Number processing and calculation: the cognitive neuroscience of number

sense

MANUELA PIAZZA

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]

« 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

« 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.

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.

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

[Simon et al., Neuron 2002]

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 ...

AIP LIP VIP Eye-centered

Head-centered Hand-centered

The crucial role of parietal cortex in calculation: evidences

(1) A crucial site for ACALCULIA

Reduced gray matter and abnormal activation in Turner’s syndrome (Molko et al., Neuron, 2003)

Classical lesion site for acalculia

(Dehaene et al., TICS, 1997)

acquired

Reduced gray matter in premature children with dyscalculia

(Isaacs et al., Brain, 2001)

developmental

z = 44 x = 39 x = - 48 50 %

22 %

z = 49

HIPS

(2) A site systematically active ACTIVE during symbolic number processing and calculation

L R

Dehaene, S., Piazza, M., Pinel, P., & Cohen, L. (2003). Cognitive Neuropsychology

• 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 than to other categories of objects (e.g. letters, colors, animals…) - Its activation varies according to a semantic metric (numerical distance, number size)

A supramodal number representation in human intraparietal cortex (Eger et al, Neuron 2003)

• Subjects are asked to detect infrequent targets (one digit, one letter, one color) • Digit, letter and color stimuli are presented in the visual or the auditory modality • Only non-targets are analyzed

Numbers: a « special » semantic category

Dissociable from other categories – Double dissociation (for ex. in degenerative disorders: Butterworth et al.,

Nature Neuroscience 2001, Delazer et al. Neuropsychologia, 2006)

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)

With reproducible neural substrate: parietal cortex

Semantically defined along one main dimension: QUANTITY

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, … )

Number is spontaneously attended by untrained monekys

Jordan, Brannon, Logothetis and Ghazanfar (2005) Current Biology

Macaque monkeys spontaneously match number across sensory modalities

(preferential looking paradigm)

[Izard et al., PNAS 2009]

12

4

Number is spontaneously extracted in newborns (cross-modal matching)

48 Newborns Age = 49 hours [7-100 h]

5 objects enter And they are covered by a screen 5 new objects enter

The screen opens up and uncovers, …

10 objects enter 5 objects exit

Number is spontaneously mentally combined in arithmetical operations [26 babies. Average age = 9 months] see video

[McCrink & Wynn., Psych Sci 2004]

Wrong result

Correct result Correct result

Wrong result

And they are covered by a screen

Tem

po d

i fis

sazi

one

(sec

ondi

)

Demonstration

Which set contains more dots?

Two sets of different number

Ratio (S/L) = 0.5 Less errors and faster reaction times

Ratio = ~0.9 More errors and slower reaction times

12 24

5 10

22 24

10 11

Weber law

Weight Loudness Brightness Numerosity

A psychophysical law describing the relationship between the physical and the perceived magnitude of a stimulus. It states that the threshold of discrimination (also referred to as ‘smallest noticeable difference’) between two stimuli increases linearly with stimulus intensity. Weber’s law can be accounted for by postulating a logarithmic relation between the physical stimulus and its internal representation.

8 16 32 640

20

40

60

80

100

0.5 1 20

20

40

60

80

100

8 16 32 640

20

40

60

80

100

Deviation ratio (log scale)

Test numerosity (log scale)

Ref = 16 Ref = 32

Test numerosity (linear scale) 3 exemplars of a given number (16 or 32; « ref »)

Followed by a single test number (8-32 and 16-64; « test »)

Weber law in numerosity judgements

On a log scale the two curves have the same width !!! This indicates that numerosity is mentally represented on a compressed scale

Ref = 16 Ref = 32

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]

The Approximate Number Sense (ANS) is universal: across species

The Munduruku (indigenous tribe in the Amazon - Brasil) have number words only up to 4. -They have a perfectly normal non-verbal magnitude system, even for very large quantities

-They have a spontaneous capacity for estimation, comparison, addition

-They fail in tasks of exact calculation

[Pica, Lemer, Izard, & Dehaene, Science, 2004]

The ANS is universal: across

cultures

Approximate addition and comparison

[Pica, Lemer, Izard, & Dehaene, Science, 2004]

+

n1 n2

n3

Approximation addition and comparison

adul

ts

child

ren

B,I

M,NI

M,NI

B,I

B,NI

M,I

Ratio of n1+n2 and n3 (L/S)

All Munduruku

French controls

[Pica, Lemer, Izard, & Dehaene, Science, 2004]

Internal representation of numerosity: a model

0

1 2 3 5 8 4 6 7 9…

w

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

Numerosity

Activation

Log scale

[Izard et al., PNAS 2009]

Human newborns Human adults

Same or different numerosity?

