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This year, Google and a corporation associated with NASA acquired for study an experimental computer that appears to make use of quantum properties to deliver results sometimes 3,600 times faster than traditional supercomputers. The maker of the quantum computer, D-Wave Systems of Burnaby, British Columbia, counts Mr. Bezos as an investor.

he devoted much of a postsummit news conference to boasting about his steadfast defense of British interests, particularly a multibillion-dollar rebate that Britain receives each year on its payments.

Today, Germany grudgingly does the minimum needed to keep the euro area intact. If exit to emulate Greece becomes an attractive proposition, Germany will be put on the spot—and the magnanimity it shows in place of its current miserliness will be the ultimate test of how much it values the euro area.

And the same forces that sent money sloshing into Turkey also make the world economy as a whole highly vulnerable.

If this is a good description of our situation, and I believe it is, we now have a world economy destined to seesaw between bubbles and depression

In France, the leader of the far-right National Front, Marine Le Pen, recently acknowledged that her party had received a loan for 9 million euros, or about $11 million, from a Kremlin-linked bank.

Faced with punishing sanctions, a petro-economy pushed to the brink by plunging oil prices and the wildly gyrating value of the ruble, Mr. Putin this month halted the project.

Geological surveys suggested that Bulgaria could be sitting atop an underground ocean of natural gas, enough to be self-sufficient for years, enough to eclipse the advantages of South Stream.

On April 4, 2014, soon after Mr. Putin annexed Crimea, Bulgaria’s Parliament gave initial passage to a bill that effectively exempted South Stream from a number of European Union regulations, most important, the one that would have forced Gazprom to allow non-Russian gas to flow through the pipeline.

“If I hear one more word about competition, I’m going to freeze your you-know-whats off,” Mr. Putin reportedly shouted.

The anti-fracking movement became so broad that in January 2012, Parliament banned not only the extraction of shale gas, but even exploration that would quantify the country’s reserves.

When the Bulgarian government refused, the European Union cut off tens of millions of euros in regional development funds.

In desperate need of the European funds, the prime minister announced the next day that South Stream would be halted until it had full European Union approval.

While “he overreached, and he underestimated the response” to his intervention in Ukraine, said Mr. Gray, the former American diplomat, the Russian leader has been “quite effective” in countries like Bulgaria.“He won a great deal by getting Nabucco stopped,” Mr. Gray said. “Ultimately, his goal is to keep as much control over the former parts of the Soviet empire as possible.”

Carefully taught, the assignments can help make math more concrete. Students don’t just memorize their times tables and addition facts but also understand how arithmetic works and how to apply it to real-life situations. But in practice, most teachers are unprepared and children are baffled, leaving parents furious.

On national tests, nearly two-thirds of fourth graders and eighth graders are not proficient in math. More than half of fourth graders taking the 2013 National Assessment of Educational Progress could not accurately read the temperature on a neatly drawn thermometer.

On the same multiple-choice test, three-quarters of fourth graders could not translate a simple word problem about a girl who sold 15 cups of lemonade on Saturday and twice as many on Sunday into the expression “15 + (2×15).” Even in Massachusetts, one of the country’s highest-performing states, math students are more than two years behind their counterparts in Shanghai.

A 2012 study comparing 16-to-65-year-olds in 20 countries found that Americans rank in the bottom five in numeracy.

One study that examined medical prescriptions gone awry found that 17 percent of errors were caused by math mistakes on the part of doctors or pharmacists.

“I’m just not a math person,” Lampert says her education students would say with an apologetic shrug.

In the 1970s and the 1980s, cognitive scientists studied a population known as the unschooled, people with little or no formal education.

For instance, many of the workers charged with loading quarts and gallons of milk into crates had no more than a sixth-grade education. But they were able to do math, in order to assemble their loads efficiently, that was “equivalent to shifting between different base systems of numbers.”

Studies of children in Brazil, who helped support their families by roaming the streets selling roasted peanuts and coconuts, showed that the children routinely solved complex problems in their heads to calculate a bill or make change.

The cognitive-science research suggested a startling cause of Americans’ innumeracy: school.

Only when the company held customer focus groups did it become clear why. The Third Pounder presented the American public with a test in fractions. And we failed. Misunderstanding the value of one-third, customers believed they were being overcharged. Why, they asked the researchers, should they pay the same amount for a third of a pound of meat as they did for a quarter-pound of meat at McDonald’s. The “4” in “¼,” larger than the “3” in “⅓,” led them astray.

