| Summary: | Aims and rationale for the studies: The overall aim of this thesis was to analyse the relationships between cognitive abilities and number skills in children and adults. Examining the links between number skills and cognitive abilities is important both to improve our theoretical knowledge and to inform practitioners who are assessing and teaching children who have number skills difficulties. One important theoretical debate that can be informed by this work is whether normally developing individuals solve problems involving numbers using distinct cognitive modules that are specialised for such work or whether they utilise more generalpurpose cognitive systems. If weaknesses in particular number skills are associated with particular cognitive deficits, it will support the hypothesis that people utilise their general cognitive architecture. Although research into the interactions between children's cognitive profiles and their responses to different teaching programmes is in the early stages, some studies have suggested that tailoring teaching to a child's cognitive profile can be effective. Therefore identifying groups of children with number skills difficulties that have homogeneous cognitive profiles may help in the design of future intervention strategies. Scope Three main areas of investigation were conducted, all of which examined the links between cognitive abilities and number skills. • An examination of the relationships between three number skills (number fact recall, counting speed and place value understanding) and three cognitive abilities (non-verbal reasoning, auditory-verbal-sequential short-term memory and visual-spatial short-term memory) in normally developing children. • An examination of the cognitive and number skills profiles of children with specific arithmetic abilities (SAD). These children had poor arithmetic attainment, but much better reading attainment. The assessment of these children's cognitive and attainment profiles was comprehensive. The children's verbal, non-verbal and spatial abilities were assessed as well as their psychomotor, visual-spatial memory and auditory-verbal memory abilities. Particular attention was paid to the balance of verbal and spatial abilities in these children as previous research has indicated that children with specific arithmetic difficulties share a homogenous ability profile with poor spatial ability, but better verbal ability. • An examination of the number skills profiles of children and adults with dyslexia. A wealth of previous research has indicated that dyslexic individuals have working memory weaknesses (Hulme, 1981; Shankweiler, Liberman, Mark, Fowler & Fisher, 1979). Three number skills (number fact recall, counting speed and place value understanding) were assessed in dyslexic children, to determine whether a diagnosis of dyslexia was associated with a particular number skills profile. As children with dyslexia had a specific difficulty with number fact recall, the number fact recall of dyslexic adults was compared with non-dyslexic adults, to determine whether this difficulty persisted into adulthood. Structure of the thesis The thesis is divided into nine chapters. Chapter 1 describes the aims of the thesis and gives an outline of its content. Chapter 2 describes and evaluates the two major models of normal adult numerical processing. Chapter 3 describes current knowledge about how children develop number skills; particular emphasis is placed on the interplay between conceptual understanding and procedural skills. Chapter 4 describes and evaluates previous research into the attainment, cognitive and psychosocial strengths and weaknesses of children with arithmetic difficulties. The limitations of the various research methodologies utilised in previous studies are examined. Chapter 5 provides an overview of how dyslexia is defined; current knowledge about the cognitive profiles of dyslexic individuals is also discussed. Research into the number skills of dyslexic children is described and evaluated. Chapter 6 describes and evaluates Study 1, which had three main aims: to produce norms for some new computerised tests of number skills; to examine how place value understanding, counting speed and number fact recall develop injunior age children; to examine the relationships between cognitive and number skills junior aged children. Chapter 7 reports the results of Studies 2 and 3. The aim of Study 2 was to examine the ability profiles of children with specific arithmetic abilities. The results indicated that children with large verbal/spatial ability discrepancies were over-represented in the group with specific arithmetic difficulties. The cognitive profiles of the children with specific arithmetic abilities were examined in Study 3. The children were divided into four groups: low general conceptual ability; non-verbal learning difficulty; low verbal reasoning; and specific memory weakness. An illustrative case study of a child in each group is provided. Chapter 8 describes and evaluates Study 4, in which the counting speed, number fact recall and place value understanding of children with SAD and children with dyslexia was compared to a randomly selected sample of children attending mainstream schools. The children with dyslexia showed weaknesses on the test of number fact recall and one test of counting speed, but they had unimpaired place value understanding. In contrast the children with specific arithmetic difficulties were impaired both on the tests of place value understanding and number fact recall. Chapter 9 describes and evaluates Study 5, in which the number fact recall of a group of dyslexic students was compared to a group of non-dyslexic students who were matched on intellectual ability. The adults with dyslexia were slower and less accurate at recalling number facts. Chapter 10 draws together the results of the five studies. The findings are discussed in reference to models of adult numerical processing and Rourke's non-verbal learning difficulty classification (Rourke & Del Dotto, 1994). A multiple-route model of arithmetic difficulties is proposed and methods that could be used to evaluate the model are described. Recommendations for the diagnostic assessment of children with arithmetic difficulties and for cognitively tailored teaching are made.
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