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ways of working are critical in mathematical work and when they are taught and valued, many more students contribute, leading to higher achievement

mathematics education we suffer from the widespread, distinctly American idea that only some people can be “math people.” This idea has been disproved by scientific research showing the incredible potential of the brain to grow and adapt. But the idea that math is hard, uninteresting, and accessible only to “nerds” persists. 

harsh stereotypical thinking—mathematics is for select racial groups and men. This thinking, as well as the teaching practices that go with it, have provided the perfect conditions for the creation of a math underclass.

online platform explaining research evidence on ability and the brain and on good mathematics teaching, for teachers and parents. The course had a transformative effect. It was taken by 40,000 people, and 95 percent said they would change their teaching or parenting as a result.

does not simply test a mathematical definition, as the first does. It requires that students visualize a triangle, use transformational geometry, consider whether different cases satisfy the mathematical definition, and then justify their thinking.

There is a good reason for this: Justification and reasoning are two of the acts that lie at the heart of mathematics. They are, in many ways, the essence of what mathematics is.  Scientists work to prove or disprove new theories by finding many cases that work or counter-examples that do not. Mathematicians, by contrast prove the validity of their propositions through justification and reasoning.

The young people who are successful in today’s workforce are those who can discuss and reason about productive mathematical pathways, and who can be wrong, but can trace back to errors and work to correct them.

American idea that those who are good at math are those who are fast. Speed is revered in math classes across the U.S., and students as young as five years old are given timed tests—even though these have been shown to create math anxiety in young children. Parents use flash cards and other devices to promote speed, not knowing that they are probably damaging their children’s mathematical development

The fact of being quick or slow isn't really relevant

gives more time for depth and exploration than the curricula it has replaced by removing some of the redundant methods students will never need or use.

And in many cases, though not all, people living in areas threatened by severe natural hazards do so by choice. If a risk has not been imposed on us, we take it much less seriously. Though Schulz’s piece certainly made a splash online, it is hard to imagine a mass exodus of Portlanders and Seattleites in response. Hey, they like it there.

They don’t have much to compare the future earthquake to. After all, there hasn’t been an earthquake or tsunami like it there since roughly 1700. Schulz poeticizes this problem, calling out humans for their “ignorance of or an indifference to those planetary gears which turn more slowly than our own.” Once again, this confounds our emotional connection to the risk.

The belief that an unlikely event won’t happen again for a while is called a gambler’s fallacy. Probability doesn’t work like that. The odds are the same with every roll of the dice.

But our “temporal parochialism,” as Schulz calls it, also undoes our grasp on probability. “We think probability happens with some sort of regularity or pattern,” says Ropeik. “If an earthquake is projected to hit within 50 years, when there hasn’t been one for centuries, we don’t think it’s going to happen.” Illogical thinking works in reverse, too: “If a minor earthquake just happened in Seattle, we think we’re safe.”

For individuals and government alike, addressing every point of concern requires a cost-benefit analysis. When kids barely have pencils and paper in schools that already exist, how much is appropriate to invest in earthquake preparedness? Even when that earthquake will kill thousands, displace millions, and cripple a region’s economy for decades to come—as Cascadia is projected to—the answer is complicated. “You immediately run into competing issues,” says Slovic. “When you’re putting resources into earthquake protection that has to be taken away from current social needs—that is a very difficult sell.”​

There are things people can do to combat our innate irrationality. The first is obvious: education. California has a seismic safety commission whose job is to publicize the risks of earthquakes and advocate for preparedness at household and state policy levels.

Another idea is similar to food safety ratings in the windows of some cities’ restaurants. Schulz reports that some 75 percent of Oregon’s structures aren’t designed to hold up to a really big Cascadia quake. “These buildings could have their risk and safety score publicly posted,” says Slovic. “That would motivate people to retrofit or mitigate those risks, particularly if they are schools.”

science points to a hard truth. Humans are simply inclined to be more concerned about what’s immediately in front of us: Snakes, fast-moving cars, unfamiliar chemical compounds in our breakfast cereal and the like will always elicit a quicker response than an abstract, far-off hazard.

How big do the numbers have to be for insensitivity to begin? Not very, it turns out.

