![Number space associations
0 "Position number 64" 100
Kindergarten 6 years old 7 years old
[Siegler & Booth, 2004]](https://image.slidesharecdn.com/piazza3lecture-110411030240-phpapp01/85/Piazza-3-lecture-23-320.jpg)
![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]](https://image.slidesharecdn.com/piazza3lecture-110411030240-phpapp01/85/Piazza-3-lecture-24-320.jpg)
![Number to space associations
[Geary et al., 2008]](https://image.slidesharecdn.com/piazza3lecture-110411030240-phpapp01/85/Piazza-3-lecture-53-320.jpg)
![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]](https://image.slidesharecdn.com/piazza3lecture-110411030240-phpapp01/85/Piazza-3-lecture-54-320.jpg)
![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]](https://image.slidesharecdn.com/piazza3lecture-110411030240-phpapp01/85/Piazza-3-lecture-55-320.jpg)
![“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)](https://image.slidesharecdn.com/piazza3lecture-110411030240-phpapp01/85/Piazza-3-lecture-59-320.jpg)
Piazza 3 lecture
- 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
- 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?
- 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
- 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)
- 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]
- 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.”
- 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
- 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?