Lecture 4 Flashcards
(22 cards)
Define numerical cognition
basic capacitys, innate and not just education
Define number abstraction
counting, estimating objects, representations of the amount/quantity
Define number reasoning
understanding the principles of how manipulations affects sets.
explain ‘adult number sense is rato bound’
- adults: good at making distinctions between amounts, quantities and rapid decisions between symbolic numbers.
- able to make quick judgements of more or less but when numbers close together e.g. 3:4, this is harder.
explain the ratio bound effect
if the difference between two digits is large = reaction time is fast. if digit difference is slow = slower reaction = distance effect.
Ratio = Bigger number / smaller number
explain subsitizing: enumerating without consciously counting
- takes 400ms to register/count objects
- by 4 objects: have to count explicitly = subsitizing = slower.
- innate capacity (as adults can make big distinctions between quantities and make rapid numerations of small sets).
Explain distance error: judging numerical inequality
Moyer & Landauer: distance effect = errors more common when numerical distance between digits is smaller
What is Piagetian perspective on childrens understanding of numbers
- concrete operations (7-12yrs) as fail tests of logic till 7yrs old.
- children must have general logic incl: conservation (change in appearance doesnt mean change in amount), class inclusion and seriation (ordering in sequence).
- number concept acquisition: domain-general process.
- no innate sense of number
What are challenges to piagetian perspective on childrens understanding of numbers
Mcgarrigle & Donaldson:
- naughty teddy conservation task: before 16% 5yrs old passed vs 63% passed when naughty teddy accidentally moves counters. as extra linguistical features can influence interpretation of question.
- conservation tasks aren’t valid as don’t get at way children reason correctly: Mehler & Bever: 200 children M&M vs clay conditions. all children picked the row with more M&Ms for the eating condition yet wrong row for the clay. Failed conservation test but understood bottom row had more M&Ms thus isnt good test.
Explain principles of counting
- learning to count = crucial to understand numbers/representations
- 2.5yrs: children exhibit number principles
- 3.5yr: would correct puppet counting when wrong
- guided by innate abstract principles - domain specific view of numerical cognition (Nativist)
what are challenges to principles of counting:
- constructivists argue not innate and children derive principles from experience
- cardinality: something we learn so counting routine not linked to number concepts.
explain pre schoolers apply principles of counting:
GELMAN & GALLISTER
1) one to one principle (each item one tag)
2) stable order principle (stable, repeatable order)
3) Cardinal principle (final tag represents quantitys)
4) Abstraction principle (any events can be counted)
5) Order irrelevance (can start counting from wherever)
Explain infants numerical abilities: Starkey & Cooper
looked at if infants can discriminate small numbers of items in subitizing range. found sig. dishabituation in small number condition but not large number. suggests after 3, counting is required.
Explain infants numerical abilities: Violations of Expecations (Wynn)
are children able to recognise simple addition and subtraction? when see impossible outcome = look longer showing infants do basic arithmetic (innate capacity)
What are challenges to Infants numerical abilities? - Clearfield & Mix (1999)
- found continuous variables e.g. area, contour are correlated with number - maybe no numerical reasoning going on but able to distinguis about amount of space covered (perceptual explanation).
Why are some people good at maths?
1) genetic influences on maths (overlap on IQ, maths and english based on twin studies showing heritability)
2) Genetic influences on number sense (0.32 heritability but hereitability can change with age - substantial role for enviro)
3) Do early experiences matter (SES strong predictor of maths ability when starting at primary school: enviro/early learning experiences)
Explain number sense hypothesis
brain: innate mechanism (evolutionary) for apprehending numerical qulalities. evidence: adults/infants/animals othr cuktures with limited education can apprehend numerosities. survival value = food
- number sense: approximate, imprecise and subject to limits. is what allows us to develop formal mathematical ablities.
Within the number sense hypothesis, define…
- Analogue magnitude representatioon system
- Exact number system
- Analogue magnitude representatioon system: number sense born with. allows us to make quick estimations of quantitys within ratios incl subsitization.
- Exact number system: education and culture born. can only engage with concept beacause of our innate abiltieis e.g. visual arabic number system and auditory verbal word system.
What is some support for the number sense hypothesis?
- same part of brain active in studies (HIPS in mathematical problems)
- number sense active throughout lifespan. across cultures and across development.
- acuity of number sense is correlated with performance on tests of mathematics. r: 0.24 - weakly correlated with maths ability
describe longitduinal evidence for number sense hypothesis
- results mixed
- one study showed number sense acuity predicted maths and another study showed t did not predict maths when controlling for number knowledge.
- adding in extra variables e.g. IQ explain variance
- argued infant research on detecting numerosity tells us little about how children reason about numbers
describe Xu and Spelkes study on infants numerical abilities.
- used arrays to large to be handled by attending to specific objects and controlled perceptual confounds. used same space covered but dif number of objects.
- compared large differences in magnitude. infants can discriminate between them same way as adults can but struggle when ratios are closer.
- suggests infants have basic number ratio dependent approximation system but fails with smaller sets.
explain core systems of number - so infants have numerical abiltiies?
Feigenson 2004:
- basic approximation of numerical magnitude in infants.
- imprecise and subject to ratio limits
- can handle large numerosities
- changes as gets older e.g. 1:2 not 2:3 at 6 months but at 10 months can do 2:3.
- harder with small numerosities when tasks control for perceptual features.
- not limited to visual arrays - also done with sounds
