De Smetd & Gilmore (2011). Defective number module or impaired access. Numerical magnitude processing in first graders with mathematical difficulties.” – Article summary

Children with mathematical difficulties have particular impairments in understanding and processing numerical magnitudes. The defective number module hypothesis states that a highly specific deficit of an innate capacity to understand and represent quantities leads to difficulties in learning number and arithmetic. The access deficit hypothesis states that mathematical difficulties originate from impairments in accessing numerical meaning (i.e. their quantity) from symbols rather than from difficulties in processing numerosity.

Infants and young children are able to understand and manipulate numerical magnitude information by means of non-symbolic representations (e.g. discriminate between large sets of dots). Non-symbolic representations are characterized by a distance effect. This holds that the numerical difference (i.e. distance) between two sets that need to be compared is small, performance is slower and less accurate than when the distance is large (e.g. comparing 7 to 442 is easier than comparing 23 to 22). This effect may occur due to overlapping internal representations of numerical magnitudes. A magnitude that is closer to another has a larger representational overlap and this makes it more difficult to compare. The size of the distance effect is an indicator for the preciseness of representations of numerical magnitudes and this effect decreases over development.

Mathematics development requires children to map symbolic representations (e.g. Arabic numerals) onto pre-existing non-symbolic representations of magnitudes.  Children’s representations of magnitude (i.e. number line estimation task) is predictive of their learning of answers to novel mathematics problems. Representations of magnitude are impaired in children with mathematical difficulties.

In the symbolic comparison task, children indicate the numerically larger of two simultaneously presented numbers. In the non-symbolic comparison task, the same happens but with dots rather than Arabic numerals. The symbolic approximate addition task refers to a task where, through a short story, an approximation between two numbers is made to assess which number is larger. In the non-symbolic approximate addition task, the same happens but with dots rather than numerals.

Children with mathematical difficulties have impairments in the ability to access numerical magnitude information from symbolic representations. However, they are not compared on the non-symbolic task. This provides evidence for the access deficit hypothesis. It appears as if the access to representations of magnitude from symbolic numbers is impaired rather than the representation of magnitude by itself.

Children with severe mathematical difficulties score lower on tests of symbolic representation but not on tests of non-symbolic representations. Children with mild forms of mathematical difficulties have a similar but less severe performance pattern. Children with severe mathematical difficulties perform more poorly and develop at a slower rate than their typically achieving peers. Children with mild forms of mathematical difficulties perform more poorly but developed at the same rate as their typically achieving peers.

Children who have difficulties in accessing numerical meaning from symbols are at risk of developing more immature counting strategies and may acquire more arithmetic facts without meaning. This makes it more difficult to retrieve it from long-term memory. Schools should provide opportunities where children learn to connect symbols and the quantities they represent in meaningful ways.