piazza 3 lecture

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).

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.

Q1. Why preschoolers?

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 !!!!!

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

Performance moyenne sur !"#$%&!!& de '()%#'()*+,&-400 450 500 550 600

Shanghai­China 600Singapore 562

Hong Kong­China 555Korea 546

Chinese Taipei 543Finland 541

Liechtenstein 536Switzerland 534

Japan 529Canada 527

Netherlands 526Macao­China 525New Zealand 519

Belgium 515Australia 514Germany 513Estonia 512Iceland 507

Denmark 503Slovenia 501Norway 498France 497

Slovak Republic 497Austria 496

OECD average 496Poland 495Sweden 494

Czech Republic 493United Kingdom 492

Hungary 490Luxembourg 489United States 487

Ireland 487Portugal 487Spain 483Italy 483

Greece 466Israel 447Turkey 445Chile 421

Mexico 419

Q2. Does Math excellence goes beyond preschool?

Trends in International Mathematics and Science Study (based in USA)

Performance significantly higher then USA

Performance non significantly different from USA

9 YEARS 13 YEARS

Performance significantly lower then USA

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’éducationnationale”

./*0)12&1vue des #!34&- sur leurs 5&!()*/0-1(4&$1!&-1professeurs

85

66

67

79

79

78

53

62

80

88

0 50 100

Je m’entends bien avec la plupart de mes professeurs

La plupart de mes professeurs s’intéressent à mon bien­être

La plupart de mes professeurs sont réellement à l’écoute de ce que j’ai à dire

Si j’ai besoin d’aide supplémentaire, mes professeurs me l’apporteront

La plupart de mes professeurs me traitent correctement

OECD average Finland France Germany Japan

Q3. Are there other cultural factors accounting for success in Asian Countries?

29

32

28

19

25

36

44

36

24

37

8

10

7

13

9

0 25 50 75 100

Les élèves n’écoutent pas ce que dit le professeur

Il y a du bruit et de l’agitation

Le professeur doit attendre un long moment avant que

les élèves se calment

Les élèves ne peuvent pas bien travailler

Les élèves ne commencent à travailler que bien après le

début du cours

OECD average Finland France Germany Japan

%

./*0)12&1vue des #!34&- sur le $!*'()2"(665&0)*--(7&

Les choses suivantes se produisent à la plupart, voire à tous les cours

./,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#'/$5()*&-16(5)*$*6()*4&-B1*!1$/04*&0)12&1'*&,?1*0</5'&51!&16,9!*$1-,51!(1+,(!*)#12&-1-&54*$&-16,9!*+,&-

Q4. Pourquoi veut­on mieux enseigner ? Jusqu’à quel point ? Quel prix serait­on prêts à payer ?

Performance moyenne sur !"#$%&!!& de $/'65#%&0-*/0 de !"#$5*)400 450 500 550 600

Shanghai­China 556Korea 539Finland 536

Hong Kong­China 533Singapore 526Canada 524

New Zealand 521Japan 520

Australia 515Netherlands 508

Belgium 506Norway 503Estonia 501

Switzerland 501Poland 500Iceland 500

United States 500Liechtenstein 499

Sweden 497Germany 497Ireland 496France 496

Chinese Taipei 495Denmark 495

United Kingdom 494Hungary 494

OECD average 493Portugal 489

Italy 486Slovenia 483Greece 483Spain 481

Czech Republic 478Slovak Republic 477

Israel 474Luxembourg 472

Austria 470Turkey 464Chile 449

Mexico 425

Q5. Is math superiority specific?

Q6. Should we change our linguistic system for teaching maths (l’académie Aix­Marseille 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 …

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 …

! Dissociations between operations and associated symptoms in brain lesioned patients

Paziente MAR

Lesione: lobo parietale (inferiore)

Sintomi associati (confusione dx­sx e

angosia digitale):

Calcolo: deficit in sottrazioni(moltissimi errori e lenta) ma in grado

di recuperare risultati di tabelline

Paziente BOO

Lesione: lobo frontale inferiore (insula)

Sintomi associati: afasia di BrocaCalcolo: deficit in moltiplicazioni(tabelline) (errori e lentissima) ma

perfettamente in grado di eseguire

calcoli complessi

Dehaene and Cohen, 1997

Blingual subjects exposed to new arithmetical facts in one of the two languages. After learning, researchers verified the

GENERALIZATION1. To the other language2. 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?

