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date: 10 December 2019

(p. 1154) Index

(p. 1154) Index

Note: Page numbers in italics refer to figures and tables. Footnotes are indicated by the suffix ‘n’ followed by the note number, for example 34n1.

(p. 1155) A
abacus experts, functional imaging studies 850–1
abstract code model 141–2
abstraction principle 241
abstractness of arithmetic 140
acalculia, acquired 635, 710, 829, 881
diagnosis and assessment 825–6
functional imaging studies 840–1, 855–6
natural recovery 827
neural correlates 824–5
rehabilitation 827–8, 855–6
selective impairment
arithmetic conceptual knowledge 816–18
arithmetic fact retrieval 819–20
encylopedic knowledge 818
executive functions 820–1
non-symbolic number processing 815–16
symbolic quantity processing 814–15
transcoding errors 818–19
access deficit hypothesis 334, 337–8
accumulator neurons 446
adaptation studies 567–8
adaptive software, use in dyscalculia 655–7, 656
addition
carry operation 114, 119
infants' abilities 259–63
language-specific effects 128–9
MD&A diagrams 1044
multi-digit number processing 117–18
semantic types 159–60
additive composition error, multi-digit number processing 113
age-related changes 344–5, 350–1, 359–60
in approximate number system 345
in arithmetic fact retrieval 348–9, 350
in computation 349–50
in counting 348
in exact number system 345–7
in neural correlates of number processing 494
research directions 357–9
in strategy distribution 356–7
in strategy execution 354–6
in strategy repertoire 352–3
in strategy selection 353–4, 355
in subitizing 347, 348
algebraic analysis, perceptual grouping effects 151
Alzheimer's disease (AD) 881
arithmetic fact retrieval 349
computation impairment 350
counting speed reduction 348
decline in subitizing 347
strategy distribution 357
strategy execution 355
strategy repertoire 352
strategy selection 354
amblyopia, estimation skills 798
American coots, numerical discrimination 215
amphibians
numerical discrimination 225
quantity discrimination 218, 221–2
analogue magnitude system see approximate number system
anatomo-functional calculation model 490–1
anchoring strategies, age-related differences 355, 356
angular gyrus 490, 507, 714, 811, 841–2
DCE studies of calculation 602–4
deficits in 664, 667–8
and Gerstmann syndrome 604
and mathematical competence 618, 852–3
role in arithmetic 503, 508–9, 513, 616, 636, 650, 709
role in auditory processing of numerals 544, 545–6
role in learning 846, 848–9, 857
role in reading 709–10
TMS studies 587, 591, 594, 595, 597
animal number representations 204, 214–17, 237–8, 566
counting principles and symbolic representation 240–2
differences form humans 206–7
future research areas 232
numbers versus continuous quantities 217–23
ontogeny 228–31
subitizing, non-primate species 223–7
see also monkeys
anterior temporal lobe (ATL)
changes in dyscalculia, brain morphometry 736
involvement in arithmetic 503, 510
ants, numerical discrimination 221
(p. 1156) approximate arithmetic 310
extent of ability 311–14
influencing factors 318–20
mechanisms of 314–18
and school mathematics 320–6
supporting evidence 310–11
approximate number acuity 897
approximate number system (ANS, analogue magnitude system) 227, 248, 249, 250, 258, 259, 330, 333, 422, 492, 709
age-related changes 345
in animals 204–5, 207, 223–4
and approximate arithmetic 309–10, 313–14, 317–18
and CAI games 757–8
developmental models 339
developmental studies 204–6, 207, 223–4, 263
impairment of 334, 776
relationship to mathematical ability 208–10, 321–2, 323–6, 335–7, 869–70
and statistical learning 266, 267–8
supporting evidence for 224–5
Arabic-speaking children, place-value understanding 420
arcuate fasciculus, changes in reading difficulties 711
arithmetic
abstract nature of 140
approximate see approximate arithmetic
chicks' abilities 229–30
cognitive architectures 141–4
componential nature 880–3, 899
relationship between components 883–5
studies in advanced mathematicians 890
studies in preschool children 888–9
conceptual metaphors
motion along a path 384–7, 385, 389, 390–5
object collection 383–4, 385–7, 388–9, 393
core neurocognitive processes 503–4
deficits in MLD and low achievement
fact retrieval 775–7
procedural development 773–5
functional dissociations 510–11, 512
infants' abilities 259–63, 265–8, 269
inter-individual differences 851–3
knowledge requirements 896
neural correlates 447–8, 495–6, 636, 650–2, 709
in the adult brain 504–14, 837, 838–20, 840–22
DCE studies 602–4, 605–6
in the developing brain 514–23
in exceptional expertise, 849–51
learning studies 842–9
TMS studies 598–9, 605–6
numeral format effects 144–50
role of language 151–3
semantic alignment effects 150–1
strategy use and individual differences 511, 513
arithmetical facts processing 810
arithmetical sign understanding, selective impairment 820
arithmetic errors, relationship to numerical format 148
arithmetic fact retrieval, age-related changes 348–9, 350
arithmetic procedures
selective impairment 813
and word problem-solving 958–61
arithmetic skills
development of, applied neuroscience 615–16
relationship to estimation and derived fact strategies 883–5
SFON as a predictor of 284
arithmetic training, functional imaging studies 513–14
art, mathematical aspects of 15–16
Asperger's syndrome 12
association neurons 471, 472
ATOM (A Theory of Magnitude) 397n6, 449, 552–3
development of magnitude processing 557
and metaphorical theories 558–9
number–action link 555–6
predictions of 554
problems to solve 559–61
space and time perception 556
attention, associations and dissociations with number system 822
attentional capacities 872
individual differences 900–2
attention deficit hyperactivity disorder (ADHD) 664, 902
associated mathematical difficulties 669–70
genetic studies 1007
attention deficits 820
in Down syndrome 727–8
in dyscalculia 668–70
impact of Numbers Count initiative 1111–12
attitudes towards mathematics 3, 5–6
early interventions 195
heritability 1000
impact of Numbers Count initiative 1110–13, 1111
promotion of maths 6–16
relationship to achievement 193–4, 435, 436, 934–5, 984
and word problem-solving 966–7
young children 1056, 1058
audio-visual processing 540
auditory event estimation (AEE) task, studies in blind individuals 796–7
auditory processing of numerals 544
auditory referents of numerals 539–40
autistic savants 880
automatic processing 47, 185–6, 857, 909
individual differences 906–7
see also primitives
(p. 1157) automatization deficits, in reading difficulties 703–4
avoidance constellation 873–4
B
basal ganglia
changes in dyscalculia 740
developmental studies 516
involvement in arithmetic 503
Basic programming language 749
Battle of Britain, role of maths 13
bees, numerical discrimination 221
beetles, numerical discrimination 224–5
behavioural educational interventions 620
behavioural inhibition, relationship to mathematical ability 921–2, 924
beliefs, influence on word problem-solving 967–9
Big Bang Fair 9–10
bilingual mathematics learning 1083
language-switching costs 617
number-knower levels 299–300
binary model of operations 960
birds, numerical discrimination 215–16, 219–20, 224, 225
bisection tasks see number interval bisection task
blindness 641, 802–3
compensatory mechanisms 798
finger-counting 801
future research areas 803–4
number representation 790–5
numerical estimation 795–801
spatial abilities 790, 795
Blocks and Water task 301
board games
home experiences 1142–3
linear number board game intervention 1143–8
preschool interventions 1088
body part counting system, Oksapmin 372–3
borrowing operation 112, 114, 811, 944
error sources 775
relationship to basic numerical representations 119
bottom-up saliency 186–7
brain lesion studies 824–5, 881
angular gyrus 710
Gerstmann syndrome 78
number interval bisection task 93–5
parietal lobe 505, 507
perisylvian cortex, arithmetic impairment 153
Building Blocks curriculum 1060, 1063, 1086–7, 1149
C
calculation
DCE studies 602–4, 605–6
neural correlates 447, 495–6
TMS studies 598–9, 605–6
studies in adults 837, 838–40, 840–2
see also arithmetic
calculation models 486–91, 490
recent modifications and extensions 491–2
canonical circuit for arithmetic 505, 506
captain's problem 957
capuchin monkeys
numerousness judgements of food items 246
ordinality judgements 244–5
cardinality principle 241, 291, 294
children's understanding of 293, 294, 297, 304–5
cardinal numbers, relationship to natural numbers 28–9
carry operation 114, 124, 130
language-specific effects 128–9
relationship to basic numerical representations 119
role of working memory 942–23
Catch Up Numeracy 1101
cats, quantity discrimination 218
