Piazza 1 lecture

Cognitive and neural basis of
numerical abilities

Manuela Piazza

Course plan
Topic More detailed Reading list
- The hypothesis of cultural recycling
- General introduction. - Dehaene and Cohen, Neuron 2007.
of cortical maps.
- The notion of
- Ethnological aspects. -
number.
1 class - Revkin et al., Pyschological Science
2009.
- The perception of
- Numerical estimation and subitizing.
number.
- Piazza, Trend In Cognitive Sscience,
2010.

- Number neurons in - Piazza et al., Neuron 2004.
humans and other - Brief introduction to neuroimaging
- Nieder., Neuron 2003.
2 class animals. Parietal and neurophysiological methods and
cortex and numerical discussion of studies. - Piazza and Izard, The neuroscientist,
quantity coding. 2009.

- Dehaene et al., Cognitive
- Cerebral circuits of Neuropsychology, 2003.
simple arithmetic.
- Handbook of Mathematical
Cultural variability and - Discussion of studies using fMRI,
Cognition (2004), J.Campbell. Ed.:
3 class universal mechanisms. EEG, behavioral, and
Chapter 10. Learning Mathematics in
Numbers and its neuropsychological methods.
China and the United States.
relation to language,
and spatial abilities. - Hubbard et al., Nature Reviews
Neuroscience, 2005

- Ansari, Nature Reviews
Neuroscience, 2008.
- Normal and abnormal
- Piazza et al., Cognition 2010.
development of - Discussion of behavioral and
4 class
elementary numerical neuroimaging studies. - Handbook of Mathematical
abilities Cognition (2004), J.Campbell. Ed.:
Chapter 10. -Developmental
Dyscalculia.

Introducion
The hypothesis of cultural
“recylcing” of pre-exising neural
circuits
Or: cultural traditions are such becuase they fund
adequante “neuronal nich” in our brains

The Primate human’s specificity for cultural acqusition

A thirteen-month-old chimpanzee traces curves on a Composition produced by an adult chimpanzee living
graphic tablet (Tanaka et al., 2003) semi-independently in the Mefou Forest Reserve in
Cameroon (© Canadian Ape Alliance)

- Other primates are able to acquire new abilities, the use of new tools (Iriki, 2005), and even of
symbols like Arabic digits (Matsuzawa, 1985)
- They also show some rudiment of cultural skills that are locally trasnmitted (Whiten et al.,
1999).

Which are the biological basis of human culture?

- Which are the brain architectures that support extraordinary cultural inventions (such
as reading and arithmetic)?
- Which are the new caracteristics in brain architectures that makes the human species
the only real “cultural species”, the only one capable of real cultural inventions?

According to some positions, and in particular by scholars in the classical social
sciences, brain architecture is completely irrelevant when it comes to
understanding “high level” cultural acquisitions.

Many scientists implicitely or explicitely agree with the idea of a “generalized brain
plasticity”, or “cultural relativism”, which reminds the idea of “tabula rasa” of
Aristotelian memory (according to which humans are born with no innate abilities,
and thus are equipotential learners).

Does the brain constrain cultural acquisitions?
Brain is irrelevant
• Constructivism in neurobiology
e.g. Quartz & Sejnowski (1997) (“The neural basis of
cognitive development: a constructivist manifesto”)
: «cerebral cortex during development is equipotential
and free to devote specific structures to different
Education
domains»
and cognition
• Functionalism in psychology:
e.g. Fodor, Johnson-Laird:
« the physical nature of the huan brain does not impose Behaviour
any constraint on thought»
Cortical regions
Cortical columns

Neurons

Synapsis

Our long term objective
Receptors
is to characterize the
conversion laws between
incapsulated levels

Neurons and culture. Which links?

• Technique of non invasive neuro-immaging allow us to investigate the neural
mechanisms involved in the manipulation of cultural tools.
• For both reading and mathematics, even though there are racross cultures a large
variability, there are in all individuals brain regions specialized for each of these
abilities.
• These results may seem paradoxical, since the natural evolution does not seem to
have had the time sufficient to select brain architectures specifically to support recent
cultural objects.

The “neural recycling” model (“exaptation”: modification of the function of a pre-selected given trait –
concept already introduced by Darwin):

• The architecture of our brain is limited.
• It is determined by genetic/epigenetic laws, but they include a certain variability and
plasticity (itself genetically determined).
• The new cultural acquisitions are possible only inhasmuch they capitalize on this
variability. Each cultural object is such because it finds its NEURONAL NICHE.
• Our brain is NOT a tabula rasa, but it adapts to the cultural environment to which it is
exposed to by a minimal riconversion or “recycling” of pre-existing systems, even
thought they were selected for a different function.

