Numerical synaesthesia and spatial representation of numbers

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Arithmetic education is accompanied by another important change: the learning of systematic links between numbers and space. The intuition of the spatial scale of numbers plays an essential role in mathematics, from the notion of measurement (geo-metry) to the study of irrational numbers, the real line, Cartesian coordinates or the complex plane. Numerous studies have demonstrated the automaticity of the link between numbers and space in adult humans: the simple presentation of an Arabic numeral is enough to bias motor responses and visuo-spatial attention, towards the right for large numbers and the left for small numbers. This "SNARC effect"(spatial-numerical association of response codes) depends on attentional and cultural variables such as the direction of writing - the direction of the effect tends to reverse in cultures that read from right to left (Dehaene, Bossini, & Giraux, 1993).

The number-space association itself seems to derive from the very close anatomical links between the middle intraparietal region, involved in number coding, and neighboring regions involved in space coding. In particular, a circuit links the LIP and VIP areas, which are involved in eye movement and the encoding of relevant positions in space. With Ed Hubbard, based on a detailed review of these issues, I proposed that this VIP-LIP circuit be partly recycled for elementary arithmetic (Hubbard, Piazza, Pinel, & Dehaene, 2005). Together with André Knops and Mariagrazia Ranzini, we have obtained functional MRI, evoked potentials and behavioral data that point in this direction. Indeed, the simple presentation of a number induces lateralized activation of posterior parietal regions (a plausible homologue of the LIP area in humans). In addition, addition and subtraction automatically evoke movement to the right and left respectively. These spatial-numerical links remain unconscious in most test subjects. However, in rare individuals, their access to consciousness could perhaps explain the phenomenon of digital synesthesia. In fact, these individuals claim to literally see numbers in well-defined spatial positions on an internal spatial scale.

Although intuitive and automatic, the correspondence between numbers and space is not entirely fixed. It changes considerably over the course of development and as a function of education. Siegler and Opfer (2003) presented a task in which children were asked to arrange the numbers they were told on a segment labeled, for example, with the number 1 on the left and the number 100 on the right. The youngest children (CP) place numbers in correspondence with space in a monotonous but non-linear way. They place much more emphasis on small numbers, and tend to follow a compressive, logarithmic scale, with 10 in the middle of 1 and 100. Responses only become linear in older children, between 8 and 10 years of age. With Pierre Pica, Véronique Izard and Elizabeth Spelke, we have shown that education plays an essential role: among the Mundurucus Indians of Amazonia, adult people with no mathematical education respond logarithmically to numbers between 1 and 10 presented verbally or non-symbolically (Dehaene, Izard, Spelke, & Pica, 2008). The logarithmic similarity scale, which derives from Weber's law, therefore seems to be one of mankind's fundamental intuitions (it being understood that this intuition is approximate and does not include abstract mathematical properties of the logarithmic function such as the transformation of sums into products). There is much evidence to suggest that, in the educated adult, the logarithmic representation has not really disappeared. Logarithmic and linear scales seem to coexist and to be selected according to instructions and task context.

Conclusion

Human arithmetic intuition consists of a complex network of knowledge ranging from the ability to quickly estimate the approximate cardinality of a set, to anticipating the result of an addition, judging that 8 is greater than 3, or seeing numbers in space and evaluating that 3 is closer to 1 than to 10. The core of this numerical knowledge consists of a log-Gaussian representation of approximate numerosity. This kernel of knowledge is already present in very young children and many animal species, and is associated with a brain circuit located in the bilateral intraparietal region. Learning the symbols of formal arithmetic relies heavily on this early number sense, although our understanding of how this is modified by education remains highly imperfect. This will be one of the major issues of future research. A key challenge will be to make better use of this knowledge to improve arithmetic readings and better understand the origins of dyscalculia.