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Number processing and calculation:the cognitive neuroscience of number sense MANUELA PIAZZA
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Introducion:The hypothesis of cultural “recycling” of pre-existing neural circuits Or: cultural traditions are such becuase they fund an adequate “neuronal nich” in our brains [ S. Dehaene and L. Cohen. Neuron 2007]
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« Exaptation » – « cooption »- « preadaptation » Terms used in the theory of evolution (Darwin, S.J. Gould, …) to indicate the shifts in the function of a trait during evolution. A trait can evolve because it served one particular function, but subsequently it may come to serve another. Classic examples: – feathers, initially evolved for heat regulation, were co-opted for use in bird flight – Social behavioural: subdominant wolves licking the mouths of alpha wolves (or dogs to humans), as deriving from wolf pups licking the faces of adults to encourage them to regurgitate food
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« Exaptation » – « cooption »- « preadaptation » We can think of cultural learning, at least in some domains (e.g., reading, arithmetic, ) as a form of exaptation. It is based on the re-use (or re-cycle) of neural systems selected by evolution for performing a given evolutionary-relevant functions.
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Some basic facts Natural evolution does not seem to have had the time sufficient to select brain architectures specifically to support recent cultural abilities such as reading or arithmetic. Writing -- invented around 5400 years ago by the Babylonians. Positional numeration -- in India around the 6th century A.D. For both reading and arithmetic there is high cross- individuals and cross-cultural consistency in the brain circuits involved. This clearly speaks against the idea that the human brain is a TABULA RASA, an equipotential learning device, which architecture is irrelevant when it comes to learning, and suggests that there is something in the architecture of our brains that make particular regions apt as being reconverted to novel cultural-based functions.
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Arithmetic Bilateral regions around the mid intraparietal sulcus rispond consistently across subjects and across cultures to numbers, and they are crucial for calculation. This region is embedded in a mosaic of regions specialized in coding quantitative aspects of the self and the environment for action planning Hand-centered [Simon et al., Neuron 2002]AIP LIP VIP Head-centered Eye-centered Their homologous in macaque monkeys are parietal regions implicated in space and quantity coding and in complex vector additions to transform sensory coorinates into motor- coordinates ...
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The crucial role of parietal cortex in calculation: evidences(1) A crucial site for Reduced gray matter in premature children with dyscalculiaACALCULIA (Isaacs et al., Brain, 2001) developmental acquired Classical lesion site for acalculia Reduced gray matter and abnormal activation (Dehaene et al., TICS, 1997) in Turner’s syndrome (Molko et al., Neuron, 2003)
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(2) A site systematically active ACTIVE during symbolic number processing and calculation 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:- It responds to number in various formats (Arabic digits, written or spoken words), more thanto other categories of objects (e.g. letters, colors, animals…)- Its activation varies according to a semantic metric (numerical distance, number size) Dehaene, S., Piazza, M., Pinel, P., & Cohen, L. (2003). Cognitive Neuropsychology
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A supramodal number representation in human intraparietal cortex (Eger et al, Neuron 2003)• Subjects are asked to detectinfrequent targets (one digit, oneletter, one color)• Digit, letter and color stimuli arepresented in the visual or theauditory modality• Only non-targets are analyzed
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Numbers: a « special » semantic category Semantically defined along one main dimension: QUANTITY Dissociable from other categories – Double dissociation (for ex. in degenerative disorders: Butterworth et al., Nature Neuroscience 2001, Delazer et al. Neuropsychologia, 2006) With reproducible neural substrate: parietal cortex Based on an ancestral « sense » of numerosity – Several animal species (for ex. Jordan et al., Current Biology 2005) – Babies (for ex. Xu & Spelke, Cognition 2000) – Populations without words for numbers (for ex. Pica et al., Science 2004)
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NUMEROSITY : the number of objects in a set • A property that characterizes any set of individual items • Abstract as independent from the nature of the items and invariant from the substitution of one or several items• Not dependent upon language as extracted by primates and many other animal species as well as human babies in an approximate fashion (strong adaptive value: social behavior, feeding, reproductive strategies, … )
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Number is spontaneously attended by untrained monekys Macaque monkeys spontaneously match number across sensory modalities (preferential looking paradigm) Jordan, Brannon, Logothetis and Ghazanfar (2005) Current Biology
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Number is spontaneously extracted in newborns (cross-modal matching)48 NewbornsAge = 49 hours [7-100 h] 12 4 [Izard et al., PNAS 2009]
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Number is spontaneously mentally combined in arithmetical operations [26 babies. Average age = 9 months] see video 5 objects enter And they are covered by a screen 5 new objects enter 10 objects enter And they are covered by a screen 5 objects exit Wrong result Tempo di fissazione (secondi) Wrong result Correct result Correct result The screen opens up and uncovers, …[McCrink & Wynn., Psych Sci 2004]
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Demonstration Two sets of different numberWhich set contains more dots?
