Mother of All Inventions

Minds on fire. That's what we need in the US. The kids who rip into math and science, who chew it up and turn it into silk. We must find them and cultivate them.

How to Fire Up U.S. Innovation

We need more hands-on tech education for American children, but we also need to keep attracting the best talent from abroad.

Whether it's the latest tablet computer, electric sports car or other cool new product, Americans get very excited about innovation—and more often than not these innovations are brought to market by engineers working in technology hubs like Silicon Valley.

An innovation engine has many moving parts and all of them have to mesh properly for the engine to run smoothly. In Silicon Valley, and elsewhere in the United States, the engine requires sources of trained professionals (engineers, scientists, business people), sources of capital (venture capitalists, fluid stock markets), and new and existing companies that form a mutually reinforcing ecosystem.

Universities such as Stanford, the University of California at Berkeley and San Jose State supply a continuous flow of trained talent. Venture capital companies line Sand Hill Road in Menlo Park, adjacent to legal firms midwifing the birth of new companies. Like small villages, everyone seems to know everyone else, and individuals move from company to company, or in and out of partnerships.

It is sometimes thought that research in universities or corporate laboratories produces technology that then transfers seamlessly into products and services. But technology doesn't transfer on its own—it is the people who have the knowledge in their heads that do the transferring. One of the keys to Silicon Valley successes is the transfer of professionals into the marketplace and the ability of researchers to start new companies. Universities that allow faculty members to consult a day a week on average seed the process of business innovation, as can be readily recognized by tallying the number of companies started by Stanford or Berkeley faculty—to say nothing of the students who start new companies.

What conditions give rise to innovation and facilitate its transforming effects? Contributing factors include the freedom to pursue ideas, the freedom to fail, and the freedom of access to information in the broadest sense. Occasional business failure in the U.S. is a mark of experience, while in other cultures it may be a permanent scar. Information sharing is generally considered a powerful means towards progress, hence the strong influence that the American university system has had on the economy.

One cannot escape the observation, however, that the incidence of intelligence is uniform in all populations around the world. There are absolutely more smart people outside the U.S. than there are living here. It is in our best interest to attract talent from anywhere in the world to participate in our innovation engine. Even if visitors return to their homelands after attending an American university, we will benefit from their contributions while they were here and, in all likelihood, even after they have returned home.

Despite our well-developed college and post-college system, America simply is not producing enough of our own innovators, and the cause is twofold—a deteriorating K-12 education system and a national culture that does not emphasize the importance of education and the value of engineering and science. The American public focuses more on sports and entertainment figures and less on the scientists and engineers whose innovations make our lives easier, safer, healthier and more productive.

Since 1990, U.S. scientists and engineers have invented the lithium-ion battery that powers all manner of devices from tablet computers to electric cars, developed GPS for civilian use to keep us on the right path to our destinations, and created both remote-controlled military aircraft (drones) to keep our soldiers safe overseas and robots that keep our floors clean at home. But how many among us know the names of the creators of the lithium-ion battery at Bell Laboratories, or the founder of iRobot Corp. and inventor of the Roomba robotic vacuum cleaner now sold around the world?

By contrast, Japan, Spain, Norway, Sweden and many European countries shine a much brighter national spotlight on international science and technology breakthroughs. In northern Spain, the Prince of Asturias Awards for science and technology is a multi-day affair, as is the Japan Prize ceremony for contributions to the progress of science and technology. And of course the Nobel prizes draw international attention and renown.

So what's America to do?

Young people should understand and experience the thrill of science and discovery. We need to help them do real science, not just read about it, through collaborative tools that help mentors and students to interact through programs such as the Institute of Electrical and Electronics Engineers' tryengineering.org. Children learn best by seeing and doing, not by memorizing.

It's also important to reintroduce to the American culture a higher regard for engineers and scientists. The winners of our National Medals of Science and Technology deserve more public attention. Our successful scientists and engineers should be made more visible and their voices heard more often. Most important, however, is the need to refresh and invigorate interest in and regard for science and engineering in our youth.

School and extracurricular opportunities for young people to work with experienced scientists and engineers should be expanded. Successful examples include the FIRST robotics program established by Dean Kamen (entrepreneur and inventor of the Segway PT), Google's recently launched global Science Fair, and the 50-year partnership between NASA and the National Science Teachers Association. By elevating interest in math and science, we will foster the innovation and ingenuity that will move this nation forward into a better future.

The Sanity of Irrational Numbers

Division and Its Discontents
By STEVEN STROGATZ

There’s a narrative line that runs through arithmetic, but many of us missed it in the haze of long division and common denominators. It’s the story of the quest for ever-more versatile numbers.

The “natural numbers” 1, 2, 3 and so on are good enough if all we want to do is count, add and multiply. But once we ask how much remains when everything is taken away, we are forced to create a new kind of number — zero — and since debts can be owed, we need negative numbers too. This enlarged universe of numbers called “integers” is every bit as self-contained as the natural numbers, but much more powerful because it embraces subtraction as well.

A new crisis comes when we try to work out the mathematics of sharing. Dividing a whole number evenly is not always possible … unless we expand the universe once more, now by inventing fractions. These are ratios of integers — hence their technical name, “rational numbers.” Sadly, this is the place where many students hit the mathematical wall.

There are many confusing things about division and its consequences, but perhaps the most maddening is that there are so many different ways to describe a part of a whole.

