Lecture 4 - Number Concept

Question 1

List of counting principles:

Answer 1

1. One-to-one
2. Stable Order
3. Cardinal
4. Order Irrelevance
5. Abstraction

Question 2

One-one-one principle

Answer 2

One tag belongs to only one item in a counting set

Question 3

Stable Order principle

Answer 3

Tags must be used the same way (for example saying 1, 2, 3 rather than 1, 3, 2)

Question 4

Cardinal principle

Answer 4

The final tag in the set represents the total number of items

Question 5

Order irrelevance principle

Answer 5

Result is the same regardless of the order you count the items in

Question 6

Abstraction principle

Answer 6

Numbers can be applied to tangible and intangible objects unlike other levels eg cat.

Question 7

Age children understand number concept

Answer 7

Some achievable by 3 but all principles are attainable by age 5

Question 8

Depth of knowledge of principles

Answer 8

Implicit knowledge of principles but they can’t articulate this knowledge. They can still follow the rules

Question 9

Gelman and Meck (1983)

Answer 9

Tested 3-5 year olds on one-to-one, stable order and cardinal.

Children monitored puppet performance so they didn’t count themselves.

Question 10

Gelman and Meck (1983) - one-to-one principle trials

Answer 10

3 trials: correct, in-error (skipped or double counted) and pseudoerror

Question 11

Gelman and Meck (1983) - stable order trials

Answer 11

2 trials: correct and in-error (reversed 1, 2, 3, 4; random order 3, 1, 4, 2; skipped tags 1, 3, 4)

Question 12

Gelman and Meck (1983) - cardinal trials

Answer 12

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)

Question 13

Gelman and Meck (1983) - Results on correct trials

Answer 13

One-to-one - 100%
Stable order - 96% and higher
Cardinal - 96%

Question 14

Gelman and Meck (1983) - results on incorrect trials

Answer 14

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)

Question 15

Gelman and Meck (1983) pseudoerrors results

Answer 15

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.

Question 16

Gelman and Meck (1983) success and set size

Answer 16

Older children performed better but success rates not affected by set size (how much they can count) – even for young children

Question 17

Gelman and Meck (1983) - Conclusions

Answer 17

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

Question 18

Baroody (1984) - Argument

Answer 18

Gelman and Meck (1983) did not answer for cardinality. Children will struggle to understand counting in a different order.

Question 19

Baroody (1984) - Experiment

Answer 19

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

Question 20

Baroody (1984) - Results

Answer 20

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

Question 21

Baroody (1984) - Conclusion

Answer 21

Understanding of order-irrelevance develops with age. Young children’s understanding of principles overestimated

Question 22

Gelman, Meck and Merkin (1986) - Argument

Answer 22

Task failure was due to a misinterpretation of instructions rather than a lack of understanding (the questions placed self-doubt)

Question 23

Gelman, Meck and Merkin (1986) - Conditions

Answer 23

1. Broody replication
2. Count 3x: 3 opportunities to count first
3. Altered question: “How many will there be” or “what will you get” (making sure kids don’t doubt their first answer as wrong)

Question 24

Gelman, Meck and Merkin (1986) - What makes a cardinal principle ‘knower’?

Answer 24

Can solve flexibly across sets, not restricted and really understands how counting works, evidenced across a variety of tasks

Question 25

Empiricism definition

Answer 25

Knowledge comes from experience, develops gradually

Question 26

Nativism definition

Answer 26

Innate understanding of some aspects of number concept, ‘core knowledge’

Question 27

Habituation studies example

Answer 27

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.

Question 28

Xu and Spelke (2000)

Answer 28

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.

Question 29

Wynn (1992) - Participants

Answer 29

32 5 month old infants

Question 30

Wynn (1992) - Experiment

Answer 30

Demonstrated adding and taking away with puppets and shown either correct or incorrect trials. Babies looked longer at incorrect trials

Question 31

Wynn (1992) - Hypotheses

Answer 31

1. Infants compute precise results of simple addictions/subtractions.
2. Infants expect arithmetical operation to result in numerical change (no expectation of size/or direction of change)

Question 32

Wynn (1992) - Conclusions

Answer 32

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

Question 33

Wakeley et al (2000) - Experiments

Answer 33

Replication of Wynn (1992) and subtraction 3-1 either being 1 or 2.

Question 34

Wakeley et al (2000) - Results

Answer 34

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)

Question 35

Wakeley et al (2000) - Conclusions

Answer 35

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

Question 36

Wynn’s response

Answer 36

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

Question 37

Animal studies with number concept

Answer 37

Consistent findings with the baby studies that they show interest in difference

Question 38

Modern view: Nativist?

Answer 38

Nativist view dominant
• Born with some innate ability, which expands with age/experience? (ex: Carey, 2009)
• This inborn ability is shared with other animals

Question 39

Experience/culture in number concept

Answer 39

• 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)

Question 40

Conclusions of all studies

Answer 40

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.