1.
Q1. Does learning symbolic arithmetic inhibit the innate non- symbolic approximate abilities? (Sarah) No! Quite on the contrary!The precision of numerical discrimination(JND or Weber fraction) increases withage. Round numbers accurately discriminated 2 1:2 1 0.8 2:3 0.6 0.4 3:4 0.2 4:5 0 10 20 30 40 50 5:6 Infants (Izard et al., 2009; Xu & Spelke, 2000; Xu & Arriaga, 2007) 0 1 2 3 4 5 6 7 10 Piazza et al., Cognition 2010; Chinello et al., submitted. Age in years Piazza et al., 2004 Pica et al., 2004 Halberda et al., 2008 Power function fit
2.
Does matheducation affect the ANS ? (disentangling maturation from education factors)The Munduruku is an indigenouspopulation of the Amazon (Brasil)- They have number words only up to 5.- They fail in tasks of exact calculation- They have a spontaneous capacity forapproximate estimation, comparison,addition- As a group, they have a normal non-verbalmagnitude system, even for very largequantities [Piazza, Pica, Dehaene, in preparation]
3.
36 Munduruku subjects Performance of Munduruku adults [aged from 4 to 67] Uneducated (n=7) Some education (n=13) 12 Completely uneducated % larger responses 100 10024 Received some education 80 80 w = 0.288 w = 0.177 60 60 “choose the larger” 40 40 * 20 20 0 0 0.7 1 1.4 0.7 1 1.4 Ratio of n1 and n2 (log scale) Weber fraction Munduruku, uneducated Weber fraction 0.5 Munduruku, some education 0.5 Italian participants (group means) r²=26.8%, p=0.001 0.4 0.4 Math education starts 0.3 0.3 0.2 0.2 0.1 0.1 0 0.0 0 10 20 30 40 50 60 70 0 1 2 3 4 5 6 7 8 Age Years of Education
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• Symbolic and non-symbolic competences go hand in hand during development and enhance one another in a form of circular or spiral causality • More exact Symbolic number • More linearIntra Parietal Sulcus codeneuronal populations Non- symbolic number • More approximate code • More compressed
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Q2. Can non-symbolic numerical abilities be trained ? Which kindsof games/manipulations can be used to enhance them? (Timothée)• We just completed a training study on kindergarteners of 4 to 6 years of age!• The games was a “matching card game”, whereby children were given a card andhad to match it with the card containing the same number of items among severaldistracting cards. It was a small group training – suited for real classroom!• Results: after a ½ hour group training every week for 4 weeks the acuity of theapprox number system is significantly higher then in a group trained on the samestimuli but on items’ shapes recognition memory.• Future project: investigate the impact on learning symbolic numbers (could notbe done because research in Italy and was only funded for one year…thus cannotdo a longitudinal follow-up).• Starting project ion training Mundurukus involving 4 groups:1. training with approximate quantities (no symbols)2. training with exact quantities (no symbols/one-to-one correspondence)3. training with exact verbal symbols (verbal sequence/verbal counting)4. training with exact visuo-spatial symbols (abacus-like/visual shape recognition)
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Q3. Does making kids aware of their existing abilities help themfeeling learning of symbolic arithmetic less complicated? Role ofmeta-cognition and self-esteem in learning (Muriel)• Having the pupils experiencing that they can COUNT ON THEIR INTUITIONSshould be extremely useful and important to boost their motivation and self-confidence.• In domains classically treated as being “hard” such that of mathematics, thereseems to be a strong (and largely unconscious) effect of STEREOTYPETHREAT. (Italian study of north vs. south stereotype in math abilities)• Math performance heavily influenced by gender stereotypes (i.e., you do noteven need to know it, your teacher “shows it” … )
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Q4. Questioning the current educational system: are we introducingsymbolization and symbolic calculation rules too early? When shouldwe start teaching symbolic maths? Shall we train the pre-existingapproximate abilities first, or shall we train approximate abilities andexact calculation at the same time? (Muriel, Marie, Théophile, Asma)According to the results of the present research, we should propose to:1. Make children aware that by relying on their intuition of magnitude they can get very accurate, even though sometimes only approximate, answers to symbolic number problems that may seem very complex (and, in passing, that there are NO gender difference in these basic abilities!).2. Train children in performing approximate calculation3. Train them to make calculation in the more intuitive way (e.g, subtractions starting from the large number and not from the units, using decomposition), and only MUCH LATER introducing calculation procedures such that of carrying.This WILL AVOID the presence of major but frequent calculation ERRORS (e.g., theresult of a subtraction is larger than the subtraend) due to bad understanding of thecalculation procedures, and withdrawing from math BECAUSE of no or littleunderstanding of calculation procedures.
