Lecture 4 - Number Concept Flashcards
List of counting principles:
- One-to-one
- Stable Order
- Cardinal
- Order Irrelevance
- Abstraction
One-one-one principle
One tag belongs to only one item in a counting set
Stable Order principle
Tags must be used the same way (for example saying 1, 2, 3 rather than 1, 3, 2)
Cardinal principle
The final tag in the set represents the total number of items
Order irrelevance principle
Result is the same regardless of the order you count the items in
Abstraction principle
Numbers can be applied to tangible and intangible objects unlike other levels eg cat.
Age children understand number concept
Some achievable by 3 but all principles are attainable by age 5
Depth of knowledge of principles
Implicit knowledge of principles but they can’t articulate this knowledge. They can still follow the rules
Gelman and Meck (1983)
Tested 3-5 year olds on one-to-one, stable order and cardinal.
Children monitored puppet performance so they didn’t count themselves.
Gelman and Meck (1983) - one-to-one principle trials
3 trials: correct, in-error (skipped or double counted) and pseudoerror
Gelman and Meck (1983) - stable order trials
2 trials: correct and in-error (reversed 1, 2, 3, 4; random order 3, 1, 4, 2; skipped tags 1, 3, 4)
Gelman and Meck (1983) - cardinal trials
2 trials: correct and in error (total + 1, less than the last number counted, irrelevant feature said as total of the object eg the colour)
Gelman and Meck (1983) - Results on correct trials
One-to-one - 100%
Stable order - 96% and higher
Cardinal - 96%
Gelman and Meck (1983) - results on incorrect trials
One-to-one – 67% (3 years) and 82% (4 years)
Stable order – 76% (3 years) and 96% (4-5 years)
Cardinal – 85% (3 years) and 99% (4-5 years)
Gelman and Meck (1983) pseudoerrors results
Pseudoerrors were detected as peculiar but not correct (only 95% accuracy).
Children were able to say why in some places and show understanding or order irrelevance.
Gelman and Meck (1983) success and set size
Older children performed better but success rates not affected by set size (how much they can count) – even for young children
Gelman and Meck (1983) - Conclusions
Children as young as 3 understand the principles (even though they can’t articulate them)
Understanding demonstrated even in set sizes too big for children to count
Children show implicit knowledge of these principles
Baroody (1984) - Argument
Gelman and Meck (1983) did not answer for cardinality. Children will struggle to understand counting in a different order.
Baroody (1984) - Experiment
Tested order irrelevance and cardinality in 5-7 year olds. Children counted themselves and shown 8 items.
- Count them left to right and then indicate the cardinal value of set
- Then asked “Can you make this number 1”? (pointing to right-most item)
- “We got N counting this way, what do you think we would get counting the other way?” - During this, they could no longer see the array – so had to PREDICT
Baroody (1984) - Results
All but 1 child could recount in the opposite direction. But only 45% of 5 year olds, and 87% of 7 year olds were successful in prediction task
Baroody (1984) - Conclusion
Understanding of order-irrelevance develops with age. Young children’s understanding of principles overestimated
Gelman, Meck and Merkin (1986) - Argument
Task failure was due to a misinterpretation of instructions rather than a lack of understanding (the questions placed self-doubt)
Gelman, Meck and Merkin (1986) - Conditions
- Broody replication
- Count 3x: 3 opportunities to count first
- Altered question: “How many will there be” or “what will you get” (making sure kids don’t doubt their first answer as wrong)
Gelman, Meck and Merkin (1986) - What makes a cardinal principle ‘knower’?
Can solve flexibly across sets, not restricted and really understands how counting works, evidenced across a variety of tasks
Empiricism definition
Knowledge comes from experience, develops gradually
Nativism definition
Innate understanding of some aspects of number concept, ‘core knowledge’
Habituation studies example
Child exposed for a long time to 4 dots, then exposed to 2 dots. If child looks longer at 2 dots they might be able to understand numerosity and basic discrimination.
Xu and Spelke (2000)
Habituation study was babies.
6 month olds discriminated between 8 and 16 dots, 4 vs 8 and 16 vs 32. However couldn’t do 3:2 ratios (eg 8/12) until 9 months old.
Wynn (1992) - Participants
32 5 month old infants
Wynn (1992) - Experiment
Demonstrated adding and taking away with puppets and shown either correct or incorrect trials. Babies looked longer at incorrect trials
Wynn (1992) - Hypotheses
- Infants compute precise results of simple addictions/subtractions.
- Infants expect arithmetical operation to result in numerical change (no expectation of size/or direction of change)
Wynn (1992) - Conclusions
5 month olds can calculate precise results of simple arithmetical operations
Infants possess true numerical concepts – Suggests humans innately possess capacity to perform these calculations
Wakeley et al (2000) - Experiments
Replication of Wynn (1992) and subtraction 3-1 either being 1 or 2.
Wakeley et al (2000) - Results
Babies had no systematic preference for incorrect versus correct in either condition (the second condition controlled for possibility that preferred answer is always greater number of items)
Wakeley et al (2000) - Conclusions
Earlier findings of numerical competence not replicated – Review of literature = inconsistent results
Infants’ reactions are variable–Numerical competencies not robust
Gradual and continual progress in abilities with age
Wynn’s response
Procedural differences affected infant’s attentiveness:
• Use of computer program versus experimenter to determine start
• Didn’t ensure infant saw complete trial
• Exclusion of “fussy” infants higher in Wynn’s (and other) studies
Animal studies with number concept
Consistent findings with the baby studies that they show interest in difference
Modern view: Nativist?
Nativist view dominant
• Born with some innate ability, which expands with age/experience? (ex: Carey, 2009)
• This inborn ability is shared with other animals
Experience/culture in number concept
- Cross-culturally: Language, counting practices impact representation and processing of number (ex. Gobel et al., 2011)
- Within-cultures: Number-talk from parents predicts CP knowledge, related to later performance in school (ex. Gunderson & Levine, 2011)
Conclusions of all studies
Children as young as 3 seem to have implicit knowledge of counting principles.
Evidence of innate ability of numerosity and arithmetical operations.
Also evidence for accumulation of knowledge and being born with limited ability that then expands (Carey, 2009)
Task and procedure have a large effect on results.
