James' World 2

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You probably haven’t had to do longhand math in years, but you do mental math every day. Or maybe you google math problems ten times a day, because you’ve forgotten how to do any math beyond your basic multiplication tables. Here are some shortcuts that will help you do more math in your head.

Calculate percentages backward

X% of Y = Y% of X. You can always swap those percentages if doing the math is easier the other way around. So 68% of 25 = 25% of 68 = 68/4 = 17.

That makes a lot of calculations easy, once you’ve memorized the percentages that equal basic fractions:

• 10% = 1/10
• 12.5% = 1/8
• 16.666…% = 1/6
• 20% = 1/5
• 25% = 1/4
• 33.333…% = 1/3
• 50% = 1/2
• 66.666…% = 2/3
• 75% = 3/4

Subtract without borrowing digits

Mental subtraction is easiest when you can subtract each digit without having to borrow from the next place up. If the second number has some bigger digits than the first, it gets more complicated. To avoid borrowing places, you want to get rid of those bigger digits. Here’s how:

 

 

Say you’re calculating 925-734. That tens place makes things a little complicated. It’d be easier to calculate 925-724, and then subtract that extra 10 separately: 925-724 = 201, and 201-10 = 191. There’s your answer.

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Let me think…

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I teach people how to teach math, and I’ve been working in this field for 30 years. Across those decades, I’ve met many people who suffer from varying degrees of math trauma – a form of debilitating mental shutdown when it comes to doing mathematics.

When people share their stories with me, there are common themes. These include someone telling them they were “not good at math,” panicking over timed math tests, or getting stuck on some math topic and struggling to move past it. The topics can be as broad as fractions or an entire class, such as Algebra or Geometry.

The notion of who is – and isn’t – a math person drives the research I do with my colleagues Shannon Sweeny and Chris Willingham with people earning their teaching degrees.

One of the biggest challenges U.S. math educators face is helping the large number of elementary teachers who are dealing with math trauma. Imagine being tasked with teaching children mathematics when it is one of your greatest personal fears.

Math trauma manifests as anxiety or dread, a debilitating fear of being wrong. This fear limits access to life paths for many people, including school and career choices. There are many reasons people may develop negative associations with mathematics. The way students are positioned as “good at math” is often based on non-mathematical characteristics such as gender, race, language, or socioeconomic status. For example, Ebony McGee, an education researcher at Vanderbilt University, describes both fragile and robust mathematics identities that Black engineering and mathematics college students developed in response to negative stereotypes about their ability to learn and do mathematics.

While math trauma has multiple sources, there are some that parents and teachers have power to influence directly: outdated ideas of what it means to be good at math. These include speed and accuracy, which were important in decades past when humans were actual computers.

But research has confirmed what many people share with me anecdotally: Tying speed with computation debilitates learners. People who struggle to complete a timed test of math facts often experience fear, which shuts down their working memory. This makes it all but impossible to think, which reinforces the idea that a person just can’t do math – that they are not a math person.

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Even some teachers suffer from anxiety about math. Undrey/shutterstock.com

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You can find Dolomite all over the world. It’s a calcium magnesium carbonate found in the Dolomite Mountains in northern Italy (obviously), but also at Niagara Falls in North America and the White Cliffs of Dover in the U.K. Altogether, this useful construction mineral makes up some 2 percent of the Earth’s crust.

But in contrast to its relative natural abundance, scientists have failed to recreate dolomite in the lab for nearly two centuries, leading to what experts call the “Dolomite Problem.” But new work from scientists at the University of Michigan (UM) and Hokkaido University in Sapporo, Japan seems to have finally solved this geologic conundrum by leveraging proprietary software and dissolving crystalline defects with an electron beam. The results of the study were published in the journal Science this past November.

“In the past, crystal growers who wanted to make materials without defects would try to grow them really slowly…our theory shows that you can grow defect-free materials quickly, if you periodically dissolve the defects away during growth,” Wenhao Sun, UM scientist and the corresponding author, said in a press statement. “If we understand how dolomite grows in nature, we might learn new strategies to promote the crystal growth of modern technological materials.”

