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Typical and Atypical Development of Number Processing and Arithmetic

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Typical and Atypical Development of Number Processing and Arithmetic Good and bad at numbers: typical and atypical development of number processing and arithmetic Teresa Iuculano University College London Thesis submitted for the degree of Doctor of Philosophy Originally submitted February 2012 1 Statement of Originality__________________________________________________ I, Teresa Iuculano, confirm that the work presented in this thesis is my own. Where information has been derived from other sources, I confirm that this has been indicated in the thesis. Please note that the work presented in this thesis on symbol learning with brain stimulation (Chapter VIII) was a collaborative project led by Dr. Roi Cohen Kadosh. 2 Abstract_______________________________________________________________ This thesis elucidates the heterogeneous nature of mathematical skills by examining numerical and arithmetical abilities in typical, atypical and exceptional populations. Moreover, it looks at the benefits of intervention for remediating and improving mathematical skills. First, we establish the nature of the ‘number sense’ and assess its contribution to typical and atypical arithmetical development. We confirmed that representing and manipulating numerosities approximately is fundamentally different from the ability to manipulate them exactly. Yet only the exact manipulation of numbers seems to be crucial for the development of arithmetic. These results lead to a better characterization of mathematical disabilities such as Developmental Dyscalculia and Low Numeracy. In the latter population we also investigated more general cognitive functions demonstrating how inhibition processes of working memory and stimulus- material interacted with arithmetical attainment. Furthermore, we examined areas of mathematics that are often difficult to grasp: the representation and processing of rational numbers. Using explicit mapping tasks we demonstrated that well-educated adults, but also typically developing 10 year olds and children with low numeracy have a comprehensive understanding of these types of numbers. We also investigated exceptional maths abilities in a population of children with Autism Spectrum Disorder (ASD) demonstrating that this condition is characterized by outstanding arithmetical skills and sophisticated calculation strategies, which are reflected in a fundamentally different pattern of brain activation. Ultimately we looked at remediation and learning. Targeted behavioural intervention was beneficial for children with low numeracy but not in Developmental Dyscalculia. Finally, we demonstrated that adults’ numerical performance can be enhanced by neural stimulation (tDCS) to dedicated areas of the brain. This work sheds light on the entire spectrum of mathematical skills from atypical to exceptional development and it is extremely relevant for the advancing of the field of mathematical cognition and the prospects of diagnosis, education and intervention. 3 Table of Contents_______________________________________________________ Statement of Originality___________________________________________________________ 2 Abstract________________________________________________________________________ 3 Table of Contents________________________________________________________________ 4 List of Tables___________________________________________________________________ 9 List of Figures__________________________________________________________________ 10 Acknowledgements______________________________________________________________ 12 Publications____________________________________________________________________ 14 Invited Talks___________________________________________________________________ 14 Conference Abstracts____________________________________________________________ 15 Poster Presentations_____________________________________________________________ 15 Individual Contribution to Research Chapters_______________________________________ 16 I. Introduction 20 1.1. Introduction 21 1.2. The heterogeneity of numbers 22 1.3. The ‘number sense’ and the development of arithmetic 24 1.4. Neural correlates of maths abilities 28 1.5. Typical development 29 1.5.1. The ‘mental number’ line in typical development 31 1.5.2. The typical acquisition of arithmetic 33 1.5.3. Neural correlates of typical development 34 1.6. Atypical development 37 1.6.1. Terminology and diagnosis of atypical development 37 1.6.1.1. Terminology 37 1.6.1.2. Diagnosis 38 1.6.2. Developmental Dyscalculia and Low Numeracy 40 1.6.3. The ‘mental number’ line in atypical development 43 1.6.4. The ‘atypical’acquisition of arithmetic 44 1.6.5. Neural correlates of atypical development 45 1.6.6. Remediation studies of atypical development 47 1.7. Individual differences in maths 49 1.8. Exceptional development 50 1.8.1. Autism Spectrum Disorder: a model for exceptional mathematical abilities 53 1.8.2. Neurobiology of Autism Spectrum Disorder 