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“Number neurons” in macaque [Nieder, Science 2002]
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Recordingneuronal firing example: Neurons in motor cortex coding the direction of the arm movement Raster-plot Each line corresponds to a trial Each train is an ACTION POTENTIAL (spike) The you calculate the mean across trials (spike rate), and compare spike rates of a given (set of) neuron(s) in different conditions . So you derive responce functions (“tuning curves”). A “tunig curve” for a given stimulus parameter (here movement direction) is a curve describing how the neuron(s) responds to different values of that parameter: Spikes/sec 0 4 3 2 1 8 7 6 5 4 directions
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Single neuronsrecording.inmonkeysperforming thenumerositycomparison task “Tuning curves” for numerosity
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Weber law fornumerositycoding at the levelof single neurontuning curves
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Multiple regions contain neurons coding for number. Which does what? Responce latency (ms) Number is initially extracted from parietal neurons and then the information is transmitted to prefrontal cortex neurons.
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Key function of PARIETAL CORTEX = DYNAMIC REMAPPING OF SPACEPairetal cortex CONTAINS MULTIPLE REPRESENTATIONS OF SPACEin different egocentric frames of reference FOR ACTION PREPARATIONSpatial location of stimuli areremapped from the coordinate of the RECEPTOR SURFACE (retina, coclea, ...) To the coordinates of the EFFECTOR (eyes, head, hands, ...) • Highly plastic (tool use changes the receptive field of MIP arm-centred neurons) • Perform operations equivalent to vector addition
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Putative homologies in parietal cortex maps of man and monkeysMacaque monkey VIP (visual-tactile-vestibular-mutlisensory head centered - NUMEROSITY) LIP (visual - saccades – eye centered) AIP (motor-tactile- grasping- hand centered)Human VIP (multisensory – face – NUMEROSITY?) LIP (saccades - eyes) AIP (grasping, hand)
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How to study the “neural code” in humans? Principles of fMRI Control condition (funtional magnetic resonance)-NUCLEAR MAGNETIC RESONANCE consists in the absorbtion by protons ofIdrogen of electromagnetic waves of given frequency (MegaHz ), in the presenceof a magnetic field. Protons’s spins are usually randomly distributed, while in theabsence of a strong magnetic field align to the directions generated by theelectromagnetic field.- If we give an electromagnetic impulse at an adequate frequency (dipendent uponthe magnetic field) spin change their rotatio axes. Then they go back to their initialstate. The retourn to the initial equilibrium generates the emission ofelectromagnetic waves measurable at distance, which constants of relaxations(T1, T2) are dependent upon the tissue in which the atom is embedded into.How to make the RMN signal sensitive to the CEREBRAL ACTIVITY? Activity condition- Deoxi-emoglobin is paramagnetic thus perturbs the RMN signal (effect on T2apparent, o T2*)- Brain activity generates:- Increased oxigen consumption and increased blood supply.- Oxi/deoxi emoglobine ratio increase- Magnetic susceptibility decreases- T2* parameter increases- RM signal increases↑Neural activity ↑ blood flux ↑ oxi-hemoglobin ↑ T2* ↑ BOLD signal
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Since BOLD (blood oxygen level dependent) signal is linked to changes in blood flow BOLD response is:1. SLOW compared to the neural response2. DELAYED compared to the neural response BOLD seconds stimulus
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This link is studied by Still quite *&^%$#@ clueless This link is studied by MR neurophysiology and is here! physics and approximatelyapproximately understood understood
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BUT….. LUCKILY …Simultaneous measures of electric NEURAL and fMRI BOLD signals demonstrate that the two ARE HIHGLY CORRELATED!!!!!!!Example: BOLD variation withstimolus intesity STRONG CORRELATION NETWEEN BOLD and elettrophysiological measures (1. average on action potentials over multiple neurons (MUA), and 2. Local field potential (LFP) on under threashold activity).
