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Then, in November, she posted on YouTube a video about doodling in math class, which married a distaste for the way math is taught in school with an exuberant exploration of math as art .

At first glance, Ms. Hart’s fascination with mathematics might seem odd and unexpected. She graduated with a degree in music, and she never took a math course in college.

The ensuing attention has come with job offers and an income. In one week in December, she earned $300 off the advertising revenue that YouTube shares with video creators. She is also happy that, unlike in her early efforts, which drew an audience typical of mathematics research — older and male, mostly — the biggest demographic for her new videos, at least among registered users, are teenage girls.

That’s because the American system of teaching these subjects is broken. For all the reform campaigns over the years, most schools continue to teach math and science in an off-putting way that appeals only to the most fervent students.

The system is alienating and is leaving behind millions of other students, almost all of whom could benefit from real-world problem solving rather than traditional drills.

the most important factor that predicted math success in middle school and upward was an understanding of what numbers are before entering the first grade. Having “number system knowledge” in kindergarten or earlier — grasping that a numeral represents a quantity, and understanding the relationships among numbers — was a more important factor in math success by seventh grade than intelligence, race or income.

Many years ago, people learned to weld in a high school shop class or in a family business or farm, and they came up through the ranks and capped out at a certain skill level. They did not know the science behind welding,” so could not meet the new standards of the U.S. military and aerospace industry. “They could make beautiful welds,” she said, “but they did not understand metallurgy, modern cleaning and brushing techniques” and how different metals and gases, pressures and temperatures had to be combined.

Welding “is a $20-an-hour job with health care, paid vacations and full benefits,” said Tapani, but “you have to have science and math. I can’t think of any job in my sheet metal fabrication company where math is not important. If you work in a manufacturing facility, you use math every day; you need to compute angles and understand what happens to a piece of metal when it’s bent to a certain angle.” Who knew? Welding is now a STEM job — that is, a job that requires knowledge of science, technology, engineering and math.

even before the Great Recession we had a mounting skills problem as a result of 25 years of U.S. education failing to keep up with rising skills demands, and it’s getting worse.

his view of mathematics has utterly changed since childhood.

he ancient art of mathematics, Tao has discovered, does not reward speed so much as patience, cunning and, perhaps most surprising of all, the sort of gift for collaboration and improvisation that characterizes the best jazz musicians.

He studied math and physics at Yale, got a masters in physics and was working on his Ph.D. at UC Berkeley. Then he realized his real love was music, and his Ph.D. turned into the study of music perception and cognition.

Nor is it clear that the math we learn in the classroom has any relation to the quantitative reasoning we need on the job. John P. Smith III, an educational psychologist at Michigan State University who has studied math education, has found that “mathematical reasoning in workplaces differs markedly from the algorithms taught in school.”

It’s not hard to understand why Caltech and M.I.T. want everyone to be proficient in mathematics. But it’s not easy to see why potential poets and philosophers face a lofty mathematics bar. Demanding algebra across the board actually skews a student body, not necessarily for the better.

Instead of investing so much of our academic energy in a subject that blocks further attainment for much of our population, I propose that we start thinking about alternatives. Thus mathematics teachers at every level could create exciting courses in what I call “citizen statistics.”

She is also frustrated by the predictable nature of many of the “dirty dozen,” the teachers’ nickname for the basic lab exercises now recommended by the College Board. In one that her class did last fall, the students looked at pre-stained slides of onion root tips to identify the stages of cell division and calculate the duration of the phases. She and her students, who historically score 4’s and 5’s on the exam, were one of several schools asked by the College Board to road test one of the proposed new labs to see if it brought back the “Oh, wow!” factor. The basic question: What factors affect the rate of photosynthesis in living plants? The new twist: Instead of being guided through the process, groups of two or three students had to dream up their own hypotheses and figure out how to test them. Caroline Brown, a senior who stages the school’s plays, connected the lab to her passion for theater. She borrowed green, sky blue and “Broadway pink” filters from the playhouse to test how different shades of light affected photosynthesis in sunken spinach leaves.

