psyc30018 numerical processing (lectures 7-10)

numbers are special because…

• we use number all the time

• number is the basis of civilisation

• low numeracy is a problem (dyscalculia)

• we are not effective at improving numeracy

what do we mean by number?

• exact representations of objects

• approximate number

• symbolic representations of number (Arabic numerals)

• non-symbolic number

evidence for number reasoning not simply based on language processes

• neuropsychological evidence (dyslexia v. dyscalculia)

• number perception is evident in other animals

• number is processed differently from language

• preverbal children

• cultures with limited language for numbers

Ai & son, Ayumu (chimpanzees)

• trained to performed symbolic numerical tasks

• first non-human to count with Arabic numerals

• training through observation resulted in son outperforming mother in computer tasks without training & humans

evidence in preverbal children

• in preferential looking tasks (that reflect novelty), babies look longer at dot displays that change in number (Starkey, 1980)

• 6mo babies can also distinguish large dot numerosities (Xu & Spelke, 2000)

Wynn (1992)

• preverbal children may understand basic arithmetic

• character stared longer at the dolls once the screen was removed if the result did not match ‘two’

Munduruku tribe

• do not use numerals in a counting sequence

• do not refer to precise quantiles

• e.g. the sentence ‘I want fish for six people’ does not exists

• numbers 1-4 exist

Pica (2004)

• Munduruku can map quantity to a spatial position using a number line task

• they map number to position in a logarithmic manner, like western children (though western adults are typically linear)

Butterworth et al. (2008)

• Australian Aborigines had a precise sense of number without words for those exact numerosities

simple core mechanisms of numerical competence

• to identify, order and compare quantities of objects (non-symbolic)

enumeration

• exact number of objects on display

number comparison

• approximate sense of numerosity of two groups of objects on display

steps of enumeration

• encoding of visual information into objects of interest

• combining the instances of an object into a total or sum

• may be strategy dependent (e.g. serial, ‘groupitising’)

subitising

• good at counting small sets of objects

• rapid & accurate for set sizes <4ish (Kaufman et al. 1949)

the approximate number system

• not as good at counting larger sets of objects

• slow and imprecise for set sizes >4ish

distance effect

• symbolic number comparison

• slower RT to prompt of which two Arabic numerals was larger when close in numerical distance (e.g. 2/3 vs. 4/1)

• suggests that neural mechanisms are ordered in a functional way (e.g. mental number line)

Weber fraction (ratio effect)

• non-symbolic number comparison

• errors depend on the ratio of the magnitudes

• implied higher sensitivity to small differences in ratio

• imprecise in the subitising range (1-4)

• tuning for numerical similarity is linear on a log scale

brain areas implicated in number processing in primates

• parietal and front regions

  • intraparietal sulcus (IPS)

  • lateral prefrontal cortex (PFC)

  • superior parietal lobule (SPL)

  • ventral parietal area (VIP)

brain areas associated with acquired acalculia

• parieto-occipital junction

• frontal lobes

how to measure neuronal responses during number task

• record electrical activity in single neurons

  • measure action potentials

  • search for neurons that respond to the ‘number’ in a display

neurons that code for numerical quantity

• useful to estimate number

• might be used to compute exact number

neuronal codes for proportional representations

• useful for judgements of relative size

• does not require symbolic representation

symbolic number neural codes

• necessary for communication

• language

numerosity tuning can be determined using…

a delayed match-to-sample task

numerosity tuned neurons are abundant in…

later PFC and IPS

• number selective cells are not all clustered together

• average latencies of IPS neurons shorter than PFC (suggesting number could be processed in the IPS and then passed on to the PFC)

dyscalculia

• specific & severe disability in learning arithmetic

• may co-occurs with developmental disorders (e.g. dyslexia, ADHD)

• persists into adulthood

• mathematical abilities have high specific heritability

consequences of dyscalculia

• earn less

• are more likely to be sick

• more likely to be in trouble with the law

• need more help in school

what dyscalculics find difficult

• everyday number usage (e.g. remembering phone no., assigning number to distance)

• everyday number tasks that require simple arithmetic (e.g. counting change)

• simple number comparison and addition tasks (e.g. using fingers to keep count)

• even approximate estimate tasks are effortful (e.g. count all the symbols on two playing cards to say which is the larger)

• number sequence tasks

dyscalculia & deficits in basic numerosities

• subitising

• number comparison

• poor magnitude comparison

core deficit hypothesis

• problem perceiving (non-symbolic) quantities is the cause of dyscalculia

access deficit hypothesis

• dyscalculics only have problems with the processing of numerical symbols

• they have an inability to associate numbers with the underlying magnitude representation (e.g. Arabic numerals)

structural anomalies in young dyscalculics

• reduced grey-matter density in left IPS in adolescent dyscalculics (contains neurons ass w/ neural processing / fewer neurons = less capacity for performing numerical tasks)

• reduced right IPS grey-matter density in 9-year-old dyscalculics

• reduced probability of connections from right fusiform gyrus to other parts of the brain, including the parietal lobes