Geometry in the Amazon: Dr. Veronique Izard Sheds Light on How We Learn Math

Are mathematics concepts innate, or are they learned? That's a question that Dr. Veronique Izard from Paris' Descartes University hopes to answer. She, along with language expert Pierre Pica, are studying the Mundurucu tribe in the Amazon to see how they grasp mathematical concepts they couldn't have learned in schools. Results so far have been surprising, suggesting that when it comes to math we may know more than we think. Schools offering Mathematics degrees can also be found in these popular choices.

Veronique Izard

Dr. Veronique Izard. Photo courtesy of Will Nunally, Harvard University.

Though she initially earned a degree in engineering at France's Ecole Polytechnique, Veronique Izard later developed an interest in cognitive science that led her to achieving a Ph.D. in the field. Her dissertation explored how our intuitions about numbers relate to physical numeric symbols. That research serves her well in her new project, an attempt to document how and why we comprehend certain geometric concepts, even if we've never heard of them before. When did you first become interested in the notion that geometric knowledge may be universal? Were there any initial 'eureka' moments that convinced you that you had to pursue this?

Veronique Izard: The topic of geometry is actually quite new for me. In my first studies, I was mostly focused on numbers. For example, in my first collaboration with Pierre Pica, we asked the Mundurucu people to compare the number of dots in different images. We showed that they have a very good sense of numbers for quantities up to 80 dots despite the fact that their language has words for numbers only up to five (with some possibility to combine these words to express quantities like ten or 15, but by no means up to 80). Having come to cognitive science from a fascination with mathematics, I was naturally interested to see if something similar was taking place in other mathematical fields: are we endowed with foundational intuitions for geometry like we are for numbers?

In our first study (published in 2006), we looked at visual perception in the Mundurucu and showed that they're sensitive to the geometric properties of visual forms. This time, we wanted to take the test one step further by seeing whether they could grasp concepts of geometry that go beyond what can ever be perceived. Indeed, it's not possible for our visual system to perceive planes or lines that are infinite or to check that infinite lines may sometimes never cross (the property of parallelism). Despite the fact that their vision could not inform them of these properties and the fact that they'd never studied formal geometry in school, pretty much all our Mundurucu participants agreed on abstract statements such as the existence of parallels. What made you decide to conduct your studies in the Amazon? How did you pick the people you use as control groups?

VI: In order to test the hypothesis of universality, we needed to find a group of people with a drastically different experience of space compared to people in industrialized societies. The Mundurucu do not study formal geometry in school; they speak a language that has only few words with geometric content (for example, they have no word for 'parallel' or 'right angle'), but at the same time they engage in complex navigation tasks to find their way in the jungle.

In the present case, our research with the Mundurucu was building on a long-lasting collaboration with Pierre Pica. To establish contacts with an indigene community in a remote area and organize research field trips is not an easy task, and Pierre has been working for years to make it possible with the Mundurucu.

As for the control groups, there were several. We first had two groups of adults (from the U.S.) and children (from France). The children were matched in age and gender with the Mundurucu. The U.S. adults were a mixture of students and members of the Cambridge community of various ages. Furthermore, we also included a last group of young U.S. children aged five or six years, when they haven't yet studied geometry in school. You told National Public Radio you weren't sure yet whether geometric knowledge is innate or learned at a very early age. Do you have a suspicion one way or the other?

VI: Our data do not enable us to conclude either way. One possibility would be that geometric knowledge is innate but emerges only later during childhood (much like the beard in human males is innate but emerges only during puberty). Another possibility could be that geometric knowledge is learned on the basis of the children's experience with space during the first years of life. Both alternatives raise very difficult challenges. If geometry is innate, why is it that it doesn't emerge until later in childhood? If geometry is learned, how is it that we can learn abstract knowledge about concepts that can never be perceived (infinite planes and lines, impossible configurations) on the basis of the limited input of perception? Your research has shown a distinct difference between knowledge of numeracy and knowledge of geometry. Why do you think that is?

VI: The results about numeracy are complex. On one hand, we found that the Mundurucu had a good intuition of numbers. On the other hand, this intuition has a profound limitation: perception of numbers only approximates numbers. That means that Mundurucu can easily perceive the difference between numbers that are far apart (20 and 40), but they can't perceive very fine differences, such as 20 versus 21. That numbers can be defined exactly, up to the precision of one unit, is an essential property of the natural numbers. People in industrialized society are no better able to perceive the difference between 20 and 21 at first glance; however, our culture provides us with a tool (counting) that enables us to assess quantities precisely.

In a second research project, we investigated whether the Mundurucu could use a spatial metaphor to refer to numbers. We asked them to place quantities on lines, where one extremity of the line corresponded to one quantity (one), the other extremity corresponded to a different quantity (ten), and, as we told them, 'all other quantities belong somewhere on this line.' We observed that the Mundurucu were able to understand the metaphor and to place the quantities along the line according to their size: the smaller quantities closer to the one extremity, the larger quantities closer to the ten extremity.

The metaphor of mapping numbers to space is universal: it can be grasped by people without specific instruction or experience with rulers and measurement, and only after minimal explanations. However, there was a crucial difference between Mundurucu and U.S. controls. The Mundurucu did not place the numbers regularly on the line, as do adults in industrialized societies: instead, they allotted a lot of room for the small numbers, and compressed all large numbers together on one extremity of the line. Children in the U.S. do exactly the same thing until second grade. Again, even though the mapping of numbers to space is intuitive and universal, this intuition is then transformed in our society in contact with culture-specific tools and under the effect of education.

To sum up, our research on numeracy shows that number concepts are a mosaic of universal intuitions and culture-specific knowledge. In our first research on geometry, we observed that the basic Euclidean concepts are present in the Mundurucu. However, this does not exclude that other aspects of geometry are culture-specific (and probably there are a lot). What results do you hope your research has on the world at large? How might it alter mathematics education, and education in general, around the globe?

VI: I would be careful of drawing hastened conclusions here. I would love it if this research can inspire teachers to develop new curricula, for example a new way of introducing geometry. However, I'm not a specialist of education myself. What our research also shows is that children are probably not ready to hear about geometry before the age of seven.

Another impact of our research is for the Mundurucu community. Under the inspiration of our findings, the encouragements of my collaborator Pierre Pica and those of Andre Ramos from FUNAI (the National Indian Foundation - Brazil), the Mundurucu associations have made a request to the Brazilian government to develop a program of differentiated education. We hope to be able to help them in this process.

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