The image of Aylan lying face down on the beach captivated the world’s attention and even, in short order, resulted in refugee policy changes in countries as far away as the United States. But 14 Syrian children drowned in the Aegean Sea the next day. Did you notice? Did you care?

demonstrated that “compassion fade” can occur when an incident involving a single person expands to as few as two people

In addition to psychic numbing, there is another psychological disposition at work, called “pseudoinefficacy.”

We found that people might be inclined to send money to an individual person in need, but that if they heard that a second person also required aid but could not be helped, they were less inclined to donate to the first person. Meeting that need no longer felt as satisfying. Similarly, when the need for assistance was described as part of a large-scale relief effort, potential donors would experience a demotivating sense of inefficacy arising from the thought that the help they could provide was but “a drop in the bucket.”

It seems that we are psychologically wired to help only one person at a time. And we don’t even care to do that if we sense that there are others we cannot help.

In addition, we believe that yet another psychological tendency, the “prominence effect,” explains why genuinely well-meaning people (and their governments) so often fail to intervene to prevent genocides and other large-scale abuses. “Prominent” actions or objectives are those that are easily justified, though they may not match our stated social values.

Decisions that protect national security or satisfy our attachment to near-term comforts and conveniences are easily justified. Such choices, as Paul Slovic explains in a recent University of Illinois Law Review article, are likely to trump decisions to protect people or the environment, especially when the humans in need or the environmental phenomena in jeopardy (species, habitats, the planet’s climate) are so vast in scale as to seem distant and abstract.

argued decades ago, in their book “New World New Mind,” that our minds had failed to keep up with the times — that we were, in a sense, cave men and cave women, struggling to deal with modern problems, like nuclear annihilation, to which our minds were not suited. They called for a “conscious evolution” in how we processed information about the modern world, meaning an intentional change in our cognitive habits.

We need to be alert to how psychic numbing, pseudoinefficacy and the prominence effect lead us to act in ways contrary to our values. Doing so can help us to improve our reactions to information about a complex and often upsetting world

No test is perfectly accurate; all generate false positives and false negatives. How many varies by the test — the new Abbott Labs test, for example, has a false negative rate of about 3 percent and a false positive rate of about 1.5 percent.

We’d also require public understanding that test results are a risk-assessment tool rather than a definitive answer. And shifting the public’s thinking might be harder than shifting Trump.

Doing that would also require fundamentally changing our thinking.

For mass testing to work, people will need to understand that even an event where all attendees just tested negative isn’t necessarily covid-free, because some false negatives might have slipped through

Giving hundreds of millions of citizens this crash course in elementary statistics would be very difficult. Teaching people how to think about a positive result would be even harder — especially if mass testing does its job of reducing caseloads. Because as case numbers fall, false positives will become a bigger and bigger problem.

Say we’re testing 200 people, 10 of whom have covid-19. Given its rate of false negatives, the Abbott test would probably catch all 10. But with false positives, it could also tell one or two people who don’t have covid-19 that they’re infected. That’s not ideal — as good citizens, they’d have to go home and quarantine. But a more sensitive follow-up test could substantially mitigate this problem, shortening such quarantines to a day or so. It would be worth it to get those 10 true positives out of circulation.

Doing this, we might eventually reduce the share of the population that’s infected to, say, 0.5 percent from 5 percent. Unfortunately, the false positive rate won’t budge, so now for every one true positive uncovered, roughly three false positives would still be generated.

That’s inherently costly: Every false positive means skipping your flight, rehearsal or birthday party. But it would be catastrophic if people didn’t understand that a positive covid-19 test is a guideline, not a guarantee

People who think that they’ve had covid-19, when they haven’t, are apt to go out and engage in risky behavior, maybe a lot of it. And if mass testing ever becomes common, there could be a lot of those people.

So mass testing isn’t just a matter of getting a test that is cheap enough and plentiful enough; the administration and the public must be educated to use this bounty wisely. That’s a hard messaging problem. Given the alternative, though, it’s the kind of problem we’d really like to have.

On the other hand, if you’re talking about 99.8 percent of a common event which translates into some very large number . . . well, suddenly it’s not great, Bob:

2,900,000 passengers fly on airlines each day. If 99.8% of them made it safely to their destinations, 5,800 people a day would be dying in crashes and people like you would never step on an airplane ever again.July 25th 20211 Retweet4 Likes