Lifespan A: Children's understanding of number, WEEK 5 Flashcards
1
Q
Number cognition
A
- Number abstraction: very basic ability to count + quantify things (intuitive) > can make rough estimations w/o counting > non-symbolic reasoning
- Numerical reasoning: product of our education (as not all species use maths) > explicit reasoning skill to solve maths problems + manipulations > symbolic reasoning as we use symbols our culture created to solve things
2
Q
Number sense & ratio
A
- can guess when things are more or less w/o counting, but depends on ratios
- if the ratio is far apart, we can easily identify or estimate accurately > when the ratio is close together, it will take longer to form an accurate estimation > number sense is ratio bound
- e.g: can see difference between 2 and 8 marbles but it is harder to see difference between 7 and 8 at first glance (number sense)
3
Q
Judging number inequalities w/ number symbols
A
- Plot our understanding of quantity onto symbolic numbers > we reason w/ numbers learnt from culture
- Make quick judgements about whether there is more or less in terms of dots/objects but about numbers learnt in ed too > it is more obvious that 9 is bigger than 2 so we react quicker but less obvious that 6 is bigger than 5 so react slower
- Errors are more common when numerical distance between digits is smaller
4
Q
Subitizing
A
- Rapidly enumerating a set of objects + is quick judgement > when objects are between 1-3 we subitize (know number w/o counting)
- Research shows beyond 3 we start counting + slow down reactions
- Some argue subitizing is an innate ability
- Positive correlation between time + amount of dots > more dots = more time taken
5
Q
Piagetian perspective
A
- Basic logical development forms basis for numerical reasoning > domain general process
- Piaget argues even if children can count verbally, this doesn’t show understanding of number
- Kid’s do not understand meaning of number until they real concrete operations > no innate understanding of number
- To understand number, you must be able to conserve, have class inclusion + seriation (concrete ops)
- Conservation = change in appearance doesn’t = change in quantity > requires understanding of quantity, space + amount
6
Q
Challenges to Piagetian perspective
A
- Basic understanding of no. > more/less > Piaget’s study requires ability to understand language
- McGariggle + Donaldson naughty teddy task > W3 cue cards
- Mehler & Behver split ppts into 2 groups > one used clay pellets + other m&ms
- were asked which row has more, are they the same + take the row you want to eat
- Children were right more in m&m conditions > Mehler & Bever argue kids have implicit understanding of number > before they learnt to count, they understood more vs less in a motivated task (they wanted to eat m&ms > wanted to eat row with more m&m)
7
Q
Implication’s of Piagetian perspective
A
- Piaget’s work suggested kid’s were not ready to reason numerically until age 7 (concrete ops) > lead to education systems not emphasising numerical reasoning up until this point
- Could be detrimental to maths ed + impacts society
8
Q
Gelman & Gallistel (1986)
Principles of counting
A
- Most researchers focused on conservation + dismissed counting but Gelman studied counting itself + counting behaviour in children to see if it shows how they reason about numbers > kids between 3.5-4.5yrs
- Argue learning to count is guided by innate abstract principles (schemes) guiding acquisition of no. concept + counting > due to these principles, greater attention is given to words about numbers
- Domain-specific view of numerical cognition > focuses only on number itself + is nativist
9
Q
Principles of counting
A
- one to one principle > set of objects tagged w/ a number (1,2,3,4)
- stable order p > order sequence (1-5) > repeatable
- Abstraction principle > anything can be counted w/ no.
- Order irrelevance > can count even if numbers aren’t in order or start at a random point
- Cardinal principle > full understanding of no. is when you recognise the final number in a set is equal to the amount of objects in that set
- Principles 1-4 = procedure while 5 is the concept of no.
- Gelman argues children’s behaviour during counting shows they understand basic principles of counting no.