In the process, she gave them an opportunity to realize, on their own, why their answers were wrong.

At most education schools, the professors with the research budgets and deanships have little interest in the science of teaching

The answer-getting strategies may serve them well for a class period of practice problems, but after a week, they forget. And students often can’t figure out how to apply the strategy for a particular problem to new problems.

Some of the failure could be explained by active resistance.

A year after he got to Chicago, he went to a one-day conference of teachers and mathematicians and was perplexed by the fact that the gathering occurred only twice a year.

More distressing to Takahashi was that American teachers had almost no opportunities to watch one another teach.

In Japan, teachers had always depended on jugyokenkyu, which translates literally as “lesson study,” a set of practices that Japanese teachers use to hone their craft. A teacher first plans lessons, then teaches in front of an audience of students and other teachers along with at least one university observer. Then the observers talk with the teacher about what has just taken place. Each public lesson poses a hypothesis, a new idea about how to help children learn.

The research showed that Japanese students initiated the method for solving a problem in 40 percent of the lessons; Americans initiated 9 percent of the time.

Finland, meanwhile, made the shift by carving out time for teachers to spend learning. There, as in Japan, teachers teach for 600 or fewer hours each school year, leaving them ample time to prepare, revise and learn. By contrast, American teachers spend nearly 1,100 hours with little feedback.

“Sit on a stone for three years to accomplish anything.”

In one experiment in which more than 200 American teachers took part in lesson study, student achievement rose, as did teachers’ math knowledge — two rare accomplishments.

Examining nearly 3,000 teachers in six school districts, the Bill & Melinda Gates Foundation recently found that nearly two-thirds scored less than “proficient” in the areas of “intellectual challenge” and “classroom discourse.”

Shell is working with Gazprom on natural gas extraction in Russia; Exxon has a multibillion dollar exploration partnership with Rosneft, a major oil producer controlled by the Russian government, and BP owns nearly 20 percent of Rosneft.

"Shell and Exxon have physical assets in Russia," said Gheit. "But pound-for-pound, BP has the biggest exposure in Russia." Although BP may have the most to lose from an economic tug-of-war between Moscow and the West, tough lessons that BP learned in Russia—through a defunct partnership with Rosneft called TNK-BP—also make BP best equipped for any future fallout, said Nicholas Spiro, managing director of Spiro Sovereign Strategy.

Spiro said that several German firms have also steeled themselves for possible fallout from friction between the Russia and the West. "German companies are huge here," said Spiro, naming BASF, energy firm RWE, and Siemens as companies with operations in Russia. BASF is working to finalize a deal with Gazprom that would give it a stake in Siberian oil fields; RWE has reached a preliminary deal to sell its natural gas subsidiary to Russian billionaires Mikhail Fridman and German Khan, and Siemens has a partnership with state-run railroad monopoly Russian Railways. Late last month, Siemens CEO Joe Kaeser made a trip to Moscow to meet with Russian President Vladimir Putin at his residence and voice support for a "trusting relationship" with Russian companies.

We know that if the West's resolve starts to crumble, it will almost certainly start in Germany,

"That's the canary in the coal mine."

"It starts with Germany and works its way down," said Hogan. "They have the most trade back-and-forth, and Germany gets the highest percentage of its energy from Russia."

Alcoa owns aluminum fabrication facilities in Russia, and Boeing has a design center in Moscow, as well as a joint venture with VSMPO-Avisma, the world's largest titanium producer.

Members of the Russian parliament have also proposed charging international payments companies like Visa and Mastercard with pre-emptive "security fees," with the stated aim of preparing for future financial disruptions.

Air travel was governed by inter-governmental deals that dictated which airlines could fly where, how many seats they could offer and, in many cases, what fares they could charge. The result was inefficiency and losses.

Low-cost carriers, such as SouthWest and Ryanair, introduced cut-throat rivalry on short-haul routes. Former flag-carriers struggled with the legacy of older fleets, large networks, uppity unionised workforces and vast pension liabilities. Low-cost carriers devastated their model of feeding short-haul passengers onto more lucrative long-haul services.

As well as stiff competition from their rivals, airlines face the problem that there is little competition in the industries that supply them. Two firms—Airbus and Boeing—provide the majority of the planes, and airports and air-traffic control are monopolies