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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.

calculus lets you take a lot of simple models and integrate them into one big picture.” Similarly, Coifman believes that modern mathematics — notably geometry — can help identify the underlying global structure of big datasets.

The key to the technique’s success is a concept known as sparsity, which usually denotes an image’s complexity, or lack thereof. It’s a mathematical version of Occam’s razor: While there may be millions of possible reconstructions for a fuzzy, ill-defined image, the simplest (sparsest) version is probably the best fit. Out of this serendipitous discovery, compressed sensing was born.

Using compressed sensing algorithms, it is possible to sample only 100,000 of, say, 1 million pixels in an image, and still be able to reconstruct it in full resolution — provided the key elements of sparsity and grouping (or “holistic measurements”) are present. It is useful any time one encounters a large dataset in which a significant fraction of the data is missing or incomplete.

“So,” Juárez Correa said, “what do you want to learn?”

His defining principle: “The children are completely in charge.”

if you’re not the one who’s controlling your learning, you’re not going to learn as well,”

Peter Gray, a research professor at Boston College who studies children’s natural ways of learning, argues that human cognitive machinery is fundamentally incompatible with conventional schooling. Gray points out that young children, motivated by curiosity and playfulness, teach themselves a tremendous amount about the world. And yet when they reach school age, we supplant that innate drive to learn with an imposed curriculum. “We’re teaching the child that his questions don’t matter, that what matters are the questions of the curriculum. That’s just not the way natural selection designed us to learn. It designed us to solve problems and figure things out that are part of our real lives.”

He squatted next to her and asked why she hadn’t expressed much interest in math in the past, since she was clearly good at it. “Because no one made it this interesting,” she said.