Mathematics performance in preschool is strongly predictive of later mathematics achievement through high school. Initial knowledge in domains like reading is also positively related to future learning, but the relationship is unusually strong and persistent for mathematics. Several Asian countries and regions like Shanghai, Singapore, Hong Kong, and Korea consistently demonstrate mathematics performance well above the OECD average, suggesting cultural factors may also contribute to their success.

1. Q1. Why preschoolers?
Why preschoolers?
Proficiency in mathematics at the beginning of kindergarten is
strongly predictive of mathematics achievement test scores
years later: in elementary school, in middle school, and even in
high school (Duncan et al., 2007; Stevenson & Newman, 1986).
In other domains initial
knowledge is positively related to
learning (Bransford, Brown, &
Cocking, 1999; e.g. reading), but
the relations in math are
unusually strong and persistent
!!!!!
Duncan, G. J., Dowsett, C. J., Claessens, A., Magnuson, K., Huston, A. C., Klebanov, P, et al. (2007). School readiness
and later achievement. Developmental Psychology, 43, 1428 – 1446.
Stevenson, H. W., & Newman, R. S. (1986). Long­term prediction of achievement and attitudes in mathematics and reading.
Child Development, 57, 646 – 659.

2. The Program for International Student Assessment (PISA) organized by OECD.
More then 400 000 students (15 years old) from 57 countries
Objectifs de PISA:
2. Apporter à chaque pays des informations sur:
! Les caractéristiques et performances de son système éducatif et sur
ses évolutions dans le temps
! Les caractéristiques et performances d’autres pays
! Les caractéristiques des systèmes éducatifs des pays les plus
performants
! Les réformes mises en œuvre dans d’autres pays

3. Q2. Does Math excellence goes beyond preschool?
Performance moyenne sur !"#$%&!!& de '()%#'()*+,&-
400 450 500 550 600
Shanghai­China 600
Singapore 562
Hong Kong­China 555
Korea 546
Chinese Taipei 543
Finland 541
Liechtenstein 536
Switzerland 534
Japan 529
Canada 527
Netherlands 526
Macao­China 525
New Zealand 519
Belgium 515
Australia 514
Germany 513
Estonia 512
Iceland 507
Denmark 503
Slovenia 501
Norway 498
France 497
Slovak Republic 497
Austria 496
OECD average 496
Poland 495
Sweden 494
Czech Republic 493
United Kingdom 492
Hungary 490
Luxembourg 489
United States 487
Ireland 487
Portugal 487
Spain 483
Italy 483
Greece 466
Israel 447
Turkey 445
Chile 421
Mexico 419

4. Trends in International Mathematics and Science Study (based in USA)
9 YEARS 13 YEARS
Performance significantly higher then USA
Performance non significantly different from USA
Performance significantly lower then USA

5. You may look in the database, yourself, and get
answers of some of your questions here:
http://www.pisa.oecd.org/pages/0,2987,en_32252351_
32235731_1_1_1_1_1,00.html
See following example from one presentation by the
“Conseil scientifique du Ministère de l’éducation
nationale”

6. Q3. Are there other cultural factors accounting for success in Asian Countries?
./*0)12&1vue des #!34&- sur leurs 5&!()*/0-1(4&$1!&-1
professeurs
OECD average Finland France Germany Japan
85
Je m’entends bien avec la plupart de mes
78
professeurs
66
La plupart de mes professeurs
53
s’intéressent à mon bien­être
67
La plupart de mes professeurs sont
62
réellement à l’écoute de ce que j’ai à dire
79
Si j’ai besoin d’aide supplémentaire, mes
80
professeurs me l’apporteront
79
La plupart de mes professeurs me traitent
88
correctement
0 50 100

7. ./*0)12&1vue des #!34&- sur le $!*'()
2"(665&0)*--(7&
Les choses suivantes se produisent à la plupart, voire à tous les cours
OECD average Finland France Germany Japan
29
Les élèves n’écoutent pas ce
36
que dit le professeur
8
32
Il y a du bruit et de l’agitation 44
10
Le professeur doit attendre 28
un long moment avant que 36
les élèves se calment 7
19
Les élèves ne peuvent pas
24
bien travailler
13
Les élèves ne commencent à 25
travailler que bien après le 37
début du cours 9
0 25 50 75 100
%

