Piazza 3 lecture
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Transcript of 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”?