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5 10 10 11 12 24 22 24Ratio (S/L) = 0.5 Ratio = ~0.9Less errors and faster reaction More errors and slower reactiontimes times
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Weber lawA psychophysical law describing the relationship between the physical and theperceived magnitude of a stimulus.It states that the threshold of discrimination (also referred to as ‘smallestnoticeable difference’) between two stimuli increases linearly with stimulusintensity.Weber’s law can be accounted for by postulating a logarithmic relation between thephysical stimulus and its internal representation. Weight Loudness Brightness Numerosity
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Weber law in 100 80numerosity judgements 60 40 20 Ref = 16 Ref = 32 03 exemplars of a given number (16 or 32; « ref ») 8 16 32 64 Test numerosity (linear scale) 100 80 60 Followed by a single test number 40 (8-32 and 16-64; « test ») 20 Ref = 16 Ref = 32 0 8 16 32 64 Test numerosity (log scale) 100 80 On a log scale the two curves 60 have the same width !!! This 40 indicates that numerosity is 20 mentally represented on a 0 compressed scale 0.5 1 2 Deviation ratio (log scale)
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The Approximate Number Sense(ANS) is universal: across species Rats The number of presses produced as a function of the number of presses requested [Mechner, 1958] Humans Errors in a dots comparison task as a function of the different reference numbers [Van Oeffelen and Vos, 1982]
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The ANS is universal: across culturesThe Munduruku (indigenous tribe inthe Amazon - Brasil) have numberwords only up to 4.-They have a perfectly normal non-verbal magnitude system, even forvery large quantities-They have a spontaneous capacity forestimation, comparison, addition-They fail in tasks of exact calculation [Pica, Lemer, Izard, & Dehaene, Science, 2004]
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Approximation addition and comparison French controls adults M,NI B,NI B,I + childrenn1 n2 All Munduruku M,NI M,I B,I n3 Ratio of n1+n2 and n3 (L/S) [Pica, Lemer, Izard, & Dehaene, Science, 2004]
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Internal representation of numerosity: a model 1 2 3 4 5 6 7 8 9… NumerosityActivation w 0 Log scale w (Internal Weber fraction) = sd of the gaussian distribution of the internal representation of numerical quantity (on a log scale!). The larger w the poorer the discriminability between two close numbers. w is a measure of the precision of the internal representation of numerosity
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ANS undergoes maturationHuman newborns Human adults [Izard et al., PNAS [Piazza et al., Neuron 2004] 2009] A sample number (16 ) A test number (8,10,13,16,20,24,32) Same or different numerosity? 100 80 60 Weber fraction Weber fraction (∆x/x) = 2 40 = 0.15 20 0 0.5 1 2 8 10 13 16 20 24 32 Test number
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The ANS acuity developmental trajectory The precision of numerical discrimination(JND or Weber fraction) increases with age. 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
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Conclusion:• A system for extracting the approximate number (ANS) – present universally in the animal world – active early during development in humans – represents number independently from the stimulus mode (simultaneous or sequential) – represents number independently from the stimulus modality (visual, auditory, motor, ...) – is used to perform approximate arithmetical operations (comparison, additions, subtractions, ...) WHAT IS ITS NEURAL BASIS, AND WHAT (IF ANY) IS IT’S ROLE IN NUMERACY ACQUISITION?
