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# (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.

abacus experts, functional imaging studies 850–1

abstract code model

*141–*2abstraction principle 241

abstractness of arithmetic 140

acalculia, acquired 635, 710, 829, 881

diagnosis and assessment 825–

*6*natural recovery 827

neural correlates 824–5

accumulator neurons 446

adaptation studies 567–8

addition

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 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

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

anatomo-functional calculation model 490–1

angular gyrus 490,

*507*, 714, 811, 841–2DCE studies of calculation 602–4

and Gerstmann syndrome 604

role in reading 709–10

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

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, 709age-related changes

*345*and CAI games 757–8

developmental models 339

supporting evidence for 224–5

*see also*approximate arithmetic

Arabic-speaking children, place-value understanding 420

arcuate fasciculus, changes in reading difficulties 711

arithmetic

abstract nature of 140

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

core neurocognitive processes

*503*–4inter-individual differences 851–3

knowledge requirements 896

neural correlates 447–8, 495–6, 636, 650–2, 709

in the developing brain 514–23

in exceptional expertise, 849–51

learning studies 842–9

numeral format effects 144–50

role of language 151–3

semantic alignment effects 150–1

arithmetical facts processing 810

arithmetical sign understanding, selective impairment 820

arithmetic errors, relationship to numerical format 148

arithmetic skills

development of, applied neuroscience 615–16

relationship to estimation and derived fact strategies 883–5

SFON as a predictor of 284

*see also*mathematical ability

arithmetic training, functional imaging studies 513–14

art, mathematical aspects of 15–16

Asperger's syndrome 12

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*attention deficit hyperactivity disorder (ADHD)

*664*, 902associated 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

promotion of maths 6–16

and word problem-solving 966–7

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

avoidance constellation 873–4

B
Basic programming language 749

Battle of Britain, role of maths 13

bees, numerical discrimination 221

beetles, numerical discrimination 224–5

behavioural educational interventions 620

beliefs, influence on word problem-solving 967–9

Big Bang Fair 9–10

binary model of operations 960

blindness 641, 802–3

compensatory mechanisms 798

finger-counting 801

future research areas 803–4

number representation 790–5

numerical estimation 795–801

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, 944error sources 775

relationship to basic numerical representations

*119*bottom-up saliency 186–7

brain lesion studies 824–5, 881

C
angular gyrus 710

Gerstmann syndrome 78

number interval bisection task 93–5

perisylvian cortex, arithmetic impairment 153

calculation

*see also*arithmetic

captain's problem 957

cardinal numbers, relationship to natural numbers 28–9

carry operation

*114*, 124, 130language-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*Change problems 159–60

Chased Chicken Sona pattern 15

children

differences in mathematical competence 618–19

finger counting 73–5

functional imaging studies 485–6

learning of number words 74

Children's Math Worlds Project 1037

chimpanzees, use of symbols 241

Chinese ideograph processing, neural substrates 543

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 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

colour–number synaesthesia 89

Combine problems 160

Common Core State Standards 1050–1

comparison distance effect 577

comparison of procedures 1127

Components Model 50–1

computation, age-related changes 349–50

computer-assisted interventions (CAI) 745–6

development of 747–50

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 components of number processing 121

conceptual knowledge 809–10, 1119

instructional methods 1126–7

measures of 1120–3

rehabilitation 828

conceptual metaphor 378, 380–2

role in conceptualization of numbers and arithmetic 383–7

use in teaching 390

confidence, impact of Numbers Count initiative 1112–13

consistency effect, multiplication

*114*continuous quantities, neuronal representation 467–

*8*coping strategies 961–2

core neurocognitive processes for arithmetic

*503*–4cortical 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

people with reading difficulties 700

role in number sense 27

(p. 1159)
and SFON 276

counting strategies 357–8

critical learning periods 519–21

cross-cultural comparisons 420–2, 882–3

informal learning activities 1136–7

number line estimation task 410–11

place-value understanding 419–20

strategic processing 904

cross-place congruency effect

multi-digit number processing

*113*relationship to basic numerical representations

*119*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

D
data capture, educational software 1069–70

decomposition processes 773

DeFries-Fulker extremes analysis 996

derived fact strategy use 883–5

developmental dyscalculia

*see*dyscalculia, developmental; mathematical difficulties; mathematics learning disabilitydevelopment