caudate nucleus, order-selective neurons 474
Celtic Knots 15
central executive 167, 517–18, 981
role in mental arithmetic 939–41
cerebellum 491–2, 561, 842, 850
cerebellar theory of reading difficulties 698, 703
Change problems 159–60
Chased Chicken Sona pattern 15
chicks
arithmetic abilities 229–30
ordinal abilities 228–9, 230
children
differences in mathematical competence 618–19
finger counting 73–5
functional imaging studies 485–6
learning of number words 74
number representations 402–4
estimation as an index of conceptual representation 410–12
influence of language 404–8, 416, 418, 421–2, 426
mental number line 408–10
neural correlates 493–6
Children's Math Worlds Project 1037
chimpanzees, use of symbols 241
Chinese ideograph processing, neural substrates 543
Chinese number naming system 417, 902
Chinese-speaking children 426
mathematical ability 421–2
choking under pressure 944, 946
reduction of effect on performance 947
cingulate cortex
changes in dyscalculia, brain morphometry 736
order-selective neurons 474
classical number theory 17–18
(p. 1158) Class Learning Path Model 1037–8
conclusion and future research areas 1050–1
mathematically-desirable and accessible (MD&A) methods 1042–8
teaching phases 1038–9, 1040–1
teaching support 1048–50
class learning zone 1038
class sizes, impact on performance 435
climate change, mathematical aspects of 13–15
cognitive ability, relationship to performance 192
cognitive control processes, functional imaging studies 517–18
cognitive development, Vygotsky's cultural–historical perspective 367–8
cognitive linguistics 378
conceptualizing numbers and arithmetic 382–7
conceptual metaphor 380–2
conclusion and future prospects 396–7
counting 379–80
fictive motion 382
support for embodied conceptualization of number 387–96
Cognitive Reflection Test (CRT) 182–3
collaboration, in educational software 1061–2, 1069
Collecting gestures 393, 395
colour–number synaesthesia 89
Combine problems 160
Common Core State Standards 1050–1
commutativity 160–1, 884, 958
children's application of 960, 961
knowledge of 368, 809, 816
comparison distance effect 577
Comparison problems 160
comparison of procedures 1127
compensatory mechanisms 621
in developmental dyscalculia 740–1
in older people 355, 358–60
in Turner syndrome 686, 687, 690
in visual deprivation 791, 795, 798, 802–3
in Williams syndrome 724, 725
Components Model 50–1
computation, age-related changes 349–50
computational modelling 450–1, 452–3
computer-assisted interventions (CAI) 745–6
development of 747–50
in dyscalculia 638–9, 655–7
in early education
current situation 1062–3
kindergarten children 1091
potential of 1058–62
software development 1063–74
educational significance 752–4
effectiveness on numerical skills learning 750–2, 753
effects of research methods on results 754–5
who benefits? 755–6
key challenges 761
neuroscientific approaches 756–7
approximate number system 757–8
number line representations 758–9
number sets and virtual manipulatives 760
terminology 746–7
computer-enriched instruction (CEI) 747
computer-managed instruction (CMI) 747
conceptual coding account of numbers 98
strengths and weaknesses 99
conceptual components of number processing 121
conceptual knowledge 809–10, 1119
instructional methods 1126–7
measures of 1120–3
rehabilitation 828
relationship to procedural knowledge 1123–6
future directions 1127–9
selective impairment 812, 816–18
conceptual metaphor 378, 380–2
role in conceptualization of numbers and arithmetic 383–7
use in teaching 390
conceptual-role bootstrapping hypothesis 207, 293, 294
supporting evidence 298–300
confidence, impact of Numbers Count initiative 1112–13
congruency effect
number processing 51–2
studies in blind individuals 792–4
consistency effect, multiplication 114
continuous quantities, neuronal representation 467–8
coping strategies 961–2
core neurocognitive processes for arithmetic 503–4
corona radiata
changes in reading difficulties 711
DTI studies 497
cortical mapping 561
counting 905
in addition and subtraction problems 1082
age-related changes 348
in blind individuals 801
capacities required 383
cognitive linguistics 379–80
development of 1081
conceptual-role bootstrapping hypothesis 293–4
Give-N (Give-A-Number) task 292
influence of language 421
learning of number words 74
number concept development 300–5
number-knower levels 292–300
procedural /conceptual knowledge gap 292
reciprocal nature 283
Numbers Count initiative 1104–5, 1106
people with reading difficulties 700
role in number sense 27
(p. 1159)
and SFON 276
counting principles 240–1, 291
counting skills
prediction of mathematical ability 905–106, 917
relationship to SFON 280–2, 283
counting strategies 357–8
critical learning periods 519–21
cross-cultural comparisons 420–2, 882–3
informal learning activities 1136–7
international surveys of mathematical achievement 430–41, 1024
number line estimation task 410–11
numerical representations 404–8, 416, 418
place-value understanding 419–20
strategic processing 904
cross-place congruency effect
multi-digit number processing 113
relationship to basic numerical representations 119
cultural line of development, Vygotsky 367–8, 369
cultural practices, role in number development 372–3
Curious Incident of the Dog in the Night-time, The, Mark Haddon 12
Current Learning Activities, Numbers Count initiative 1105–6
curriculum reform 1023–4
curriculum type, influence on mathematical development 982–3, 988
D
data capture, educational software 1069–70
decade number effects 115
relationship to basic numerical representations 119
decimal processing 116, 119
decomposed processing 108–10, 132n1
decomposition processes 773
DeFries-Fulker extremes analysis 996
delayed match-to-sample task 457, 458
dementia
semantic 510, 821
derived fact strategy use 883–5
development
of counting 1081
conceptual-role bootstrapping hypothesis 293–4
Give-N (Give-A-Number) task 292
influence of language 421
learning of number words 74
number concept development 300–5
number-knower levels 292–300
procedural /conceptual knowledge gap 292
reciprocal nature 283
of number concepts
cardinality and equinumerosity 304–5
cardinality and the successor function 304
discrete quantification 300–2
linguistic influences 404–8, 416–26
numerosity versus quantity 302–4
understanding of numerosity changes 302
of numerical skills 330–2, 338–40
embodied learning 380–2
Piaget's interpretation 368–9
relationship to finger gnosis 73–5
role of cultural practices 372–3
role of language 369–70
developmental models 338–40, 488–9, 490, 733
developmental studies 447–8, 452
of arithmetic 514–23
future research areas 496–8
of memory-based strategies 521–3
neural correlates of number processing
in childhood 493–6
in non-verbal infants 492–3
developmental trajectories
characteristics of 976–8, 987–9
Class Learning Path Model 1037–41
mathematically-desirable and accessible (MD&A) methods 1042–8
classroom effects 983
in fragile X syndrome 683
future research areas 986–7
growth curve modelling 985–6
influence of curriculum 982–3
diffusion tensor imaging (DTI) 497
studies of developmental dyscalculia 737–8
direct cortical electrostimulation (DCE) 450, 583–4, 600–1
qualitative analysis of errors 604–5
studies of calculation 602–4, 605–6
discrete quantification 300–2
discrete-trial self-control test 247
distance effect 26, 49, 248, 332–3, 566–7, 809, 868
age-related changes 345–7, 346
in blind individuals 790, 791
developmental changes 536
for fractions 52–3, 54, 115
for multi-digit numbers 107–9
in neglect patients 95, 96
and numerosity-selective neurons 464
practical and diagnostic implications 111
relationship to mathematical ability 868–9, 871
for symbolic numbers 533
versus non-symbolic representations 536–7
types of 576–7
division
asymmetric nature of 164
infants' abilities 265–8
semantic types 160–1, 958
word problem-solving 958, 960–21
see also arithmetic
(p. 1160) dogs, numerical discrimination 219, 220, 224, 225
dolphins, quantity discrimination 219
dorsal visual stream, involvement in arithmetic 506, 507–8
dorsolateral prefrontal cortex (DLPFC) 671
changes in dyscalculia 736
involvement in arithmetic 503
TDCS studies 585
Dots2Track software 1063
double-digit numbers, as primitives 48–52, 60–1
double dissociations 100, 487, 635, 883
between automaticity and numerical learning 585
in bisection tasks 94–5, 576
between dyscalculia and dyslexia 666
in genetic developmental disorders 722, 725
in grey matter density 854
between number and semantic knowledge 821
TMS studies 586, 593
in Williams' syndrome and Down syndrome 725
Down syndrome 642, 729
attention deficits 727–8
language abilities 723
number abilities 724–6
