Numeracy development Flashcards
How do we know that we have an innate sense of numeracy? Research?
Found basic numerosity in non-humans –> must be innate and not taught
Church & Meck (1984) - rats could discriminate between 2 and 4 light flashes
According to Cooper (1984) what features can 10 month-olds and 14 month-olds detect?
10 month-olds can detect equality
14 month-olds can detect ‘less than’
Which researcher/s found that infants are sensitive to ratios?
Xu and Spelke (2000)
What study did Wynn (1992) do to investigate infants’ numerical understanding?
- Hand puts a teddy in a box, then a screen goes up
- Hand puts a 2nd teddy behind the screen
- Screen goes down and there is only 1 teddy
What did Wynn (1992) measure? What would it indicate?
How long infants looked at the teddy for
If they looked longer, it indicated that their expectations had been violated
What did Wynn (1992) find?
Infants looked longer when there was only 1 teddy behind the screen vs. 2 –> it had violated their expectation
Who criticised Wynn’s (1992) study and why?
Simon et al. (1995) claimed that the results could be explained by the infant’s knowledge of physical objects in the world (= object permanence) rather than numerical understanding
What study was done to test the validity/reliability of Wynn’s (1992) research?
Simon, Hespat and Rochat (1995) repeated the study using a clown instead - 2 teddies were put behind the screen but when the screen went down, it would show 1 clown toy
What conditions were used in Simon, Hespat and Rochat’s (1995) study?
POSSIBLE – screen comes down and shows 2 teddies
ARITHMETICALLY IMPOSSIBLE – screen comes down and shows 1 teddy
IDENTITY IMPOSSIBLE – screen comes down and shows 1 teddy + 1 clown
IDENTITY AND ARITHMETICALLY IMPOSSIBLE – screen comes down and shows 1 clown
What did Simon, Hespat and Rochat (1995) find?
Both arithmetically impossible conditions had longer looking times than arithmetically possible ones
There was no difference in looking times between arithmetically impossible vs. identity and arithmetically impossible
Infants looked at identity and arithmetically impossible for longer than identity impossible
What did Simon, Hespat and Rochat (1995) conclude?
Wynn (1992) was correct - infants can do simple additive reasoning but only of quantities up to 3 (= subitising) – they can accurately see small numbers without having to count (instantly recognisable)
What numbers of dots did Xu, Spelke and Goddard (2005) find that infants could/couldn’t discriminate between?
Infants could discriminate between 8 and 16 dots but not 1 and 2 dots
What is Analogue Magnitude Representation (Dehaene, Dehaene-Lambertz and Cohen, 1998)?
A mental representation of continuous quantities
Perceptual discrimination depends on the similarity of the stimuli intensity (ratio-sensitive)
Which researcher/s found that adults are more precise at deciding if there are 12 dots in a display when there are 4 or 20 dots than when there are 10 or 11 dots?
van Oeffelen and Vos (1982)
What is the symbolic distance effect?
Moyer (1973) - when people are presented with simple physical stimuli (e.g. dots, straight lines), the greater the difference between the two amounts/lengths, the easier the decision (of which is more/less/shorter/longer) and the shorter time taken to respond
Of what is the symbolic distance effect a marker of?
Analogue coding
Who proposed analogue coding?
Moyer and Landauer (1967)
What is analogue coding?
An internal representation (mental image) of an external stimulus that is a copy of the stimulus
Xu and Spelke (2000) found that 8 vs. 16 dots (1:2) was easier for infants to discriminate than 8 vs. 12 dots (2:3). What concept explains this finding?
Analogue Magnitude Representation
By 3 years old, how far can most children count to?
5
Which researcher/s proposed the 5 principles of counting?
Gelmen and Gallistel (1978)
What are the 5 principles of counting?
- One-to-one correspondence
- Ordinality
- Cardinality
- Abstraction
- Order irrelevance
What is one-to-one correspondence?
Child must understand and ensure that each item only receives 1 tag (i.e. only count each item once)
What skills are required for one-to-one correspondence?
- physical/mental tracking of counted and to-be-counted items
- tagging (applying distinct names/tags one at a time and tracking them)
- recognising that tags are abstract and unrelated to the item
What problems may affect one-to-one correspondence?