A sample number (16 )

A test number (8,10,13,16,20,24,32)

0.5 1 20

20

40

60

80

100

[Piazza et al., Neuron 2004]

8 10 13 16 20 24 32 Test number

ANS undergoes maturation

Weber fraction (∆x/x) = 2

Weber fraction = 0.15

10 20 30 40 500

0.2

0.4

0.6

0.8

1

2

Infants (Izard et al., 2009; Xu & Spelke, 2000; Xu & Arriaga, 2007) Piazza et al., Cognition 2010; Chinello et al., submitted.

Halberda et al., 2008 Pica et al., 2004 Piazza et al., 2004 Power function fit

The precision of numerical discrimination (JND or Weber fraction) increases with age.

Age in years

Estim

ated

web

er fr

actio

n

Round numbers accurately discriminated

0 1 2 3 4 5 6 7 10

5:6

4:5

3:4

2:3

1:2

Age in years

The ANS acuity developmental trajectory

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?

“Number neurons” in macaque

[Nie

der,

Sci

ence

200

2]

“Tuning curves” for numerosity

Single neurons recording in monkeys performing the numerosity comparison task

Number is initially extracted from parietal neurons and then the information is transmitted to prefrontal cortex neurons.

Responce latency (ms)

Multiple regions contain neurons coding for number. Which does what?

« 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

Two pathways in vision : dorsal pathway / ventral pathway [Mishkin & Ungerleider, 1982; Milner and Goodale]

The most important function of parietal cortex is the DYNAMICAL REMAPPING OF THE MULTISENSORY SPACE

Parietal cortex contains MULTIPLE REPRESENTATIONS OF SPACE

EACH 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).

• 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

Macaque’s brain

Subtraction task

Ocular saccade

Grasping task

a

b

AIP LIP VIP

Putative homologies in the parietal lobe

Monkey brain

Human brain

Simon, Mangin, Cohen, LeBihan, and Dehaene (2002) Neuron Hubbard, Piazza, Pinel, Dehaene (2005) Nature Reviews Neuroscience

NUMBER NEURONS

Number and shape

Rare deviant stimuli (10%)

Number only

10 (medium) 13 (close) 16 (same) 20 (close) 24 (medium) 32 (far) 8 (far)

Is there a response to approximate number in

human IPS?

Piazza, Izard, Pinel, Le Bihan & Dehaene, Neuron 2004

Habituation to a fixed quantity

(e.g. 16 dots)

16 16

16

16 16

Deviant 32

Regions responding to a

change in number

L R

Response to numerosity change in the bilateral intraparietal sulcus

0.5 1 2-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5 Parietal activation

Same shape

Shape change

Log ratio of deviant and habituation numbers

Regions that respond to a change in SHAPE

Left intraparietal cortex Right intraparietal cortex

8 16 32 64-0.4

-0.2

0

0.2

0.4

Deviant numerosity (linear scale)

Nhabit 16 Nhabit 32

Deviant numerosity (linear scale) 8 16 32 64

-0.4

-0.2

0

0.2

0.4

Nhabit 16 Nhabit 32

8 16 32 64-0.4

-0.2

0

0.2

0.4

8 16 32 64-0.4

-0.2

0

0.2

0.4

Deviant numerosity (log scale)

Nhabit 16 Nhabit 32

Nhabit 16 Nhabit 32

Deviant numerosity (log scale)

0.5 1 2-0.4

-0.2

0

0.2

0.4

0.5 1 2-0.4

-0.2

0

0.2

0.4

Deviation ratio (log scale)

w = 0.252

Deviation ratio (log scale)

w = 0.183

z = 42

Weber’s law in the intraparietal sulcus

F(1,11)= 14.4, p<0.001 F(1,11)= 17.2, p<0.001

L R

First replication by Cantlon et al (2005). (Number change > Shape change). Since then,

MANY replications (e.g., Hyde 2010, etc …)

8 16 32 640

20

40

60

80

100

0.5 1 20

20

40

60

80

100

8 16 32 640

20

40

60

80

100

Deviation ratio (log scale)