drawings, use in word problem-solving 172–4
drill learning 845
computer-assisted 745, 747–22, 755, 1062–23
functional imaging studies 846
in rehabilitation 828, 855
TRNS studies 585
dual number naming systems 423–5
dual process theories 963
dual representation, word problem-solving 957
duration of stimuli, TMS studies 589
dyscalculia, developmental 268, 487n1, 642–3, 662–3, 732, 1084
applied neuroscience 613–15
attention 668–70
characteristics of 648–50, 663–4
comorbidities 734
dyslexia 666–8
comparison with math learning difficulties 701
compensatory mechanisms 740
definition of 879–80
general contributory cognitive factors 639–40
heritability 648
importance of 647–8
indicators of 649
interventions 638–9, 653–7
brain function changes 741
in children with reading difficulties 712–13
future research areas 657
neural correlates 447, 495, 652–3, 734
brain connections 737–8
brain function 738–41
brain metabolism 738
brain morphometry 736–7
summary 735
neuroanatomical model 671–2
research areas 657–8
and SFON 277
subtypes 640
symbolic versus non-symbolic representations 664–6
terminology 487n1
underlying deficits 637
Weber fraction 335–6, 338, 339
working memory capacity 637–8, 668–0
dyslexia 664
comorbidity with dyscalculia 640–1, 648, 666–8
E
early education
children's mathematical minds 1056–7
content of 1057–8
need for improvement 1055–6
pedagogical principle 1056
software use 1074–5
current situation 1062–3
development process 1063–74
potential of 1058–62
research and development 1075
early experiences
of board games 1142–3
learning advantages 907–8
numerical activities in the home 1137–40
early interventions 1086
first grade interventions 1091–3
kindergarten interventions 1089–91, 1090
preschool interventions 1086–8
curricula 1148–9
linear number board games 1143–8
early numeracy, prediction of mathematical ability 916–18, 925–6, 1027–8
early screening 1084–6
East Asian number naming systems 417, 422–4, 425–6
influence on mathematical learning 420–2
influence on number representation 416, 418
influence on place-value understanding 419–20
education
Class Learning Path Model 1037–41
conceptual and procedural knowledge 1119–20
instructional methods 1126–7
measures of 1120–3
relationship between 1123–6, 1127–9
curriculum reform 1023–4
effects on intuitive strategy use 193
grouping of school pupils 1025–7, 1029–30
importance of learning mathematics 1021–2
interventions in dyscalculia 653–7
understanding individual differences 1025–30
use of conceptual metaphors 390
(p. 1161)
use of research 1050–1
educational neuroscience 451–2, 612–13
ecological validity 623–4
interdisciplinary collaboration 624–5
neurointervention 620–3
neuroprediction (neuroprognosis) 619–20
neurounderstanding 613–19
unrealistic expectations 625–6
electroencephalography (EEG) studies
of arithmetic strategies 616
in blind individuals 791
of infants 262–3
of SFON 280
elephants, numerical discrimination 224
embodied learning 388–90
embodied number representations 38, 67–8, 387
anthropological perspective 68–72
CAI games 759
neural correlates 78–82
and touch screen technology 1060
use by adults 75–8
embodied reasoning 390–5
emotional factors 7, 873–4, 908, 933
choking under pressure 944
interest and motivation 934–5
mathematics anxiety 936
impact on performance 941–3
reduction of effect on performance 946–8
self-efficacy 935–6
stereotype threat 945–6
in word problem-solving 965–6
encoding-complex model 39, 143–4
encylopedic knowledge 810
selective impairment 812, 818
endophenotypes 924
English-speaking children, number representation 406, 416, 418
entity theory of intelligence 937
environmental influences 994
molecular genetic studies 1011
Episodic Situation Model 163–4
equinumerosity, children's understanding of 304–5
equiprobability bias 189–90, 191
effects of education and training 193
ESpaN model 50–1, 60
estimation skills 1140
blind individuals 795–801, 803
genetic studies 1003
individual differences 886–7
relationship to arithmetic performance and derived fact strategies 883–5
Euclid, Elements 17
event-related potentials (ERP) studies 616
in blind individuals 791, 793–4
in dyscalculia 665, 669
in infants 493
of learning 843
event-related spectral perturbation (ERSP), studies in blind individuals 791, 792
Every Child Counts initiative 1101
exact number system 795
age-related changes 345–7
deficits in MLD 771
exceptional arithmetic expertise 880
genetic studies 1006, 1007–8
neural correlates 849–51
executive attention, individual differences 900–1
executive functions 167–8, 739, 778–19, 811
age-related changes 349–50, 352, 353, 354, 355, 357
associations and dissociations with number system 822, 823
in children 517–8
deficits in dyscalculia and MLD 668–0, 739–20
selective impairment 640, 813, 820–1
explicit reasoning 185
exploration of problems 1127
expressed representations, definition 532
F
factor analytic studies 882
fact retrieval 773, 841
age-related differences 355, 357
deficits 813, 819–20
in MLD and low achievement 775–7
neural correlates 708
in people with reading difficulties 699, 702–3, 707
rehabilitation 828
functional imaging studies 521–2, 710
role of angular gyrus 636
Fechner's law 464
feedback, from educational software 1061, 1068–9
fictionalism 21
fictive motion 378, 382
figural-spatial training, functional imaging studies 848–9
finger discrimination studies 80–1
finger representations 67–8
in adults 75–8
anthropological perspective 68–72
in blind individuals 801
in children 73–5
counting direction preferences 76
and mathematical competence 649
multiplication methods 71–2
neural substrate 78–82
TMS studies 591
(p. 1162) first grade interventions 1091–3
fish
numerical discrimination 215, 220, 222–3
ontogeny of numerical abilities 231
quantity discrimination 225–6
fluency effects 182–3, 184
footstep estimation task, studies in blind individuals 798–9
formative research, educational software design 1070–1
forms 1037–8
formulae
for average temperature of the Earth 14
beauty of 7–8
relevance of 11
four-step-developmental model 488, 489, 733
fractions
as primitives 52–5, 61–2
processing of 109–10, 115, 119
understanding of in MLD and LA 777–8
fragile X syndrome 642, 691
functional imaging studies 685
MLD phenotype 682–3
number knowledge 684
syndrome overview 682
working memory demands 685–6
frontal eye field (FEF), order-selective neurons 474
frontal lobe 811
changes in dyscalculia 671, 739–40
role in arithmetic 636
role in exact calculation tasks 152
frontal lobe lesions 664, 825
frontoparietal cortex 447
role in numerosity encoding 280
functional imaging studies 470, 881–2
adaptation studies 567–8
of adult learning 838–40, 841–2, 843–9
in children, specific issues 485–6
in developmental dyscalculia 738–41
developmental studies 447–8, 452
differences from TMS findings 449
of exact versus approximation tasks 152
of exceptional arithmetic expertise 849–51
of finger representations 79–82
in fragile X syndrome 685
of proportion processing 470
of ratio and fraction processing 55
of rehabilitation 855–6
of SFON 280
in Turner syndrome 688, 690
fusiform gyrus, changes in dyscalculia
brain morphometry 736
functional studies 740
G
gambler's fallacy (negative recency effect) 189
gambling behaviour 194
games
educational use 1061, 1069
effect on mathematical skills 620, 621
home experiences 1142–3
kindergarten interventions 1091
Numbers Count initiative 1103–4
preschool interventions 1088
use in dyscalculia 638–9, 655–7, 656
gender differences
in CAI studies 755–6
genetic studies 1008–9
gene–environment interactions 1011
generalized fraction (GeF) 53, 61
genetic disorders 722
developmental approach 722–3, 728–9
language abilities 723
number abilities
children and adults 724–5
in infants 725–6
visual and attention abilities 727–8
genetics 1012
factors in RD and MD 712
of high and low ability 1005–8
influences on MLD and low achievement 769–70
molecular genetic studies 1009–11
quantitative methodologies 993–7
relationship between mathematics and cognitive traits 1001–4
relationship between mathematics and motivational traits 1000–1
sex differences 1008–9
studies of phenotypic stability in mathematical ability 1004–5
twin studies 997–1000
Genome Wide Association Studies (GWAS) 1009–10
Gerstmann syndrome 78, 594, 601, 604, 649, 733
gestures, metaphorical 392–5
giftedness 880
Give-N (Give-A-Number) task 292, 296, 539
goal-drive activities, computer-assisted 1059–60, 1065
grammatical knowledge, relationship to mathematical ability 903
GraphoGame Math game 638, 655, 1091
grasping movements, effect on number magnitude processing 70
grey matter density
changes due to training 637, 840, 853–5
in dyscalculia 637, 652–3, 671, 672, 735, 736, 737
