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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.

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.”

“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.

In the 38 years since the Hungarian architecture professor Erno Rubik invented his cube, it has alternately been regarded as an object of fun, art, mathematics, nostalgia and frustration

“You can use Rubik’s Cube to teach engineering, you can use it to teach mathematics, and you can use it to talk about the interplay between design and engineering and mathematics and creativity,”

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.