The concept of number

Listen to audio

The concept of number covers a wide range of contents. So it's important to introduce a few essential terminological distinctions at the outset. Our education has accustomed us to numerical symbols: numbers written in base 10 using Arabic numerals, such as "1053", or expressed using number names such as "one thousand fifty three". However, cognitive psychology shows the importance, in animals and very young children, of non-symbolic perception of number, where the quantity 13 can be presented in the concrete form of a cloud of 13 dots or a sequence of 13 sounds. Here, number is seen as the property of a set, often referred to by the technical terms cardinality or numerosity. Finally,ordinality refers to the rank of an element in an ordered series. We refer to it verbally through ordinal numbers (e.g. "thirteenth"). The concept of number, in the educated adult, consists in the harmonious integration of these different symbolic and non-symbolic facets of number.

It goes without saying that the mathematician would like to extend this list to include integers, fractions, real or complex numbers, even quaternions or matrices... For him, any mental object that can be manipulated according to certain coherent operations can be considered a number. For the moment, however, cognitive psychology has paid little attention to these higher-level mathematical concepts. We shall therefore confine ourselves to the non-symbolic perception of numbers, and its connection with written or spoken number symbols.

Number perception

Adults have at least three distinct cognitive processes forenumerating, i.e. apprehending the numbering of a set of objects:

Subitizing refers to the immediate apprehension of small numbers (one, two or three objects).
Estimation enables us to evaluate, in an approximate way, the numbering of a set of arbitrary size. Véronique Izard's laboratory research has shown that an untrained adult can quickly and efficiently estimate even very large sets. However, the perceived number is not always linearly related to the number actually presented: underestimation is frequent, and a power law links the two quantities (Izard & Dehaene, 2008).
Finally, counting, the principles of which were studied by Gelman and Gallistel (1978), enables the precise enumeration of any set. It involves matching, one by one, each of the objects enumerated with a reference list, which may be verbal (number names) or non-verbal (fingers, body parts).

Recent research confirms that the three processes of subitization, estimation and counting are dissociable. The distinction between subitization and counting is clear when we measure the time it takes to name the numbers in a set: beyond three objects, the response time shows a sudden linear increase. Brain imaging detects, at this boundary, a sudden amplification of activity in a vast cerebral network linked to verbal counting, including bilateral posterior parietal regions associated with attentional movements. The activity of these regions is so characteristic that it makes it possible to distinguish, virtually trial by trial, whether the subject has counted or not and how many objects have been counted (Piazza, Giacomini, Le Bihan, & Dehaene, 2003). A dissociation between subitization and counting is also observed in some patients who, following brain damage, develop simultanagnosia (Dehaene & Cohen, 1994).

Until very recently, it was possible to argue that estimation and subitization were simply the reflection of one and the same process. Indeed, estimation is characterized by Weber's law: the imprecision of the estimate, measured by its standard deviation, increases linearly with the number estimated, according to a constant of proportionality called the coefficient of variation or Weber fraction. For very small numbers, Weber's law could explain subitization, as it results in sufficient precision to discriminate and name the numbers 1, 2 or 3 with virtually no error. Subitization would thus be reduced to a kind of "precise estimation". However, Susannah Revkin's research in the laboratory shows that this explanation is not sufficient (Revkin, Piazza, Izard, Cohen, & Dehaene, 2008). The precision with which we detect the presence of one, two or three objects exceeds that expected according to Weber's law, as it far exceeds that with which we discriminate 10, 20 or 30 objects. Thus, subitization seems to call upon a dedicated system, perhaps that linked to the apprehension of discrete objects ("object tracking" or "object files" system, Feigenson, Dehaene, & Spelke, 2004).