predictive value 619
in reading difficulties 710
grip adjustment, effect of number magnitude 69–70, 556
growth curve modelling 874, 985–6, 988
growth mixture modelling 986
guppies, ontogeny of numerical abilities 231
(p. 1163) H
handwriting ability, relationship to mathematical ability 922
handwritten digits, spatial properties 96
Hans the horse 237–8
heuristics 41, 181, 188, 191, 962–23
representativeness heuristic 188–9
see also intuition
hippocampus 496
developmental studies 516
and fact retrieval skills 619
repetition enhancement 514
homework, impact on performance 436
hyenas, numerical discrimination 214–15
I
identifying descriptions of numbers 27–8
implicit reasoning 185
IMSSS project 748–9, 751
incremental theory of intelligence 937
individual differences 867–8, 878–80, 895–7, 1135
age of emergence 885–7
applied neuroscience 617–19
and approximate arithmetic skills 869–70
functional imaging studies 851–3
genetic and environmental factors 997–1000
giftedness 880
growth curve modelling 874
in infants 887–8, 1080
link to cognitive capacities 897
attentional processes 900–2
learning experiences 904–8
linguistic knowledge 902–3, 979–80
magnitude comparison skills 868–9
memory problems 980–2
numerical quantity knowledge 897–900
strategic knowledge 903
neuroscientific approach 984–5
and number knowledge 870–2
in numerical magnitude knowledge 1141–3
relationships between calculation, estimation, and derived fact strategies 883–5
research areas 908–9
research trends 875–6
role of early home environment 1137–40
role of emotional factors 873–4
school grouping practices 1025–7
within-school variation 1025, 1026
social factors 436, 983–4, 1136–7
syndrome research 678–9, 680
in trajectories 977
in word problem-solving 953–5
knowledge base differences 955–61
see also genetics
individual internal forms (IIFs) 1037–8, 1048–9
'tuning' of 1049
infants 268–9
addition and subtraction 259–63
individual differences 887–8
multiplication and division 265–8
neural correlates of number processing 492–3
number sense 258–9, 1080
numerical change detection 204–5, 330–1
object-file system 206
ordering 264–5
quantity discrimination 222
subitizing 886
inference control, relationship to mathematical ability 923
inferior frontal gyrus, changes in dyscalculia 736
inferior longitudinal fasciculus
DTI studies 497
relationship to mathematical ability 737
inferior parietal lobe, TMS studies 588, 592
informal learning activities
linear number board games 1143–8
numerical activities in the home 1137–40
sociocultural perspective 1136–7
inquiry learning-path teaching 1040–1, 1042
insula 492
role in arithmetic 503
salience processing 517–18
intelligence 732–3
in MLD and low achievement 778
theories of 936–7
intelligent tutoring systems (ITS) 752
interest in mathematics
relationship to achievement 934–5
interference effects 150–1, 555
international surveys of mathematical achievement 430–3
causality issues 435, 438
contextual factors 435–6
implications 439–40
information sources 441
key findings 433–5
outcomes, variation over time 438–9
policy lessons 436–7
rankings 437–8, 1024
Internet, maths resources 9
interventions 1086, 1100–1
early 1086
first grade 1091–3
kindergarten 1089–91, 1090
preschool 1086–8, 1143–9
influence of curriculum type 982–3
Numbers Count initiative 1101–7
discussion 1113–15
impact of 1107–13
(p. 1164) intraparietal sulcus (IPS) 269, 507, 811, 841–2
association neurons 471, 472
deficits in 664
developmental studies 447, 515–17, 519
in dyscalculia 614, 637, 652–3, 671, 736
and mathematical competence 618–19
non-numerical magnitude processing 589–90
notation effects 495
numerical symbol processing 540–3
numerosity processing 490–1, 573–4
numerosity-selective neurons 78–9, 459, 460–1, 462
role in arithmetic 503, 507–9, 511, 599, 616, 636, 650–2, 709
role in finger representations 80
TMS studies 588–90, 592, 594, 599
intrusion errors 143, 148, 640, 776–7, 819
intuition 41, 180–1, 196
effect of fluency 182–3
effect of symmetry 183–4
effect of temporal contiguity 184
implications for research and education 194–5
overlearned strategies 181–2
primary and secondary 190–1
problems with probability and randomness 187–91
role in word problem-solving 963–4
spatial numerical associations 389, 390
see also heuristics
intuitionism 20
intuitive rules 185–7
intuitive strategy use, influencing factors 192–4
inversion errors 113
long-term consequences 129–31
relationship to language 125, 126–8, 407
inversion principle, use of 511, 896
invertebrates, numerical discrimination 221, 224–5
IQ 732–3
see also intelligence
isomorphism of measures relations 160
iterative view, conceptual and procedural knowledge 1124–6
J
Japanese number naming systems 404, 417
Japanese-speaking children 425
number representation 405, 416, 418
place-value understanding 419, 421
K
key press estimation (KPE) task, studies in blind individuals 796–7
key word strategies, word problem-solving 170–1, 962
kindergarten interventions 1089–91, 1090
Knowledge Forum© 1062
Korean number naming systems 417, 422–4, 423
Korean-speaking children 426
place-value understanding 419
L
language
associations and dissociations with number system 822, 823
and ATOM 560
bilingual mathematics learning 617
influence on numerical representations 404–8, 416, 418
influence on place-value understanding 418–20
place identification 124–6
place-value activation 126–8
place-value computation 128–9
nature of relationship with mathematical learning 420–2
number naming systems 422–5, 883, 904–5
relationship to mathematical ability 918–19, 926, 979–80, 987
genetic studies 1001, 1007
role in arithmetic 151–3
role in number development 369–70, 425–6
young English language learners 1083
language skills, relationship to arithmetic skill 318, 872–23, 902–3
'last resort' hypothesis, animal numerical discrimination 218–19
lateral prefrontal cortex
numerosity-selective neurons 459, 460–1
learning
applied neuroscience 615–17
see also education
learning experiences, as cause of individual differences 904–8
learning studies 513–14, 842–3, 846, 856–8
complex arithmetic 843–5
educational software 1071–2
effect of TDCS 856
effects of different learning methods 845–6
operation-specific changes 846–7
simple arithmetic 843
specificity of brain activation changes 848–9
strategy use, effects on brain activation 636–7
structural brain changes 853–5
time course of effects 847
transfer effects 847–8
length discrimination, TMS studies 590
lexical errors 120, 818–19
linear number board games 1143–8
line lengths, neuronal representation 467–8
lions, numerical discrimination 214
logarithmic numerical coding 446
logarithmic scaling, mental number line 370–1, 389, 408–9
(p. 1165) logical thinking, prediction of mathematical ability 916–17
logicism 23–4
Logo programming language 749
longitudinal research see developmental trajectories
low achievement (LA) 779–80
deficits
in arithmetic 773–8
domain-general 778–9
in number representation 770–3
in rational number understanding 777–8
diagnosis 768
etiology 769–70
prevalence 769
M
macaques
numerical discrimination 219, 239
ordinality judgements 243
magic tricks, mathematics-based 12–13
magnetic resonance spectroscopy (MRS), study in developmental dyscalculia 738
magnitude
interference effects 555
semantic representation 120
magnitude comparison tasks
multi-digit number processing 117–18
relationship to mathematical ability 868–70
TMS studies 587–9
magnitude effect (size effect) 26, 248, 333, 566–7
and numerosity-selective neurons 464
magnitude representation
developmental models 339
development of 557
shared 816
magnocellular deficit theory 697
manipulative materials
role in word problem-solving 171
value in dyscalculia 654
virtual 760, 1060, 1065
mapping problem, and ATOM 559–60
Match-to-Sample task 303
MathemAntics software 1064–74
mathematical ability
impact of Numbers Count initiative 1107–9
individual differences 617–19
influencing factors 435–6
emotional 934–6
language 420–2
international surveys 430–41
and number-line estimation 409
prediction of 925–7
early numeracy 916–18
language issues 918–19
motor, visual perceptual, and visuomotor skills 919–21
number sense 924–5
phonological awareness 704–5
phonological loop function 705–6
RAN tasks 706
working memory, inhibition, naming speed, and inference control 921–4
relationship to approximate number system 208–10, 321–2, 323–6, 335–7
relationship to brain activation 495, 511, 513, 851–3
mathematical communication 6–8
ongoing activities 8–10
pitfalls 10–11
popular image 3, 5
problems 4–5
in schools 5–6
successes 12–16
mathematical communicators 10
mathematical difficulties 701, 779–80, 915–16, 1028–30, 1099–100