- rhythm of counting can determine the speed rather than the number of items
- inaccurate finger pointing
What is ordinality?
Child must use the same order in different situations
What skills are required for ordinality?
- must memorise a long, abstract list of words that initially don’t mean anything
- rhythm and intonation may help
What problems may affect ordinality?
Child hasn’t committed name order to memory
Bryant and Nunes (2002) claim that ordinality is not just about the order of the number names, but it is also about…
…using an ordered scale of magnitude (e.g. knowing that ‘3’ represents something larger than ‘2’)
What is cardinality?
Knowing that the final number represents the size of the set
How do Fuson and Hall (1983) say that cardinality is developed?
- Reciting the final number without linking it to quantity, but realising that it is the ‘correct answer’
- Understanding that the last number relates the quantity of the set
- Understanding the progressive nature of cardinality (that the last number they count represents the quantity of that set)
What problems may affect cardinality?
- child hasn’t committed name order to memory
- too simplistic (Bryant and Nunes, 2002)
Bryant and Nunes (2002) claim that cardinality is also about…
…relations between sets of numbers (knowing that a set of 8 is larger than a set of 8) and being able to extrapolate info from one set to use in another
What is abstraction?
Understanding that both real and imagined things can be counted
What skills are required for abstraction?
Understanding that abstract things can be counted (e.g. ideas, events)
What problems may affect abstraction?
Child can only conceive of physical objects
What is order irrelevance?
Knowing that the order that items are counted in doesn’t matter
What skills are required for order irrelevance?
- must understand the 4 other principles first
- recognise that the counted items are ‘things’, not a ‘one’, ‘two’, etc.
- understand that name tags are temporary
- recognise that order irrelevance doesn’t affect cardinality
How do children develop number sense according to the Nativist account?
It is an innate ability
How do children develop number sense according to the Empiricist account?
It is environmental - children learn
How do children develop number sense according to the Interactionist account?
We have some innate ability that develops in the environment
Nativist theories state that we have 2 innate abilities that give rise to number sense. What are they?
- Analogue system - we have access to it from birth, it lets the child make judgements about numerical quantities
- Innate knowledge of the counting principles - number words attached to quantities is derived from the analogue system
What is a criticism of Nativist theories of number sense?
It is unclear if/how the Analogue System is linked to the system based on counting that allows us to understand maths – it partly explains infancy but doesn’t explain all development of number sense
Empiricist theories state that there are 3 ways young children represent number sense (Carey, 2004). What are they?
- Analogue system
- Parallel individuation system
- Set-based quantification
According to Carey (2004), what does the Analogue System do?
- sees number sense as a separate system that adults also use
- disagrees with the Nativist view that it has a role in counting
According to Carey (2004), what is the Parallel Individuation System?
- lets infants exactly represent small numbers
- it can learn number words
- it can link number with the counting system through the ‘bootstrapping’ (induction) mechanism
- children realise that numbers bigger that 3 are hard to discriminate and that they need to learn how to count
According to Carey (2004), what is Set-based Quantification?
- depends on language
- we start learning terms like ‘a’ vs. ‘some’
- we learn that there are quantities and magnitudes
What are criticisms of Empiricist theories of number sense?
X too much emphasis on induction and the role of language
X bootstrapping hypothesis is a circular argument – depends on number knowledge, which is what it is supposed to explain
How did Gelman and Butterworth (2005) criticise Empiricist theories?
Children with poor language still understand large quantities – it isn’t the case that language pulls up (bootstraps) their understanding of number
What does Piaget’s (1952) Interactionist theory of number development claim about our ability to understand relations between quantities?
Understanding relations between quantities is gained through ‘schemes of action’
Actions are initially reflexes (sucking, grasping, etc.)
- representations of actions can be applied to any object
- children predict outcomes of their actions based on their experiences
- action schemes form the basis for children’s understanding of number
Which core insights does the Interactionist theory claim that children must understand?
- Equivalence
- Order
- Class inclusion (= the whole is the sum of its parts)
What can we conclude about these 3 theories of number development?
Nothing - no evidence has contrasted these accounts well; no account is superior
Piaget’s account has prompted the most experimental discovery so Nunes and Bryant (2011) consider it the most successful
Is learning to count to 100 the same as learning 100 words in a particular order?
