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Special Issue: Space, Time and Number
Foundational numerical capacities and the origins of dyscalculia

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One important cause of very low attainment in arithmetic (dyscalculia) seems to be a core deficit in an inherited foundational capacity for numbers. According to one set of hypotheses, arithmetic ability is built on an inherited system responsible for representing approximate numerosity. One account holds that this is supported by a system for representing exactly a small number (less than or equal to four4) of individual objects. In these approaches, the core deficit in dyscalculia lies in either of these systems. An alternative proposal holds that the deficit lies in an inherited system for sets of objects and operations on them (numerosity coding) on which arithmetic is built. I argue that a deficit in numerosity coding, not in the approximate number system or the small number system, is responsible for dyscalculia. Nevertheless, critical tests should involve both longitudinal studies and intervention, and these have yet to be carried out.

Section snippets

Why are people bad at learning arithmetic?

Low numeracy is a serious handicap for individuals and a major cost for nations (see [1] for data relevant to the UK). It makes individuals less employable, is a risk factor for depression in adulthood and significantly reduces lifetime earnings. In the UK, approximately 25% of adults have poor functional numeracy [2]. Low arithmetic attainment has been attributed in the past to a deficit in general cognitive abilities such as working memory (WM) [3] and executive function [4], and there is

Domain-specific foundational capacities for arithmetic

Here I briefly outline proposals for a domain-specific capacity for numbers before discussing whether this capacity is foundational for acquiring arithmetic ability.A foundational capacity for numbers is revealed in the ability of human infants to discriminate on the basis of the numerosity of a display [15] and to match numerosity across modalities [16], which suggests that the capacity is not tied to one modality and implies a relatively abstract understanding of numerosity Box 2.

There is

The approximate number system (ANS)

Although there is little doubt that we share with many nonhuman species a system for estimating and comparing approximate numerosities 26, 32, the role this system plays in the development of arithmetic remains to be clarified. The ANS is one system of core knowledge of numbers [32] According to this approach, number-abstraction processes extract some kind of summary statistics from a scene (which, in principle, could be in a modality other than visual) that is separate from the processes

The small numerosity system

Arithmetic is about exact numbers, and to be foundational, representations of exact numbers need to be developed. How do approximate representations (of the type the ANS hypothesis proposes) develop into a sequence of numerosities, each with a unique successor?

One possibility is to exploit our ability to represent small numerosities without serial enumeration and with a high degree of precision; this is called subitizing (for a review see [43], but see [44]). It has therefore been proposed that

Numerosity coding

Piaget maintained that the concept of number, by which he meant cardinal number, is based on sets [31]. However, he thought that conservation of number under numerosity-irrelevant transformations was only possible at approximately the age of 4 years, when a particular stage in logical reasoning had been reached [31].

Since then, many studies have indicated that human infants can use the numerosity of visual arrays as a discriminative stimulus [15]. Moreover, infants can select collections of

Role of language

The counting words, or what Carey calls the integer list, is held to play a special role in the emergence of exact arithmetic ability during child development [46]. Thus, a critical test of the role of language in the development of arithmetic ability is whether language impairments cause arithmetic disabilities. It is claimed that the role of language is to refine approximate representations through bootstrapping and regular association of a counting word with a particular approximate

Neural basis of dyscalculia

Can studies of neural differences in structure or activation in dyscalculic subjects be used to decide among the foundational hypotheses? Structurally, reduced grey matter in dyscalculic individuals has been observed in areas involved in basic numerical processing, in the left IPS [69], in the right IPS [70] and in the IPS bilaterally (Figure 3) [71]. Moreover, there seem to be differences in connectivity between relevant regions as revealed by diffusion tensor imaging tractography [71].

Intervention

The efficacy of interventions designed to strengthen purported foundational capacities would constitute a critical test of the hypotheses discussed in preceding sections. For the ANS, Piazza and colleagues note that their ‘findings lend support to remediation programs for developmental dyscalculia that include exercises aimed at retraining the core non-symbolic sense of number and to cement its links to the symbols used to denote it’ [79]. The intervention they cite as appropriate is the Number

Concluding remarks

In summary, although the evidence is not yet conclusive, it seems that the ANS and small number systems are not sufficient to support the typical development of arithmetic skills. A system that represents sets, their numerosities and the effects of transformations on these sets seems to be required. Numerosity coding is such a system and there is extensive evidence that young humans possess this. However, we have only just begun to collect critical evidence from longitudinal and intervention

Acknowledgements

I am grateful for helpful comments by and discussions with Randy Gallistel, Justin Halberda and Sashank Varma, and for comments on the manuscript by three anonymous referees.

Glossary

  • Approximate numerosity tasks: tasks involving clouds of dots (or other objects) typically too numerous to enumerate exactly in the time available. One common task is to compare two clouds of dots. Addition and subtraction tasks for which the solution is compared with a third cloud of dots are also used. (The term

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