Number Concept Flashcards
Explain the 5 Counting Principles
- Gelman and Gallistel (1978)
- Five principles govern counting:
1. One-to-one principle
2. Stable-order principle
3. Cardinal principle
4. Order irrelevance principle
5. Abstraction principle
Explain 5 Counting Principle’s One-to-One Principle
- One and only one tag or “counting word” for each item in the set
- ex.
Explain 5 Counting Principle’s Stable-Order Principle
- Tags must be used in the same way
- ex. 1, 2, 3 vs 1, 3, 2
Explain 5 Counting Principle’s Cardinal Principle
- The tag of the final object in the set represents the total number of items
- ex. Knowing the word ‘two’ refers to sets of two entities
Explain 5 Counting Principle’s Order-Irrelevance Principle
- Result is the same regardless of order you count the items in
- ex. cardinality is constant regardless of order
Explain 5 Counting Principle’s Abstraction Principle
- These principles can be applied to any collection of objects (including intangible objects)
At what age are the 5 counting principles typically attained?
- Attainable by the age of 5
- Some achievable by the age of 3
Explain how Gelman and Meck’s (1983) Error Detection Task support The 5 Counting Principles
Tested 3-5yo on the following principles:
- One-to-one Order
- Stable order
- Cardinal
Children monitor the performance of a “puppet” and so, they dont have to count themselves (relieves possible restriction of performance demands)
Explain how Gelman and Meck’s (1983) Error Detection Task support ONE-TO-ONE ORDER
3 types of trials:
1. Correct
2. In-error (skipped or double-counted)
3. Pseudoerror
Results:
- 100% accuracy on correct trials
- 67% (3yo), 82% (4yo) on incorrect trials
- 95% accuracy on pseudoerrors + some ability to articulate why
Explain how Gelman and Meck’s (1983) Error Detection Task support STABLE ORDER
2 types of trials:
1. Correct
2. In-error
*Reversed -> 1, 3, 2, 4
*Randomised -> 3, 1, 4, 2
*Skipped tags -> 1, 3, 4
Results:
- 96% accuracy on correct trials
- 76% (3yo), 96% (4yo) on incorrect trials
Explain how Gelman and Meck’s (1983) Error Detection Task support CARDINALITY
2 types of trials:
1. Correct
2. In-error (Nth value + 1; Less than N; Irrelevant feature of object, e.g. colour)
Results:
- 96% accuracy on correct trials
- 85% (3yo), 99% (4yo) on incorrect trials
Explain how Baroody’s (1984) findings support The 5 Counting Principles
(Principles Tested, Task Procedure, Results)
Tested 5-7yo on the following principles:
- Order-irrelevance
- Cardinality
(Argued that understanding tags does not imply understanding of order-irrelevance and cardinality)
Task Procedure:
1. Children count items themselves; Given 8 items
2. Asked to count left to right and indicate the cardinal value
3. Asked to recount in opposite direction
4. Asked to predict value if the count started from a different item; Can no longer see the array
Results:
- All but 1 child could recount in the opposite direction
- Prediction task: 45% of 5yo and 87% of 7yo
- Conclusion: understanding of order-irrelevance develops with age
- Young children’s understanding of principles overestimated with age
Explain how Gelman, Meck & Merkin (1986) replicated Baroody’s (1984) study
- Argued that the task could affect how the children perform
Task Procedure:
- Replicate exactly as Baroody’s study
- Same but add in count 3x: 3 chances to count first, to let children gain confidence
- Altered-question: Can you start with N? How many will there be? What will you get?
Results:
- Steep increase of correct children from Baroody replication to Count 3x to Altered Qs
Define empiricism
Knowledge comes from experience, develops gradually
Define nativism
Innate understanding of some aspects of number concept; “Core knowledge”
Explain Xu and Spelke’s (2000) habituation study and how it informs nature vs nurture
- 6mo discriminated between 8 and 16 dots (in diff patterns to ensure its not the pattern/shape they’re getting used to)
- Replicated w/ 4vs8 and 16vs32
- Ability to detect more precise ratios continues with development
Explain Wynn’s (1992) Addition and Subtraction procedure
(Task Procedure, Results, Conclusions)
Task Procedure:
- 32 5mo infants
- Looking time procedure
- Shown different mathematical operations; Possible and impossible events
- Intended to test if its discrimination or numerical concept
- Experiments 1 (+), 2 (-), 3 (+; 2/3)
Results:
- Pre-test trials: no difference in looking times
- Test trials: Infants looked longer at the “incorrect” result
Conclusions:
- 5mo can calculate precise results of simple arithmetical operations
- Infants possess true numerical concepts -> Suggests humans innately possess capacity to perform these calculations
Explain Wakeley et al’s (2000) study that replicated Wynn’s (1992)
(Task Procedure, Results
Task Procedure:
- 3 Experiments
- Replications of Wynn (1992) exps 1 & 2
- Subtraction counterpart to Wynn’s Exp 3 (-; 1/2) -> Controls for possibility that preferred answer is always greater no of items
Results:
- No systematic preference for incorrect vs correct
- Earlier findings of numerical competence not replicated -> Inconsistent results
- Infants’ reactions are variable; Numerical competence not robust
- Gradual and continual progress in abilities with age
How did Wynn respond to the Wakeley et al (2000) study
- Procedural differences affected attentiveness of infants (Wakeley’s use of a computer program vs experimenter to determine start)
- Even though it is less prone to human error, use of comp program leads to possibility that the infants weren’t actually paying attention or saw the complete trial
- Exclusion of “fussy” infants higher in Wynn’s (and other) studies
What is the evidence that animals can develop numerical concepts?
- Primate researchers in Japan taught chimpanzees how to count
- Chimpanzees were able to order the numbers in the correct order on a screen
- They also demonstrated impressive photographic memory in ordering the numbers