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“Number neurons” in macaque [Nieder, Science 2002]
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Single neuronsrecording inmonkeysperforming thenumerositycomparison task “Tuning curves” for numerosity
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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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Two pathways in vision : dorsal pathway / ventral pathway [Mishkin & Ungerleider, 1982; Milner and Goodale] « WHERE?» It transforms information into spatial coordinates useful for programming movement « WHAT?» It transforms the information in rich representations of objects shapes useful for recognition
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The most important function of parietal cortex is the DYNAMICAL REMAPPING OF THE MULTISENSORY SPACE Parietal cortex contains MULTIPLE REPRESENTATIONS OF SPACEEACH WITH DIFFEREENT REFERENCE FRAMES, which are necessary to PREPARE ACTION. Object’s position is remapped from the receptor co-ordinates (retina, coclea, ) into the effector co-ordinates (eyes, mouth, hands, feet). Macaque’s brain • It is highly plastic (receptive fields in AIP centred on the hand are modified after tool use to integrate the tool space) • It performs operation that are equivalent to vector combination
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Putative homologies in the parietal lobe NUMBER NEURONSMonkey brain a AIP LIP VIP Subtraction taskHuman brain b Ocular saccade Grasping task Simon, Mangin, Cohen, LeBihan, and Dehaene (2002) Neuron Hubbard, Piazza, Pinel, Dehaene (2005) Nature Reviews Neuroscience
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Is there a response to 16 Deviant approximate number in 32 16 16 human IPS? 16 16 Habituation to a fixed quantity (e.g. 16 dots) Rare deviant stimuli (10%) 8 (far) 10 (medium) 13 (close) 16 (same) 20 (close) 24 (medium) 32 (far) Number only Numberand shape Piazza, Izard, Pinel, Le Bihan & Dehaene, Neuron 2004
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Response to numerosity change in the bilateral intraparietal sulcus Regions that respond to a change in SHAPE 0.5 Parietal activation 0.4 Regions 0.3responding to a 0.2 change in 0.1 number 0 -0.1 -0.2 Same shape -0.3 Shape change L R -0.4 0.5 1 2 Log ratio of deviant and habituation numbers
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Weber’s law in Left intraparietal cortex F(1,11)= 14.4, p<0.001 Right intraparietal cortex F(1,11)= 17.2, p<0.001 the intraparietal 0.4 0.4 0.2 0.2 sulcus 0 0 -0.2 Nhabit 16 -0.2 Nhabit 16 Nhabit 32 Nhabit 32 z = 42 -0.4 -0.4 8 16 32 64 8 16 32 64 Deviant numerosity (linear scale) Deviant numerosity (linear scale) 0.4 0.4 0.2 0.2 0 0 -0.2 Nhabit 16 -0.2 Nhabit 16 Nhabit 32 Nhabit 32 -0.4 -0.4 8 16 32 64 8 16 32 64 Deviant numerosity (log scale) Deviant numerosity (log scale) 0.4 w = 0.183 0.4 w = 0.252 0.2 0.2 0 0 L R -0.2 -0.2 First replication by Cantlon et al (2005).(Number change > Shape change). Since then, -0.4 0.5 1 2 -0.4 0.5 1 2 MANY replications (e.g., Hyde 2010, etc …) Deviation ratio (log scale) Deviation ratio (log scale)
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Weber’s law in numerical behavior 100 100 80 80 60 60Three samples of a given numerosity (16 or 32) 40 40 20 Nhabit 16 20 Nhabit 16 Nhabit 32 Nhabit 32 0 0 8 16 32 64 10 16 32 48 Deviant numerosity (linear scale) Deviant numerosity (linear scale) Followed by a single deviant: 100 100 80 80 60 60 40 40 20 Nhabit 16 20 Nhabit 16 Nhabit 32 Nhabit 32 0 0 8 16 32 64 10 16 32 48 Deviant numerosity (log scale) Deviant numerosity (log scale) Same or different Larger or smaller numerosity? numerosity? w = 0.170 w = 0.174 100 100 80 80 60 60 40 40 20 20 0 0 0.5 1 2 0.7 1 1.4 Deviation ratio (log scale) Deviation ratio (log scale)
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Numerosity coding in 3 months old baby brains. EEG A. Experimental design… … Possible test stimuli:
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Risposta alla numerosità nel cervello di bebè già a 3 mesi !!! Tecnica dell’EEGStesso numero Diverso numero Stesso numero Stesso numero Emisfero DeDiversa forma Stessa forma Diversa forma Stessa forma
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• NICE ... SO WHAT ? IS THAT ANY INFORMATIVE FOR EDUCATION ?• WHAT IS THE ROLE OF THE PARIETAL APPROXIMATE NUMBER SYSTEM IN NUMERACY ACQUISITION ?• Hp: the non-verbal SENSE of NUMERICAL QUANTITY (ANS) GROUNDS our capacity to understand numbers and arithmetic. it is a domain specific “START-UP TOOL”
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• Criteria for a start-up function / “precursor map” (see prediction from the neuronal recycling hypothesis): (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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• 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 during the initial stages of learing Gilmore et al., Nature 2007