Risults

More evidence favouring a verbal code for arithmetical facts

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

EXACT CALCULATION :Increased response to a language change

Left Inferior frontal gyrus (Broca)

Left Angular Left posterior parietal

APPROX. CALCULATION :Increased response to a language change

Left Inferior frontal gyrus (Broca)

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)

Human

LIP (saccades ­ eyes)

AIP (grasping ­ hand)

VIP (multisensory ­ face ­ numerical quantity)

Activity to number processing is embedded in and partially overlapping with circuits involved in action preparation via complex mechanisms of spatial co­ordinate coding and transformations.

At the same time there are evidence that spatial abilities might be very important in calculation and mathematics

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

CAUSES OR EFFECTS OF NUMBER­SPACE ASSOCIATIONS IN THE SCHOOL PROGRAMS?

Massive use of the number line in elementary schools …

Learning numbers…

DRAMATICALLY CHANGES THE PRE­EXISTING REPRESENTATIONS OF QUANTITY

1) APPROXIMATE " EXACT2) COMPRESSED " LINEAR

NOTION OF “ONE” AS THE MINIMAL FIXED DIFFERENCE BETWEEN ANY TWO FOLLOWING INTEGERS

Number ­ space associations

0 100"Position number 64"

[Siegler & Booth, 2004]

Kindergarten 6 years old 7 years old

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]

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 1 2 3 4 5 6 7 8 9 10 END 21

IMPROVEMENTS ALSO IN:1) Numeral identification2) Number comparison3) Counting

Psychological Science, 2008

Kindergarteners

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

Erhsson et al., J Neurophysiol, 2001

« Grasping » numerical magnitude

25 cm

?

Andres et al., Cortex, 2008

Cinematic of grasping

2.2. Planning vs on­line control of grasping

1 1 1Small digits

Large digits

2 2 2

8 8 8

9 9 940 mm 50 mm 60 mm

Description of wooden blocks that subjects have to grasp

Results:A large number written on a block induces a larger grip aperture during grasping trajectory but only in an initial phase of grasping.

Num

erical difference (in mm)

Time(normalized)

Amplitude of grip apertue

Time(normalized)

Interference between numerical magnitide and grasping aperture

Hp 1 : numerical magnitude interferes directly withgrasping aperture during grasping

# After the alaboration of a small number subjectsshould 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 subjectsshould OVERESTIMATE THE MAX size of a graspableobject compared to a neutral condition

2 28

?

Exp 1 : Graspability judgements(max dimension of a graspable item)

12.3 cm 11.8 cm12.6 cm

$

? Longer than X ?

28$2 28

Exp 2 : Perceptual judgements

13.2 cm 13.2 cm 13.2 cm

! SUMMARY

! Numerical magnitude interferes with the estimation of the dimension of the to be grasped object, which then influences the graspabilityjudgement.

! 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 sianorappresentazioni almeno parizialmente comuni dellagrandezza numerica e di altre grandezze, come la grandezza fisica …

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…

! 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

Developmental dyscalculia

! http://dyscalculiaforum.com/viewpage.php?page_id=10

YouTube:! http://www.youtube.com/watch?v=rSb­PD­Od7o&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&feature=related

! 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!! ??

– 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…)

“Symptoms”

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

Llanderl, Bevan & Butterworth, 2004 (8­9 years old kids)

In simple calculation (1 digit numbers) kids with DD make more errors and are slower

Observed difficulties

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 countingLong term memory retrieval

Norm = non dyscaclulicsDC = dyscalculics

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

% di errori

Observed difficulty: strategies?