Dolomite is usually found in rocks older than 100 million years, meaning that this mineral take a long time to form. According to the researchers, this slow growth rate can be attributed to how dolomite’s crystalline structure forms. The mineral’s growth edge is made of alternating rows of calcium and magnesium, and in water, these elements randomly attach to the wrong places and prevent dolomite from forming. While the Earth has nearly infinite patience to wait out this slow growth (like, only one dolomite layer produced per 10 million years slow), humans—with their comparatively infinitesimally small lifespans—do not.

To figure out how to speed up this natural process, scientists needed to understand how these defects attach to the dolomite surface. Usually, this would take thousands of hours of supercomputing, but new UM software leverages a novel technique to complete these simulations “in 2 milliseconds on a desktop,” according to one researcher.

“Our software calculates the energy for some atomic arrangements, then extrapolates to predict the energies for other arrangements based on the symmetry of the crystal structure,” UM associate research scientist and co-author Brian Puchala, one of the software’s lead developers, said in a press statement.

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There is a rip in the armpit of Orsola de Castro’s jumper. She raises her hand high in the air so I can see it: a slash of pale skin peeks from tomato-red wool. This “memory hole”, as De Castro describes it, tells the story of the jumper’s long life. It was owned by her cousin, then her daughter. “It is very old Benetton, from when Benetton was still made in Italy. You can’t see it on Zoom, but this is really nice wool,” she says, arm still aloft.

De Castro, 54, is an activist, a lecturer, a former designer, and a co-founder of not-for-profit movement Fashion Revolution. With the release of her book Loved Clothes Last, she has also become a kind of anti-Marie Kondo. She advocates “radical keeping”, not decluttering. “The only antidote to throwaway culture is to keep. So I am an obsessive keeper,” she says.

The book is full of startling facts about fashion’s impact on the planet and its people. It is “as much about mending systems as mending clothing”, says De Castro. She had just four months to write it, so her daughter Elisalex de Castro Peake, who runs the independent sewing-pattern label By Hand London, and her colleague Bronwyn Seier helped with the research. “That meant that all I had to do was vomit words, which were nestling inside me, quite pumping to get out.” Those words are still pumping in today’s interview: she talks rapidly and lyrically, her eyes shining behind thick-rimmed cat-eye glasses, her salt-and-pepper curls trembling as she gestures energetically.

De Castro grew up in Rome. Her mother is an artist and runs a traditional printmaking school in Venice. Her father, who died when she was two, was a businessman. Her Venetian grandmother, whom she calls Nonna Stanilla, taught her how to crochet at the age of six. She moved to London at 16, did her A-levels, then had the first of her four children at 18. She made clothes for years, first with a small line of upcycled hats, then printed textiles.

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Embellish your imperfections. Photo by Iryna Khabliuk / EyeEm

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Maybe you know the routine. Every so often, I go through my refrigerator, check labels on the items, and throw out anything that’s a month, or a week, or maybe a few days past the date on the label. I might stop to sniff, but for my whole adult life, I’ve figured that the problem was obvious — my jam or almond milk or package of shredded Italian cheese blend had “expired” — and the fix was simple: Into the garbage, it goes.

This habit is so ingrained that when I think about eating food that’s gone past its date, I get a little queasy. I’ve only had food poisoning once or twice in my life, always from restaurants, but the idea is still there in my head: past the date, food will make me sick. You’ll probably never catch me dumpster-diving.

I know, on some intellectual level, that throwing away food is probably wrong. The statistics are damning. Forty percent of food produced in America heads to the landfill or is otherwise wasted. That adds up. Every year, the average American family throws out somewhere between $1,365 and $2,275, according to a landmark 2013 study co-authored by the Harvard Food Law and Policy Clinic and the Natural Resources Defense Council. It’s a huge economic loss for food growers and retailers, who often have to ditch weirdly shaped produce or overstocked food that didn’t sell.

Environmentally it’s bad, too. The study found that 25 percent of fresh water in the US goes toward producing food that goes uneaten, and 21 percent of input to our landfills is food, which represents a per-capita increase of 50 percent since 1974. Right now, landfills are piled high with wasted food, most of which was perfectly fine to eat — and some of which still is.

On top of this, I know that in the same country that throws away so much food, about 42 million people could be living with food insecurity and hunger. Yet state-level regulations often make it difficult to donate past-date food to food banks and other services.

