Euclidean geometry describes space using planes, spheres, lines, points, etc. However, do all human beings intuitively grasp geometric concepts, even in the absence of formal training in geometry? To help answer this question, cognitive science researchers devised two experiments to evaluate an individual’s performance in geometry, whatever the level of instruction. The first test consists of replying to questions about abstract properties of straight lines, such as their infinite nature and parallelism. The second test involves completing a triangle by indicating the position of its vertices as well as the angle of the vertices.

For the study to be successful, it was necessary for participants to have had no formal training in geometry, since the objective was compare their aptitudes at such tests with those of individuals who had studied the discipline. The researchers selected a group of Mundurucu Indians living in an isolated Amazonian region. The group consisted of 22 adults and eight children between the ages of 7 and 13. Some of the participants had never been educated, while others had received several years of education but had not studied geometry. In order to present geometry to the Mundurucus, the researchers asked them to imagine two worlds, one flat ("a plane") and the other round ("a sphere"), on which were located villages (corresponding to "points" in Euclidean geometry) and paths ("lines"). They then asked them a series of questions illustrated by geometric figures presented on a computer screen. The same tests were given to some 30 adults and children from France and the United States who, unlike the Mundurucus, had studied geometry at school.

The result: the Mundurucu Indians showed themselves to be quite capable of resolving geometry problems, and performed particularly well in plane geometry. For example, when asked the question, "Can two paths never cross?" a large majority responded "Yes". Their responses to the second test, that of the triangle, demonstrated their "intuitive" understanding of an essential property of plane geometry, namely that the sum of the angles of the vertices of a triangle is constant (equal to 180°). In a spherical universe, the Amazonian Indians responded better than the French and Americans, who, by learning geometry at school, acquired a greater familiarity with plane geometry than with spherical geometry. Another interesting observation: American children aged 5-6 (who had not yet learnt geometry at school) had mixed results on the tests, which would indicate that acquisition of geometry starts at 6-7 years of age.

The researchers suggest that all human beings are able to understand Euclidean geometry, regardless of culture or level of education. Individuals who received little or no instruction were therefore able to grasp geometrical concepts such as points or parallel lines. Such intuition may be innate; it appears to emerge at a specific age, i.e., somewhere between 6 and 7 years old. If, on the other hand, such intuition results from learning (acquired between birth and the age of 6-7), then it must be based on experience common to all human beings.

Bibliography

Flexible intuitions of Euclidean geometry in an Amazonian indigene group. Véronique Izard, Pierre Pica, Elizabeth S. Spelke, and Stanislas Dehaene. Proceedings of the National Academy of Sciences of the United States of América, week of 23 May 2011.

[1] The two CNRS researchers involved in this study are Véronique Izard from the Laboratoire psychologie de la perception (CNRS / Université Paris Descartes) and Pierre Pica from the Structures formelles du langage unit (CNRS/Université Paris 8). They collaborated with Stanislas Dehaene, professor at the Collège de France and director of the Neuroimagerie cognitive unit at NeuroSpin (Inserm/CEA/Université Paris-Sud 11) and Elizabeth Spelke, professor at Harvard University.