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School maths’ achievement correlates with accuracy in symbolic approximatecalculation tasks Approximate calculation tasks Number line = approx location of a number on a line Measurement = apprix length of a line in inches (“this is a line 1 inch long. Draw a 3,6,8,9 inches line”) Numerosity = approx number of candies in a jar Computational = approx additions ( “Is 34 + 29 closest to 40, 50, or 60?”) [Booth & Siegler, 2006]
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(1) Traces of the ANS in symbolic number processing - behavioural Same 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 0,95 THE ENTIRE LIFE-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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(1) ANS correlates with symbolic number processing throughout life-span …Correlates with math scores up to 10 years earlier ... at 8 yoa at 14 yoa w measured at 14 years of age … … …[Halberda et al., Nature 2009]
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(1) ANS acuity higher in adult mathematics vs. psychology university students “choose the larger” * [Ranzini and Girelli, under revision]
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(1) ANS in kindergarteners predicts performance in calculation in 1 grade (longitudinal) TEMA: counting, reading/writing 2 digits number, additions and divisions withconcrete sets, symbolic number comparison, 1 digit additions and multiplications [Mazzocco et al., PlOsONE, 2011]
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(1) Traces of the ANS in symbolic number processing - neural 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 [Piazza et al., Neuron 2007] 1010 close close Left Parietal Peak Right Parietal PeakNumber adaptation protocol 8 8 far far(brain response to a change in number) DEVIANTS 66 20 Activation (betas) HABITUATION 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
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(1) Convergence towards a quantity code in the IPS in adults [Eger et al., Curr Biol., 2009] MVPA trained on digits accurately predicts dots but not the reverse Symbolic number code Non- symbolic number code
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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.
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(2) ANS maturation may account for number 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
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(2) ANS maturation may account for number lexical acquisition patternIn the NUMBER domain, lexical acquisition before the discovery of thecounting principles is a slow and strictly 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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• 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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(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 1Estimated weber fraction 1 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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(3) Impaired ANS in dyscalculia (replications …) Dyscalculics Low maths Typical maths High mathsMath: Test of Early Mathematics Ability (TEMA), andthe Woodcock-Johnson Calculation subtest (WJR- [Mazzocco et al., Child Development, 2011] Calc) [Mussolin et al., Cognition 2010]
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(3) ANS parietal system is ipoactive in dyscalculia [Price et al., Current Biology, 2007]
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Correlations do 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 modified in turn 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 What is the role of education? 1 R2=0.74 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]
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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]
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36 Munduruku subjects Performance of Munduruku adults [aged from 4 to 67] Uneducated (n=7) Some education (n=13) % larger responses 12 Completely uneducated 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
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We need to re-think learningas a deeply iterative process … Pre-existing abilities New cultural abilities (e.g., the ANS) (e.g., calculation skills) Other cognitive domains where we observe a spiral causality link between basic perception and cultural acquisitions :(1) Phonological abilities, visual acuity reading skills [Bradley, Morais, Dehaene, …] (2) Colour perception colour naming [Regier, Kay, ...]
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ConclusionsThe evolutionary ancient parietal system for approximate number groundsthe human cultural acquisition of numbers and calculation, and there is along lasting cross-talk between innate approximate number sense andacquired symbolic arithmetical abilities.From approximate non-symbolic quantity to exact number: a MAJOURCONCEPTUAL STEP. The acquisition of symbols and their connection to the representation of thecorresponding quantities deeply modify the mental representation of quantity : - It becomes PRECISE even for large numbers (analogic digital) -The internal scale becomes LINEAR (logarithmic linear) - How does the brain support these modifications?