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A scene like this greets most supermarket shoppers in the US. But much of what we think we know about food shopping needs an overhaul. Steve Pfost/Newsday/Getty Images

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Was ever a word so misused as “sustainable”? “Healthy” comes close, and indeed the two are often bandied around together, in trite “good for you, good for the planet” taglines that often appear on foods which are anything but. The question of what we should eat to help combat climate change and environmental degradation has never been more important – nor so confusing. While research on the subject is ongoing, there are definitely some foods which, with caveats, you can scoff with a clear conscience.

“Good eating starts at home, and one of the most important things we can do for the future of the planet is to minimize food miles – so our staples should be foods that can grow perfectly well in this country,” advises Patrick Holden, chief executive of the Sustainable Food Trust. Another basic principle is to do your best to understand the story behind what you’re eating – be it plant or animal: “If you know who produced your food, they are accountable to you, and more likely to care.”

Grass-fed beef and lamb

These meats are the most controversial, complex, and heavily caveated inclusion in this list, but Holden, one of the earliest proponents of regenerative agriculture (which involves rearing livestock within a mixed farming system in order to restore organic matter – and with it, carbon – to the soil) makes a case for eating them. Soil is an invaluable carbon sink; yet the separation of crops and livestock farming has left half of the country dependent on artificial fertilizers, the application of which “reduces organic matter and microbial diversity”, he says, resulting in the leaching of carbon. By rotating livestock with crops (as was done for centuries before the intensification of agriculture), farmers can “build soil carbon and so offset livestock emissions” – and make the most of grass, a plant we can’t eat, but which grows in abundance in the UK.

Consumed in moderation, red meat is highly nutritious and also increases the bioavailability of nutrients in plant foods. “There is a good reason humans have coevolved alongside animals which eat grass,” says Carolyn Steel, author of Sitopia: How Food Can Save the World. “Grass is rich in nutrients, but we can’t digest it. So we eat animals that can.”

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Wakame seaweed salad with sesame and green tea. Photograph: Westend61/Getty Images

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On 27 August 1883, the Earth let out a noise louder than any it has made since.

It was 10:02 a.m. local time when the sound emerged from the island of Krakatoa, which sits between Java and Sumatra in Indonesia. It was heard 1,300 miles away in the Andaman and Nicobar islands (“extraordinary sounds were heard, as of guns firing”); 2,000 miles away in New Guinea and Western Australia (“a series of loud reports, resembling those of artillery in a north-westerly direction”); and even 3,000 miles away in the Indian Ocean island of Rodrigues, near Mauritius* (“coming from the eastward, like the distant roar of heavy guns.”) In all, it was heard by people in over 50 different geographical locations, together spanning an area covering a thirteenth of the globe.

Think, for a moment, just how crazy this is. If you’re in Boston and someone tells you that they heard a sound coming from New York City, you’re probably going to give them a funny look. But Boston is a mere 200 miles from New York. What we’re talking about here is like being in Boston and clearly hearing a noise coming from Dublin, Ireland. Traveling at the speed of sound (766 miles or 1,233 kilometers per hour), it takes a noise about 4 hours to cover that distance. This is the most distant sound that has ever been heard in recorded history.

So what could possibly create such an earth-shatteringly loud bang? A volcano on Krakatoa had just erupted with a force so great that it tore the island apart, emitting a plume of smoke that reached 17 miles into the atmosphere, according to a geologist who witnessed it¹. You could use this observation to calculate that stuff spewed out of the volcano at over 1,600 miles per hour—or nearly half a mile per second. That’s more than twice the speed of sound.

This explosion created a deadly tsunami with waves over a hundred feet (30 meters) in height. One hundred sixty-five coastal villages and settlements were swept away and entirely destroyed. In all, the Dutch (the colonial rulers of Indonesia at the time) estimated the death toll at 36,417, while other estimates exceeded 120,000.

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A lithograph of the massive 1883 eruption of Krakatoa. From The eruption of Krakatoa, and subsequent phenomena, 1888; Parker & Coward; via Wikipedia.

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Even in an age of smartphones, crunching numbers quickly — and getting good at it — can give you snappy, useful answers to pressing real-life math questions such as, Is this shirt a good deal? Or How much should I tip my Uber driver? Here are some good mental math shortcuts to keep under your hat.

