Is understanding geometry innate?

A new study suggests that even four-year-old children have a grasp of basic geometric concepts.
A new study suggests that even four-year-old children have a grasp of basic geometric concepts.
Tom Herde/File photo 2002

Is our understanding of basic geometric concepts innate? For centuries, psychologists and philosophers have mulled the question, in forums ranging from Socratic dialogue to tests of an isolated, unschooled Amazonian tribe whose members are capable of the same basic geometric reasoning as people who learned to use protractors and compasses in school in Boston.

For years, researchers at a child development laboratory at Harvard University have been probing the question by trying to understand how geometric reasoning abilities develop in the first place.

In a study published Monday, those researchers tested 4-year-old childrens’ ability to use a map to locate a particular spot in a triangular enclosure and found the children could use two different types of geometric information to pinpoint the spot, each related to a skill with deep evolutionary roots, shared by other animals. By subtly altering the triangle environment, researchers could test which of two different sets of spatial reasoning skills the children were using—drawing a direct line to more ancient geometric reasoning skills.

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“Either humans have something altogether new that allows them to have these abstract thoughts about geometric intuitions, or these abstract abilities have an origin that comes from our evolutionary ancestors—we’re building on something that was already there to begin with,” said Moira Dillon, a graduate student at Harvard who led the work, published in the Proceedings of the National Academy of Sciences.

In a laboratory in Cambridge, 45 four-year-olds were given a series of different geometric tests. Those included tests of how well they oriented themselves in a rectangular fenced-off area after being blindfolded and spun around, and a computer test of their ability to recognize different types of geometric patterns.

Then, the children stood in the center of an area that was fenced off in the shape of a right triangle with white foam walls. In one scenario, the triangle was missing all three corners and had only three opposing walls. In another, the triangle was missing its sides, but had three corners. In both tasks, the researcher showed the children the same map, depicting a dot at a particular spot along the edge of the triangle, where the child was asked to place a stuffed animal.

Other studies had shown that humans and animals use two kinds of geometric information—either situating themselves by examining distances and direction, or by using angles and the relative length of objects in the environment.

What the researchers found was that in the two different navigation tasks, children used the two different systems that animals have, but not simultaneously. The children who were best at using distance and direction information from their environment were also best at figuring out where to place the stuffed animal in the triangle that lacked corners. Those that were best at a computer-based task that asked them to understand various geometric patterns were the best when the triangle was made up only of corners.

That suggests humans’ earliest geometric abilities are the same as the ones animals use to navigate their environments. Then, around age two and a half, children begin to be able to abstract those principles to read maps, which animals can’t do—adapting the type of information they use depending on what the situation calls for, as in the two triangle scenarios. That leaves one big development gap that researchers are now hoping to probe more deeply.

“We need to figure out how we get from a shape in a map and relating it to the world at age 4, to reasoning about Euclidean geometry at ages 12 to 13,” Dillon said. “Between 4 and age 12, they are relying on abilities with evolutionary origins, but we don’t know the process by which these things are really coming together.”