One may thus propose a simple scenario for the acquisition of elementary arithmetic in humans. Evolution endowed the primate parietal lobe with a coarse representation of numerosity, which was presumably useful in many situations in which a set of objects or congeners had to be tracked through time. This primitive number representation is also present in humans. It emerges early on in infancy, although its precision is initially quite mediocre and matures during the first year of life (Lipton and Spelke 2003). It provides children with a minimal foundation on which to build arithmetic: the ability to track small sets of objects, to estimate coarse numerosity, and to monitor increases or decreases in numerosity. In the first year, this knowledge is entirely non-verbal, but around three years of age, it becomes connected with symbols, first with the counting words of spoken language (and their surrogate, the fingers), then with the written symbols of the Arabic notation.

An interesting question is whether and how the availability of these cultural symbols affects the arithmetic competence of the human primate. A severe limit of animal arithmetic is that, beyond the numbers 1, 2, 3, arithmetic performance is imprecise: no animal, for instance, has ever shown any ability to precisely discriminate 10 from 11 objects, or to compute 9-8. Humans, on the other hand, are able to perform arithmetic calculations with arbitrary precision (although they remain able to approximate, for instance when verifying a grossly false arithmetic problem such as 13+19 = 92). By allowing reference to discrete numbers in a categorical way, the labels provided by number words and Arabic digits may support a crucial transition from approximate to exact arithmetic.

Recently, my team and I had an opportunity to test this hypothesis directly in a population of humans deprived of verbal labels for large numbers. With Pierre Pica, we studied numerical cognition in the Munduruku, an Amazonian group whose language includes very few words for numbers (see also Gordon 2004; Pica et al. 2004; see also Gordon 2004). Munduruku essentially has number words for one through five, plus some quantifiers such as few or many (Fig. 2, top). Using a battery of computerized tasks, we first demonstrated that even those few number words are only used to refer to approximate quantities. When asked to name the numerosity of a set of dots, the Munduruku used their number words fuzzily, for instance using the word for "four" (ebadipdip) when the actual quantity ranges from three to eight. In spite of this lexical limitation, our participants gave evidence of an excellent understanding of large numbers. They could decide which of two sets of dots was the more numerous, even with numbers ranging up to 80, and even in the presence of considerable variation in non-numerical parameters such as object size or density. They could even perform approximate calculation: when successively shown two sets of objects being hidden in a jar, they could estimate their sum and compare it to a third number. Amazingly, these isolated and non-educated Indians, with a limited language, were as accurate as educated French adults in this non-symbolic approximation task (Fig. 2, middle).

Where they differed, however, is in exact calculation. We presented them with concrete depictions of very simple subtraction problems such as 6-4, by hiding six objects in a jar and then drawing four out. The final result was always 0,1 or 2, easily within the Munduruku naming range. In one test, we asked participants to name the result, and in another to point to the correct outcome (zero, one or two objects in the jar). In both cases, the Munduruku failed to calculate the exact result. They performed relatively well with numbers below three (e. g., 2-2, 3-1), but they failed increasingly frequently as the numbers got larger, not faring better than 50% correct as soon as the initial number exceeded five (Fig. 2, bottom).

We concluded that linguistic labels are not necessary to master the major concepts of arithmetic (quantity, larger-smaller relations, addition, subtraction) and to perform approximate operations. Linguistic coding of numerals, however, may be essential to go beyond this evolutionarily ancient system of approximate arithmetic and to perform exact calculations. If our interpretation is correct, what limits the Munduruku is not a lack of conceptual knowledge - and thus, our experiments do not provide support for the Whorfian hypothesis that language determines conceptual structure (contra Gordon 2004). Rather, the linguistic coding of numbers is a "cultural tool" that augments the panoply of cognitive strategies available to us to resolve concrete problems. In particular, the mastery of a sequence of number words enables us to count, in a quick and routinize fashion, any number of objects. The Munduruku do not have a counting routine. Although some possess a rudimentary ability to count on their fingers, it is rarely used. By requiring an exact one-to-one pairing of objects with the sequence of numerals, counting may promote a conceptual integration of approximate number representations, discrete object representations, and the verbal code (Carey 1998; Spelke and Tsivkin 2001). Around the age of three, Western children exhibit an abrupt change in number processing as they suddenly realize that each count word refers to a precise quantity (Wynn 1990). This "crystallization" of discrete numbers out

Fig. 2. Explorations of arithmetic in an Amazonian culture with few number words. The Munduruku language only has number words up to five, and these words mostly refer to an approximate range rather than a precise quantity (top). Munduruku participants were almost as precise as educated French subjects in a task of approximate numerosity addition (middle). However, they failed in an exact calculation task as soon as the numbers exceeded three (bottom).

of an initially approximate continuum of numerical magnitudes does not seem to occur in the Munduruku.

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