Perception of numerical
quantity

Humans have two systems for
perceiving numerical quantity:

1. The approximate number
system

2. The object tracking system

The approximate number
system
Allows estimation of the
approximate number of items in
collections

Demonstration

Two sets of
different
number

Which set contains more dots?

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

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

Weber law in 100

80

numerosity judgements 60

40

20 Nhabit 16
Nhabit 32
0
3 exemplars of a given number (16 or 32) 8 16 32 64

Deviant numerosity (linear scale)

100

80

60

Followed by a single deviant number 40

(8-32 and 16-64) 20 Nhabit 16
Nhabit 32
0
8 16 32 64

Deviant 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)

The ANS is universal: across
species
The ability to discriminate between two quantities depends upon their ratio,
according to Weber’s law.

This law holds for humans but also for animals’ numerosity judgments (here,
macaque monkeys vs. humans).

The 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
reference number [Van
Oeffelen and Vos, 1982]

The ANS is
universal: across
cultures

The Munduruku have number words
only up to 5.

-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]

Mundurúkú Territory

Transamazonian Rio Tapajos
highway Jacarecanga
Sai Cinza Katõ
Rio das tropas

Missão Velha

Rio Juruena Rio Cururu
Brasil

Rio Teles Pires
40
km

Adults Children
n=9 (55.5 y) n=9 (4.7 y) monolingual, no instruction
n=10 (59.3 y) bilingual, no instruction
n=7 (8.6 y) monolingual, with instruction
n=7 (38.7 y) n=13 (9.6 y) bilingual, with instruction
+ 10 French adults [Pica, Lemer, Izard, & Dehaene, Science, 2004]

pug ma = one
Numerosity naming xep xep = two
100.0
ebapug = three
ebadipdip = four
90.0
pug põgbi = one hand
response frequency

80.0 xep xep põgbi = two hands
70.0 adesu/ade gu = some, not many
ade/ade ma = many, really many
60.0 ~

50.0

40.0

30.0

20.0

10.0

0.0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
stimulus numerosity


Approximate addition and comparison


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)


Internal representation of numerosity: a model

1 2 3 4 5 6 7 8 9… Numerosity

Activation

w

0 Log scale

w = sd of the gaussian distribution of the internal representation of numerical
quantity (on a log scale!). The larger w the worst the discriminability between two
close numbers.

w is a measure of the precision of the internal representation of numerosity

The ANS is active in infants
Human newborns spontaneously match approximate number
(violation of expectation paradigm)

SENSITIVE TO A MINIMAL
48 Newborn babies 1:3 RATIO
Mean age= 49 h [7-100 h]

12

4

[Izard et al., PNAS 2009]

The ANS is active in infants
Also older babies (6 months) spontaneously match number

Habituation 8 16

ou
1:2 RATIO

Test 16 8

8 vs. 16 but not 8 vs. 12, nor 4 vs. 5…

F. Xu, E. S. Spelke, Cognition 74, B1-B11 (2000).

The ANS is active in children

5 years old children

4:5 RATIO

Barth (2005)

The ANS acuity improves during
development

ANS acuity (Weber fraction) Round numbers accurately discriminated
2

1:2

1

0.8

0.6
2:3
0.4

0.2

0 3:4
10 20 30 40 50
4:5
Age in years
5:6

0 1 2 3 4 5 6 7 10

Age in years

The ANS acts on numerical quantity independent
from the nature of the stimulus
(1/9) (8/9)
Match SEQUENCIAL to SIMULTANEOUS

(S/L)

Match AUDITORY to VISUAL

(Example: Jordan et al., 2008)

Adds AUDITORY to VISUAL

Vedi video:
http://www.duke.edu/web/mind/level2/faculty/liz/xmodal.htm

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 simple operations (comparison, additions,
subtractions, ...)

 WHAT (IF ANY) IS IT’S ROLE IN NUMERACY ACQUISITION?

