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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.

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.”

The Maastricht Treaty took it for granted that only the public sector can produce chronic deficits. It assumed that financial markets would always correct their own excesses

For instance, they treated the euro crisis as if it were a purely fiscal, i.e., budgetary, problem. But only Greece qualified as a genuine fiscal crisis. The rest of Europe suffered largely from banking problems and a divergence in competitiveness, which gave rise to balance of payments problems.

The result was rapid growth combined with significant inflation: between 2000 and 2008, the prices of goods and services produced in Spain rose by 35 percent, compared with a rise of only 10 percent in Germany. Thanks to rising costs, Spanish exports became increasingly uncompetitive, but job growth stayed strong thanks to the housing boom.

Then the bubble burst.

But the flood of red ink

was a result, not a cause, of Spain’s problems.

The nation’s core economic problem is that costs and prices have gotten out of line with those in the rest of Europe. If Spain still had its old currency, the peseta, it could remedy that problem quickly through devaluation — by, say, reducing the value of a peseta by 20 percent against other European currencies. But Spain no longer has its own money, which means that it can regain competitiveness only through a slow, grinding process of deflation.

Eurobonds and a solidarity fund could promote growth and stabilize the interest rates faced by governments in crisis. Lower interest rates, for example, would free up money so that even countries with tight budget constraints could spend more on growth-enhancing investments.

Even well-managed banking systems would face problems in an economic downturn of Greek and Spanish magnitude; with the collapse of Spain’s real-estate bubble, its banks are even more at risk.

Europe’s leaders did not recognize this rising danger, which could easily be averted by a common guarantee, which would simultaneously correct the market distortion arising from the differential implicit subsidy.

If these countries made a mistake, it was only that, like Germany today, they were overly credulous of markets, so they (like the United States and so many others) allowed an asset bubble to grow unchecked.

Moreover, Germany is on the hook in either case: if the euro or the economies on the periphery collapse, the costs to Germany will be high.

While structural problems have weakened competitiveness and GDP growth in particular countries, they did not bring about the crisis, and addressing them will not resolve it.CommentsView/Create comment on this paragraph

In the areas that are sound, it is mainly that TPP members will probably have to approach the US norms faster than desirable, and possibly faster than they can effectively administer. But there are also areas in which the TPP is not in the interests of most non-US members.

However, it is generally accepted that TTIP is more important to Europe than to the US, which greatly strengthens the US’s hand in negotiations.

it is widely accepted that the deeper intra-European integration fostered by the Single Market initiative was a major contributor to European prosperity between 1992 and 2007

he US has strongly promoted Investor-State Dispute Arbitration in which foreign-owned private firms can seek settlements against governments for taking actions that are not prohibited by the agreements but which reduce the value of investments that the firms have made in member countries.

For states that do not have a lot of, say, social or environmental legislation at the time TPP is signed, Investor-State Arbitration threatens to make progress in these dimensions difficult.

This reads very much like an agreement to cooperate to make sure that outcomes in the trading system are as the US and EU want them – and with around half of world GDP between them and a further 15% in the rest of TPP, it suggests that the choice facing other will be capitulation vs. exclusion. I fear the latter.

Champions of the multilateral system must be much more explicit about its virtues and value – and among these I include Europe (middle-sized countries with a strong belief in negotiated outcomes and order) and China (which has been a massive beneficiary of open markets and non-discrimination to date).

urope had better get on with an internally driven liberalisation, especially of services and utilities markets, to stimulate the recovery quite independent of the outside pressures of a trade negotiation;

During the years of easy money, wages and prices in the crisis countries rose much faster than in the rest of Europe. Now that the money is no longer rolling in, those countries need to get costs back in line.

Now that Greece and Germany share the same currency, however, the only way to reduce Greek relative costs is through some combination of German inflation and Greek deflation. And since Germany won’t accept inflation, deflation it is.

Until recently, most analysts, myself included, considered a euro breakup basically impossible, since any government that even hinted that it was considering leaving the euro would be inviting a catastrophic run on its banks. But if the crisis countries are forced into default, they’ll probably face severe bank runs anyway, forcing them into emergency measures like temporary restrictions on bank withdrawals. This would open the door to euro exit.

