Geometry Is Supposed to Be Easy

For anyone who struggled through high school geometry, here's some depressing news: Even people in one of the most remote cultures on the planet are able to master it intuitively.

It seems the ability to solve geometric problems is universal among human beings, spanning all cultures, and begins at a very young age.

Scientists reached that conclusion after comparing children and adults in a remote region of the Amazon with the privileged kids and educated adults in the Cambridge-Boston area. It turns out that the Munduruku people, who live along Brazil's Cururu River, are about as well-equipped to handle geometry as the Bostonians.

That's pretty amazing, considering that the Munduruku don't even have words for such geometrical figures as triangles and points and lines. But what they have, according to the researchers, is an intuition to do geometry, and the research strongly suggests that all people have that basic foundation.

"The Munduruku are probably as different from us as any culture could be," says Elizabeth Spelke, professor of psychology at Harvard University and one of the authors of a paper in the current issue of the journal Science. They have no formal education, and no obvious reason to struggle with geometry, but they did about as well on a series of tests as the children in Boston, although they lagged behind the Bostonian adults.

This finding comes from an interdisciplinary team of scientists who have spent years trying to work out a problem they all share. They want to know, as Socrates and Plato wondered in ancient Greece, how it is that we can do mathematics? How do we understand numbers and geometrical relationships?

The team includes Stanislas Dehaene of the College de France in Paris, who is a cognitive neuroscientist with a background in mathematics; Veronique Izard and Pierre Pica, an expert on comparative linguistics; and Spelke, who for 30 years has been researching the foundational abilities that allow us to do mathematics.

Dehaene had done previous research involving the Munduruku, so it was a natural setting for the team to seek the answers to some pretty basic questions. Could children as young as 6 years old pick out the oddball among a set of geometrical figures? Are adults any better at it than children? And is there any real difference between those who are educated and those who aren't?

"Our studies show these abilities are universal, and they develop in the absence of instruction," says Spelke. "You don't need to take a class in geometry, and you don't need to learn from a teacher what a right triangle is in order to show this sensitivity."

The team did two types of experiments. One was designed to test the ability to recognize dissimilar geometric symbols. The other was to see if people who had never seen maps could intuitively use a map to locate objects in the real world.

As they sat in front of a solar-powered laptop computer deep in the Amazonian jungle, the Munduruku participants where shown a series of six geometrical figures, including triangles, lines that cross, lines that are parallel, and so forth.

For example, "five triangles will have one right angle, and the sixth will not," Spelke says. "They differ in size, and orientation, but the one property five have in common is they all have one right angle.

"We simply ask people to pick out the one that's different from the others."

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."