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 and4 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
Which are the biological basis of human culture?- Which are the brain architectures that support extraordinary cultural inventions (suchas reading and arithmetic)?- Which are the new caracteristics in brain architectures that makes the human speciesthe 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 neurobiologye.g. Quartz & Sejnowski (1997) (“The neural basis ofcognitive development: a constructivist manifesto”): «cerebral cortex during development is equipotentialand free to devote specific structures to different Educationdomains» and cognition• Functionalism in psychology:e.g. Fodor, Johnson-Laird:« the physical nature of the huan brain does not impose Behaviourany 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 neuralmechanisms involved in the manipulation of cultural tools.• For both reading and mathematics, even though there are racross cultures a largevariability, there are in all individuals brain regions specialized for each of theseabilities.• These results may seem paradoxical, since the natural evolution does not seem tohave had the time sufficient to select brain architectures specifically to support recentcultural 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 andplasticity (itself genetically determined).• The new cultural acquisitions are possible only inhasmuch they capitalize on thisvariability. 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 isexposed to by a minimal riconversion or “recycling” of pre-existing systems, eventhought they were selected for a different function.
Humans have two systems forperceiving numerical quantity:1. The approximate number system2. The object tracking system
The approximate number system Allows estimation of theapproximate number of items in collections
Demonstration Two sets of different numberWhich set contains more dots?
5 10 10 11 12 24 22 24Ratio (S/L) = 0.5 Ratio = ~0.9Less errors and faster reaction More errors and slower reactiontimes times
Weber lawA psychophysical law describing the relationship between the physical and theperceived magnitude of a stimulus.It states that the threshold of discrimination (also referred to as ‘smallestnoticeable difference’) between two stimuli increases linearly with stimulusintensity.Weber’s law can be accounted for by postulating a logarithmic relation between thephysical stimulus and its internal representation. Weight Loudness Brightness Numerosity
Weber law in 100 80numerosity judgements 60 40 20 Nhabit 16 Nhabit 32 03 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 speciesThe 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 culturesThe Munduruku have number wordsonly up to 5.-They have a perfectly normal non-verbal magnitude system, even forvery large quantities-They have a spontaneous capacity forestimation, 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 Childrenn=9 (55.5 y) n=9 (4.7 y) monolingual, no instructionn=10 (59.3 y) bilingual, no instruction n=7 (8.6 y) monolingual, with instructionn=7 (38.7 y) n=13 (9.6 y) bilingual, with instruction + 10 French adults [Pica, Lemer, Izard, & Dehaene, Science, 2004]
pug ma = oneNumerosity 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 [Pica, Lemer, Izard, & Dehaene, Science, 2004]
Approximation addition and comparison French controls adults M,NI B,NI B,I + childrenn1 n2 All Munduruku M,NI M,I B,I n3 Ratio of n1+n2 and n3 (L/S) [Pica, Lemer, Izard, & Dehaene, Science, 2004]
Internal representation of numerosity: a model 1 2 3 4 5 6 7 8 9… NumerosityActivation 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 SENSITIVE TO A MINIMAL 1:2 RATIO Test 16 88 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 SENSITIVE TO A MINIMAL 4:5 RATIOBarth (2005)
The ANS acuity improves during development ANS acuity (Weber fraction) Round numbers accurately discriminated 2 1:2 10.80.6 2:30.40.2 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 VISUALVedi 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
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 inError 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
SUBITIZINGThis phenomenon:1) Does not depend on the spatial layout of the items (no “patter recognition”) II III IIIIIIIIII2) Does not depend on the nature of the items OX OXO OXOXOX3) It is not altered by tasks tapping on verbal memoryBUT1) 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 discriminableratios)PREDICTIONS:(A) We should find a similar discuntinuity in RTs and errors for similar ratios in thelarge number quantification 1 2 3 4 …8, 10 20 30 40 ... 80(B) Subitizing SPAN (number of objects that can be readily quantified with noerrors) 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 individualswith their basic features, like color shape position) multiple objects at once.PREDICTIONS:(A) Subitizing SPAN should correlate with other measures of multiple objecttracking (e.g., visuo-spatial working memory span)(B) Subitizing SPAN should be altered by a concurrent visuo-spatial task tapping onsimilar 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 Time200 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 interfereselectively with each other.TASK 4 – Dual task : Enumeration + VWM task. Results: Subitizing range is reduced by increasing working memory loadTASK 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 generalarchitecture of our visuo-attentive system, which cannot individuate more then 3-4 itemsat once. It is a separate function from the ANS WHAT (IF ANY) IS IT’S ROLE IN NUMERACY ACQUISITION?