is, or what it feels like to learn from a good teacher. They watch a Khan video and think "That guy is doing it at least as well as my teacher (often a lot better) and I can play through his explanation as often as I need." And they are right.

Vinod is probably like me. We learned in spite of not being taught well. Some of us figured out early in our education that the most efficient way to progress was to skip, or at least pay little attention to, classes we found boring or pedestrian, or even incomprehensible, and "teach ourselves," seeking out help from more advanced colleagues or, in my case, the teacher whose classes I largely ignored.

But for all Sal's charisma and instructional talent (and I am using his first name, since I sought him out when he was just becoming widely known and got to know him, since I recognize a valuable talent when I see it), what he is delivering is instruction -- a one-way information feed. And instruction is just part of teaching and learning. Watching videos of people playing golf will surely help you learn to play, but you won't get very far without going out on the fairway, frequently, and doing so with a good coach who can watch what you do and correct your inevitable errors. Not once but many times, over a long haul. That's teaching. It's interactive (bi-directional). And it's very human.

The traditional, instructional blackboard lecture should perhaps have been relegated to an occasional part of teaching with the invention of the photocopy machine, though maybe some people preferred a dynamic delivery by a friendly person. In which case, resources like Khan Academy should now put to rest forever math classes that consist primarily of blackboard instruction followed by "do all the odd numbered exercises on page 156." (BTW, Khan himself recognizes the importance of the teacher, and advocates using his videos as part of a "flipped classroom" model of teaching, a concept that goes back well before YouTube was launched.)

In the flipped model, teachers devote most of their class-time to the important activity that no technology can provide (at least today): helping students to learn in the same way a golf coach helps beginners (and not-so-beginners) to learn how to play golf.

Meanwhile, it will help if those who provide the technology platforms, people like Khosla, apply to education the same principles they adopt instinctively in running their businesses: seek advice from the experts. Sure, there are education experts who are still living in the past. But the same is true in technology: Xerox, IBM, and Nokia, to name just three of many, all had painful experiences as a result of not listening to those who could see that "the times they are a-changing." There are plenty of knowledgeable mathematics educators who use modern platforms and can provide good advice. Dan Meyer, Karim Ani, and Marilyn Burns are three that I know personally, but there are many more. A Silicon Valley (or Redmond, Washington) executive who wants to make a useful contribution to education could do well to spend an evening checking out just those three sources.]]>