The componential nature of arithmetical cognition: some important questions

Abstract

Research on typically developing children and adults and people with developmental and acquired dyscalculia converges in indicating that arithmetical ability is not unitary but is made up of many different components. Categories of components include non-symbolic quantity representation and processing; symbolic quantity representation and processing; counting procedures and principles; arithmetic operations; arithmetical knowledge and understanding; multiple forms and applications of conceptual knowledge of arithmetic; and domain-general abilities such as attention, executive functions and working memory. There is much evidence that different components can and often do show considerable functional independence, not only in developmental and acquired dyscalculia, but in typically achieving children and adults. At the same time, it is possible to find complex interactions and bidirectional relationships between the different components, including between domain-specific and apparently domain-general abilities. There is a great deal that still needs to be discovered. In particular, we need to learn more about the origins in infancy of subitizing and approximate magnitude comparison, the extent to which these interact, the extent to which they may be further divisible, and the extent and ways in which they themselves may develop with age and the extent to which they may influence later-developing components. There also needs to be a lot more research on exactly how domain-general and domain-specific abilities contribute to mathematical development, and how they interact with one another.

Developmental Dyscalculia in Relation to Individual Differences in Mathematical Abilities

“…One key issue in determining the nature and frequency of dyscalculia is that arithmetical cognition is not a single entity but is composed of multiple components, according to Dowker [1,40,41]. Children most commonly have difficulties in only some of these components, rather than in all aspects of arithmetic [42][43][44], Gifford and Rockliffe [44] (p. 21) found that, in one group, "no pure cases [of dyscalculia] were found, although the children presented complex patterns of learning difficulties and compensatory strategies.…”

“…In fact, while patients do tend to perform worse in mathematics than healthy controls, dissociations can be found in apparently typical individuals if one looks for them. For example, Dowker [ 1 , 40 ] reported a healthy undergraduate student in a science subject, with an A grade in A-level mathematics, who generally performed very well in tests of calculation but showed a specific inability to carry out subtraction problems involving borrowing, somewhat similar to Carota et al’s patient [ 30 ]. Dowker (2005) also reported a highly educated adult participant who had struggled with school mathematics but did have the equivalent of O-level mathematics [ 1 ].…”