classroom effects 983
deficits
arithmetic 773–8
number representation 770–3
rational number understanding 777–8
diagnosis 768
domain-general deficits
intelligence 778
working memory 778–9
early identification 873, 1084–6
etiology 769–70
genetic studies 1005–8
growth curve modelling 985–6
importance of 767–8
interventions 1086, 1100–1
first grade 1091–3
influence of curriculum type 982–3
kindergarten 1089–91, 1090
Numbers Count initiative 1101–13
preschool 1086–8
prevalence 769, 879
mathematical drawings 1042
for addition and multiplication 1044
expanding cultural maths forms to become pedagogical forms 1046–8
for multidigit computation 1043–6
representing problem situations 1043
mathematically-desirable and accessible (MD&A) methods 1042
for addition and multiplication 1044
expanding cultural maths forms 1046–8
(p. 1166)
for multidigit computation 1043–6
representing problem situations 1043
mathematicians, arithmetic abilities 890
mathematics anxiety 874, 908, 933, 936, 946, 965
impact on performance 941–3
reduction of effect on performance 946–8
mathematics books 8–9
mathematics learning disability (MLD)
longitudinal studies 681
subtypes 680–1
syndrome research
fragile X syndrome 682–6
MLD phenotypes 679–80
models of individual difference 678–9
Turner syndrome 686–91
Mathematics Millennium Project 9
Mathematics Recovery 1100–1
Maths Busking 9
media, coverage of mathematics 6, 8
medial temporal lobe memory system
developmental studies 515–16
involvement in arithmetic 503
mediation, Vygotsky's views 367–8
memory-based strategies, development of 521–2
memory networks
numeral format effects 149–50
memory structures, individual differences 906–7
mental arithmetic
cognitive architecture 39–40
finger-based representation 67–83
intuitive reasoning 41
role of spatial coding 97
role of working memory 938–41
word problem-solving 40–1
see also arithmetic
mental disciple doctrine 1021–2
mentalism 20–1
mental models, use in word problem-solving 164–6
mental number line 36, 37, 46, 90, 97, 121, 593–4, 1140–1
and CAI games 758–9
children's scalings 408–10
logarithmic scaling 370–1, 389, 408–9
representation of 2-digit numbers 48–9
representation of fractions 53
representation of zero 58, 59
strengths and weaknesses 98–9
TMS studies 594–8
mental representations, definition 531–2
metacognitive strategies 963, 1057
metaphor 380–2
and ATOM 558–9, 560
in conceptualization of arithmetic 382–7
embodied learning 388–90
motion along a path, arithmetic 384–7, 385, 389, 390–5
microworlds 1059–60
mild cognitive impairment, numerical distance effect 346
mode-control model of numerosity detection 465, 466
Model task, SFON assessment 279
modularity 487, 552–3
implications of ATOM 559
monkeys
mathematical abilities 238–40, 251–2
counting principles and symbolic representation 240–2
mechanisms 248–50
neuroscience 250–1
numerical discrimination 219, 239
numerosity and self-control 247–8
numerousness judgements of food items 245–6
numerousness judgements of non- food items 246–7
ordinality judgements 242–5
use of symbols 241
single neuron studies 446, 452, 567
evidence for number-sensitive coding 570–1
of numerical rank processing 473–5, 476
of numerical rules encoding 477, 478–9
of numerosity-selective neurons 459–73
motivation 654, 657
relationship to achievement 193–4, 935
young children 1056, 1058
motor cortex
order-selective neurons 474
role in number processing 82
motor skills, relationship to mathematical ability 919–21, 922, 926
movement tracking, numerosity-selective neurons 461
multi-digit number processing 38–9, 106–7, 131–2, 577
definition and clarifications 111–12
distance effect 107–9
early understanding of, long-term consequences 129–31
language modulations 124–9, 404–8, 416, 418
maths drawings and MD&A methods 1042–8
place-value processing 123–9
practical and diagnostic implications 111
representations and processes involved 112, 120–3
specific effects 113–16
tasks used in research 112, 117–18
multi-digit numbers, commonalities and differences 110–11
multi-linear effects, number line estimation task 116
(p. 1167) multiple proportions relations 160
multiplication
consistency effect 114
infants' abilities 265–8
MD&A diagrams 1044
multi-digit number processing 117–18
semantic problem types 160, 161, 958
word problem-solving 958, 960–21
multiplication consistency effect, relationship to basic numerical representations 119
multiplicative composition error, multi-digit number processing 113
multivariate genetic analysis 995–6
of cognitive and motivational factors 1000–4
of phenotypic stability 1004–5
MundurukÚ 422
N
naming speed, relationship to mathematical ability 922
natural line of development, Vygotsky 367
natural number bias 41, 181–2
natural numbers 378, 1023
conceptual-role bootstrapping hypothesis 207
relationship to finite cardinal numbers 28–9
negative numbers 36, 48
as primitives 56–7, 61
negative recency effect (gambler's fallacy) 189
neglect patients
asymmetrical distance effect 95, 96
number interval bisection task 93–5, 815
neural filtering model of numerosity detection 465, 466
neural networks
in development of number skills 491
model of symbolic representation 534–6
studies of numerical discrimination 221
neural representations
of number 79, 83, 269, 333–5
multi-digit numbers 108
of primitives 62
neurointervention 451, 620–3
neuroprediction (neuroprognosis) 451, 619–20
neuroscience
application to education 451–2, 453, 612–26
ATOM 449
computational modelling 450–1
developmental studies 446–8
future research areas 452–3
monkey single neuron studies 446, 457–79
neurounderstanding 451, 613–19
New Math 1023–4
nominalism 18–20
non-numerical magnitude processing, TMS studies 589–91
non-symbolic number processing 448, 495, 809
definition 532
in dyscalculia 665
pathways 568, 572–4
selective impairment 812, 815–16
stages of 572, 573
non-verbal learning disability 708
notation effects, neural correlates of number processing 494–5
number, Euclid's definition of 17
number–action link 555–6
Number Bonds game 656
number comparison tasks 26, 331, 333–4
in ADHD 902
age effects 494
computer-assisted interventions 638, 655, 757, 1091
in diagnosis of acalculia 826
in dyscalculia 665, 671
influence of language 407–8
in mathematics anxiety 943
in prediction of mathematical disabilities 917, 924–25
ratio dependence 204–5, 223
symbolic versus non-symbolic representations 494–5, 899
tDCS studies 622
TMS studies 594–8, 596
number concepts 377–8
development of
cardinality and equinumerosity 304–5
cardinality and the successor function 304
discrete quantification 300–2
linguistic influences 404–8, 416–26
numerosity versus quantity 302–4
understanding of numerosity changes 302
number-encoding neurons 269
number form processing 503
number interval bisection task 100–1
in blind individuals 790–1
decade crossing effect 115
evidence for spatial imaging of numbers 93–5
multi-digit number processing 118
in neglect patients 93–5, 99, 815
TMS studies 594–5
verification version 122
number-knower levels 292–5
age ranges for each level 296–7
within-child, cross-linguistic consistency 299–300
versus counting or estimation 295–6
relationship to vocabulary development 298–9
number knowledge, relationship to mathematical ability 870–2
(p. 1168) number line
software representations 1060, 1067
number line estimation task 1141
children's scalings 408–10
cross-cultural comparisons 410
as an index of conceptual representation 410–12
multi-digit number processing 118
multi-linear effects 116
operational momentum effect 116
prediction of mathematical ability 925
number line training 639
number magnitude
effect on grip adjustment 69–70
processing, effect of grasping movements 70
understanding of, preschool children 1081–2
number-matching task 150–1
number module theory 760
number naming systems 60, 422–5, 883, 902–3, 904–5, 918
English, Chinese, Japanese, and Korean 417
influence on numerical representations 404–8, 416, 418
learning of number words 74
role in conceptual understanding 369–70
number naming tasks, multi-digit number processing 117–18
number operations, use by preschool children 1082–3
Number Race game 638–9, 655, 757–8, 1063, 1091
number representation 35–6, 45–6, 402–4, 566–8
deficits in MLD and low achievement 770–3
developmental studies 516–17
influence of language 404–8
multi-digit numbers 38–9, 106–18, 120–32, 404–8, 577
non-symbolic number processing 572, 573
number-sensitive and number-selective coding 569–71
object location map 574–5
processing pathways 568, 572–4
spatial associations 36–7, 90–101, 121, 575–6
task-specific components 576–7
number reversal see inversion errors
Numbers Count initiative 1101–3
curriculum 1102