Deviant numerosity (log scale)

Nhabit 16 Nhabit 32

Nhabit 16 Nhabit 32

w = 0.170

Deviant numerosity (linear scale)

10 16 32 480

20

40

60

80

100

0.7 1 1.40

20

40

60

80

100

10 16 32 480

20

40

60

80

100

Deviation ratio (log scale)

Deviant numerosity (linear scale)

Deviant numerosity (log scale)

Nhabit 16 Nhabit 32

Nhabit 16 Nhabit 32

w = 0.174

Three samples of a given numerosity (16 or 32)

Followed by a single deviant:

Weber’s law in numerical behavior

Same or different numerosity?

Larger or smaller numerosity?

Possible test stimuli:

…

…

A. Experimental design

Numerosity coding in 3 months old baby brains. EEG

Risposta alla numerosità nel cervello di bebè già a 3 mesi !!! Tecnica dell’EEG

Stesso numero Diversa forma

Stesso numero Diversa forma

Diverso numero Stessa forma

Stesso numero Stessa forma

Emisfero De

• 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”

• 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.

• 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.

Numbers are treated as representing APPROXIMATE

QUANTITIES during the initial stages of learing

Gilmore et al., Nature 2007

(1) Traces of the ANS in symbolic number processing - behavioural

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]

School maths’ achievement

correlates with accuracy in

symbolic approximate

calculation tasks

(1) Traces of the ANS in symbolic number processing - behavioural

Same Ratio-dependent responses in non-symbolic and symbolic number

processing

0,75

0,8

0,85

0,9

0,95

1

1 2 3

Acc

urac

y

* “choose the larger”

* “choose the larger”

12 16

Symbolic comparison

Non-symbolic comparison

1.1 1.3 1.6 Ratio (bigger/smaller set)

[Chinello et al., under revision]

ADULTS

Numbers are treated as analogical APPROXIMATE

QUANTITIES THROUGHOUT THE ENTIRE LIFE-SPAN !!!!!

w measured at 14 years of age … … …

at 14 yoa at

8 y

oa

…Correlates with math scores up to 10 years earlier ...

(1) ANS correlates with symbolic number processing throughout life-span

[Halberda et al., Nature 2009]

(1) ANS acuity higher in adult mathematics vs. psychology university students

[Ranzini and Girelli, under revision]

* “choose the larger”

[Mazzocco et al., PlOsONE, 2011]

(1) ANS in kindergarteners predicts performance in calculation in 1 grade

(longitudinal)

TEMA: counting, reading/writing 2 digits number, additions and divisions with concrete sets, symbolic number comparison, 1 digit additions and multiplications

Deviant number

17, 18, o 19

47, 48, o 49

20 50

close far

far close

Adaptation number

NUMBER

Deviant format

Dots

Arabic digits

dots digits

same different

different same

Adaptation format

FORMAT

2 CRITERA DEFINITIONAL For a SEMANTIC representation: •INVARIANCE TO ENTRY FORMAT

=

=

•SEMANTIC METRIC

<

>

(1) Traces of the ANS in symbolic number processing - neural

(1) Convergence towards a quantity code in the IPS in adults

50

18 19 19

HABITUATION 20

DEVIANTS

or

Number adaptation protocol (brain response to a change in number)

[Piazza et al., Neuron 2007]

-8

-6

-4

-2

0

2

4

6

8

10

Act

ivat

ion

(bet

as)

closefar

Right Parietal Peak

DOTS (among

dots)

DOTS (among arabic)

ARABIC (among arabic)

ARABIC (among

dots)

same different -8

-6

-4

-2

0

2

4

6

8

10

Act

ivat

ion

(bet

as)

closefar

DOTS (among

dots)

DOTS (among arabic)

ARABIC (among arabic)

ARABIC (among

dots)

Left Parietal Peak

Symbolic number code

Non-symbolic number code

(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

• 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.

10 20 30 40 500

0.2

0.4

0.6

0.8

1

2

(2) ANS maturation may account for number lexical acquisition pattern

Infants (Izard et al., 2009; Xu & Spelke, 2000; Xu & Arriaga, 2007) Piazza et al., Cognition 2010; Chinello et al., submitted.