Finger countingVerbale countingLong term memory retrieval

Norm = non dyscaclulicsDC = 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 self­esteem, 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.”

“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.”

Consequences in adults

Lewis et al.(1994): 1056 kids UK9­10 years oldPREVALENCE: 3.6% (of which 64% Dyslexia)(3.9% Pure dyslexia)

Barbaresi (2005): 5718 kids USA6 ­19 years oldPREVALENCE 5.9 % (of which 43% Dyslexia)Ratio male ­ female 2:1

Gross­Tsur, Manor & Shalev (1996): 3029 kids Israel10 years oldPREVALENCE: 6.5 % (of which 17% Dyslexia and 26% ADHD)Ratio male ­ female 1:1.1

Prevalence & co­morbidity

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

! Number comparison

– Distance effect

– Number­size interference

! Number to space associations

Basic numerical abilities (symbolic)

2 9 7 9

0 105

Llanderl, Bevan & Butterworth, 2004.

7 9

DD are slower in comparing numerical magnitudes but not physical magnitudes

Size comparison Number comparison

Number to space associations

0 100"Position number 64"

Siegler & Booth, 2004

Inter­individual differences in this task correlate with math tasks performance

Kindergarten 6 years old 7 years old

[Geary et al., 2008]

Number to space associations

0.7 1 1.40

20

40

60

80

100

4­6 years 8­11 years Adultsw=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)

% resp «n2 is larger»

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

RESULTS (non dyscalculic subjects)

*

“choose the larger”

n1 n2

[Piazza et al., Cognition 2010]

Basic numerical abilities (non­symbolic)

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

non­dyscalculics

power function (R2 = 0.97)

Age (years)

Estimated weberfraction

[Piazza et al., Cognition 2010]

Estimated w

Distribution Estimates

­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 errors in num

ber com

parison

tasks

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

reading)

In dyscalculic children the ANS is substantially impaired:

*

“choose the larger”

n1 n2

dyscalculics

Basic numerical abilities (non­symbolic)

Causes of dyscalculia! Open ongoing debate! Most popular view is that there can MULTIPLE FORMS, and therefore multiple CAUSES:

– CORE DEFICIT dyscalculia– VERBAL dyscalculia– SPATIAL dyscalculia– EXECUTIVE dyscalculia…

NUMBER SENSE(HIPS)

VERBAL ASSOCIATIONS(left AG)

VISUO­SPATIAL REPRESENTATIONS(PSPL)

NUMBER SENSE(HIPS)

VERBAL ASSOCIATIONS(left AG)

VISUO­SPATIAL REPRESENTATIONS(PSPL)

…Our neurocongitive model can help us formulating hypothesis on the pattern of associations/dissociation and their neural basis …

“Core deficit” dyscalculia " HP (1): problems in understanding the meaning of

numbers (comparison, quantification, approx. calculation)

7 9 Internal w

Distribution Estimates

­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

*

“where are more dots?

n1n2

Developmental Gerstman syndrome [parietal sindrome]: ­Left­right disorientation­Digital agnosia­Dysgrafia­Dyscalculia:

LIP (spazio visivo)

AIP (afferrare, mano)

VIP (quantità numerica)

“Core deficit” dyscalculia " HP (2): often observed in the context of Developmental Gerstman syndrome? Associated also with dyspraxia?

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 sub­cortical … ??

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).

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 …

Dyscalculia “core deficit”" malformations/malfunctions at the level of the mid­anterior IPS? " Molko et al., 2007 (Sindrome di Turner –monosomia x

– associata a discalculia)

Anormale PROFONDITA’ dell’ HIPS destro

Anormale densità di sostanza grigia in HIPS destro

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?)

1) Have a good model2) Develop fine diagnostic tests3) Experiment different treatments (rehab within the number domain but also the associated deficitary domains ...

“core deficit” body schema, finger, quantities;“language” language/reading; “spatial deficit” visuo­spatial abilities). Is there transfer of training?

How to diagnose?" How to “rehabilitate”?