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Hypotheses• 1. Connexion between quantity representations and numerical symbols (visual and verbal digitalisation) and creation of a verbal network of arithmetical facts ( verbal arithmetical facts)• 2. Connection between quantity representations and spatial representations ( linearisation number line)
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The brain architecture for mental calculation Before children learn to perform calculation, the major systems for - numerical quantity representation (in parietal areas), - visuospatial attention (in posterior parietal areas), - visual object processing (in occipito-temporal areas), - speech processing (in left peri-sylvian and temporal areas), seem to be already in place. In order to calculate, interfaces must be created between number-sense, language, and space processing Pronunciation Representation ofand articulation numerical quantities « # # » « two », Spatial operations ordering « arithmetical / zooming / remapping facts » 123 Visual object processing number form « 2 »
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Three parietal circuits for number processing: meta- analysis CS Left hemisphere Right hemisphere IPS HORIZONTAL SEGMENT OF THE Seen from top INTRAPARIETAL SULCUS (HIPS) hVIP? -Number comparison -Ratio effect -Approximate calculation LEFT ANGUALR GYRUS (Left AG) -Retrieval of arithmetical facts (multiplications, additions) POSTERIOR SUPERIOR PARIETAL LOBE (PSPL) vLIP? -Subtractions -Complex additions -Approximate calculation [Dehaene, Piazza et al.,2003]
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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 retrieval2. Arithmetical tasks performed in the scanner and activation correlated with subsequent 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 calculation• 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 inarithmetical computations: neglect Modello Copia del pazienteRegions typically damaged Typical drawing Line mark test Line bisection test
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Evidence for a spatial code in arithmetical computations: neglect Numerical bisection test :“What is the number between 2 and 6?” “Answer: 5” RIGHT BIAS! Zorzi et al., Nature 2002
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12 subjects in a dark room produced 40 numbers in an order “as random aspossible”. Eye movements analyzed inthe window in the 500ms PRECEEDING number production
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Spatial code in numberrepresentations: the mental number line (SNARC effect) Shaki et al., 2009 (Psych Bull Rev)
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Number - space associations0 "Position number 64" 100 Kindergarten 6 years old 7 years old [Siegler & Booth, 2004]
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Psychological Science, 2008 Kindergarteners Across subjects, and in both populations, deviation from linearity correlates with number of errors in solving simple additions
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Number to space associations in dyscalculia[Geary et al., 2008]
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Developmental dyscalculia• Called “Mathematics disorder” (DSM-IV Diagnostic and Statistical Manual of Mental Disorders ) « impairment in numerical and arithmetical competences in children with a normal intelligence without acquired neurological deficits»• Criteria: – Numeracy < expected level accoring to age, intelligence, and scolarity – Interferes significantly with everyday life of school achievement – Not linked to a sensory deficit
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Early observed difficulties– Problems in acquiring counting principles– Problems in understanding and using strategies for solving simple arithmetical problems (es. in additions –counting on from the largest number ....– Problems in memorizing arithmetical facts (tables)– Continuous use of “immature” strategies (finger counting…)
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Observed difficulties: wrong strategies?• Geary e Brown, 1991: Dyscalculic kids of 6-7 years, in simple calculation (e.g., 3+2) use more immature strategies such as verbal or finger counting and much less then facts retreival% trials Finger counting Verbal counting Long term memory retrieval Norm = non dyscaclulics DC = dyscalculics
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Observed difficulties: wrong strategies?• Those strategies (verbal and finger counting) have a LARGE COST, because they are at the origin of many errors% errors Finger counting Verbal counting Long term memory retrieval Norm = non dyscaclulics DC = dyscalculics
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Observed difficulties• In reading numbers (epsecially multidigits) linked to difficulties in understanding the positional system• In number decomposition (e.g. recognizing that 10 is the result from 4 + 6)• In learning and understanding procedures in complex calculation• Anxiety or negative attitude in maths
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Consequences in adults• Infuences professional choices (lower salaries)• Difficulties in managing money• Difficulties in understanding stats, proportions, probabilities,nel comprendere la statistica, le proporzioni (impact on decision making)• Low self-esteem, anxiety, refuse socialization, … “I have always had difficulty with simple addition and subtraction since young, always still have to ‘count on my fingers quickly’ e.g. 5+7 withoutanyone knowing. Sometimes I feel very embarrassed! Especially under pressure I just panic.”
