Arithmetical ability is made up of many components

In order to study the nature of the arithmetical difficulties that children experience, and thus to understand the best ways to intervene to help them, it is important to remember one crucial thing: arithmetic is not a single entity: it is made up of many components, including knowledge of arithmetical facts; ability to carry out arithmetical procedures; understanding and using arithmetical principles such as commutativity and associativity; estimation; knowledge of mathematical knowledge; applying arithmetic to the solution of word problems and practical problems; etc.

Experimental and educational findings with typically developing children (Ginsburg, 1977; Dowker, 1998) and adults (Geary and Widaman, 1992) have shown that it is possible for individuals to show marked discrepancies between almost any two possible components of arithmetic. For example, Dowker (1998) studied calculation and arithmetical reasoning in 213 unselected children between the ages of 6 and 9. She reported (p. 300) that “(1) individual differences in arithmetic are relatively marked; (2) that arithmetic is indeed not unitary and that it is relatively easy to find children with marked discrepancies [in either direction] between [almost any two] different components; and that (3) in particular it is risky to assume that a child “does not understand maths” because he or she performs poorly in some calculation tasks”.

Studies of adults who have arithmetical difficulties as a result of brain damage (Dehaene, 1997; Butterworth, 1999) show that almost any component of arithmetic can be selectively impaired: e.g. patients can show double dissociations between estimation and calculation; memory for facts and following procedures; written versus oral arithmetic; different arithmetical operations such as subtraction versus multiplication; etc. Detailed case studies of children with mathematical difficulties (usually not associated with obvious brain damage) have also shown extreme discrepancies between different types of mathematical ability. For example, Temple (1991) reports one child who could carry out arithmetical calculation procedures correctly but could not remember number facts, and another child who could remember the facts but not carry out the procedures.

Macaruso and Sokol (1998) studied 20 adolescents with both dyslexia and arithmetical difficulties, and found that the arithmetical difficulties were very heterogeneous, and that factual, procedural and conceptual difficulties were all represented. Desoete, Roeyers and De Clercq (2004) studied 37 Belgian third grade pupils (8 to 9-yearolds) with mathematical difficulties, as demonstrated in both their school performance, and

scores of at least two standard deviations below the mean on at least one mathematics test. Many showed discrepancies in their performance on tests of number knowledge and mental arithmetic; memory for number facts; and word problem solving. Only when children were given all three of the tests, were all 37 identified as having mathematical difficulties. Children, with and without mathematical difficulties can indeed have strengths and weaknesses in almost any area of arithmetic.

Ginsburg (1972, 1977) and his colleagues carried out several individual case studies of children who were failing in school mathematics. Such children typically combined significant strengths with specific weaknesses. Some had a good informal understanding of number concepts, but had trouble in using written symbolism and standard school methods. Some had particular difficulties with the language of mathematics. Some children appeared to have very limited number understanding at first sight, but still had a good understanding of counting techniques and principles. Though some patterns of strengths and weaknesses were more common than others, some children showed unusual and distinctive patterns. In view of this variability of patterns of strengths and weaknesses, Ginsburg recommends the use of clinical interview techniques as part of assessments, in order to understand a child's  specific strengths and weaknesses, and the reasons for their errors: "Standard tests...usually provide only vague characterizations of a child's performance. They show perhaps that he or she does well or poorly. But usually you already know that...(C)hildren's mathematical thinking is complex. You need to understand their intuitions, their errors, their invented strategies.” Further evidence for variable patterns of strengths and weaknesses comes from the work of Denvir and Brown (1986a), who worked with 7-to 9-year-old children who were low attainers in arithmetic.

The major aspects included in the diagnostic tests included: (i) strategies for adding and subtracting small numbers in 'sums' and word problems; (ii) commutativity of addition; (iii) enumerating grouped collections; (iv) strategies for adding larger numbers, and dealing with place value; and (v) Piagetian tasks including number conservation and class inclusion. Each aspect included numerous specific items. There was an approximate order of difficulty of items,  ranging from "makes 1:1 correspondence" to "mentally carries out two-digit 'take away' with regrouping". When each of the skills was ordered according to facility and each of the pupils ordered by overall raw score, it was possible to group the skills into 'levels' defined by a particular range of facility so that every pupil who had succeeded in 2/3 of the skills at any level had succeeded in 2/3 of the skills at every preceding level. However, it was not possible to establish an exact hierarchy of skills, such that the more advanced skills invariably followed on from the easier skills. Some skills did indeed seem to be prerequisite to other skills (e.g. counting collections grouped in tens for tasks that involved adding tens and units) but many seemed almost independent of each other, and some tended to develop in one order, but could occur in the reversed order (e.g. most children who could carry out 2-digit addition without regrouping

mentally could also carry out 2-digit addition with regrouping using base ten apparatus; but there were some who could do the former without the latter.) Thus, although the study does establish approximate hierarchies of skills, it also study supports the componential theory of arithmetic, with the possibility of double dissociations/discrepancies between at least some of the components.