discussion 1113–15
impact of 1107–13
lesson structure 1103–7, 1104
number-selective coding 450, 569–70
number sense 26–7
preschool children 1080–1
relationship to mathematical ability 924–5, 978, 987
genetic studies 1003
role of vision 788–9
number sense deficits 1083–4
early screening 1084–6
interventions 1086
first grade 1091–3
kindergarten 1089–91, 1090
preschool 1086–8
Number Sense Screener (NSS) 1084–5
number-sensitive coding 450, 570
supporting evidence 570–1
Number Sets Test 772–3
number–space interactions see spatial–numerical associations
number system
executive functions 811
relationship to other cognitive systems 821–4
selective impairments 812–13
arithmetic conceptual knowledge 816–18
arithmetic fact retrieval 819–20
encylopedic knowledge 818
executive functions 820–1
non-symbolic number processing 815–16
number reading and writing 818–19
symbolic quantity processing 814–15
semantic 808–10
verbal 810
number theory 1021
classical 17–18
Number Worlds curriculum 1148
numeracy learning, task demands 698
numeracy levels 976
numeral format effects 144–5
arithmetic errors 148
arithmetic memory networks 149–50
on brain activation 447
and problem-size effect 145–6
on strategy choice 146–8
on zero and one problems 146
numerals 532–3
auditory processing of 544
auditory referents of 539–40
semantic processing of 540–3
semantic referents of 533–9
numerical adaptation paradigm 542–3
numerical coding, logarithmic scaling 446
numerical discrimination
infants 258–9, 330–1
innate nature of 379
numerical distance effect see distance effect
numerical magnitude knowledge, relationship to socio-economic status 1141–3
numerical magnitude system 492–3, 1140–1
numerical priming task 537
(p. 1169) numerical quantity knowledge
individual differences 898–900
types of 897–8
numerical rank see ordinality
numerical rule encoding, monkey single neuron studies 477, 478–9
numerical skills development 330–2, 338–40
embodied learning 380–2
Piaget's interpretation 368–9
relationship to finger gnosis 73–5
role of cultural practices 372–3
role of language 369–70
numerical symbol processing 448
numerosity-selective neurons 79, 269, 332, 459, 460–1, 567
behavioural significance 463–4
coding of continuous and discrete quantities 467–8
implementation models 465–7
latencies 461–3
locations 460–1
simultaneous versus sequential enumeration 459, 461
tracking of numbers of movements 461
O
object collection metaphor 383–4, 388–9, 393
object-file system 206, 207, 223–4, 227, 258, 422
role of vision 788
supporting evidence 225, 249, 261
object location map 569, 574–5
occipital cortex 447
occipito-parieto-precentral network, role in finger representations 80
occipito-temporal cortex 811
lesion study 825
Oksapmin body part counting system 372–3
one to one correspondence principle 240
ontogeny of numerical abilities 228–31
operand errors 819
operational momentum effect 116, 391–2
relationship to basic numerical representations 119
operation errors 819
oral verbal estimation task, studies in blind individuals 798–9
ordinality judgements 898
infants 264–5
monkeys 242–5
non-primate animals 228, 230
rank order-sensitive neurons 269
research areas 547
single neuron studies 473–5, 476
TMS studies 590–1
ordinality principle 241
overlap of brain areas 505
overlearned strategies 181–2
P
pair-coding neurons 473
paired associate learning deficits 703
parahippocampal gyrus, changes in dyscalculia 736, 740
parallel individuation system 492
parents, provision of number-related activities 1137–8
parietal lobe 250–1, 269, 452, 507, 811, 841–2
age-dependent activation shift 494
anatomy 507, 541
ATOM 449, 560
changes in dyscalculia 447, 652–3, 739
competency-dependent activation shifts 496
developmental studies 519
neural circuits 490
numerosity-selective neurons 78–9, 459, 460–1, 461–2
role in approximation tasks 152
role in arithmetic 503, 505, 506, 507–10, 511, 636, 650, 709
role in finger representations 80–1, 82
TDCS studies 585
TMS studies 588–9, 592, 594–5, 595–6, 597
parietal lobe lesions 505, 507, 824–5
parity effects, multi-digit number processing 113, 117
parrots
numerical discrimination 216, 224
use of symbols 241
Path gestures 393, 394, 395
Pathways Model of development 872
pedagogical agents 1061, 1067–8
performance prediction
effects of cognitive capacity and cognitive effort 192
effects of education and training 193
effects of thinking styles and thinking dispositions 193–4
philosophy of number
classical view 17–18
fictionalism 21
logicism 23–4
mentalism 20–1
neo-classical views 21–3
nominalism 18–20
numbers as set-sizes 24–8
phlogiston 25
phonetic recoding in working memory 705–6
phonological awareness 1083–4
relationship to mathematical ability 704–5, 903
phonological deficit hypothesis 703
phonological difficulty, relationship to mathematical ability 979–80
phonological loop 900, 981
individual differences 901
prediction of mathematical ability 705–6
role in mental arithmetic 939, 940
(p. 1170) phonological processing skills 704–6
association with mathematical ability 706–7
phonological recoding in lexical access 706
Piaget, J. 368–9
Piagetian logical thinking, prediction of mathematical ability 916–17
pictures
use in word problem-solving 172–4
pigeons, numerical discrimination 215–16
PISA (Programme for International Student Assessment) 431–2
causality issues 435, 438
contextual factors 435–6
implications 439–40
information sources 441
key findings 433, 434–5
minimum level 648
outcomes, variation over time 438–9
policy lessons 436–7
rankings 437–8
pitch, association with number magnitude 391
placeholder concept 293–4
place identification 123–6
place-value, as a primitive 51–2
place-value activation 124, 126–8
place-value congruency effect 114
relationship to basic numerical representations 119
place-value computation 124, 128–9
place-value representation 122–3
place-value understanding
influence of language 124–9, 405–7, 418–20
people with reading difficulties 700
plasticity
effect of education 620–1
functional imaging studies 513–14
Plato, number theory 18
PLATO project, computer-assisted interventions 749
polarity coding account of numbers 97–8
strengths and weaknesses 99
policy-making, lessons from international surveys 436–7, 440
positional digit repetition effect
multi-digit number processing 113
relationship to basic numerical representations 119
Positive Ending activities, Numbers Count initiative 1106–7
Positive Start activities, Numbers Count initiative 1103–4
posterior parietal cortex
developmental studies 515
involvement in arithmetic 505, 506, 509–10, 511, 709
neuroanatomy 507
numerosity-selective neurons 78–9, 459, 460–1, 461–2
TDCS studies 585
TMS studies 588–9, 592, 594–5, 595–6, 597
posterior superior parietal sulcus (PSPS) 490
posteromedial cortex, repetition enhancement 514
prediction of ability 192–4, 925–7
early numeracy 916–18, 925–6, 1027–8, 1083
grey matter density 619
language issues 918–19
logical thinking 916–17
motor, visual perceptual, and visuomotor skills 919–21
neuroprediction 451, 619–20
number comparison tasks 917, 924–5
number sense 924–5
and phonological awareness 704–5
and RAN tasks 706
and SFON 284
working memory, inhibition, naming speed, and inference control 921–4
prefrontal cortex 446, 811
association neurons 471–3, 472
developmental studies 515–17, 519, 521
encoding of mathematical rules 477
involvement in arithmetic 503, 504, 509–10, 511
involvement in memory-based strategies 521, 522
numerosity-selective neurons 459, 460–1, 461–2
temporal order processing 475, 476
Pre-K Mathematics curriculum 1148–9
premotor cortex
involvement in arithmetic 503, 505, 506
role in number processing 82
preschool children, early screening 1084–6
preschool experiences, early learning advantage 907–8
preschool interventions 1086–8, 1089–91, 1090
curricula 1148–9
linear number board games 1143–8
preschool number skills 1080–3
deficits and learning difficulties 1083–4
predictive value 1027–8, 1083
Pressey, S.L., teaching machines 748
pressure situations 944, 946
reduction of effect on performance 947
pre-supplementary motor areas (pre-SMA), order-selective neurons 474
primary intuitions 190–1
primates, non-human
numerical discrimination 217, 219
processing of ratios and fractions 55
see also monkeys
priming effects 113, 117, 119
priming distance effect 537, 567, 577
evidence for number-sensitive coding 571
TMS studies 588
primitive models of operations 959–61
primitives 36, 37
1-digit numbers 46–8
double-digit numbers as 48–52
fractions as 52–5
(p. 1171)
general discussion 60–2
negative numbers as 56–7
zero as 57–60
probabilistic reasoning
equiprobability bias 189–90
representativeness heuristic 188–9