8. Q4. Pourquoi veut­on mieux enseigner ? Jusqu’à quel point ?
Quel prix serait­on prêts à payer ?
./,5+,/*12&1)&!!&-1'&-,5&-8
!Les !"#$%&$&'!("!)*+,-.%'sont un des
65&'*&5-1$/'6/-(0)-12,19,27&)12&-1:)()-
!;(16&5</5'(0$&12&-1-=-)3'&-1#2,$()*<-12/*)1
>)5&1('#!*/5#&1(<*012&1'*&,?1-()*-<(*5&
! La demande sociale
! Limiter les déficits publiques
!Améliorer la qualité des services rendus
! @(0-1!&-12#'/$5()*&-1'/2&50&-1A2#'/$5()*&-1
6(5)*$*6()*4&-B1*!1$/04*&0)12&1'*&,?1*0</5'&51
!&16,9!*$1-,51!(1+,(!*)#12&-1-&54*$&-16,9!*+,&-

9. Q5. Is math superiority specific?
Performance moyenne sur !"#$%&!!& de $/'65#%&0-*/0 de !"#$5*)
400 450 500 550 600
Shanghai­China 556
Korea 539
Finland 536
Hong Kong­China 533
Singapore 526
Canada 524
New Zealand 521
Japan 520
Australia 515
Netherlands 508
Belgium 506
Norway 503
Estonia 501
Switzerland 501
Poland 500
Iceland 500
United States 500
Liechtenstein 499
Sweden 497
Germany 497
Ireland 496
France 496
Chinese Taipei 495
Denmark 495
United Kingdom 494
Hungary 494
OECD average 493
Portugal 489
Italy 486
Slovenia 483
Greece 483
Spain 481
Czech Republic 478
Slovak Republic 477
Israel 474
Luxembourg 472
Austria 470
Turkey 464
Chile 449
Mexico 425

10. Q6. Should we change our linguistic system for teaching maths (l’académie Aix­
Marseille suggesting we should teach maths in Chinese)?
Q7. Should we conclude that the role of parents/culture with respect to
General expectations?
The role of “hard work”?
Early education?
Q8. What do bilingual kids do? e.g., If the have the choice between Chinese and
French? … always chose one preferred language, most probably the one first
and most consistently taught at school? ? ? Do not know …

11. Language plays an important role during the early
stages of numeracy acquisition
but
there are plenty of evidences that (at least some
forms of) calculation is dissociable from language …

12. ! Dissociations between operations and
associated symptoms in brain lesioned
patients
Paziente MAR Paziente BOO
Lesione: lobo parietale (inferiore) Lesione: lobo frontale inferiore (insula)
Sintomi associati (confusione dx­sx e Sintomi associati: afasia di Broca
angosia digitale): Calcolo: deficit in moltiplicazioni
Calcolo: deficit in sottrazioni (tabelline) (errori e lentissima) ma
(moltissimi errori e lenta) ma in grado perfettamente in grado di eseguire
di recuperare risultati di tabelline calcoli complessi
Dehaene and Cohen, 1997

13. More evidence favouring a verbal code for arithmetical facts
Risults
Blingual subjects exposed to new
arithmetical facts in one of the
two languages. After learning,
researchers verified the
GENERALIZATION
1. To the other language
2. To similar arithmetical facts
Exact calculation:
Quanto fa ventiquattro più
trentasette? Sessantuno o
cinquantasette?
Approximate
calculation
Ventiquattro più trentasette fanno
più o meno… Sessanta o
quaranta?