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

developmental studies 447–8, 452

of arithmetic 514–23

future research areas 496–8

of memory-based strategies 521–3

developmental trajectories

classroom effects 983

in fragile X syndrome 683

future research areas 986–7

growth curve modelling 985–6

influence of curriculum 982–3

discrete quantification 300–2

discrete-trial self-control test 247

distance effect 26, 49, 248, 332–3, 566–7, 809, 868

developmental changes 536

for multi-digit numbers 107–9

and numerosity-selective neurons 464

practical and diagnostic implications 111

types of 576–7

dolphins, quantity discrimination 219

dorsolateral prefrontal cortex (DLPFC) 671

changes in dyscalculia 736

involvement in arithmetic

*503*TDCS studies 585

Dots2Track software 1063

double dissociations 100, 487, 635, 883

between automaticity and numerical learning 585

between dyscalculia and dyslexia 666

in grey matter density 854

between number and semantic knowledge 821

in Williams' syndrome and Down syndrome 725

drawings, use in word problem-solving 172–

*4*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

E
applied neuroscience 613–15

attention 668–70

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*, 734brain 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

early education

children's mathematical minds 1056–7

content of 1057–8

need for improvement 1055–6

pedagogical principle 1056

early experiences

of board games 1142–3

learning advantages 907–8

numerical activities in the home 1137–40

early screening 1084–6

East Asian number naming systems

*417*, 422–4, 425–6influence on mathematical learning 420–2

influence on place-value understanding 419–20

education

Class Learning Path Model 1037–41

curriculum reform 1023–4

effects on intuitive strategy use 193

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

*see also*finger representations

embodied reasoning 390–5

emotional factors 7, 873–4, 908, 933

choking under pressure 944

interest and motivation 934–5

reduction of effect on performance 946–8

self-efficacy 935–6

stereotype threat 945–6

in word problem-solving 965–6

endophenotypes 924

entity theory of intelligence 937

Episodic Situation Model 163–4

equinumerosity, children's understanding of 304–5

estimation skills 1140

genetic studies 1003

individual differences 886–7

relationship to arithmetic performance and derived fact strategies 883–5

*see also*approximate number system

Euclid,

*Elements*17Every Child Counts initiative 1101

executive attention, individual differences 900–1

explicit reasoning 185

exploration of problems 1127

expressed representations, definition 532

F
factor analytic studies 882

fact retrieval 773, 841

role of angular gyrus 636

Fechner's law 464

fictionalism 21

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

footstep estimation task, studies in blind individuals 798–

*9*formative research, educational software design 1070–1

forms 1037–8

fractions

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

functional imaging studies 470, 881–2

G
adaptation studies 567–8

in children, specific issues 485–6

in developmental dyscalculia 738–41

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

gambler's fallacy (negative recency effect) 189

gambling behaviour 194

games

home experiences 1142–3

kindergarten interventions 1091

Numbers Count initiative 1103–4

preschool interventions 1088

gene–environment interactions 1011

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

gestures, metaphorical 392–5

giftedness 880

grammatical knowledge, relationship to mathematical ability 903

grasping movements, effect on number magnitude processing 70

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

homework, impact on performance 436

hyenas, numerical discrimination 214–15

I
identifying descriptions of numbers 27–8

implicit reasoning 185

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

link to cognitive capacities 897

attentional processes 900–2

learning experiences 904–8

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

in trajectories 977

*see also*genetics

infants 268–9

addition and subtraction 259–63

individual differences 887–8

multiplication and division 265–8

neural correlates of number processing 492–3

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

informal learning activities

linear number board games 1143–8

numerical activities in the home 1137–40

sociocultural perspective 1136–7

interest in mathematics

