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“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.

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.

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.

By discovering hidden mathematical patterns and regularities in nature that we call equations of physics, we have gotten progressively better at predicting things — from tomorrow’s weather to tomorrow’s technology. The planet Neptune, the radio wave and the Higgs boson were all predicted mathematically before they were observed.

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.

Like many children with autism spectrum disorders, Ravi is fascinated by trains and buses, entranced by their motion and predictability.

Now, the museum, and others like it, are moving beyond accommodating the enthusiasm for trains and buses among children with autism and trying to use it to teach them how to connect with other people — and the world.

The link between trains and autism is well documented. Autism refers to a spectrum of disorders that typically includes impairment in social interaction and sometimes includes stereotyped interests, like trains. People with autism have difficulty processing and making sense of the world, so they are drawn to predictable patterns, which, of course, trains run by.