role of intuition 187–8
probability estimation 116
infants 266–7
problem size effect 39, 652, 896
fractions 115
impact of training 843
and numeral format 145–6
problem-solving
people with reading difficulties 700
procedural components of number processing 122
procedural development, in MLD and low achievement 773–5
procedural errors 821
procedural knowledge 1119–20
instructional methods 1126–7
measures of 1123
relationship to conceptual knowledge 1123–6
future directions 1127–9
procedural subtype, MLD 681
and Turner syndrome 687–8
prodigies 880
genetic studies 1006, 1007–8
neural correlates 849–51
'product of measures' relations 160
Program for International Student Assessment (PISA) 1024
programming languages 749
properties
knowledge of 25–6
reality of 25
proportionality, overuse of 182
proportioning strategies 357–8
proportions, neuronal representation 469–70
prosthetic dimensions, definition 562n2
prototypical counting numbers (PCNs) 378, 379, 380
psychometric intelligence, variability 732–3
public perceptions of mathematics 181
Q
quantitative trait loci (QTL) 1008, 1009
quantitative numerical capacities 872
quantity discrimination, animals 217–19
quantity representation 809
R
random digit generation task 95–6
random events, perception of 188, 194
rankings, international 437–8
rank order-sensitive neurons 269
rapid automatized naming (RAN) tasks 706, 707
ratio-based reasoning, infants 267–8
ratio bias problem 186–7
rational number understanding 52, 181–2
deficits in fragile X syndrome 683
deficits in MLD and low achievement 777–8, 780
ratios
as primitives 55
see also fractions
rats
numerical discrimination 215
ordinal abilities 228
reading ability
interaction with numerical symbol processing 547
relationship to mathematical ability 922, 979–80, 987
genetic studies 1001–2, 1006–7
reading difficulties 697–8, 1083–4
associations and dissociations with number system 822
association with mathematical difficulties 680–1, 698, 713–14
causal explanations 703–7
interventions 712–13
neural abnormalities 708–11
pure reading difficulties versus reading and maths difficulties 701–3
task-specific difficulties 699–701
comorbidities 712
heritability 711–12
heterogeneity of 700–1
phonological processing skills, association with mathematical ability 706–7
selective impairment 813, 818–19
see also dyslexia
reading skills, role in word problem-solving 169–70
redeployment (recycling) hypothesis, symbolic number representation 471
referents, definition of 532
rehabilitation 827–8
functional imaging studies 840
remedial education
effects of 621
repetition priming, functional imaging studies 513–14
repetitive pulse transcranial magnetic stimulation (rpTMS) 586
representativeness heuristic 188–9, 191
effects of education and training 193
Rescue Calcularis game 621, 639, 758
research-based software 1063, 1070–3, 1075
(p. 1172) retrieval strategies 773, 841
age-related differences 355, 357
deficits 813, 819–20
in MLD and low achievement 775–7
neural correlates 708
in people with reading difficulties 699, 702–3, 707
rehabilitation 828
functional imaging studies 521–2, 710
role of angular gyrus 636
retrieval usage, numeral format effects 146–8
reversed-reward contingency task 248
revision stage, educational software design 1071
rhesus monkeys
numerical discrimination 239
numerousness judgements of food items 246
numerousness judgements of non- food items 246–7
ordinality judgements 242, 244–5
relative numerousness judgement 244
Royal Institution, Mathematics Masterclasses 9
rule-encoding units 477
S
salamanders
numerical discrimination 225
quantity discrimination 218, 221–2
salience network 517–18
saliency 186–7
stimulus-dependence 222
task-dependence 222–3
saliency maps 574–5
scaffold learning 1060–1, 1066–7, 1068
scalability of educational software 1073–4
scaling problem, functional imaging studies 251
scanning abilities, in Williams' syndrome and Down syndrome 728
schema-based instruction (SBI) 171
schemata, use in word problem-solving 162–3
schizophrenia, number interval bisection 94
science fairs 9–10
science museums, maths exhibits 10
Scratch programming language 1062
secondary intuitions 190–1
Seeing Number As Pitch (SNAP) effect 391
selective impairment 812–13
arithmetic fact retrieval 819–20
number reading and writing 818–19
self-confidence 984, 1056, 1113
self-control test, monkeys 247–8
self-efficacy 193–4
impact of Numbers Count initiative 1112–13
relationship to achievement 935–6
and word problem-solving 967
self-evaluation 1106–17
genetic influences 1000–1
self-explanation 1127
self-initiated practice see Spontaneous Focusing on Numerosity (SFON)
semantic alignment effects 150–1
semantic dementia 510, 821
semantic memory
associations and dissociations with number system 823
deficits 708
semantic memory subtype, MLD 680–1
semantic number system 533–4, 808–10
in blind individuals 790–5
neural substrates 540–3
selective impairment 812
arithmetic conceptual knowledge 816–18
encylopedic knowledge 818
non-symbolic number processing 815–16
symbolic quantity processing 814–15
symbolic versus non-symbolic representations 534–9
sense-making, word problem-solving 967–8
sequential enumeration, neuronal representation 459–61, 462
serial order effect
multi-digit number processing 114
relationship to basic numerical representations 119
sets, numbers as 22–4
set-sizes, numbers as 24–8, 898
setting of school pupils 1026
sex differences see gender differences
sex-limitation models 996–7
single neuron studies 446, 452, 567
of number-sensitive coding 570–1
of numerical rank processing 473–5, 476
of numerical rules encoding 477, 478–9
of numerosity-selective neurons 459–73
single nucleotide polymorphisms (SNPs) 1009–11
single photon emission computed tomography (SPECT) studies 841
single pulse transcranial magnetic stimulation (spTMS) 586
situation models, use in word problem-solving 163–4
Situation Strategy First model 166
size congruity effect (SiCE) 47–8, 49, 61, 116, 122–3
in 2-digit numbers 50–1
in comparisons with zero 58–9
in fractions 53–4
in negative numbers 56–7
relationship to basic numerical representations 119
TMS studies 588–9
size effect (magnitude effect) 26, 248, 333, 566–7
and numerosity-selective neurons 464
socio-economic status
and approximate arithmetic skills 318–19
and early numeracy skill 907, 1083
and effectiveness of CAI 756
(p. 1173)
impact on development of numerical knowledge 1136–7, 1139
impact on performance 436, 983, 988
and numerical magnitude knowledge 1141–3
Sona designs 15
spatial abilities
blind individuals 790, 795, 802
relationship to mathematical ability, genetic studies 1002
spatial judgements, link to time perception 556, 557
Spatial Numerical Association of Response Codes (SNARC) effect 37, 47, 91, 100, 121, 391, 410, 575, 1141
age-related changes 346–7
automatic nature of 90–1
in blind individuals 790, 792, 803
flexibility of 91–2
and ordinal information 92–3
TMS studies 595
underlying spatial codes 92
spatial–numerical associations 36–7, 89–90, 121, 575–6, 593–4, 789–90
evidence for 90–7
explanatory theories 97–9
hybrid account of 100–1
TMS studies 594–8
spatial–numerical priming experiment 556
Spatial Quantity Association of Response Code (SQUARC) effect 554–5
spiders, numerical discrimination 221
Spontaneous Focusing on Numerosity (SFON) 275–7, 286–7
assessment of 277–9
brain imaging studies 280
enhancement of 285–6
individual differences 277, 887
as a mechanism of self-initiated practice in number skills 284–5
relationship to numerical skills 280–4
squirrel monkeys
numerical discrimination 240, 247
ordinality judgements 243
stable-order principle 240
and finger counting 73
statistical heuristics 191
statistical learning, infants 266–8
statistical methods 867
stereotype threat 945–7
STIMULUS programme 9, 10
story problems see word problem-solving
strategic components of number processing 121
strategic knowledge, relationship to mathematical ability 903–4
strategy-based learning, functional imaging studies 846
strategy distribution, age-related changes 356–7
strategy repertoire, age-related changes 352–3
strategy use 351–2
age-related changes 353–4, 354–6, 355
derived fact strategies 883–5
EEG studies 616
functional imaging studies 511, 513
in MLD and low achievement 774
models of 358
numeral format effects 146–8
proportioning versus counting 357–8
sequential difficulty effects 358
in word problem-solving 959–61
heuristics 962–3
metacognitive strategies 963
superficial coping strategies 961–2
streaming of school pupils 1025–6
structural representation of number system 122–3
subitizing 897–8
age-related changes 347, 348
in infancy 886
innate nature of 379
non-primate species 223–7