Halberda et al., 2008 Pica et al., 2004 Piazza et al., 2004 Power function fit

The precision of numerical discrimination (JND or Weber fraction) increases with age.

Age in years

Estim

ated

web

er fr

actio

n

Round numbers accurately discriminated

0 1 2 3 4 5 6 7 10

5:6

4:5

3:4

2:3

1:2

Age in years

(2) ANS maturation may account for number lexical acquisition pattern

In the NUMBER domain, lexical acquisition before the discovery of the counting principles is a slow and strictly serial process.

2 years of age

3 years of age

4 years of age

Number words refer to quantities

Understand “one” Understand

“two” Understand “three”

Counting principles “discovered”

Understand “four”

• 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.

0.7 1 1.40

20

40

60

80

100

4-6 years 8-11 years Adults w=0.15

0.7 1 1.40

20

40

60

80

100

n1/n2 (log scale)

w=0.34

0.7 1 1.40

20

40

60

80

100 w=0.25

n1/n2 (log scale) n1/n2 (log scale)

% re

sp «

n2

is la

rger

»

4 groups of subjects (1) 8-11 years old dyscalculic (diagnosis: Italian standardized test), no neurological problems (2) 8-11 years old matched for IQ and cronological age (3) 4-6 years old (4) Adults

RESULTS (non dyscalculic subjects)

*

“choose the larger”

n1 n2

(3)The necessity of ANS for numeracy development: dyscalculia

[Piazza et al., Cognition 2010]

0,10

0,15

0,20

0,25

0,30

0,35

0,40

0,45

0,50

0 5 10 15 20 25 30

non-dyscalculics

power function (R2 = 0.97)

Age (years)

Est

imat

ed w

eber

frac

tion

[Piazza et al., Cognition 2010]

Estimated w

Dis

tribu

tion

Est

imat

es

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

1

2

3

4

5

6

7

adults10 yo5 yo10 yo dyscalculics

R2 = 0,17P=0.04

0

0,5

1

1,5

2

2,5

3

3,5

4

0,1 0,3 0,5 0,7

Estimated wN

erro

rs in

num

ber c

ompa

rison

ta

sks

Impairment in the ANS predicts symbolic number impairement but not performance in other domains (word

reading)

(3)The necessity of ANS for numeracy development: dyscalculia

In dyscalculic children the ANS is substantially impaired:

*

“choose the larger”

n1 n2

dyscalculics

[Mazzocco et al., Child Development, 2011]

Dyscalculics Low maths Typical maths High maths

Math: Test of Early Mathematics Ability (TEMA), and the Woodcock-Johnson Calculation subtest (WJR-

Calc)

(3) Impaired ANS in dyscalculia (replications …)

[Mussolin et al., Cognition 2010]

(3) ANS parietal system is ipoactive in dyscalculia

[Price et al., Current Biology, 2007]

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.

10 20 30 40 500

0.2

0.4

0.6

0.8

1

2

Development of ANS

[Piazza & Izard, The Neuroscientist , 2009]

Estim

ated

web

er fr

actio

n

Power function: Exponent = -0.43 R2=0.74 p=0.001

Infants (Izard et al., 2009; Xu & Spelke, 2000; Xu & Arriaga, 2007) Piazza et al., Cognition 2010; Chinello et al., submitted.

Halberda et al., 2008 Pica et al., 2004 Piazza et al., 2004 Power function fit

Age in years

The precision of numerical discrimination increases with age. What is the role of maturation? What is the role of education?

Does math education affect the

ANS ? (disentangling maturation from

education factors)

The Munduruku is an indigenous population of the Amazon (Brasil) - They have number words only up to 5. - They fail in tasks of exact calculation - They have a spontaneous capacity for approximate estimation, comparison, addition - As a group, they have a normal non-verbal magnitude system, even for very large quantities [Piazza, Pica, Dehaene, in preparation]

0

0.1

0.2

0.3

0.4

0.5

0 10 20 30 40 50 60 70

Munduruku, uneducated

Munduruku, some education

Italian participants (group means)

Age

Weber fraction Weber fraction

0.0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4 5 6 7 8 Years of Education

r²=26.8%, p=0.001

36 Munduruku subjects [aged from 4 to 67]

12 Completely uneducated 24 Received some education

0.7 1 1.40

20

40

60

80

100

0.7 1 1.40

20

40

60

80

100

Uneducated (n=7) Some education (n=13)

Performance of Munduruku adults

w = 0.288 w = 0.177

% larger responses

Ratio of n1 and n2 (log scale)

* “choose the larger”

We need to re-think learning as a deeply iterative process …

Pre-existing abilities (e.g., the ANS)

New cultural abilities (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, ...]