If you cut a chocolate layer cake right down the middle into two equal pieces, you could certainly say that each piece is “half” the cake. Or you might express the same idea with the fraction 1/2, meaning 1 of 2 equal pieces. (When you write it this way, the slash between the 1 and the 2 is a visual reminder that something is being sliced.) A third way is to say each piece is 50 percent of the whole, meaning literally 50 parts out of 100. As if that weren’t enough, you could also invoke decimal notation and describe each piece as 0.5 of the entire cake.

This profusion of choices may be partly to blame for the bewilderment many of us feel when confronted with fractions, percentages and decimals. A vivid example appears in the movie “My Left Foot,” the true story of the Irish writer, painter and poet Christy Brown. Born into a large working-class family, he suffered from cerebral palsy that made it almost impossible for him to speak or control any of his limbs, except his left foot. As a boy he was often dismissed as mentally disabled, especially by his father, who resented him and treated him cruelly.

A pivotal scene in the movie takes place around the kitchen table. One of Christy’s older sisters is quietly doing her math homework, seated next to her father, while Christy, as usual, is shunted off in the corner of the room, twisted in his chair. His sister breaks the silence: “What’s 25 percent of a quarter?” she asks. Father mulls it over. “Twenty-five percent of a quarter? Uhhh … That’s a stupid question, eh? I mean, 25 percent is a quarter. You can’t have a quarter of a quarter.” Sister responds, “You can. Can’t you, Christy?” Father: “Ha! What would he know?”

Writhing, Christy struggles to pick up a piece of chalk with his left foot. Positioning it over a slate on the floor, he manages to scrawl a 1, then a slash, then something unrecognizable. It’s the number 16, but the 6 comes out backwards. Frustrated, he erases the 6 with his heel and tries again, but this time the chalk moves too far, crossing through the 6, rendering it indecipherable. “That’s only a nervous squiggle,” snorts his father, turning away. Christy closes his eyes and slumps back, exhausted.

Aside from the dramatic power of the scene, what’s striking is the father’s conceptual rigidity. What makes him insist you can’t have a quarter of a quarter? Maybe he thinks you can only take a quarter of a whole or of something made of four equal parts. But what he fails to realize is that everything is made of four equal parts. In the case of something that’s already a quarter, its four equal parts look like this:

Since 16 of these thin slices make the original whole, each slice is 1/16 of the whole — the answer Christy was trying to scratch out.

A version of the same kind of mental rigidity, updated for the digital age, made the rounds on the Internet a few years ago when a frustrated customer named George Vaccaro recorded and posted his phone conversation with two service representatives at Verizon Wireless. Vaccaro’s complaint was that he’d been quoted a data usage rate of .002 cents per kilobyte, but his bill showed he’d been charged .002 dollars per kilobyte, a hundredfold higher rate. The ensuing conversation climbed to the top 50 in YouTube’s comedy section.

About halfway through the recording, a highlight occurs in the exchange between Vaccaro and Andrea, the Verizon floor manager:

V: “Do you recognize that there’s a difference between one dollar and one cent?”
A: “Definitely.”
V: “Do you recognize there’s a difference between half a dollar and half a cent?”
A: “Definitely.”
V: “Then, do you therefore recognize there’s a difference between .002 dollars and .002 cents?”
A: “No.”
V: “No?”
A: “I mean there’s … there’s no .002 dollars.”

A few moments later Andrea says, “Obviously a dollar is 1.00, right? So what would .002 dollars look like? I’ve never heard of .002 dollars… It’s just not a full cent.”

The challenge of converting between dollars and cents is only part of the problem for Andrea. The real barrier is her inability to envision a portion of either.

From first-hand experience I can tell you what it’s like to be mystified by decimals. In 8th grade Ms. Stanton began teaching us how to convert a fraction into a decimal. Using long division we found that some fractions give decimals that terminate in all zeroes. For example, 1/4 = .2500…, which can be rewritten as .25, since all those zeroes amount to nothing. Other fractions give decimals that eventually repeat, like

5/6 = .8333…

My favorite was 1/7, whose decimal counterpart repeats every six digits:

1/7 = .142857142857….

The bafflement began when Ms. Stanton pointed out that if you triple both sides of the simple equation

1/3 = .3333…,

you’re forced to conclude that 1 must equal .9999…

At the time I protested that they couldn’t be equal. No matter how many 9’s she wrote, I could write just as many 0’s in 1.0000… and then if we subtracted her number from mine, there would be a teeny bit left over, something like .0000…01.

Like Christy’s father and the Verizon service reps, my gut couldn’t accept something that had just been proven to me. I saw it but refused to believe it. (This might remind you of some people you know.)

But it gets worse — or better, if you like to feel your neurons sizzle. Back in Ms. Stanton’s class, what stopped us from looking at decimals that neither terminate nor repeat periodically? It’s easy to cook up such stomach-churners. Here’s an example:

0.12122122212222…

By design, the blocks of 2 get progressively longer as we move to the right. There’s no way to express this decimal as a fraction. Fractions always yield decimals that terminate or eventually repeat periodically — that can be proven — and since this decimal does neither, it can’t be equal to the ratio of any whole numbers. It’s “irrational.”

Given how contrived this decimal is, you might suppose irrationality is rare. On the contrary, it is typical. In a certain sense that can be made precise, almost all decimals are irrational. And their digits look statistically random.

Once you accept these astonishing facts, everything turns topsy-turvy. Whole numbers and fractions, so beloved and familiar, now appear scarce and exotic. And that innocuous number line pinned to the molding of your grade school classroom? No one ever told you, but it’s chaos up there.