Number Concept

Question 1

Explain the 5 Counting Principles

Answer 1

• 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

Question 2

Explain 5 Counting Principle’s One-to-One Principle

Answer 2

• One and only one tag or “counting word” for each item in the set
• ex.

Question 3

Explain 5 Counting Principle’s Stable-Order Principle

Answer 3

• Tags must be used in the same way
• ex. 1, 2, 3 vs 1, 3, 2

Question 4

Explain 5 Counting Principle’s Cardinal Principle

Answer 4

• 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

Question 5

Explain 5 Counting Principle’s Order-Irrelevance Principle

Answer 5

• Result is the same regardless of order you count the items in
• ex. cardinality is constant regardless of order

Question 6

Explain 5 Counting Principle’s Abstraction Principle

Answer 6

• These principles can be applied to any collection of objects (including intangible objects)

Question 7

At what age are the 5 counting principles typically attained?

Answer 7

• Attainable by the age of 5
• Some achievable by the age of 3

Question 8

Explain how Gelman and Meck’s (1983) Error Detection Task support The 5 Counting Principles

Answer 8

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)

Question 9

Explain how Gelman and Meck’s (1983) Error Detection Task support ONE-TO-ONE ORDER

Answer 9

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

Question 10

Explain how Gelman and Meck’s (1983) Error Detection Task support STABLE ORDER

Answer 10

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

Question 11

Explain how Gelman and Meck’s (1983) Error Detection Task support CARDINALITY

Answer 11

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

Question 12

Explain how Baroody’s (1984) findings support The 5 Counting Principles
(Principles Tested, Task Procedure, Results)

Answer 12

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

Question 13

Explain how Gelman, Meck & Merkin (1986) replicated Baroody’s (1984) study

Answer 13

• 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

Question 14

Define empiricism

Answer 14

Knowledge comes from experience, develops gradually

Question 15

Define nativism

Answer 15

Innate understanding of some aspects of number concept; “Core knowledge”

Question 16

Explain Xu and Spelke’s (2000) habituation study and how it informs nature vs nurture

Answer 16

• 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

Question 17

Explain Wynn’s (1992) Addition and Subtraction procedure
(Task Procedure, Results, Conclusions)

Answer 17

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

Question 18

Explain Wakeley et al’s (2000) study that replicated Wynn’s (1992)
(Task Procedure, Results

Answer 18

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

Question 19

How did Wynn respond to the Wakeley et al (2000) study

Answer 19

• 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

Question 20

What is the evidence that animals can develop numerical concepts?

Answer 20

• 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