Geometry Is Supposed to Be Easy

That's not as easy as it sounds, considering the limits of their vocabulary, so they were asked to pick the one that was "weird" or "ugly."

Both the children and the adults got it right about 71 percent of the time. That matches the rate for the American children, but lags significantly behind the American adults, indicating that education does improve our ability to do geometry. Big surprise, eh?

But the second finding may be the most significant. Some of the problems were more difficult than others, and it turns out that the level of difficulty was transcultural.

"You see the same profile of difficulty among the educated Bostonian adults, and the going-to-school Bostonian kids, and the adults and children in the Amazon," Spelke says. "The problems that were hard for them were hard for us."

That's primarily what led the researchers to conclude that the ability to conceptualize geometric relationships is universal among all humans. But is it something we are born with, or do we learn it at a very early age?

The research doesn't answer that question. The fact that the more educated American adults were better at solving the problems does indicate that education helps, but it doesn't tell us if the abilities are innate or learned.

Neither does the map experiment, although it is intriguing. The Munduruku participants, who had never seen a map, had no trouble locating objects in the real world after they were shown a two-dimensional diagram with objects arranged in a geometrical pattern. One of those objects contained a prize, the participants were told, and they had little trouble finding the same relationships among objects on the ground. They found the prize 71 percent of the time.

That strongly suggests that whether you're a 6-year-old kid in the Amazon jungle, or a college professor in Boston, you have sound "intuitions," as the researchers put it, that should guide you through a world filled with geometrical symbols.

But why, then, does geometry sometimes seem so difficult?

"My guess is it's difficult because it focuses on proofs," says Spelke.

It's not enough to have the intuition that we apparently all share.

"Many kids go into geometry thinking that they love it, and then they get turned off," she adds. Maybe it has something to do with memorizing all those axioms and learning all the rules.

Spelke finds encouragement among the Munduruku.

"The people in the Amazon haven't memorized anything," she says. "They don't even have words in their vocabulary that clearly pick out geometrical relationships, but they have these intuitions and they have them spontaneously.

"So it seems to me that we've all got them, and we ought to be able to build on them."

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