Unfortunately, often supervisors in charge of the failing entity in the years of the debt run up were the ones charged with uncovering the problems.

Spain was the leader in the introduction of a dynamic provision – a provisioning tool that forces banks to increase provisions without reference to any specific loan. The intention of this tool was twofold: to mitigate the bad times, and to cool the booms in the good times (Holmstrom and Tirole 1997). Dynamic provisions were endorsed as part of the Basel III standards in December 2010, in part on the strength of Spain’s experience. And indeed the existing evidence (Jiménez et al. 2012) shows that the tool worked as intended, dampening the credit boom and softening somewhat the credit crunch. However, it is clear by now that their level was not nearly enough, as their size – 3% of GDP at their highest point (2004) – was simply not of a magnitude commensurate with the credit losses.

Without the provisions, the reality of the cajas' accounts would have become much faster a concern, and would have imposed itself on the regulator

Had the Banco de España ordered an audit of the system after uncovering numerous irregularities in CCM, it would have not been able to deal with the capital shortfalls uncovered as there was no appropriate resolution regime in Spain at the time

More systematic evidence of the role played by these governance issues is provided in a 2009 paper (Cuñat and Garicano 2009b) where we showed that cajas with chief executives who had no previous banking experience (!), no graduate education, and were politically connected did substantially worse in the run up to the crisis (granting more real estate developer loan, up to half of the entire loan book in some instances) and during the crisis (with higher NPLs).

The euro was sold to us as a modernization program for Europe, and we were also told that it would push the Community toward stability. But, in reality, it has drifted apart and become a truly unstable entity.

The euro was sold to us as a modernization program for Europe, and we were also told that it would push the Community toward stability. But, in reality, it has drifted apart and become a truly unstable entity.

There is no reason why the euro should be coming under pressure. The decision to introduce it was smart and far-sighted.

thanks to the common currency, it's no longer possible, for example, to wage speculative attacks on individual currencies. This eliminates a key disruptive factor that massively destabilized markets in the past.

Still, thanks to the common currency, it's no longer possible, for example, to wage speculative attacks on individual currencies. This eliminates a key disruptive factor that massively destabilized markets in the past.

Starbatty: But that's exactly the problem! In the past, exchange rates served as a valve. Individual countries could control their economies by allowing their currencies to gain or lose value.

But that's exactly the problem! In the past, exchange rates served as a valve.

SPIEGEL: What would happen if the old currencies were reintroduced in the euro zone tomorrow? Bofinger: It would be a catastrophe. The German mark would have to appreciate significantly -- I'd say by 10 percent to 20 percent. Everything that we've worked so hard to attain in terms of competitiveness would vanish overnight.

What would happen if the old currencies were reintroduced in the euro zone tomorrow? Bofinger: It would be a catastrophe. The German mark would have to appreciate significantly -- I'd say by 10 percent to 20 percent. Everything that we've worked so hard to attain in terms of competitiveness would vanish overnight.

SPIEGEL: Would it have been better if all countries in Europe had kept their own currencies? Starbatty: Yes. A community can't function when it's made up of unequal partners who are supposed to behave as equals. With the euro, Germany has created an artificial competitive advantage for itself, which has enabled us to conquer markets all over the world.

Starbatty: Yes. A community can't function when it's made up of unequal partners who are supposed to behave as equals. With the euro, Germany has created an artificial competitive advantage for itself, which has enabled us to conquer markets all over the world.

Since 1995, there have been almost no appreciable wage increases in Germany, partly as a result of pressure brought on from increases in subcontracted labor. Politicians have done everything to relieve employers of the burden of paying social security contributions because we fell into this strange panic, believing we weren't globally competitive. With our economic policies, we placed too much of a lopsided emphasis on exports.

Politicians have done everything to relieve employers of the burden of paying social security contributions because we fell into this strange panic, believing we weren't globally competitive.

In the German Council of Economic Experts, we proposed a consolidation pact, under which each country would be required to specify a fully verifiable path that it will follow as it puts its financial house in order. It wouldn't just be a solution for Greece; it would be for everyone.

Starbatty: In my experience, speculators are only successful when political promises diverge from economic reality, as has become clear in Greece.

its 1.3 million short-time workers do not find regular employment again,

The European Central Bank would never, ever contemplate using inflation to eliminate debt.