Mechanisms of numerosity
extraction: Models
• 1. “Numerosity detector”
(Dehaene, 1993 version)

2. Object Location 3. Numerosity
1. (Visual) Input and Normalization Accumulation 4. Numerosity
Detection
(« band-pass
filters »)

1 2 3 4 5 6

Operations that occur in parallel

extraction: Models
• 1. “Numerosity detector”
(Verguts, 2006 version)

extraction: Models
• 2. “Accumulator”
(Gallistel, 2002)

Operations occurring serially

The object tracking system
(OTS)
Allows exact apprehension of the
number of items in collections (but
limited in capacity to 3-4 items)

Demonstration :

How many dots?

The OTS
• OTS is capacity limited  “subitizing”

0,5
Accuracy
1000 Mandler & Shebo, 1982, Trick &
Speed
Pylyshyn, 1994,…
0,4 900

Response Time (ms)
A discuntinuity in
Error rate

0,3 800
both accuracy and
0,2 700
response time
0,1 600

0 500
1 2 3 4 5 6 7 8
Set size

SUBITIZING
This phenomenon:

1) Does not depend on the spatial layout of the items (no “patter recognition”)
II III IIIIIIIIII

2) Does not depend on the nature of the items
OX OXO OXOXOX

3) It is not altered by tasks tapping on verbal memory

BUT

1) It disappears if the elements do not “pop-out” from the background

Count the number Count the number of
of horizontal bars horizontal red bars

2) It is altered by a concurrent taslk which occupies visuo-spatial memory

What is the origin of SUBITIZING?
1. DOMAIN-SPECIFIC HYPOTHESIS (Butterworth, ...):

Subitizing reflects the sensitivity of the ANS (1:2, 2:3, 3:4 are easily discriminable
ratios)

PREDICTIONS:
(A) We should find a similar discuntinuity in RTs and errors for similar ratios in the
large number quantification 1 2 3 4 …8,  10 20 30 40 ... 80
(B) Subitizing SPAN (number of objects that can be readily quantified with no
errors) should correlate with other measures of the ANS precision.

2. DOMAIN-GENERAL HYPOTHESIS (Phylyshyn, ...):

Subitizing is reflects our ability to “index” (individuate, or pay attention to individuals
with their basic features, like color shape position) multiple objects at once.

PREDICTIONS:
(A) Subitizing SPAN should correlate with other measures of multiple object
tracking (e.g., visuo-spatial working memory span)
(B) Subitizing SPAN should be altered by a concurrent visuo-spatial task tapping on
similar visuo-attentive resources

Characterizing the OTS
• Hp1. The OTS is “just” the ANS that is of
higher precision for small numbers?

[Revkin, et al., Psychological Science 2008]

RANGE [1-8] RANGE [10-80] RANGE [1-8] RANGE [10-80]

i The Subitizing range across subjects does NOT correlate with the ANS acuity
(measured with an independent large numerosity comparison task)

Characterizing the OTS
• Hp 2. The OTS is not number-specific, but it reflects the
capacity of the visual object individuation system, used for
processing multiple items individually

10 adult subjects; 3 tasks TASK 1 -Enumeration task
How many?
0,5 1000
Accuracy
250 ms Speed
0,4 900

Response Time (ms)
200 ms
Error rate

0,3 800

Time 0,2 700

0,1 600

0 500
1 2 3 4 5 6 7 8
Set size
[Piazza, Fumarola, and Melcher, under revision]

TASK 2-VWM task TASK 3 -Large numerosity comparison task
Same or different?

“choose the larger” 100 W = 0.19
2000 ms
80
1000 ms *
60

700 ms 40
Time

200 ms 20

7 0
Cowan’s K=S (hits - f.alarms) 16 32
Number of objects encoded (K)

6

5

4

3

2 Results
1

0
1 2 3 4 5 6 7 8
4,5
Set size

3,5

VWM capacity
(1) High correlation across capacity limited 2,5
systems, and NO correlation between both
these systems and the acuity of the ANS. 1,5
3,5 4 4,5 5
Subitizing capacity

If they tap on same mechanisms then they should interfere
selectively with each other.

TASK 4 – Dual task : Enumeration + VWM task.
Results: Subitizing range is reduced by increasing
working memory load

TASK 5 – Dual task : Large number comparison + VWM task.
Results: Weber fraction is unchanged by increasing
working memory load

OTS developmental trajectory

CONCLUSIONS:

Subitizing is based on object tracking mechanisms (OTS) and reflects the general
architecture of our visuo-attentive system, which cannot individuate more then 3-4 items
at once. It is a separate function from the ANS

 WHAT (IF ANY) IS IT’S ROLE IN NUMERACY ACQUISITION?

Piazza 1 lecture

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