14. The cerebral impact of switching language in bilingual subjects
Bilingual Anglo­Chinese were trained in exact and approx calculation in one
language, and then tested in the same or in the other language

15. EXACT CALCULATION : APPROX. CALCULATION :
Increased response to a language change Increased response to a language change
Left Inferior frontal gyrus (Broca) Left Inferior frontal gyrus (Broca)
Left Angular Left posterior parietal

16. Three subjects with large lesions to the perisilvian areas of the
dominant hemisphere with SEVERE language and in particular
GRAMMATICAL IMPAIRMENTS (performing no greater than at
a chance level on understanding reversible sentences. 2 also
performed at chance level on a written grammaticality judgments
test)

17. At the same time there are evidence that spatial
abilities might be very important in calculation and
mathematics
Activity to number processing is embedded in and partially overlapping with
circuits involved in action preparation via complex mechanisms of spatial co­
ordinate coding and transformations.
Human VIP (multisensory ­ face ­ numerical quantity)
LIP (saccades ­ eyes)
AIP (grasping ­ hand)

18. 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

21. CAUSES OR EFFECTS OF
NUMBER­SPACE ASSOCIATIONS
IN THE SCHOOL PROGRAMS?
Massive use of the number line in
elementary schools …

22. Learning numbers…
DRAMATICALLY CHANGES THE PRE­EXISTING REPRESENTATIONS OF
QUANTITY
1) APPROXIMATE " EXACT
2) COMPRESSED " LINEAR
NOTION OF “ONE” AS THE MINIMAL
FIXED DIFFERENCE BETWEEN ANY TWO
FOLLOWING INTEGERS

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

24. Task: approximate
a) the answers to addition problems (Computational “Is 34 + 29 closest to 40, 50, or 60?”)
b) the number of candies in a jar (Numerosity)
c) the length of a line in inches (Measurement “this is a line 1 inch long. Draw a 3,6,8,9 inches line”)
d) the location of a number on a line (Number line)
[Booth & Siegler, 2006]

25. 88 preschoolers (4­5.5 y)
met individually with an
experimenter for five 15­20
minute sessions within a
three­week period. “Control
task = colour board game
START END
1 2 3 4 5 6 7 8 9 10 1 2
IMPROVEMENTS
ALSO IN:
1) Numeral
identification
2) Number
comparison
3) Counting

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

28. « Grasping » numerical magnitude
Erhsson et al., J Neurophysiol, 2001

29. 2.2. Planning vs on­line control of grasping
Cinematic of grasping
?
25 cm
Andres et al., Cortex, 2008

30. Description of wooden blocks that subjects have to grasp
Small
1 1 1
digits
2 2 2
Large
8 8 8
digits
9 9 9
40 mm 50 mm 60 mm

31. Amplitude of grip apertue
Time(normalized)
Numerical difference (in mm)
Results:
A large number written on a block
induces a larger grip aperture during
grasping trajectory but only in an initial
phase of grasping.
Time(normalized)

32. Interference between numerical magnitide and grasping aperture
Hp 1 : numerical magnitude interferes directly with
grasping aperture during grasping
# After the alaboration of a small number subjects
should UNDERESTIMATE THE MAX size of a
graspable object compared to a neutral condition
Hp 2 : numerical magnitude interferes directly with the
estimation of the size of an object
#After the alaboration of a small number subjects
should OVERESTIMATE THE MAX size of a graspable
object compared to a neutral condition

33. Exp 1 : Graspability judgements
(max dimension of a graspable item)
2 $ 8
2
12.6 cm 12.3 cm 11.8 cm
?

34. Exp 2 : Perceptual judgements
2 $
2
8 8
2
13.2 cm 13.2 cm 13.2 cm
? Longer
than X ?

35. ! SUMMARY
! Numerical magnitude interferes with the estimation of the dimension
of the to be grasped object, which then influences the graspability
judgement.
! This interference likely occurs in parietal cortex (perception for action)
! Motor programming integrate non only perceptual BUT ALSO
CONCEPTUAL INFORMATION
! " E’ probabile che nella corteccia parietale vi siano
rappresentazioni almeno parizialmente comuni della
grandezza numerica e di altre grandezze, come la
grandezza fisica …

36. CAUSE OR EFFECT OF
NUMBER­SIZE ASSOCIATIONS IN
THE SCHOOL PROGRAMS?
Measurement tools (ruler, termometer …)
We may hypothesize that our culture makes massive use of
spatial metaphors for representing quantities
BECAUSE that is the best way to
SUPPORT the abstract notion of LINEAR AND EXACT
MAGNITUDES…