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Using “adaptation” we can increase spatial resolution sampled volume (voxel, typically 2X2X2 mm) tuning curves stimulus space CLASSIC SUBTRACTION METHOD ADAPTATION METHODstimulus S 1 stimulus S 2 S2 preceded by S 2 S2 preceded by S 1 total I(S1 ) = total I(S2 ) Different populations code for total I(S2, S2 ) < total I(S1,S 2) Measurable difference in activation, S1 and S2, but the total indicating that S1 and S2 are coded by activation is = for S1 and S2 different neural populations
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Using “adaptation” we can decipher neural coding schemes (“tuning curves”) Adattamento dell’attività neurale Firing rate 1 2 3 4 5 6 7 8 9… 0
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Using “adaptation” we can decipher neural coding schemes (“tuning curves”) Adattamento dell’attività neurale Firing rate 1 2 3 4 5 6 7 8 9… 0
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Using “adaptation” we can decipher neural coding schemes (“tuning curves”) Adattamento dell’attività neurale Firing rate 1 2 3 4 5 6 7 8 9… 0
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Using “adaptation” we can decipher neural coding schemes (“tuning curves”) Adattamento dell’attività neurale Firing rate 1 2 3 4 5 6 7 8 9… 0 1 2 3 4 5 6 7 8 9… Test numbers
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Using “adaptation” we can decipher neural coding schemes (“tuning curves”) Adattamento dell’attività neurale Firing rate 1 2 3 4 5 6 7 8 9… 0 1 2 3 4 5 6 7 8 9… Test numbers
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Multiple replications using the same paradigm (e.g., Cantlon et al., 2005) ADULTS 4 YEARS OLD KIDS Especially in the RIGHT HIPS!
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Risposta alla numerosità nel cervello di bebè già a 3 mesi !!! Tecnica dell’EEG A. Experimental design… … Possible test stimuli:
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Response to number change in 3 months old babies!! EEG (ERPs) RIGHTStesso numeroDiversa forma Diverso numero Stessa forma Stesso numero Stesso numero Diversa forma Stessa forma HEMISPHE
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• WHY IS THIS INTERESTING ? ? ? ? ? ???????????????????????????????
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• Hp: the non-verbal intuitions of NUMEROSITY GROUND our capacity to understand numbers and arithmetic (Butterworth, Dehaene, etc...) If we better understand the cognitive and neural basis underlying such start-up-tool we can better understand the development of numerical abilities and maybe help developing tools which improve teaching efficacy and therapeutic tools in cases of dysfunctioning systems (sia dello sviluppo che acquisite)
34.
• Criteria for a start-up function / brain region: (1)-> its integrity should be a necessary condition for normal development of symbolic number skills. (2)-> its computational constraints should predict the speed and ease of symbolic number acquisition. (3)-> some traces of its computational signatures may be present when humans process symbolic numbers.
35.
• If the ANS grounds the cultural acquisition of symbolic number skills it should guide and constrain it: (1)-> its integrity should be a necessary condition for normal development of symbolic number skills. (2)-> its computational constraints should predict the speed and ease of symbolic number acquisition. (3)-> some traces of its computational signatures may be present when humans process symbolic numbers.
