Researchers in France and at Harvard University have found that isolated indigenous peoples deep in the Amazon readily grasp basic concepts of geometry such as points, lines, parallelism and right angles, and can use distance, angle and other relationships in maps to locate hidden objects. The results suggest that geometry is a core set of intuitions present in all humans, regardless of their language or schooling.
The study of geometrical understanding among the Mundurukú, who live in remote areas along the Cururu River in Brazil, is described this week in the journal Science.
"Although there has been a lot of research on spatial maps, navigation and sense of direction, there is very little work on the conceptual representations in geometry," says co-author Stanislas Dehaene of the Collège de France in Paris. "What is meant by 'point,' 'line,' 'parallel,' 'square' versus 'rectangle'? All are highly idealized concepts never met in physical reality. Our work is a first start in the exploration of these concepts."
The work by Dehaene and colleagues suggests that such concepts are largely universal across humans.
"While geometrical concepts can be enriched by culture-specific devices like maps, or the terms of a natural language, underneath this variability lies a shared set of geometrical concepts," says co-author Elizabeth S. Spelke, a professor of psychology in Harvard's Faculty of Arts and Sciences. "These concepts allow adults and children with no formal education, and minimal spatial language, to categorize geometrical forms and to use geometrical relationships to represent the surrounding spatial layout."
Dehaene, Spelke and co-authors Véronique Izard and Pierre Pica developed and administered two different sets of tests during visits to the Mundurukú in 2004 and 2005. Their first test, designed to assess comprehension of basic concepts such as points, lines, parallelism, figure, congruence and symmetry, presented arrays of six images, one of which was subtly dissimilar. For instance, five comparable trapezoids might be matched with a sixth non-trapezoidal quadrilateral of similar size. The Mundurukú were then asked, in their own language, which of the images was "weird" or "ugly."
"If the Mundurukú share with us the conceptual primitives of geometry," the researchers write, "they should infer the intended geometrical concept behind each array and therefore select the discrepant image."
Mundurukú subjects, even those as young as six years old, chose the correct image an average 66.8 percent of the time, showing competence with basic concepts of topology, Euclidean geometry and basic geometrical figures. The performance of both Mundurukú adults and children on the task rivaled that of American children in separate testing done by the scientists, while the performance of American adults was significantly higher.
Dehaene, Spelke and colleagues also administered an abstract map test where subjects were given a simple diagram to identify which of three containers arrayed in a triangle on the ground hid an object. Both Mundurukú adults and children were able to relate the geometrical information on the map to geometrical relationships in the environment, attaining an overall success rate of 71 percent that again matched the performance of American children while lagging American adults.
The superior performance of Western adults suggests that formal education enhances or refines geometrical concepts. Nevertheless, the report concludes, "the spontaneous understanding of geometrical concepts and maps by this remote human community provides evidence that core geometrical knowledge ... is a universal constituent of the human mind."
The study of human geometrical knowledge has a long history, dating back at least to Socrates' probing of the intuitions of an uneducated slave in a Greek household, chronicled by Plato approximately 2,400 years ago.
"Many of the references in our paper are from Plato, Riemann and Poincaré," Dehaene says. "What excited us was the ability to ask experimentally some questions which belong to a very long history of questions about the foundations of geometry."
Dehaene, Izard, Pica and Spelke's work was supported by INSERM, CNRS, the National Institutes of Health and the McDonnell Foundation.
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