The development of digital perception

Piagetian tests of conservation of number and inclusion of classes initially suggested that the young child lacked invariant mechanisms of number apprehension. However, this conclusion was clearly invalidated by the advent of new methods of experimentation with very young children, based on adaptation and reaction to novelty or the violation of physical laws. Numerous studies have demonstrated a sensitivity to numbers in children aged 4-6 months. For example, a six-month-old baby detects when the numbering of a set changes from 8 to 16 objects or vice versa, even when non-numerical parameters such as density and total surface area are constant (Xu & Spelke, 2000). Children also detect the violation of addition and subtraction rules, at least in an approximate way. For example, when they see 5 objects, then another 5, disappear behind a screen, they expect to see around 10 objects appear and express surprise by looking longer when the screen lowers and reveals only 5 objects (McCrink & Wynn, 2004; Wynn, 1992).

While estimation has been well demonstrated in very young children, the perception of small numbers 1, 2 or 3 has been more debated. Some have suggested that their discrimination is due solely to confounding parameters such as the total quantity of matter (see Feigenson et al., 2004). However, very recent results indicate that children may, depending on the context, pay attention to either numerosity or non-numerical parameters (Feigenson, 2005). Cordes and Brannon (2008) go so far as to suggest that it is simpler for children to pay attention to number than to the total quantity of material, since their Weber fraction is higher in the latter case. In the laboratory, we have indeed observed, using evoked potentials, discrimination of numbers 2 and 3 in the absence of any non-numerical artifacts (Izard, Dehaene-Lambertz, & Dehaene, 2008).

Thus, subitization, estimation and approximate calculation seem to be present in the human species from the very first year of life. It's fascinating to note that this numerical intuition also exists in many animal species (pigeons, rats, lions, monkeys, dolphins...). Elizabeth Brannon, in particular, has systematically compared the arithmetic skills of adult humans and macaque monkeys, in tasks involving the approximate perception of numbers, their addition and comparison (Cantlon & Brannon, 2007). She observed very similar psychophysics, with calculation accuracy simply being better in humans.

Links with symbolic arithmetic

To what extent are these early skills for manipulating non-verbal numbers relevant to understanding the intuition of numerical symbols in adulthood? Numerous experiments suggest that, as soon as an educated adult perceives a number name or a number in Arabic notation, this symbolic input is quickly and automatically translated mentally into an approximate quantity whose internal manipulation obeys the same laws as those for the manipulation of numerosity perceived as sets of objects. Thus, the seminal experiment by Moyer and Landauer (1967) showed that, when we decide which of two numbers is larger, our response time varies as an inverse function of the numerical distance between them. Number size also influences the comparative judgment of numbers presented in symbolic form, and Weber's law captures all this data. Even more surprisingly, this is also the case when we judge whether two numbers are the same or different - the "different" response is slower for 8 against 7 than for 8 against 1. The verification of symbolic operations also demonstrates a numerical distance effect: when an obviously false operation is proposed to us, the large distance that separates the proposed result from the correct one enables us to reject it without doing the exact calculation (Ashcraft & Stazyk, 1981). Finally, a brain lesion can cause us to lose all capacity for exact calculation, while leaving intact this basic skill for approximation (Dehaene & Cohen, 1991). Thus, the approximation of numerical quantities, based on Weber's law, appears to be a fundamental skill that shows up in many symbolic tasks.

Most recently, a developmental study has confirmed that arithmetic intuition of approximate quantities precedes and underpins later learning of symbolic arithmetic (Gilmore, McCarthy, & Spelke, 2007). Gilmore et al. gave 5- and 6-year-old kindergarten children verbal problems such as "Sarah has 21 pieces of candy, and she is given 30 more. John has 34. Who has the most?" The children had received no explicit lectures on numbers of this size, or on the operations of addition and subtraction. However, whatever their socio-economic level, they responded well above chance (60-70% success rate), and their performance followed Weber's law, suggesting that they were mentally translating symbolic problems into quantities in order to exploit their non-symbolic intuition. More importantly, their performance in this assessment of arithmetic intuition correlated with their success in mathematics at school. Holloway and Ansari (2008) also reported, in slightly older children (aged 6-8), that the variability of the distance effect during numerical comparison predicted school success in mathematics, but not reading scores. Taken together, these findings suggest that Weber's Law-based understanding of approximate numbering and distance relationships between numbers plays a key role in later understanding of symbolic arithmetic.