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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.”

In late 2008, the Chinese Communist Party began the “1,000 Talents” program, aimed at these supremely talented Chinese. Through a wide variety of terrific incentives — sometimes as much as $1 million — the party has encouraged academic and research institutes, as well as municipal governments, to “bring back the best.”

Most important, it must weaken the power of academic and scientific administrators.

Similarly, in many institutions, promotion depends on your relationship with the dean or senior faculty and not your academic pedigree.

Returnees, or those who hesitate to return, often say that in China, “personal relationships are too complex” – a code for the backstabbing and petty jealousies and the need to cultivate ties with leaders in your own field.

The country of 4.7 million people climbed seven spots to 102nd in the World Bank’s annual “Doing Business” report this year, lagging behind China, Vietnam and Namibia. Moody’s Investors Service lowered its outlook on Costa Rica to negative from neutral in September, citing a rising debt burden and widening budget deficit. Moody’s rates the country Baa3, putting it in the same category as Turkey and Iceland.

In a Bloomberg survey published last month, Costa Rica was ranked fourth behind Russia, Argentina and Ukraine on a list of countries confronting the biggest loss of investor confidence. The survey cited data including the rising cost of credit default swaps and the currency’s performance against the dollar.

California-based Intel, whose processors run more than 80 percent of personal computers shipped worldwide every year, originally chose to establish operations in Costa Rica after studying sites in Indonesia, Thailand, Brazil, Argentina, Chile and Mexico, according to a 2000 case study by Harvard University’s Center for International Development. The company’s $600 million investment at the time represented about 4.2 percent of GDP, prompting the company’s then-Vice President Bob Perlman to say Intel’s arrival was like “putting a whale in a swimming pool,” according to the study.

n 2013, about 21 percent of Costa Rica’s exports of goods came from Intel, according to investment promotion agency CINDE

Today renewable energy supplies only about 6 percent of American demand. And most of that comes from water flowing through dams. Solar energy contributes next to nothing.

The arithmetic is merciless. To make it likely that the world’s temperature will rise no more than 2 degrees Celsius above the average of the preindustrial era — a target agreed to by the world’s governments in 2010 — humanity must spew no more than 900 billion more tons of carbon dioxide into the air from now through 2050 and only 75 billion tons after that, according to an authoritative new study in Britain.

The United States Energy Information Administration forecasts that global energy consumption will grow 56 percent between now and 2040.

The only way around this is to put something in coal’s place, at a reasonably competitive price. Neither the warm glow of the sun nor the restless power of the wind is going to do the trick, at least not soon enough to make a difference in the battle to prevent climate change.

In the case of driving schools, the government offers only a limited number of exams each year, and these are doled out to the driving schools depending on their success rate the year before. That fact alone gives the old guard a virtual monopoly,

calling for an overhaul of the written test, which he says goes far beyond making sure that a person knows the rules of the road.

Some studies have concluded that the French are probably paying 20 percent more than they should for the services they get from regulated professions, which include notaries, lawyers, bailiffs, ambulance drivers, court clerks, driving instructors and more.

barriers to getting a license are so high that about one million French people, who should have licenses, have never been able to get them.

Mr. Kramarz said that it often costs 3,000 euros, or about $3,900, to get a license. But others said the average was closer to 1,500 to 2,000 euros.

Although students are required to take only 20 hours of driving lessons, most end up doing double that while they wait for a chance to take the test.

In Japan, for instance, an environmental engineer has attributed a mysterious pestilence that is killing trees on Yakushima Island to pollutants from China.

But the proportion of China’s exports that are made in China has risen steadily in recent years as many companies move more of their supply chains, instead of just having final assembly work done here.