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(1) Traces of the ANS in symbolic number processing - behavioural Numbers are treated as representing APPROXIMATE QUANTITIES since the initial stages of learing Gilmore et al., Nature 2007
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(1) Traces of the ANS in symbolic number processing - behaviouralSame Ratio-dependent responses in non-symbolic and symbolic number processing ADULTS “choose the larger” “choose the larger” * 12 * 16 Numbers are treated as 1 Symbolic comparison analogical APPROXIMATE QUANTITIES throughout the life- 0,95 span Accuracy 0,9 Non-symbolic comparison 0,85 0,8 0,75 11.1 2 1.3 3 1.6 Ratio (bigger/smaller set) [Chinello et al., under revision]
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AFFERMAZIONE:EVIDENCES (behavioral):1) “EFFETTO DISTANZA” CON NUMERISIMBOLICI Tempi di risposta 63 76 25 32Ai soggetti viene presentato un numero e vienechiesto di rispondere se sia più grande o più piccolodi un numero di riferimento (ad es. 65). Più piccolo Più grande Errori I tempi di risposta e gli errori sono modulati dalla distanza (numerica) tra i numeri e questo è indice che vi sono tracce di una rappresentazione ANALOGICA dei numeri Numeri presentati
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(1) ANS correlates with symbolicnumber processing throughout life-span Number kindergarteners (3 to 6 yoa, N= Finger gnosis Comparison Visuo-spatial memory 94) and of adults (N = 36) Grasping 23 5 “dorsal” tasks: • visuo-spatial memory (Corsi) • numerosity comparison • symbolic number comparison • finger gnosis • grasping 2 “ventral” tasks (Golara et al., 2007): • face recognition memory • object recognition memory Objects [Simon et al., Neuron 2002] Faces [Chinello et al., under revision]
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(1) ANS correlates with symbolicnumber processing throughout life-span …Correlates with math scores up to 10 years earlier ... at 8 yoa at 14 yoaw measured at 14 years of age … … …
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Symbolic number cognition is associated to parietal cortex Missing gray matter in premature children with dyscalculiaPARIETAL [Isaacs et al., Brain, 2001]DYSFUNCTIONSCAUSE ACALCULIA developmental acquired Classical lesion site for acalculia Abnormal gyrification and activation [Dehaene et al., TICS, 1997] in Turner’s syndrome with dyscalculia [Molko et al., Neuron, 2003]
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Parietal cortex in symbolic number cognition PARIETAL ACTIVATION IS SYSTEMATICALLY OBSERVED IN SYMBOLIC NUMBER PROCESSING x = - 48 L z = 44 z = 49 x = 39 R 50 %HIPS 22 %• All numerical tasks activate this region (e.g. addition, subtraction, comparison, approximation, digit detection…)• This region fulfils two criteria for a semantic-level representation:- Format invariance- Quantity-related Crucial parameter coded: numerical quantity [Dehaene, Piazza, Pinel, & Cohen, Cognitive Neuropsychology 2003]
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Example of parietal activation “specific” to numbers(Eger et al, Neuron 2003) Numbers-(letters&colors)• Subjects are asked to respondto a given infrequent stimulus(number « 5 », letter « B», color« red »)• Numbers, letter, and coloursare presented visually andauditory•Only non-target stimuli areanalysed
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(1) Convergence towards a quantity code in the IPS in adults FORMAT NUMBER Deviant format Deviant number Adaptation number Adaptation format dots digits 20 50 Dots same = different 17, 18, o 19 close < far Arabic digits different = same 47, 48, o 49 far > close 2 CRITERA DEFINITIONAL For a SEMANTIC representation: •INVARIANCE TO ENTRY FORMAT •SEMANTIC METRIC
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(1) Convergence towards a quantity code in the IPS in adults 1010 close close Left Parietal Peak Right Parietal PeakNumber adaptation protocol far 8 8 far(brain response to a change in number) DEVIANTS 66 HABITUATION 20 Activation (betas) Activation (betas) 4 4 2 18 19 2 19 0 or -2 0 50 -2 -4 -6 -4 DOTS DOTS ARABICARABIC -8 -6 (among(among (among (among dots) arabic) arabic) dots) DOTS DOTS ARABIC ARABIC Symbolic -8 (among (among (among (among number code samedifferent dots) arabic) arabic) dots) Non- symbolic number code[Piazza et al., Neuron 2007]
47.
• If the ANS grounds the cultural acquisition of symbolic number skills it should guide and constrain it: (1)-> its integrity should be a necessary condition for normal development of symbolic number skills. (2)-> its computational constraints should predict the speed and ease of symbolic number acquisition. (3)-> some traces of its computational signatures may be present when humans process symbolic numbers.
48.
(2) ANS maturation may account for lexical acquisition patternThe precision of numerical discrimination(JND or Weber fraction) increases withage. Round numbers accurately discriminated 2 1:2Estimated weber fraction 1 0.8 2:3 0.6 0.4 3:4 0.2 4:5 0 10 20 30 40 50 5:6 Age in years Infants (Izard et al., 2009; Xu & Spelke, 2000; Xu & Arriaga, 2007) 0 1 2 3 4 5 6 7 10 Piazza et al., Cognition 2010; Chinello et al., submitted. Age in years Piazza et al., 2004 Pica et al., 2004 Halberda et al., 2008 Power function fit
49.
