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Studies [e.g.,  this one, and many others referenced in this symposium] have shown that games or free play are efficient ways for children to learn, and they enjoy them.

start by creating rich and social mathematical experiences that are complex (allowing them to be taken in many different directions) yet easy (making them conducive to immediate play). Activities that fall into this quadrant: building a house with LEGO blocks, doing origami or snowflake cut-outs, or using a pretend “function box” that transforms objects (and can also be used in combination with a second machine to compose functions, or backwards to invert a function, and so on).

What is learned without play is qualitatively different. It helps with test taking and mundane exercises, but it does nothing for logical thinking and problem solving.

It wasn’t the size of the data set that was daunting; by big data standards, the size was quite manageable. It was the sheer complexity and lack of formal structure that posed a problem.

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.