To sign your check in record time:

To calculate the amount of a 20 percent tip, calculate 10 percent (or remove the last digit) and double it. For instance, with a $42.50 bill, 10 percent is $4.25, and double it to get an $8.50 tip.

To ace sale shopping:

To find out how much you’ll pay for an item that’s a certain percentage off, first subtract the percent off from 100. So if it’s 30 percent off, use 70; 60 percent off, use 40, etc. Divide this number and the price by 10 and then multiply the resultant two numbers. For instance, if a $20 shirt is 60 percent off, multiply 2 by 4 for a total of $8. A $40 shirt at 30 percent off would be 4 multiplied by 7 for a total of $28 final price.

If you just want to calculate the amount that’s coming off (not the final price), divide both numbers by 10 and then multiply the resulting numbers. So 60 percent off $120 would be 6 x 12, for a total of $72 off.

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Photo by Bethany Robertson/Apartment Therapy

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Taylor Wang was deeply despondent.

A day earlier, he had quite literally felt on top of the world by becoming the first Chinese-born person to fly into space. But now, orbiting Earth on board the Space Shuttle, all of his hopes and dreams, everything he had worked on for the better part of a decade as an American scientist at NASA’s Jet Propulsion Laboratory, had come crashing down around him.

Wang was the principal investigator of an experiment called the Drop Dynamics Module, which aimed to uncover the fundamental physical behavior of liquid drops in microgravity. He had largely built the experiment, and he then effectively won a lottery ticket when NASA selected him to fly on the 17th flight of the Space Shuttle program, the STS-51-B mission. Wang, along with six other crew members, launched aboard Space Shuttle Challenger in April 1985.

On the second day of the mission, Wang floated over to his experiment and sought to activate the Drop Dynamics Module. But it didn’t work. He asked the NASA flight controllers on the ground if he could take some time to try to troubleshoot the problem and maybe fix the experiment. But on any Shuttle mission, time is precious. Every crew member has a detailed timeline, with a long list of tasks during waking hours. The flight controllers were reluctant.

After initially being told no, Wang pressed a bit further. “Listen, I know my system very well,” he said. “Give me a shot.” Still, the flight controllers demurred. Wang grew desperate. So he said something that chilled the nerves of those in Houston watching over the safety of the crew and the Shuttle mission.

“Hey, if you guys don’t give me a chance to repair my instrument, I’m not going back,” Wang said.

Exactly what happened after that may never be known. But thanks to new reporting, we may finally have some answers. And though this is an old story, it still reverberates today, four decades on, with lasting consequences into the era of commercial spaceflight as more and more people fly into orbit.

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The STS-51-B mission begins with the liftoff of the Challenger from Pad 39A in April 1985.

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Albert Einstein was one smart cookie; there’s no doubt about it. But even he knew his general theory of relativity – the 21st century’s answer to Newton’s universal theory of gravity – wasn’t perfect.

Like the second-hand car you bought using your first paycheck, it does the job for day-to-day errands. Push it too hard up a steep hill or park it near a quantum strip mall, and that engine shudders to a standstill.

Peoples’ Friendship University of Russia astrophysics grad student Hamidreza Fazlollahi’s solution is to dive under the hood and see which components aren’t as essential as they seem.

By plucking out a law conserving a mathematical quality involving energy and momentum, he believes general relativity just might make it over a few more speed bumps.

Gravity describes the tendency for things with mass to come together. Whether it’s colliding galaxies, a moon struggling against inertia in its pull towards a planet, or an apple falling to Earth from the top branches of a tree, models of gravity need to explain why masses attract.

Yet such a theory also needs to operate in a Universe where objects of identical charge repel, explain why atomic nuclei stick together with incredible force, or why neutrons spontaneously decay to form protons. It also needs to work still when masses become so dense or space-time so compact, light itself can no longer escape.

And frankly, as good as it is, general relativity just isn’t up to the task.

“The problem of non-renormalizability of Einstein’s gravity is well known. It has led to dozens of attempts to treat it as a low-energy theory,” Fazlollahi says.

Renormalization is a magical sleight-of-hand used by theoretical physicists to make frustrating infinities in quantum fields disappear. When loops of reality seem to recede forever into the distance in a confusing fractal, pull a few of these techniques out of the bag, and your model is on solid ground once again

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