(2) ANS maturation may account for lexical acquisition pattern In the NUMBER domain, lexical acquisition is a slow and serial process. Number wordsrefer to quantities Understand “one” Understand “two” Understand “three” Understand 2 years of age “four” Counting principles “discovered” 3 years of age 4 years of age
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Round numbers accurately1:2 discriminated Symbolic number Number words refer to acquisition quantities Understand “one” Understand “two” Understand “three” Understand 2 years of age “four”2:3 Counting principles “discovered” 3 years of age3:4 4 years of age4:55:6 0 1 2 3 4 5 6 7 10 Age in years OTS capacity (number of objects attended at a time) 4 The OTS reaches the adult 3 capacity by 12 months: 4 “attentional pointers” 2 already available. This does not account for the 1 lexical acquisition pattern! 0.5 1 adults Age in years
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• If the ANS grounds the cultural acquisition of symbolic number skills it should guide and constrain it: (1)-> its integrity should be a necessary condition for normal development of symbolic number skills. (2)-> its computational constraints should predict the speed and ease of symbolic number acquisition. (3)-> some traces of its computational signatures may be present when humans process symbolic numbers.
52.
(3)The necessity of ANS for numeracy development: dyscalculia 4 groups of subjects “choose the larger” (1) 8-11 years old dyscalculic (diagnosis: Italian standardized * test), no neurological problems (2) 8-11 years old matched for IQ and cronological age n1 n2 (3) 4-6 years old (4) AdultsRESULTS (non dyscalculic subjects) 4-6 years 8-11 years Adults 100 100 100 w=0.34 w=0.25 w=0.15 80 80 80% resp « n2 is larger » 60 60 60 40 40 40 20 20 20 0 0 0 0.7 1 1.4 0.7 1 1.4 0.7 1 1.4 n1/n2 (log scale) n1/n2 (log scale) n1/n2 (log scale) [Piazza et al., Cognition 2010]
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(3)The necessity of ANS for numeracy development: dyscalculia “choose the larger” Impairment in the ANS predicts 7 symbolic number impairement but not * performance in other domains (word adults 10 yo 6 4 reading) yo 5 Distribution Estimates N errors in number comparison 10 yo dyscalculics n1 n2 5 3,5 4 3 In dyscalculic children the ANS is 2,5 substantially impaired: 3 tasks 0,50 2 2 non-dyscalculics 0,45 1,5 dyscalculics 1 1Estimated weber fraction 0,40 R2 = 0,17 00,5 P=0.04 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0,35 0 Estimated w 0,30 0,1 0,3 0,5 0,7 0,25 Estimated w power function (R2 = 0.97) 0,20 0,15 0,10 0 5 10 15 20 25 30 Age (years) [Piazza et al., Cognition 2010]
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Correlations does not imply causation The “circular causality” issue• During development, attaching “meaning” to numerical symbols may entail: 1. Mapping numerical symbols onto pre-existing approximate quantity representations. 2. Refining the quantity representations• It is thus possible that the core quantity system is: –Not only fundational for the acquisition of numerical symbols and principles –But also deeply modified by the acquisition of numerical symbols and numerical principles.
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Development of ANS 2 The precision of numerical discrimination increases with age.Estimated weber fraction Power function: What is the role of maturation? Exponent = -0.43 R2=0.74 What is the role of education? 1 p=0.001 0.8 0.6 0.4 0.2 0 10 20 30 40 50 Age in years Infants (Izard et al., 2009; Xu & Spelke, 2000; Xu & Arriaga, 2007) Piazza et al., Cognition 2010; Chinello et al., submitted. Piazza et al., 2004 Pica et al., 2004 Halberda et al., 2008 Power function fit [Piazza & Izard, The Neuroscientist , 2009]
56.
Does matheducation affect the ANS ? (disentangling maturation from education factors)The Munduruku is an indigenouspopulation of the Amazon (Brasil)- They have number words only up to 5.- They fail in tasks of exact calculation- They have a spontaneous capacity forapproximate estimation, comparison,addition- As a group, they have a normal non-verbalmagnitude system, even for very largequantities [Piazza, Pica, Dehaene, in preparation]
57.
