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Unlike their northern neighbors, the countries in the zone’s southern half have difficulty placing bonds — issued to finance their national deficits — with international capital investors. Nor are these countries competitive in the global economy, as shown by their high trade deficits.

If Greece were outside the euro zone, for example, it could devalue its currency

Instead, the fiscal strictures of the euro zone are forcing the country to curtail public expenditures, raise taxes and cut government employees’ salaries, actions that may push Greece into a deep depression and further undermine its already weak international credit standing.

In short, th

e euro is headed toward collapse.

hat opportunity and pull out of the euro, it wouldn’t be alone. The same calculus would probably lure Austria, Finland and the Netherlands — and perhaps France — to leave behind the high-debt states and join Germany in a new, stable bloc, perhaps even with a new common currency.

If Germany were to take t

A strong-currency bloc could fulfill the euro’s original purpose. Without having to worry about laggard states, the bloc would be able to follow a reliable and consistent monetary policy that would force the member governments to gradually reduce their national debt. The entire European economy would prosper. And the United States would gain an ally in any future reorganization of the world currency system and the global economy.

When teachers demand rigorous work, students often rise to the occasion, whereas tracking students at different cognitive levels tends to “diminish learning and boost inequality”. Low expectations are often duly rewarded.

America’s classrooms do not fare well in this book. Against these examples of academic achievement, the country’s expensive mistakes look all the more foolish. For example, unlike the schools in Finland, which channel more resources to the neediest kids, America funds its schools through property taxes, ensuring the most disadvantaged students are warehoused together in the worst schools.

From a standing start, China now has 12% of graduates in the world's big economies - approaching the share of the UK, Germany and France put together. The incumbent superpower, the United States, still towers above with 26% of the graduates.

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.

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.

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

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.

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.”