- Experience counting refines understanding of no. through different procedure (1-4)
10
Q
Measure’s of children’s access to number
A
- Verbal counting task > how high can you count > measure the sequence of no + labels of no. understood
- Enumeration task > could you help big bird count his toys by pointing to each one? > record child + can measure 1-1 principle, order irrelevance + cardinality
- Numerical recognition > set of no. + ask which one is 2 e.g > doesn’t measure principles of counting but measures access to number, symbols + words > younger children won’t always be able to do this as it is a product of education
- Give-a-number task > Kermit asks for x amount of toys he wants from you > instead of watching the counting behaviour of children we look at if the child recognises the final number is the total amount Kermit wants
- Point-to-x task > can you point to 3? > sees if child counts the objects then points or if they do it w/o counting > tests cardinality
11
Q
Challenges to principles of counting
A
- Cannot be sure children’s counting because it’s innate > may have learnt from parents (EV) > children may derive principles after experience
- Gelman said counting is innate as she saw it in early childhood > just because you see something in early childhood doesn’t make it innate
- Wynn made a task where toys were counted when they were visible then went into a box where there were sounds + jumping in sets of 2,3,5,6 + asked how many were in the box > older children 3.5yrs counted more accurately than younger 2.5 year olds > they had to keep the number in mind + understand final number
- 3.5 year olds were more likely to use cardinality principle + understanding of cardinality is more important to understanding number
12
Q
Types of numbers
A
- Symbolic number: Abstract and exact representations, Number Words and Digits
- Non-symbolic number: Perceiving Quantities of Objects/Events, Comparing Quantities + imprecise
- We are quick at counting small sets of numbers but not big no. beyond 5 > can count non-symbolic no. quicker but is more imprecise > is this innate?
13
Q
Infant’s numerical ability: Starkey & Cooper (1980)
A
- Infant’s are non-verbal so we rely on looking beh
- If an infant recognises a visual change in the amount of something they will look longer (dishabituation)
- Starkey & Cooper look at 5.5 month old infants to see if they can subitize + discriminate between no. sets
- Found dishabituation in small number conditions when there was an extra dot added (e.g: 2 dots > 3 dots) > but no dishabituation when there were larger numbers (4-6)
- Habituation is where the infant’s looking behaviour is not surprised + more bored
- Suggests infant’s have basic subitization ability which is more advanced in adults
14
Q
Infant’s numerical ability: Wynn (1992)
A
- If the infant understands that a concept or rule has been broken, they will look longer than if it is knowledge consistent.
- Tested 5mo infants > condition 1: 1 mickey mouse puppet, cover w/ board, another puppet enters + hand in puppet leave + left w/ 2 puppet when board goes down (KC) > condition 2: same as above ^ but only 1 puppet is left
- Infant’s looked reliably longer in unexpected outcome
- Supports that infant’s aren’t perceiving the amount of space covered but counting somehow
15
Q
Infant’s numerical ability: reliable?
A
- Christodoulou wanted to see if claims were robust in meta-analysis reviewing studies replicating Wynn
- 26 studies used (550 infants)
- Found results are reliable that infant’s can detect difference in amount but cannot when it’s more than 3
- Results hold even when task factors change like number of objects, infant age, type of stimulus
- Cohen’s d- 0.34 > moderate effect of condition
16
Q
Infant’s numerical abilities: challenges
A
- Stimulus don’t differ in amount but also by contour length (amount of white space covered) > more black lines in second condition
- Maybe children are just able to detect basic differences in space, colour, or light
- Clearfield & Mix find contour length + area is correlated w/ number
- Tested if 7mo infants were sensitive to no. controlling for area (A) or to area controlling for no. (B) > increased amount of squares but w/ different sizes so amount of space covered is equal (A) > Compared 2 squares w/ the same amount in no. but increased size, bigger squares (B)
- Infant’s didn’t see a difference in condition A even though there were more squares because they were made smaller + were equal in area covered but did in condition B > challenges earlier studies > innate?
