The document discusses number processing and calculation from a cognitive neuroscience perspective. It proposes that cultural practices like reading and arithmetic may have developed by "recycling" pre-existing neural circuits in the brain. In particular, regions in the parietal cortex that evolved to process quantities and perform spatial transformations may have been adapted for numerical tasks. Evidence for this comes from studies finding that the same parietal regions are consistently activated during tasks involving numbers across individuals and cultures.

1. Number processing and calculation:
the cognitive neuroscience of number
sense
MANUELA PIAZZA

2. 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]

3. « 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

4. « 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.

5. 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.

6. 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 ...

7. The crucial role of parietal cortex
in calculation: evidences
(1) A crucial site for Reduced gray matter
in premature children with dyscalculia
ACALCULIA (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)

8. (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 than
to 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

9. 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

10. 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)

11. 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, … )

12. 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

13. Number is spontaneously extracted in
newborns (cross-modal matching)
48 Newborns
Age = 49 hours [7-100 h]
12
4
[Izard et al., PNAS 2009]

14. 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]

15. Demonstration
Two sets of
different
number
Which set contains more dots?

18. 5 10 10 11
12 24 22 24
Ratio (S/L) = 0.5 Ratio = ~0.9
Less errors and faster reaction More errors and slower reaction
times times

19. Weber law
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.
Weight Loudness Brightness Numerosity

20. Weber law in 100
80
numerosity judgements 60
40
20 Ref = 16
Ref = 32
0
3 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)

21. 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]

22. The ANS is
universal: across
cultures
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]

23. Approximate addition and comparison
[Pica, Lemer, Izard, & Dehaene, Science, 2004]

24. Approximation addition and comparison
French controls
adults
M,NI B,NI B,I
+
children
n1 n2
All Munduruku
M,NI M,I B,I
n3
Ratio of n1+n2 and n3 (L/S)
[Pica, Lemer, Izard, & Dehaene, Science, 2004]

25. Internal representation of numerosity: a model
1 2 3 4 5 6 7 8 9… Numerosity
Activation
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

26. ANS undergoes maturation
Human 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

27. The ANS acuity developmental
trajectory
The precision of numerical discrimination
(JND or Weber fraction) increases with age.
Round numbers accurately
discriminated
2 1:2
Estimated 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

28. 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?

29. “Number neurons” in macaque
[Nieder, Science 2002]

30. Single neurons
recording in
monkeys
performing the
numerosity
comparison task
“Tuning curves” for
numerosity

31. 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.

32. 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

33. 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).
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

34. Putative homologies in the parietal lobe
NUMBER NEURONS
Monkey brain
a AIP LIP VIP
Subtraction task
Human brain
b
Ocular saccade
Grasping task
Simon, Mangin, Cohen, LeBihan, and Dehaene (2002) Neuron
Hubbard, Piazza, Pinel, Dehaene (2005) Nature Reviews Neuroscience

35. 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
Number
and shape
Piazza, Izard, Pinel, Le Bihan & Dehaene, Neuron 2004

36. Response to numerosity change
in the bilateral intraparietal sulcus
Regions that respond to a
change in SHAPE
0.5 Parietal activation
0.4
Regions 0.3
responding 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

37. 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)

38. Weber’s law in numerical behavior
100 100
80 80
60 60
Three 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)

39. Numerosity coding in 3 months old baby
brains. EEG
A. Experimental design
…
…
Possible test stimuli:

40. Risposta alla numerosità nel cervello di bebè già a 3 mesi !!! Tecnica dell’EEG
Stesso numero Diverso numero Stesso numero Stesso numero Emisfero De
Diversa forma Stessa forma Diversa forma Stessa forma

41. • 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”

42. • 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.

43. • 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.

44. (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

45. School maths’
achievement
correlates with
accuracy in
symbolic
approximate
calculation 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]

46. (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]

47. (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]

48. (1) ANS acuity higher in adult mathematics
vs. psychology university students
“choose the larger”
*
[Ranzini and Girelli, under revision]

49. (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 [Mazzocco et al., PlOsONE, 2011]

50. (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

51. (1) Convergence towards a quantity
code in the IPS in adults
[Piazza et al., Neuron 2007]
1010
close
close
Left Parietal Peak
Right Parietal Peak
Number 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

52. (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

53. • 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.

54. (2) ANS maturation may account for
number lexical acquisition pattern
The precision of numerical discrimination
(JND or Weber fraction) increases with
age. Round numbers accurately
discriminated
2 1:2
Estimated 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

55. (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.
Number words
refer 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

56. • 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.

57. (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) Adults
RESULTS (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]

58. (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
Estimated 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]

59. (3) Impaired ANS in dyscalculia (replications …)
Dyscalculics Low maths Typical maths High maths
Math: Test of Early Mathematics Ability (TEMA), and
the Woodcock-Johnson Calculation subtest (WJR- [Mazzocco et al., Child Development, 2011]
Calc) [Mussolin et al., Cognition 2010]

60. (3) ANS parietal system is ipoactive in
dyscalculia
[Price et al., Current Biology, 2007]

61. 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.

62. 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]

63. 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]

64. 36 Munduruku subjects Performance of Munduruku adults
[aged from 4 to 67] Uneducated (n=7) Some education (n=13)
% larger responses
12 Completely uneducated 100 100
24 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

65. We need to re-think learning
as 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, ...]

66. 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?

67. 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)

68. 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 of
and articulation numerical quantities « # # »
 « two »,
Spatial operations ordering
« arithmetical
/ zooming / remapping
facts » 123
Visual object processing  number form « 2 »

69. 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]

70. 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

71. Left angular gyrus in arithmetical
facts retrieval
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 ]

72. 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

73. Evidence for a spatial code in
arithmetical computations: neglect
Modello Copia del paziente
Regions typically damaged
Typical drawing
Line mark test Line bisection test

74. 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

75. 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

78. Spatial code in number
representations: the mental number
line (SNARC effect)
Shaki et al., 2009 (Psych Bull Rev)

79. Number - space associations
0 "Position number 64" 100
Kindergarten 6 years old 7 years old
[Siegler & Booth, 2004]

80. Psychological Science, 2008
Kindergarteners
Across subjects, and in both populations, deviation from linearity correlates
with number of errors in solving simple additions

81. Number to space associations
in dyscalculia
[Geary et al., 2008]

82. 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

83. 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…)

84. 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

85. 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

86. 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

87. 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

88. 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 without
anyone knowing. Sometimes I feel very
embarrassed! Especially under
pressure I just panic.”

89. 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

90. 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?)

91. 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 »

92. “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 »

93. “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 »

94. How to diagnose?
 How to “rehabilitate”?
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?

95. 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.