relationship to SFON 281–2, 283
role of vision 788
studies in blind individuals 800
subset-knowers 292–3
subtraction
borrowing operation 112, 114, 119, 775, 811, 944
infants' abilities 259–63
multi-digit number processing 117–18
see also arithmetic
Successmaker ITS 752
successor function, children's understanding of 304
sum interference effect 150–1
superficial coping strategies 961–2
superior corona radiata
changes in reading difficulties 711
DTI studies 497
superior longitudinal fasciculus 737
superior parietal lobe 507
involvement in arithmetic 509
numerosity-selective neurons 461
superior temporal gyrus 545–6
supplementary eye field (SEF), order-selective neurons 474
supplementary motor area (SMA)
involvement in arithmetic 503
order-selective neurons 474
supporting functions 496
supramarginal gyrus 495, 505, 507, 511, 515, 522, 541 , 545, 603
(p. 1174) symbolic estrangement hypothesis 538
symbolic maths performance, relationship to ANS 208–10
symbolic representation 491, 495, 809
animal studies 241
auditory referents of numerals 539–40
definitions 531–2
in dyscalculia 665
evolutionary development 470–1
learning of 898–9
neural network model 534–6
neural substrates 540–6
open questions 546–8
pathways 568, 572–4
selective impairment 812, 814–15
semantic referents of numerals 533–9
single neuron studies 471–3
symmetry effects 183–4
synaesthesia 89, 559–60
syndrome research
fragile X syndrome 682–6
longitudinal studies 681
MLD phenotypes 679–80
models of individual difference 678–9
Turner syndrome 686–91
syntactic errors 120, 818
T
table and non-table errors 819
task-dependence, numerical discrimination 222–3, 226
teaching approach 1036
curriculum reform 1023–4
effects of 983, 988, 999–1000, 1030
impact of tests 1022
see also education
teaching machines
development of 747–50
technology-enhanced learning (TEL) 746–7
TEDI-MATH 931–2
temporal contiguity effects 184
temporal lobe
changes in dyscalculia 736
developmental studies 515–16
involvement in arithmetic 503, 510
studies in reading difficulties 711, 714
studies in Turner syndrome 690
temporal lobe lesions 818, 825
temporal order processing 475, 476
temporal-parietal cortex, involvement in SFON 280
tests, teaching to 1022, 1056
thalamus, changes in dyscalculia 740
thinking styles and dispositions, effects on intuitive strategy use 193–4
time events
impaired estimation of 816
spatial experience of 381–2
TMS studies of duration 589
time perception, link to spatial judgements 556, 557
TIMSS (Trends in International Mathematics and Science Study) 431–2, 983–4, 1024
causality issues 435, 438
contextual factors 435–6
implications 439–40
information sources 441
key findings 433–5
outcomes, variation over time 438–9
policy lessons 436–7
rankings 437–8
tools, digital 1060, 1065–6
top-down saliency 187
training
effects on brain activation 636–7
effects on brain structure 637
effects on intuitive strategy use 193
effects on speed and accuracy 844
exceptional arithmetic expertise 849–51
transcoding 810, 811
in diagnosis of acalculia 826
transcoding errors 39, 818–19
long-term consequences 129–31
multi-digit number processing 113, 117
rehabilitation 827–8
relationship to basic numerical representations 119, 120
relationship to language 125–6
transcranial direct current stimulation (TDCS) 584–5
educational interventions 622, 856
transcranial electric stimulation (TES) 450
educational interventions 621–3
transcranial magnetic stimulation (TMS) 449, 584, 585–6
future research areas 591–3
studies of calculation 598–9, 605–6
studies of counting tasks 82
studies of finger representations 591
studies of mental number line 594–8
studies of quantity 587–91, 592
transcranial random noise stimulation (TRNS) 585
educational interventions 622–3
transfer effects, learning studies 847–8
Transform-Sets task 302
Trends in International Mathematics and Science Study see TIMSS
triple-code model 38–9, 121, 142, 487, 545, 598–9, 709, 846, 919 (p. 1175)
recent modifications and extensions 491–2
Turner syndrome 642
functional imaging studies 688, 690
MLD phenotype 686–91
number knowledge 689
syndrome overview 686
twin studies 998–1000
of cognitive and motivational factors 1001, 1001–2, 1003
of mathematical difficulties 1005–8
methodologies 994–6
of phenotypic stability 1004
of reading difficulties 711–12
Twins Early Development Study (TEDS) 998–19
genome wide association studies (GWAS) 1010
studies of high mathematical ability 1007–8
studies of mathematical difficulties 1005–6
study of cognitive and motivational factors 1000–3
study of phenotypic stability 1004
of sex differences 1008–9
Western Reserve Reading Project for Math 999
Western Reserve Twin Project 997
two core number systems hypothesis 249–50
two-digit numbers, as primitives 48–52, 60–1
U
unary model of operations 960
unit-decade compatibility effect 49, 108–9, 113, 132n1
relationship to basic numerical representations 119
V
velocardiofacial syndrome 738
ventral intraparietal sulcus (VIPS) 269
TMS studies 596–8, 597, 599
ventral temporal-occipital cortex, involvement in arithmetic 503, 505, 506
ventral visual stream
changes in dyscalculia 736
developmental studies 519–20
involvement in arithmetic 506, 507–8
visual number form area 544
ventrolateral prefrontal cortex (VLPFC)
involvement in arithmetic 503
involvement in memory-based strategies 522
verbalizers 852
verbal numerical representations 120
verbal problems see word problem-solving
verbal skills, associations and dissociations with number system 822
verbal word frame 709
virtual manipulative materials 760, 1060, 1065
vision
importance in numerical representation 787–9
visual estimation strategy, age-related differences 354–5, 356–7
visual impairment
mathematical ability 641
see also blindness
visualizers 852
visual number form representation 112, 120, 544
visual perceptual skills, relationship to mathematical ability 919–21, 926
visual system, numerical information extraction 221
visuo-motor integration skills, relationship to mathematical ability 920–1, 926
visuospatial attention
associations and dissociations with number system 822
deficits
in dyscalculia 669
in Turner syndrome 687
in Williams' syndrome 727–8
visuospatial sketch pad (VSSP) 900, 980–1, 982
individual differences 901–2
role in mental arithmetic 940
visuospatial subtype, MLD 681
and Turner syndrome 687
vocabulary development, relationship to number-knower level 298–9
voxel-based morphometry (VBM)
learning studies 853–4
studies of developmental dyscalculia 736–7
Vygotsky, L.S., cultural–historical perspective 367–8, 369
W
Weber fractions 333, 795
relationship to mathematical ability 335–6, 338, 339, 771
Weber's law 223, 258–9, 464
Wechsler Individual Achievement Test (WIAT) 1025
within-school variation 1026
Welsh-speaking children
number comparison ability 407, 883
place-value understanding 420
Western Reserve Reading Project for Math (WRRPM) 999, 1001
Western Reserve Twin Project 997
white matter structure
changes in dyscalculia
diffusion tensor imaging 737–8
voxel-based morphometry 736
changes in reading difficulties 710–11
whole number bias 52
Williams' syndrome 642, 729, 898
language abilities 723
number abilities
children and adults 724
in infants 725–6
visuospatial impairments 727–8
(p. 1176) word problem-solving 40–1, 158–9, 174–5, 874, 953–5, 969–70
addition and subtraction problems 955–7, 959–60
affective factors 964–5
attitudes 966–7
beliefs 967–9
emotions 965–6
applied neuroscience 616–17
dual representation 957
enhancing performance 170–4
kindergarten interventions 1090–1
knowledge base differences 955–61
maths drawings 1043
mental representations 161–6
multiplication and division problems 958, 960–1
placing the question before the text 165
preschool children 1082
and reading comprehension skills 169–70
realistic considerations 967–8
reformulation of problems 171–2
role of intuitions 963–4
semantic characteristics 159–61
strategic aspects
heuristics 962–3
metacognitive strategies 963
superficial coping strategies 961–2
and working memory capacity 167–9, 170, 171
working memory 900, 938–9, 980–1
in blind individuals 799–800
central executive 939–41
deficits 820
in dyscalculia 639–40, 669
in MLD and low achievement 778–9
neural correlates 708
relationship to mathematical ability 921–4, 926
demands on, fragile X syndrome 683, 685–6
emotional influences 946
mathematics anxiety 942–3
pressure situations 944
stereotype threat 945–6
individual differences 900–2
influence on mathematical development 981–2, 987
phonetic recoding 705–6
role in word problem-solving 167–9, 170, 171
role of object location map 574–5
spatial coding 98, 99
Z
zero
and number sense 27
as a primitive 57–60, 61