37. Developmental dyscalculia
! Called “Mathematics disorder” (DSM­IV 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

38. ! http://dyscalculiaforum.com/viewpage.php?page_id=10
YouTube:
! http://www.youtube.com/watch?v=rSb­PD­
Od7o&eurl=http%3A%2F%2Fdyscalculiaforum%2Ecom
%2Fnews%2Ephp&feature=player_embedded
! http://www.youtube.com/watch?v=RD9xhI076ZI&feature
=related
! http://www.youtube.com/watch?v=ms5N1X6CYAM&feat
ure=related

39. ! In U.K. dyscalculia has been recognized as a specific
learning dysability from the Department of Education
(equivalent to the French Ministry of Education) ONLY IN
2001!!
! In Italy, the articles 9 e 10 of the law that relates to
pupils’ evaluations suggests to take into account of the
“dysabilities, if certified by the means predisposed by the
current legal rules...” (“certificate nelle forme e con le
modalità previste dalle disposizioni in vigore…”)
BUT
A standardized and universally recognized diagnostic
battery does not exist!! ??

40. “Symptoms”
– Problems in acquiring counting principles
– Problems in acquiring strategies for solving
simple arithmetical problems (es. in
additions –counting on from the largest
number ..vs. count all)
– Problems in memorizing arithmetical facts
(tables)
– Use of “immature” strategies (finger
counting…)

41. Observed difficulties
! In simple calculation:
Objects < Fingers­Verbal < 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

42. Observed difficulties
In simple calculation (1 digit numbers) kids with DD make more errors and are
slower
Llanderl, Bevan & Butterworth, 2004 (8­9 years old kids)

43. Observed difficulty: strategies?
! Geary e Brown, 1991: Dyscalculic kids of 6­7 years, in simple calculation (e.g., 3+2)
use more immature strategies such as verbal or finger counting and much less then
facts retreival
% di prove
Finger counting
Verbale counting
Long term memory retrieval
Norm = non dyscaclulics
DC = dyscalculics

44. Observed difficulty: strategies?
! Those strategies (verbal and finger counting) have a LARGE COST,
because they are at the origin of many errors
% di errori
Finger counting
Verbale counting
Long term memory retrieval
Norm = non dyscaclulics
DC = dyscalculics

45. 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

46. 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 self­esteem, 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.”

47. Consequences in adults
“I struggled through school with maths to the point the
teachers gave up on me. I can only count on my
fingers or with a calculator. I can't count out
change, no matter how small and often get flustered with
any tasks involving numbers. Despite trying hard I could
never remember my 'times tables'. Division etc just
bewildered me totally. English was one of my best subjects
at school.”
“I have no trouble whatsoever reading or writing,
understanding literary concepts and theories etc., but
spend an hour sitting in the bank trying to work out
how much money is in my cheque account! Last year
I returned to University, attempting to avoid any papers
containing mathematics, but hidden in nearly
everything are formulas and calculations.”

48. Prevalence & co­morbidity
Lewis et al.(1994):
1056 kids UK
9­10 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
Gross­Tsur, 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

49. Standardized Tests …
! Most measure performance on school­type tasks … but bad
performance can be cuse by a moltitude of factors
– Specific problems
– Inadequate teaching
– Lack of motivation (possibly related to a stereotype effect of
gender, social class, eetc ...)
! It would be important to also have tests
measuring basic capacities, less influenced by
specific training in a specific TASK

50. Basic numerical abilities (symbolic)
! Number comparison
– Distance effect
2 9 7 9
– Number­size interference
! Number to space associations
5
0 10

51. DD are slower in comparing numerical magnitudes but not physical magnitudes
7 9
Size comparison Number comparison
Llanderl, Bevan & Butterworth, 2004.