relationship to achievement 934–5

international surveys of mathematical achievement 430–3

contextual factors 435–6

implications 439–40

information sources 441

key findings 433–5

outcomes, variation over time 438–9

policy lessons 436–7

Internet, maths resources 9

interventions 1086, 1100–1

influence of curriculum type 982–3

*see also*computer-assisted interventions

(p. 1164)
intraparietal sulcus (IPS) 269,

*507*, 811, 841–2deficits in

*664*and mathematical competence 618–19

non-numerical magnitude processing 589–90

notation effects 495

numerical symbol processing 540–3

role in finger representations 80

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

*see also*heuristics

intuitionism 20

intuitive rules 185–7

intuitive strategy use, influencing factors 192–4

isomorphism of measures relations 160

iterative view, conceptual and procedural knowledge 1124–6

J
Japanese-speaking children 425

K
key press estimation (KPE) task, studies in blind individuals 796–

*7*Knowledge Forum© 1062

L
language

and ATOM 560

bilingual mathematics learning 617

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

role in arithmetic 151–3

young English language learners 1083

*see also*cognitive linguistics

'last resort' hypothesis, animal numerical discrimination 218–19

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

linear number board games 1143–8

line lengths, neuronal representation 467–

*8*lions, numerical discrimination 214

logarithmic numerical coding 446

logicism 23–4

Logo programming language 749

low achievement (LA) 779–80

M
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

magic tricks, mathematics-based 12–13

magnetic resonance spectroscopy (MRS), study in developmental dyscalculia 738

magnitude

interference effects 555

semantic representation 120

*see also*ATOM (A Theory of Magnitude)

magnitude comparison tasks

multi-digit number processing

*117–18*relationship to mathematical ability 868–70

TMS studies 587–9

magnocellular deficit theory 697

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

international surveys 430–41

and number-line estimation 409

mathematical communication 6–8

ongoing activities 8–10

pitfalls 10–11

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

diagnosis 768

etiology 769–70

genetic studies 1005–8

growth curve modelling 985–6

importance of 767–8

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

Mathematics Millennium Project 9

Mathematics Recovery 1100–1

Maths Busking 9

mediation, Vygotsky's views 367–8

memory-based strategies, development of 521–

*2*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

representation of 2-digit numbers 48–9

representation of fractions 53

strengths and weaknesses 98–9

TMS studies 594–8

mental representations, definition 531–2

microworlds 1059–60

mild cognitive impairment, numerical distance effect 346

Model task, SFON assessment 279

monkeys

mathematical abilities 238–40, 251–2

counting principles and symbolic representation 240–2

mechanisms 248–50

neuroscience 250–1

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

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

maths drawings and MD&A methods 1042–8

place-value processing 123–9

practical and diagnostic implications 111

specific effects

*113–16*multi-digit numbers, commonalities and differences 110–11

multi-linear effects, number line estimation task

*116*multiplication

consistency effect

*114*infants' abilities 265–8

MD&A diagrams 1044

multi-digit number processing

*117–18*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 numbers 378, 1023

conceptual-role bootstrapping hypothesis 207

relationship to finite cardinal numbers 28–9

negative recency effect (gambler's fallacy) 189

neural networks

in development of number skills

*491*model of symbolic representation 534–

*6*studies of numerical discrimination 221

neuroscience

ATOM 449

computational modelling 450–1

developmental studies 446–8

future research areas 452–3

stimulation studies 449–50

New Math 1023–4

nominalism 18–20

non-numerical magnitude processing, TMS studies 589–91

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*, 333–4in ADHD 902

age effects 494

in diagnosis of acalculia

*826*influence of language 407–8

in mathematics anxiety 943

tDCS studies 622

*see also*distance effect

number concepts 377–8

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*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–*9number knowledge, relationship to mathematical ability 870–2

