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If you had asked them, most of my high school teachers would have called me an unmotivated student or said that I lacked discipline and didn't take learning seriously. And yet, that abandoned storage bin told another story: with the aid of my calculator, I'd crafted narratives, drawn storyboards, visualized foreign and familiar environments and coded them into existence. I'd learned two programming languages and developed an online network of support from experienced programmers. I'd honed heuristics for research and discovered workarounds when I ran into obstacles. I'd found outlets to share my creations and used feedback from others to revise and refine my work. The TI-83 Plus had helped me cultivate many of the overt and discrete habits of mind necessary for autonomous, self-directed learning. And even more, it did this without resorting to grades, rewards, or other extrinsic motivators that schools often use to coerce student engagement.

I've now begun to see Texas Instruments graphing calculators as unique among educational technologies in that they enable learning that is couched in discovery more than formal teaching.

take the notion of "correctness." School typically assumes that answers fall neatly into categories of "right" and "wrong." As a conventional tool for computing "right" answers, calculators often legitimize this idea; the calculator solves problems, gives answers. But once an endorsed, conventional calculator becomes a subversive, programmable computer it destabilizes this polarity. Programming undermines the distinction between "right" and "wrong" by emphasizing the fluidity between the two. In programming, there is no "right" answer. Sure, a program might not compile or run, but making it offers multiple pathways to success, many of which are only discovered through a series of generative failures. Programming does not reify "rightness;" instead, it orients the programmer toward intentional reading, debugging, and refining of language to ensure clarity. This is a form of learning that privileges the process of discovery over the interventions of formal teaching. It can fuel an intrinsic desire to pursue similar learning experiences, but even more, it gradually transforms the outlook of the student

As she headed into fifth grade, she assumed she was in for more of the same—lectures, memorization, and busy work. Sergio Juárez Correa was used to teaching that kind of class. For five years, he had stood in front of students and worked his way through the government-mandated curriculum. It was mind-numbingly boring for him and the students, and he’d come to the conclusion that it was a waste of time. Test scores were poor, and even the students who did well weren’t truly engaged.

Juárez Correa didn’t know it yet, but he had happened on an emerging educational philosophy, one that applies the logic of the digital age to the classroom. That logic is inexorable: Access to a world of infinite information has changed how we communicate, process information, and think. Decentralized systems have proven to be more productive and agile than rigid, top-down ones. Innovation, creativity, and independent thinking are increasingly crucial to the global economy. And yet the dominant model of public education is still fundamentally rooted in the industrial revolution that spawned it, when workplaces valued punctuality, regularity, attention, and silence above all else.

knowledge isn’t a commodity that’s delivered from teacher to student but something that emerges from the students’ own curiosity-fueled exploration. Teachers provide prompts, not answers, and then they step aside so students can teach themselves and one another. They are creating ways for children to discover their passion—and uncovering a generation of geniuses in the process.

But this progression actually “has nothing to do with how people think, how children grow and learn, or how mathematics is built,” says pioneering math educator and curriculum designer Maria Droujkova.

The current sequence is merely an entrenched historical accident that strips much of the fun out of what she describes as the “playful universe” of mathematics

“Calculations kids are forced to do are often so developmentally inappropriate, the experience amounts to torture,” she says. They also miss the essential point—that mathematics is fundamentally about patterns and structures, rather than “little manipulations of numbers,” as she puts it.

Yet recent research has found that true experts have something at least as valuable as a mastery of the rules: gut instinct, an instantaneous grasp of the type of problem they’re up against.

Now, a small group of cognitive scientists is arguing that schools and students could take far more advantage of this same bottom-up ability, called perceptual learning. The brain is a pattern-recognition machine, after all, and when focused properly, it can quickly deepen a person’s grasp of a principle, new studies suggest.

Yet there is growing evidence that a certain kind of training — visual, fast-paced, often focused on classifying problems rather then solving them — can build intuition quickly

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.

Taking a test is not just a passive mechanism for assessing how much people know, according to new research. It actually helps people learn, and it works better than a number of other studying techniques.

non-­scientist gamers developed more-­complex proteins than biochemists did

As harder concepts are introduced, students who need more time on a level get additional problems; those who understand it move on.

93 percent of K–12 students successfully mastered concepts after only 90 minutes of gameplay, and they didn’t want to stop

used properly, testing as part of an educational routine provides an important tool not just to measure learning, but to promote it.

Various kinds of testing, though, when used appropriately, encourage students to practice the valuable skill of retrieving and using knowledge.

tests serve students best when they’re integrated into the regular business of learning and the stakes are not make-or-break, as in standardized testing.

Powerful statistical algorithms allow us to connect these patterns to individual learning success.

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.

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.

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.”

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.

“We’re trying to take kids who haven’t been engaged in school and hook them to an expanded vision of what their future might be,” he said. When they return to their own schools, the hope is that they’ll be more interested in history, math and English — and have a sense of environmental stewardship as well.

“The most important thing really is teaching kids through hands on, experiential learning,” said Rees. “Our kids do this because they’re inspired to be there every week, to work with adults and do hands on things.”

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.

This approach can even be useful for applications that are not, strictly speaking, compressed sensing problems, such as the Netflix prize.

Given the enormous popularity of Netflix, even an incremental improvement in the predictive algorithm results in a substantial boost to the company’s bottom line. Recht found that he could accurately predict which movies customers might be interested in purchasing, provided he saw enough products per person. Between 25 and 100 products were sufficient to complete the matrix.

Across every discipline, data sets are getting bigger and more complex, whether one is dealing with medical records, genomic sequencing, neural networks in the brain, astrophysics, historical archives, or social networks. Alessandro Vespignani, a physicist at Northeastern University who specializes in harnessing the power of social networking to model disease outbreaks, stock market behavior, collective social dynamics, and election outcomes, has collected many terabytes of data from social networks such as Twitter, nearly all of it raw and unstructured. “We didn’t define the conditions of the experiments, so we don’t know what we are capturing,” he said.