52. Number to space associations
0 "Position number 64" 100
Kindergarten 6 years old 7 years old
Inter­individual differences in this task correlate
Siegler & Booth, 2004 with math tasks performance

53. Number to space associations
[Geary et al., 2008]

54. Basic numerical abilities (non­symbolic)
4 groups of subjects
“choose the larger”
(1) 8­11 years old dyscalculic (diagnosis: Italian standardized
* test), no neurological problems
(2) 8­11 years old matched for IQ and cronological age
n1 n2 (3) 4­6 years old
(4) Adults
RESULTS (non dyscalculic subjects)
4­6 years 8­11 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]

55. Basic numerical abilities (non­symbolic)
“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)5 yo
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
non­dyscalculics
0,45 1,5
dyscalculics
1
Estimated weber fraction
1
0,40 R2 = 0,17
0,5
0 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]

56. Causes of dyscalculia
! Open ongoing debate
! Most popular view is that there can MULTIPLE
FORMS, and therefore multiple CAUSES:
VISUO­SPATIAL
NUMBER SENSE REPRESENTATIONS
(HIPS) (PSPL)
VERBAL
– CORE DEFICIT dyscalculia ASSOCIATIONS
(left AG)
– VERBAL dyscalculia
– SPATIAL dyscalculia
– EXECUTIVE dyscalculia…

57. …Our neurocongitive model can help us formulating hypothesis on the pattern of
associations/dissociation and their neural basis …
VISUO­SPATIAL REPRESENTATIONS
NUMBER SENSE
(PSPL)
(HIPS)
VERBAL ASSOCIATIONS
(left AG)

58. “Core deficit” dyscalculia
" HP (1): problems in understanding the meaning of
numbers (comparison, quantification, approx. calculation)
“where are more dots?
*
n2 n1
Distribution Estimates
7
adults
6 10 yo
5 yo
10 yo dyscalculics
5
4
3
7 9 2
1
0
­0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Internal w

59. “Core deficit” dyscalculia
" HP (2): often observed in the context of Developmental
Gerstman syndrome? Associated also with dyspraxia?
Developmental Gerstman syndrome [parietal sindrome]:
­Left­right disorientation
­Digital agnosia
­Dysgrafia
­Dyscalculia:
VIP (quantità numerica)
LIP (spazio visivo)
AIP (afferrare, mano)

60. A short digression on Gerstman
syndrome…
! Constellation of deficit which can also
appear in isolation … ??
! Typically following a left parietal lesion…in
some patients the lesion maybe sub­
cortical … ??

62. a. Group activations for the four
functional contrasts projected
onto flat cerebral hemispheres
are depicted on the left side
and group conjunction analysis
on the right side (L.H = left
hemisphere, R.H = right
hemisphere).
b. MDS representation of
between tasks functional
similarities for parietal lobe (left)
and middle frontal gyrus (right).

65. Conclusions on Gerstman…
! The association between different symptoms can occurr
because:
1. There are damaged fiber bundles that generate from
different and separate regions but they regroup and
follow a common trajectory towards more frontal cortical
regions.
2. The entire region develops inadequately, thus
compromizing mutliple functions
! " If, during ontogenetic development, there is a
malformation of the white or gray matter the
developmental gerstman syndrome can emerge …

66. Dyscalculia “core deficit”
" malformations/malfunctions at the level of the mid­
anterior IPS? " Molko et al., 2007 (Sindrome di Turner –monosomia x
– associata a discalculia)
Anormale densità di sostanza grigia Anormale PROFONDITA’ dell’ HIPS
in HIPS destro destro

67. Dyscalculia “verbal deficit”
" HP: problems in learning and remembering arithm tables (especially
multiplications), and in counting.
" ipoactivation o malfunctioning at the level of the angular gyrus
(association with dyslexia?)
Dyscalculia “spatial deficit”
" HP: problems in visual counting and in tasks requiting orienting on the
number line (es. number­line tests, bisection tests). Problems in written
calculation.
" ipoactivation o malfunctioning at the level of the posterior parietal cortex
(associated to vitsuo­spatial problems? dyspraxia?)
Dyscalculia “executive functions deficit”
" HP:problems in learning and applying calculation procedures
" Frontal cortex problems (associated with ADHD?)

68. 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” visuo­spatial abilities). Is there
transfer of training?