All human beings may have the ability to understand elementary geometry, independently of their culture or their level of education.

This is the conclusion of a study carried out by CNRS, Inserm, CEA, the Collège de France, Harvard University and Paris Descartes, Paris-Sud 11 and Paris 8 universities (1). It was conducted on Amazonian Indians living in an isolated area, who had not studied geometry at school and whose language contains little geometric vocabulary. Their intuitive understanding of elementary geometric concepts was compared with that of populations who, on the contrary, had been taught geometry at school. The researchers were able to demonstrate that all human beings may have the ability of demonstrating geometric intuition. This ability may however only emerge from the age of 6-7 years. It could be innate or instead acquired at an early age when children become aware of the space that surrounds them. This work is published in the PNAS.

Euclidean geometry makes it possible to describe space using planes, spheres, straight lines, points, etc. Can geometric intuitions emerge in all human beings, even in the absence of geometric training?

To answer this question, the team of cognitive science researchers elaborated two experiments aimed at evaluating geometric performance, whatever the level of education. The first test consisted in answering questions on the abstract properties of straight lines, in particular their infinite character and their parallelism properties. The second test involved completing a triangle by indicating the position of its apex as well as the angle at this apex.

To carry out this study correctly, it was necessary to have participants that had never studied geometry at school, the objective being to compare their ability in these tests with others who had received training in this discipline. The researchers focused their study on Mundurucu Indians, living in an isolated part of the Amazon Basin: 22 adults and 8 children aged between 7 and 13. Some of the participants had never attended school, while others had been to school for several years, but none had received any training in geometry. In order to introduce geometry to the Mundurucu participants, the scientists asked them to imagine two worlds, one flat (plane) and the second round (sphere), on which were dotted villages (corresponding to the points in Euclidean geometry) and paths (straight lines). They then asked them a series of questions illustrated by geometric figures displayed on a computer screen.

Around thirty adults and children from France and the United States, who, unlike the Mundurucu, had studied geometry at school, were also subjected to the same tests.

The result was that the Mundurucu Indians proved to be fully capable of resolving geometric problems, particularly in terms of planar geometry. For example, to the question Can two paths never cross?, a very large majority answered Yes. Their responses to the second test, that of the triangle, highlight the intuitive character of an essential property in planar geometry, namely the fact that the sum of the angles of the apexes of a triangle is constant (equal to 180°).

And, in a spherical universe, it turns out that the Amazonian Indians gave better answers than the French or North American participants who, by virtue of learning geometry at school, acquire greater familiarity with planar geometry than with spherical geometry. Another interesting finding was that young North American children between 5 and 6 years old (who had not yet been taught geometry at school) had mixed test results, which could signify that a grasp of geometric notions is acquired from the age of 6-7 years.

The researchers thus suggest that all human beings have an ability to understand Euclidean geometry, whatever their culture or level of education. People who have received no, or little, training could thus grasp notions of geometry such as points and parallel lines. These intuitions could be innate (they may then emerge from a certain age, as it happens 6-7 years). If, on the other hand, these intuitions derive from learning (between birth and 6-7 years of age), they must be based on experiences common to all human beings.

(1) The two CNRS researchers involved in this study are Véronique Izard of the Laboratoire Psychologie de la Perception (CNRS / Université Paris Descartes) and Pierre Pica of the Unité ?Structures Formelles du Langage? (CNRS / Université Paris 8). They conducted it in collaboration with Stanislas Dehaene, professor at the Collège de France and director of the Unité de Neuroimagerie Cognitive à NeuroSpin (Inserm / CEA / Université Paris-Sud 11) and Elizabeth Spelke, professor at Harvard University.