36 Munduruku subjects Performance of Munduruku adults [aged from 4 to 67] Uneducated (n=7) Some education (n=13) 12 Completely uneducated % larger responses 100 10024 Received some education 80 80 w = 0.288 w = 0.177 60 60 “choose the larger” 40 40 * 20 20 0 0 0.7 1 1.4 0.7 1 1.4 Ratio of n1 and n2 (log scale) Weber fraction Munduruku, uneducated Weber fraction 0.5 Munduruku, some education 0.5 Italian participants (group means) r²=26.8%, p=0.001 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0.0 0 10 20 30 40 50 60 70 0 1 2 3 4 5 6 7 8 Age Years of Education
58.
Conclusions-There is some good evidence for a fundational role of theparietal system for approximate numerosity in symbolicnumerical representations.But there is a lot to be discovered:1) A true causal role of the ANS in dyscalculia awaits confirmation (longitudinal studies)2) What are the neural mechanisms that drive the refinement of the quantity code for symbolic stimuli? Are they necessarily mediated by language?3) Which aspects of maths education enhance approximate number prepresentation acuity?
59.
THREE PARIETAL CIRCUITS FOR NUMBER PROCESSING Left hemisphere Axial slice Right hemisphereA. x = - 48 z = 44 z = 49 x = 39 50 % HORIZONTAL SEGMENT OF THE INTRAPARIETAL SULCUS (hips) 22 %B. x = - 49 z = 30 x = 54 LEFT ANGUALR GYRUSC. x = - 26 z = 61 x = 12 POSTERIOR SUPERIOR PARIETAL LOBE (more right)
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Three parietal circuits for number processing(Dehaene, Piazza et al.,2003) CS Left hemisphere Right hemisphere IPS HORIZONTAL SEGMENT OF THE INTRAPARIETAL SULCUS (HIPS) hVIP? -Number comparison -Ratio effect Seen from top -Numerical priming -Approximate calculation LEFT ANGUALR GYRUS (l AG) -Retrieval of arithmetical facts (multiplications, additions) -Simple exact calculation POSTERIOR SUPERIOR PARIETAL LOBE (more right) (PSPL) vLIP? -Subtractions -Complex additions -Approximate calculation
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Evidence for a verbal code in arithmetical facts retrieval• Interference on TRs in calculation Task1 (arithmetic): Multiplicazions or subtractions Task 2 (short term memory): Phonological (whisper a non- word) o visuo-spatial (remember the position of an object) Single task Phonological dual task Visuo-spatial dual task
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Left angular gyrus in arithmetical facts retrieval1. Training experiment: Trained to memorize complex two digits number arithmeticalfacts and measure the effects on brain activity UNTRAINED > TRAINED TRAINED > UNTRAINED Ischebeck et al., 2009
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Left angular gyrus in arithmetical facts retrieval2. Arithmetical tasks performed in the scanner and activation correlated withsubsequent subjects’ report on the strategy used (fact retreival or computation) [Grabner et al., 2009 ]
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Evidence for a spatial code in arithmetical computations• Interference on TRs in calculation Task1 (arithmetic): Multiplicazions or subtractions Task 2 (short term memory): Phonological (whisper a non- word) o visuo-spatial (remember the position of an object) Single task Phonological dual task Visuo-spatial dual task
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Evidence for a spatial code in arithmetical computationsDo spatial/motor processes interfere with calculation ? « Answer the arithmetical problems while performing a sequence of finger movements in the same time ! »
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NO MVTS MVTS * 1200 * 1100 1000 RT (msec) 900 800 700 600 MULTIPLICATION ADDITION SOUSTRACTIONIn the dual task, sequential finger movements were found to slow down responsesto additions and subtractions, whereas multiplications (matched for difficulty)were unaffected
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Evidence for a spatial code in arithmetical computations: neglect Modello Copia del pazienteRegioni corticali tipicamentedannegggiate nel neglect Tipico disegno (copia da modello) di un paziente con negelct Test della bisezione di lineeTest dello sbarramento di linee
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Evidence for a spatial code in arithmetical computations: neglectNumerical bisection test :“What is the number between 2 and 6?”“Answer: 5” RIGHT BIAS! Zorzi et al., Nature 2002
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Posterior parietalsaccade regions in calculation
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Spatial code in numberrepresentations: the mental number line (SNARC effect) is culture dependent Shaki et al., 2009 (Psych Bull Rev)
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