Counting to 100 is simplified by the understanding of our number system –> recursive, Base-10
Which 2 principles of the number system must children grasp?
Additive principle = ‘twenty one’ = twenty plus one
Multiplicative principle = ‘two hundred’ = 2 x 100
Grasping the 2 principles of the number system depends on which 2 critical insights?
Additive composition of number – any number can be seen as the total of two other numbers; this is important for understanding what number is & that it represents quantity
Multiplicative principle – we must understand that units can have different values (one, ten, hundred, etc.) & each digit’s value depends on its location, from left to right
What type of number system do the Oksapmin society of Papua New Guinea have?
A finite, non-recursive system
What type of systems are usually irregular up to 100? Give an example.
European systems
E.g. French - soixante-deux = 60 + 2 = 62 BUT soixant-douze = 60 + 12 = 72
Which culture doesn’t have names for numbers beyond 2?
Piraha of the Amazon
The Munduraku of the Amazon have inconsistent names up to which number?
5
Explain the number system of Chinese/Chinese-based languages.
Numerical names are compatible with the base 10 system
Spoken numbers correspond to the written equivalent (e.g. 15 is spoken as ‘ten five’; 57 = ‘five ten seven’)
Which countries consistently outperform others on children’s maths attainment?
Mullis et al. (2012) - Japan, South Korea & Russia outperform the USA
From which countries did Miura, Kim, Chang & Okamoto (1988) use 6-7 y/o participants from?
America, China, Japan, Korea
What did Miura, Kim, Chang and Okamoto’s (1988) study involve?
Gave 6-7 y/o 10-based blocks and single-unit blocks
TRIAL 1
Asked children to represent certain numbers using the blocks
TRIAL 2
Asked children to represent the numbers using an alternative method (using the blocks still)
What were the possible solutions that children could use in Miura, Kim, Chang and Okamoto’s (1988) study?
One-to-one collection (e.g. 34 represented by 34 single-unit blocks)
Canonical base-10 (e.g. 34 represented by 3 x 10-base blocks + 4 single-unit blocks)
Non-canonical base (e.g. 34 represented by 2 x 10-base blocks + 14 single-unit blocks)
What did Miura, Kim, Chang and Okamoto (1988) find?
Accuracy in trial 1 was similar in all countries (USA = 91%, China = 99%, Japan = 99%, Korea = 100%)
Accuracy in trial 2 was must lower for American children (34%, 96%, 93%, 99%)
% of children who used the one-to-one correspondence solution in trial 1 was highest in American children (91%, 10%, 18%, 6%)
% of children who used the canonical base-10 solution was lowest in American children (8%, 81%, 72%, 83%)
Which researcher/s replicated Miura, Kim, Chang and Okamoto’s (1988) study?
Saxton and Towse (1998)
What did Saxton and Towse (1998) add to their experiment that was not included in Miura, Kim, Chang and Okamoto’s (1988) study?
Added an extra condition where children were shown possible base-10 answers in a practice session
What did Saxton and Towse (1998) find among the children in this additional condition?
There were no differences between language groups
–> experience/instruction may play an important role
From which countries did Vasilyeva et al. (2015) use their participants from?
Korea, Taiwan (spoke Mandarin), USA, Russia
Who’s paradigm did Vasilyeva et al. (2015) use?
Saxton and Towse (1998)
What was different between Vasilyeva et al.’s (2015) study and Saxton and Towse’s (1998) study?
Vasilyeva et al. (2015) used 5-6 y/o pps instead of 6-7 y/o
What did Vasilyeva et al. (2015) find?
No difference between language groups in the use of base-10 strategies but there was a difference dependent on the instructional condition (i.e. whether they were shown a solution beforehand or not)
How does language influence learning to count?
Children start treating number words by referring to specific numerosities
What system does counting develop?
A symbolic, non-analogue number system
A developed symbolic, non-analogue number system is required for the development of which abilities?
Doing accurate calculations and arithmetic
Counting requires understanding of which concepts?
- understanding of relationships between sets
- Piagetian reversibility
- inversion (4+5=9 AND 9-5=4)
It is not about language but instead ____ that influences how children count.
It isn’t about language but the way that children are told to construct number – depends on experience/instruction
= task-dependent, not language-dependent