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*–70processing, 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*learning of number words 74

role in conceptual understanding 369–70

*see also*numeral format effects

number naming tasks, multi-digit number processing

*117–18*number operations, use by preschool children 1082–3

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

number-sensitive and number-selective coding 569–71

object location map 574–5

task-specific components 576–7

Number Sense Screener (NSS) 1084–

*5*Number Sets Test 772–3

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 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

*see also*symbolic representation

numerical adaptation paradigm 542–3

numerical coding, logarithmic scaling 446

numerical discrimination

innate nature of 379

numerical magnitude knowledge, relationship to socio-economic status 1141–3

numerical priming task 537

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,

O
*460–1*, 567behavioural significance

*463*–4coding of continuous and discrete quantities 467–

*8*implementation models 465–7

latencies 461–3

locations

*460–1*tracking of numbers of movements 461

occipital cortex 447

occipito-parieto-precentral network, role in finger representations 80

Oksapmin body part counting system 372–

*3*one to one correspondence principle 240

ontogeny of numerical abilities 228–31

operand errors 819

operation errors 819

oral verbal estimation task, studies in blind individuals 798–

*9*ordinality judgements 898

infants 264–5

monkeys 242–5

rank order-sensitive neurons 269

research areas 547

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

parallel individuation system 492

parents, provision of number-related activities 1137–8

parietal lobe 250–1, 269, 452,

*507*, 811, 841–2age-dependent activation shift 494

competency-dependent activation shifts 496

developmental studies 519

neural circuits 490

role in approximation tasks 152

TDCS studies 585

Pathways Model of development 872

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 deficit hypothesis 703

phonological difficulty, relationship to mathematical ability 979–80

phonological recoding in lexical access 706

Piaget, J. 368–9

Piagetian logical thinking, prediction of mathematical ability 916–17

pigeons, numerical discrimination 215–16

PISA (Programme for International Student Assessment)

*431–2*contextual factors 435–6

implications 439–40

information sources 441

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 representation 122–3

place-value understanding

people with reading difficulties 700

Plato, number theory 18

PLATO project, computer-assisted interventions 749

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 superior parietal sulcus (PSPS) 490