56 1.8.3. Neural correlates of math abilities in Autism Spectrum Disorder 58 1.9. ‘Measuring’, ‘decoding, and‘charging’ the brain 59 1.9.1. Functional Magnetic Resonance Imaging (fMRI) 60 1.9.1.1. Predictive analyses tools – Support Vector Machine methods 61 1.9.2. Transcranial Direct Current Stimulation (tDCS) 64 1.10. Theoretical roadmap 65 1.10.1. The modular view 66 1.10.2. The continuum view 67 1.11. Research Aims 68 II. Nature of the ‘number sense’in typical and atypical development 70 2.1. Introduction 71 Experiment 1A.The ‘number sense’ in typical development 73 2.2. Aim 1A 73 2.3 Method 1A 73 2.3.1 Participants 73 2.3.2. Experimental design 74 2.3.2.1. Standardized tasks 74 2.3.2.2. Experimental tasks 76 4 2.4. Results 1A 80 2.4.1. Correlational analyses 80 2.4.1.1. The approximate system 80 2.4.1.2. The approximate system in relation to the discrete numerosity and analogue area tasks 81 2.4.1.3. The approximate system in relation to the exact system. 81 2.4.1.4. What predicts exact arithmetic? 81 2.5. Interim Discussion 1A 82 Experiment 1B.The ‘number sense’ in Developmental Dyscalculia and Low Numeracy 83 2.6. Aim 1B 83 2.7 Method 1B 84 2.7.1 Participants 84 2.7.2. Experimental design 85 2.8 Results 1B 85 2.8.1. Differences between the Normal Achievers and the Low Numeracy group 85 2.8.1.1. Approximate tasks 86 2.8.1.2. Analogue Area task 86 2.8.1.3. Symbolic and non-symbolic numerosity tasks 86 2.8.1.4. Non-symbolic Stroop effects 87 2.8.2. Single cases with Developmental Dyscalculia 89 2.8.2.1. Subject 1 (DD1) 89 2.8.2.2. Subject 2 (DD2) 90 2.9. Interim Discussion 1B 91 2.9.1. Low Numeracy group (LN) 91 2.9.2. Developmental Dyscalculia (DD) single cases 92 2.10. Final Discussion 93 2.11. Summary 97 III. ‘Updating’ the cognitive profile of low numeracy: Working Memory abilities 99 3.1. Introduction 100 Experiment 2.WM Maintenance and Updating of different stimulus material in Low Numeracy 103 3.2. Aim 2 103 3.3 Method 2 103 3.3.1 Participants 103 3.3.2. Experimental design 105 3.3.2.1. Standardized tasks 105 3.3.2.2. Experimental tasks 105 3.4. Results 2 107 3.4.1. Span tasks 108 3.4.2. Updating tasks 108 3.5. Final Discussion 111 3.6. Summary 113 IV. The development of rational numbers: fractions and decimals 115 4.1. Introduction 116 Experiment 3A.Rational number representations in adults – The mental number line tasks 118 4.2. Aim 3A 118 4.3 Method 3A 119 4.3.1 Participants 119 4.3.2. Experimental design 119 4.3.2.1. Experimental tasks 119 4.4. Results 3A 122 4.4.1. Education 122 4.4.2. Regression analyses 123 4.5. Interim Discussion 3A 125 Experiment 3B.Rational number representations in children – The mental number line tasks 127 4.6. Aim 3B 127 4.7 Method 3B 128 4.7.1 Participants 128 5 4.7.2. Experimental design 129 4.8 Results 3B 129 4.8.1. Regression analyses 129 4.9. Interim Discussion 3B 130 Experiment 3C.Rational number processing across development 132 4.10. Aim 3C 132 4.11. Method 3C 132 4.11.1. Participants 132 4.11.2. Experimental design 133 4.12. Results 3C 133 4.12.1. Accuracy analyses 133 4.12.2. Latency analyses 134 4.13. Interim Discussion 3C 136 4.14. Final Discussion 137 4.15. Summary 140 V. Double-digit number representations in children with low numeracy 141 5.1. Introduction 142 Experiment 4A.Double-digit number representations in children with low numeracy– The mental number line tasks 143 5.2. Aim 4A 143 5.3. Method 4A 145 5.3.1. Participants 145 5.3.2. Experimental design 146 5.4. Results 4A 146 5.4.1. Regression analyses 146 5.4.2. Correlational analyses 148 5.5. Interim Discussion 4A 148 Experiment 4B.Double-digit number processing in typical and atypical development 150 5.6. Aim 4B 150 5.7. Method 4B 151 5.7.1. Participants 151 5.7.2. Experimental design 151 5.8. Results 4B 152 5.8.1. Accuracy analyses 152 5.8.2. Latency analyses 154 5.9. Interim Discussion 4B 155 5.10. Final Discussion 157 5.11. Summary 159 VI. ‘If’ and ‘how’ of exceptional maths abilities in Autism Spectrum Disorders 160 6.1. Introduction 161 Experiment 5A.Characterizing math abilities in Autism Spectrum Disorder 163 6.2. Aim5A 163 6.3. Method 5A 163 6.3.1. Participants 163 6.3.2. Experimental design 165 6.3.2.1. Standardized tasks 165 6.3.2.2. Experimental tasks 165 6.4. Results 5A 166 6.4.1. Standardized measures of mathematical abilities 166 6.4.2. IQ and maths Discrepancy scores 166 6.4.3. Working Memory abilities 166 6.4.4. Strategy assessments 167 6.5. Interim Discussion 5A 168 Experiment 5B.Neuronal signature of maths abilities in Autism Spectrum Disorder 170 6.6. Aim 5B 170 6.7. Method 5B 170 6 6.7.1. Participants 170 6.7.2. Experimental design 170 6.7.2.1. Experimental tasks 170 6.7.2.2. fMRI data acquisition 171 6.7.2.3. fMRI data analysis 171 6.8. Results 5B 174 6.8.1. Behavioural performance 174 6.8.2. Univariate differences in intensity of activation between HFA and TD 174 6.8.3. Multivariate differences in brain activation between HFA and TD 174 6.9. Interim Discussion 5B 177 Experiment 5C. Decoding maths performance from hemodynamic signal change 183 6.10. Aim 5C 183 6.11.
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