Conclusions The evolutionary ancient parietal system for approximate number grounds

the human cultural acquisition of numbers and calculation, and there is a long lasting cross-talk between innate approximate number sense and acquired symbolic arithmetical abilities.

From approximate non-symbolic quantity to exact number: a MAJOUR

CONCEPTUAL STEP.

The acquisition of symbols and their connection to the representation of the corresponding 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?

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)

The brain architecture for mental calculation

Representation of numerical quantities « # # »

Spatial operations ordering / zooming / remapping

1 2 3

Visual object processing number form « 2 »

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 and articulation « two »,

« arithmetical facts »

Left hemisphere Right hemisphere

Seen from top

LEFT ANGUALR GYRUS (Left AG) -Retrieval of arithmetical facts (multiplications, additions)

POSTERIOR SUPERIOR PARIETAL LOBE (PSPL) vLIP? -Subtractions -Complex additions -Approximate calculation

HORIZONTAL SEGMENT OF THE INTRAPARIETAL SULCUS (HIPS) hVIP? -Number comparison -Ratio effect -Approximate calculation

CS

IPS

Three parietal circuits for number processing: meta-analysis

[Dehaene, Piazza et al.,2003]

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

2. 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 ]

Left angular gyrus in arithmetical facts retrieval

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

Regions typically damaged

Typical drawing

Modello Copia del paziente

Line mark test Line bisection test

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

Evidence for a spatial code in arithmetical computations: neglect

12 subjects in a dark room produced 40 numbers in an order “as random as

possible”. Eye movements analyzed in the window in the 500ms PRECEEDING

number production

Shaki et al., 2009 (Psych Bull Rev)

Spatial code in number representations: the mental number

line (SNARC effect)

Number - space associations

0 100 "Position number 64"

[Siegler & Booth, 2004]

Kindergarten 6 years old 7 years old

Psychological Science, 2008

Kindergarteners

Across subjects, and in both populations, deviation from linearity correlates with number of errors in solving simple additions

[Geary et al., 2008]

Number to space associations in 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

Developmental dyscalculia

– 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…)

Early observed difficulties

Observed difficulties

• In simple calculation: Objects < Fingers-Verbal < Conceptual

– Counting all 3 + 8 = 1 2 3 4 5 6 7 8 9 10 11 – Counting on 3 + 8 = 4 5 6 7 8 9 10 11 – Counting min 3 + 8 = 9 10 11 – Retrieval 3 + 8 = 11 – Decomposition 3 + 8 = 10 + 1

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

%

tria

ls

Finger counting

Verbal counting Long term memory retrieval

Norm = non dyscaclulics DC = dyscalculics

• Those strategies (verbal and finger counting) have a LARGE COST, because they are at the origin of many errors

%

erro

rs

Observed difficulties: wrong strategies?

Finger counting Verbal counting Long term memory retrieval

Norm = non dyscaclulics DC = dyscalculics

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

• 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, …

Consequences in adults

“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 without

anyone knowing. Sometimes I feel very embarrassed! Especially under

pressure I just panic.”

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

Prevalence & co-morbidity

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 des quantités”

“Déficit aux systèmes de support” 1. - dyscalculie spatiale

(associé à la dyspraxie?) 2. - dyscalculie phonologique

(associé à la dyslexie?)

“Syndrome pariétale générale”

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

Representation of numerical quantities « # # »

Spatial operations ordering / zooming /

remapping

Visual object processing number form « 2 »

Pronunciation and articulation « two »,

« arithmetical facts » X X X X X

“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?)

Representation of numerical quantities « # # »

Spatial operations ordering / zooming /

remapping

Visual object processing number form « 2 »

Pronunciation and articulation « two »,

« arithmetical facts »

X X X X X

“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?)

Representation of numerical quantities « # # »

Spatial operations ordering / zooming /

remapping

Visual object processing number form « 2 »

Pronunciation and articulation « two »,

« arithmetical facts » X X X

X

1) Have a good model 2) Develop fine diagnostic tests 3) 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?

How to diagnose? How to “rehabilitate”?

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.