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Prevalence & co-morbidity Lewis et al.(1994): 1056 kids UK 9-10 years old PREVALENCE: 3.6% (of which 64% Dyslexia) (3.9% Pure dyslexia) Barbaresi (2005): 5718 kids USA 6 -19 years old PREVALENCE 5.9 % (of which 43% Dyslexia) Ratio male - female 2:1 Gross-Tsur, Manor & Shalev (1996): 3029 kids Israel 10 years old PREVALENCE: 6.5 % (of which 17% Dyslexia and 26% ADHD) Ratio male - female 1:1.1
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Calculation: relation between number sense, spatial abilities, language - Les sujets avec dyscalculie ont des difficultés dans la représentation des quantités, mais souvent aussi des déficits spatiaux et/ou de mémoire phonologique. Notre hypothèse est que selon le system cérébral atteint, nous pouvons nous attendre a différent sous-types de dyscalculie:“Déficit au système “Déficit aux systèmes de support” “Syndrome pariétale des quantités” 1. - dyscalculie spatiale générale” (associé à la dyspraxie?) 2. - dyscalculie phonologique (associé à la dyslexie?)
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Dyscalculia “core deficit” HP: problems in perception of numerical quantity, problems in associating numerical symbols to quantity, and in mental calculation. ipoactivation/malformation at the level of hIPS Pronunciation Representation of numerical and articulation quantities « # # » « two »,« arithmetical facts » XX Spatial operations ordering / zooming / X X X remapping Visual object processing number form « 2 »
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“Verbal” dyscalculia HP: problems in storing arithmetical facts (multiplications…), and in mastering counting sequence. Ipoactivations/malformations at the level of leftAG (hp: co-morbidity with dyslexia?) Pronunciation Representation of numerical and articulation quantities « # # » « two »,« arithmetical facts » Spatial operations ordering / zooming / X X remapping X XX Visual object processing number form « 2 »
89.
“Spatial” dyscalculia HP: problems in counting, in tasks requiring the use of number line, in written calculation. Ipoactivation/malformations at the level of the PSPL (hp: co-morbidity with spatial-dysorders, dyspraxia?) Pronunciation Representation of numerical and articulation quantities « # # » « two »,« arithmetical facts » Spatial operations XX ordering / zooming / X remapping X Visual object processing number form « 2 »
90.
How to diagnose? How to “rehabilitate”?1) Have a good model2) Develop fine diagnostic tests3) Experiment different treatments (rehab within the number domain but also the associated deficitary domains ...“core deficit” body schema, finger, quantities;“language” language/reading;“spatial deficit” visuo-spatial abilities). Is there transfer of training?
91.
Some ideas to offer educators – who should first test their efficacy in a controlled way• PRESCHOOL – Play with numerical and non-numerical quantities and operations with concrete sets since very early, and initially without using number words. – Offer as many occasions of « focusing on number » as possible. Respect the developmental trajectory of the ANS (there is no point in trying to teach the menaing of 4 at 2 years of age, unless the kid is ready to « see » what you mean) – Teach verbal symbols for numbers not by counting only but instantiate it may different concrete ways (« give me a number », + 1 games) and use multiple sensory modalities.• PRIMARY SCHOOL – Introduce first mental calculation and only much later on written procedures. – Teach calculation by decomposition as soon as possible. – Engage children in calculation problems as often as possible in any possible occasion, not only during math classes (engage them in organizing things for the school including estimation of time, material, space, using numbers) – Keep training approximate calculation even after having introduced exact calculation. – Play with estimation as frequently as possible (number of candies in a jar, lenghts, weight, time estimation and comparison) – For written calculation strategy keep consistent with number sense. The big numbers first, in both addition and subtraction + ask to estimate the result of any proposed calculation before enganging in the exact calculation procedure.
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