This story of discovery begins a century ago with Albert Einstein, who realized that space is not an immutable stage on which events play out, as Isaac Newton had envisioned. Instead, through his general theory of relativity, Einstein found that space, and time too, can bend, twist and warp, responding much as a trampoline does to a jumping child. In fact, so malleable is space that, according to the math, the size of the universe necessarily changes over time: the fabric of space must expand or contract — it can’t stay put. For Einstein, this was an unacceptable conclusion. He’d spent 10 grueling years developing the general theory of relativity, seeking a better understanding of gravity, but to him the notion of an expanding or contracting cosmos seemed blatantly erroneous. It flew in the face of the prevailing wisdom that, over the largest of scales, the universe was fixed and unchanging. Einstein responded swiftly. He modified the equations of general relativity so that the mathematics would yield an unchanging cosmos. A static situation, like a stalemate in a tug of war, requires equal but opposite forces that cancel each other. Across large distances, the force that shapes the cosmos is the attractive pull of gravity. And so, Einstein reasoned, a counterbalancing force would need to provide a repulsive push. But what force could that be? Remarkably, he found that a simple modification of general relativity’s equations entailed something that would have, well, blown Newton’s mind: antigravity — a gravitational force that pushes instead of pulls. Ordinary matter, like the Earth or Sun, can generate only attractive gravity, but the math revealed that a more exotic source — an energy that uniformly fills space, much as steam fills a sauna, only invisibly — would generate gravity’s repulsive version. Einstein called this space-filling energy the cosmological constant, and he found that by finely adjusting its value, the repulsive gravity it produced would precisely cancel the usual attractive gravity coming from stars and galaxies, yielding a static cosmos

There are three formal AIMS undertakings: a master’s degree program in Mathematical Sciences, research, and teacher training. The master’s program offers free tuition to accepted students and trains them in both general principles – problem formulation, the scientific method, communication – and cutting-edge math in subjects including computer science, biomathematics, and financial mathematics. Research will allow for international collaborations and advanced student training.

Traditionally, classrooms were led by an authoritative teacher who disseminated information to silent students, but Zomahoun hopes to turn that paradigm on its head. “We train people who can challenge the status quo,” he explains, “not just people who learn from books, listen to lectures, and just repeat it.” Rather, he hopes to instill qualities like “critical thinking, independent thinking, and problem solving” in order to prepare students for real-world problems.

Corrypt, upon proper exploration, revealed itself to be a brilliantly designed puzzle game, unforgiving and unwilling to accommodate players who refuse to give it their full attention. Peel back one layer, and it reveals another more surprising one.

After completing a degree in math and computer science at the University of Auckland, Brough moved to London and began working towards a Ph.D. He landed a decent-paying programming job while continuing his scholastic work, but continued making games, including the beautiful, abstract strategy game Vertex Dispenser, which even Brough admits may have been too esoteric. It combined elements of shooters and real-time strategy games with a complex puzzle system, and many players felt overwhelmed. “I just could not get my head around those concepts at the same time,” said one.

Well-designed games, he believes, can teach people how to do some things better. By simulating challenging situations, games can teach us about “managing unexpected situations… making good decisions, thinking about the costs of our actions and dealing with the consequences,” he says.

“I’m interested in seeing what design principles nature uses,” Dr. Lalvani said. “Math, perhaps; maybe physics, whatever. The whole D’Arcy Thompson-type stuff.” The biologist D’Arcy Wentworth Thompson’s book “On Growth and Form,” published in the early 20th century, was a seminal work on the subject of patterns in nature. Thompson, a Scotsman, argued that growth and the structures that resulted were governed by physical principles and could often be described in mathematical terms. He saw examples throughout nature — in the spiral shell of a nautilus, the branching veins on an insect wing and the scales of a pine cone, to name just a few.

The impact of data abundance extends well beyond business. Justin Grimmer, for example, is one of the new breed of political scientists. A 28-year-old assistant professor at Stanford, he combined math with political science in his undergraduate and graduate studies, seeing “an opportunity because the discipline is becoming increasingly data-intensive.” His research involves the computer-automated analysis of blog postings, Congressional speeches and press releases, and news articles, looking for insights into how political ideas spread.

But the computer tools for gleaning knowledge and insights from the Internet era’s vast trove of unstructured data are fast gaining ground. At the forefront are the rapidly advancing techniques of artificial intelligence like natural-language processing, pattern recognition and machine learning.