17
Q
Do infant’s actually have numerical abilities?:
Xu & Spelke (2000)
A
- Argues infant’s may just be good at making judgement’s on differences between larger quantities rather than individual small quantities
- Used arrays too large to be handled by attending to specific objects (object based attention) + controlled for perceptual confounds like size + colour
- Condition 1: can infant’s discriminate between 8 vs 16
- Condition 2: can infant’s discriminate between 8 vs 12 > like adult’s do worse on ratio’s close together (ratio bound) but better on those far apart
18
Q
Do infant’s actually have numerical abilities?:
Feigenson et Al., (2004)
A
- Argues infant’s can make distinctions between large sets but is ratio bound > as we get older we improve
- 6mo infants can differentiate between 1:2 but no 2:3 > 10mo can do 2:3
- Doesn’t have to be visual arrays, could be auditory like 3 sounds vs 2
19
Q
Number sense hypothesis: N/N
A
- Animals can detect quantities in a similar way to humans > Dehaene argues this must be an innate ability which exists across species as it has survival value
- Number sense helps understand number reasoning
- Argues infants, adults, and animals from cultures w/ limited ed can apprehend numerosities
- This number sense is rapid but approximate, imprecise + subject to limits
- Contrasts this “approximate no. system” w/ exact symbolic no. system which is a product of ed
- Argues there are 2 systems (triple code model) > Number sense (analogue, magnitude, representation)> born w/ basic ability to detect quantities > helps benefit from learning exp (N/N interaction) > basis for other systems
- Auditory no. system: develops in ed where we learn rules of addition + subtraction > visual no. system developed in ed where we learn to manipulate symbols (auditory + visual = one system)
- Experience of learning refines number sense so we become better at making estimations from exp
20
Q
Support for number sense hypothesis
A
- Active throughout lifespan
- 6 Months: 2:1 ratios > 9 Months: 3:2 ratios > 3 Years: 4:3 ratios > 6 Years: 6: 5 ratios > Adults: 11:10 ratios
- As we mature we gain more experience + refine number sense which is why discrimination between sets gets better but remain ratio dependant at all stages > developmental change
21
Q
Does the approximate number system relate to symbolic number system?
A
- Developmental possibilities
1. ANS is foundation for symbolic number knowledge.
2. Experience of symbolic number knowledge refines the precision of ANS.
3. ANS / symbolic number knowledge are reciprocally related > as one refines the other improve + vice versa
4. ANS and symbolic number knowledge are unrelated.
22
Q
Number sense & symbolic number
A
- Evidence for a reciprocal relationship
- Elliot et Al measured 193 3-5yr olds over 12 months
symbolic reasoning (counting etc) and number sense (estimation) - Children who did better on ANS task did better 6 months later on the symbolic task, 6 months after ANS improved + became refined
- Children who did better on symbolic task improved in ANS 6 months later then symbolic no. refined 6 months after that
23
Q
Number sense & mathematical ability
A
- Evidence for predictive effect > looks at if early performance on number sense correlated w/ later maths ability
- Meta analysis (17,201) performance on number sense tests correlate w/ maths performance (r= .24)
- Longitudinal findings are mixed > Libertus et Al found number sense predicted maths in 175 4 yr olds
- Gobel et Al however found in 173 6 yr olds that number sense did not predict maths when controlling for number knowledge
24
Q
Genetic origins of number sense
A
- Number sense is genetically influenced but there is still EV influence > experience of maths may refine number sense > more research needed
- Tosto et Al find in 2259 twin pairs aged 16 a modest heritability (.32) of individual differences in Number Sense > Substantial role for Non-Shared Environment (.68)
- Braham & Libertus find associations between parent number sense + child number sense > looked at 58 parent-child pairs > did same task + compare parent and child result > positive correlation between results but parents + kids share genes and EV so we don’t know what this association is a result of
25
Q
Parent-child interaction & number knowledge
A
- Informal learning experiences at home might aid the emergence of symbolic number knowledge.
- 44 parent-child dyads seen at 14, 18, 22, 26, 30 months.> Observed interacting at home for 90 minutes. > conversation coded for total amount of ‘number talk’.
- Parent Number talk increased from 14 to 30 months. > Marked individual differences in parent number talk. > Number talk predicted child ‘point-to-x’ score but not verbal ability.
- Parent’s who spoke to their children more about no. did better in number reasoning task > early exp may associate w/ reasoning skills before going to school