posteromedial cortex, repetition enhancement 514

prediction of ability 192–4, 925–7

grey matter density 619

language issues 918–19

logical thinking 916–17

motor, visual perceptual, and visuomotor skills 919–

*21*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

*Pre-K Mathematics*curriculum 1148–9

preschool children, early screening 1084–6

preschool experiences, early learning advantage 907–8

Pressey, S.L., teaching machines 748

pre-supplementary motor areas (pre-SMA), order-selective neurons 474

primary intuitions 190–1

primitive models of operations 959–61

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

procedural components of number processing 122

procedural development, in MLD and low achievement 773–5

procedural errors 821

'product of measures' relations 160

Program for International Student Assessment (PISA) 1024

programming languages 749

proportionality, overuse of 182

proportioning strategies 357–8

proportions, neuronal representation

*469*–70prosthetic dimensions, definition 562n2

psychometric intelligence, variability 732–3

public perceptions of mathematics 181

Q
quantitative numerical capacities 872

quantity discrimination, animals 217–19

quantity representation 809

R
random digit generation task 95–6

rankings, international 437–8

rank order-sensitive neurons 269

ratio-based reasoning, infants 267–8

ratio bias problem

*186–*7reading ability

interaction with numerical symbol processing 547

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

*see also*dyslexia

reading skills, role in word problem-solving 169–70

redeployment (recycling) hypothesis, symbolic number representation 471

referents, definition of 532

remedial education

effects of 621

repetition priming, functional imaging studies 513–14

repetitive pulse transcranial magnetic stimulation (rpTMS) 586

*see also*transcranial magnetic stimulation

(p. 1172)
retrieval strategies 773, 841

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

relative numerousness judgement

*244*Royal Institution, Mathematics Masterclasses 9

rule-encoding units 477

S
salience network 517–

*18*saliency maps 574–5

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

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-explanation 1127

semantic alignment effects 150–1

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

serial order effect

multi-digit number processing

*114*relationship to basic numerical representations

*119*sets, numbers as 22–4

setting of school pupils 1026

sex-limitation models 996–7

single neuron studies 446, 452, 567

of number-sensitive coding 570–1

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

*see also*transcranial magnetic stimulation

situation models, use in word problem-solving 163–4

Situation Strategy First model 166

size congruity effect (SiCE) 47–8,

*49*, 61,*116*, 122–3in 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

socio-economic status

and approximate arithmetic skills 318–19

and effectiveness of CAI 756

(p. 1173)
and numerical magnitude knowledge 1141–3

Sona designs

*15*spatial abilities

relationship to mathematical ability, genetic studies 1002

Spatial Numerical Association of Response Codes (SNARC) effect 37, 47,

*91*, 100, 121, 391, 410, 575, 1141age-related changes 346–

*7*automatic nature of 90–1

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

*see also*mental number line

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

as a mechanism of self-initiated practice in number skills 284–5

relationship to numerical skills 280–4

statistical heuristics 191

statistical learning, infants 266–8

statistical methods 867

stereotype threat 945–7

stimulation of the brain 449–50

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

derived fact strategies 883–5

EEG studies 616

in MLD and low achievement 774

models of 358

numeral format effects 146–8

proportioning versus counting 357–8

sequential difficulty effects 358

streaming of school pupils 1025–6

structural representation of number system 122–3

Successmaker ITS 752

successor function, children's understanding of 304

sum interference effect 150–1

superficial coping strategies 961–2

superior longitudinal fasciculus 737

superior temporal gyrus 545–6

supplementary eye field (SEF), order-selective neurons 474

supporting functions 496

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

semantic referents of numerals 533–9

single neuron studies 471–3

symmetry effects 183–4

syndrome research

T
fragile X syndrome 682–6

longitudinal studies 681

MLD phenotypes 679–80

models of individual difference 678–9

Turner syndrome 686–91

table and non-table errors 819

TEDI-MATH 931–2

temporal contiguity effects 184

temporal lobe

changes in dyscalculia 736

developmental studies 515–16

studies in Turner syndrome 690

temporal-parietal cortex, involvement in SFON 280

thalamus, changes in dyscalculia 740

thinking styles and dispositions, effects on intuitive strategy use 193–4

TIMSS (Trends in International Mathematics and Science Study)

*431–2*, 983–4, 1024contextual 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

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 errors 39, 818–19

long-term consequences 129–31

rehabilitation 827–8

relationship to language 125–6

transcranial magnetic stimulation (TMS) 449, 584, 585–6

future research areas 591–3

studies of counting tasks 82

studies of finger representations 591

studies of mental number line 594–8

transfer effects, learning studies 847–8

Transform-Sets task 302

triple-code model 38–9, 121,

*142*, 487, 545, 598–9, 709, 846, 919 (p. 1175)recent modifications and extensions 491–2

twin studies 998–1000

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

U
unary model of operations 960

unit-decade compatibility effect 49, 108–9,

V
*113*, 132n1relationship to basic numerical representations

*119*velocardiofacial syndrome 738

ventral visual stream

changes in dyscalculia 736

developmental studies 519–

*20*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 skills, associations and dissociations with number system

*822*verbal word frame 709

vision

importance in numerical representation 787–9

visualizers 852

visual system, numerical information extraction 221

visuospatial attention

associations and dissociations with number system

*822*visuospatial sketch pad (VSSP) 900, 980–1, 982

individual differences 901–2

role in mental arithmetic 940

vocabulary development, relationship to number-knower level

W
*298–*9Western Reserve Twin Project 997

white matter structure

changes in reading difficulties 710–11

whole number bias 52

(p. 1176)
word problem-solving 40–1, 158–9, 174–5, 874, 953–5, 969–70

Z
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

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