PSY2001 Lecture 6 - The Number Concept

Number concept

Numerosity, counting, arithmetic

The 5 Counting Principles

Gelman and Gallistel (1978)

• One-to-one principle

• Stable-order principle

• Cardinal principle

• Order irrelevance principle

• Abstraction principle

3 counting principles that define counting procedure

• One-to-one principle

• Stable-order principle

• Cardinal principle

One-to-One principle

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

Stable-order principle

Tags (counting word) must be used in the same way

• Ex. 1,2,3 vs.1,3,2

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

Order-irrelevance principle

Result the same regardless of order you count items in

Abstraction principle

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

• Not “labeling” (like the label “cat”)

Children’s knowledge of the principles

Implicit knowledge

• Can’t articulate this knowledge but follow the rules

All attainable by the age of 5

• (some achievable by 3)

Error detection task

Gelman and Meck (1983)

3 to 5-year-olds tested on 3 principles:

• One-to-one

• Stable order

• Cardinal

Children monitor performance of a “puppet”

• Don’t have to count themselves (relieves possible restriction of performance demands)

One-to-One principle

3 types of trials

• Correct

• In-error (Skipped or double-counted)

• Pseudoerror

Stable-order principle

2 types of trials

• Correct

• In-error

  • Reversed: 1,3,2,4;

  • Randomly-ordered: 3,1,4,2;

  • Skipped tags: 1,3,4

Cardinality

2 types of trials

• Correct

• In-error (Nth value + 1; Less than N; Irrelevant feature of object, e.g. colour)

One to one principle trials

Accuracy on correct trials: 100%

Accuracy on in-error trials: 67% + (3yrs); 82% + (4yrs)

Stable-order principle trials

Accuracy on correct trials: 96% and higher (+)

Accuracy on in-error: 76% + (3yrs), 96% + (4-5yrs)

Cardinal principle trials

Accuracy on correct trials: 96%+

Accuracy on in-error trials: 85% + (3yrs), 99% + (4-5yrs)

Summary of error task findings

• Pseudo-errors detected as peculiar, but not incorrect (95%+ accuracy)

  • Sometimes able to articulate why

  • Show understanding of order-irrelevance

• Children as young as 3 understand the principles– can’t articulate them

  • Understanding demonstrated even in set sizes too big for children to count

• Children show IMPLICIT knowledge of these principles

Order irrelevance - Baroody (1984)

5-7 year olds

• Are able to understand that tags can be assigned arbitrarily

This does NOT imply understanding that:

• Differently ordered counts produce the same cardinal designation

Baroody (1984) procedure

Children counting themselves (not error-detection)

• Children 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

However, in the prediction task:

• Only 45% of 5yr-olds were successful

• Only 87% of 7yr-olds were successful

Baroody (1984) Conclusion

• Understanding of order-irrelevance develops with age

• Young children’s understanding of principles overestimated

“Principles-after” concept

Gelman, Meck & Merkin (1986)

• Task affects how children perform:

• Failure is due to misinterpretation of instructions, not lack of understanding

Procedure: 3 groups

• Baroody replication

• Count 3x: 3 opportunities to count first

• Altered-question: “How many will there be” or “What will you get”

Results – Gelman et al (1986)

• Baroody group - most children were incorrect

• Count 3 times group - more children were correct

• Altered question group - most children correct

Give ‘N’ task and knower levels

Sarnecka & Carey (2008)

• Child asked to give ‘N’ number of items

• Up to ‘4-knowers’ called ‘subset’ knowers

  • Only know how a subset of numbers work

• Switch to Cardinal Principle-knower

  • Can solve flexibly across sets, not restricted

  • Really understand how counting works, evidenced across a variety of tasks

Knower level increases with age

Habituation studies

Used with very young infants to gauge innate knowledge

Procedure

• Habituation to 4 dots

• Followed by exposure to 2 dots

Results

• Looked longer at two dots

Conclusion

• Understand numerosity?

• Basic discrimination?

Example habituation study

Xu and Spelke (2000)

Larger numbers, controlled for other properties of the arrays.

6-month-olds discriminated between 8 and 16 dots

• Replicated with 4 vs 8 and 16 vs 32

• BUT, infants can’t do 3:2 ratios until 9 months (ex: 8 vs.12)

Ability to detect more precise ratios continues with development

Addition and Subtraction study

Wynn (1992) - 32 5-month old infants

Looking time procedure (Violation of Expectation), shown different mathematical outcomes.

• Control condition: no difference in looking times to 1 or 2 object

• Test trials: Infants looked longer at the “incorrect” result

  • 1+1 group looked longer at 1, than 2 puppets

  • 2-1 group looked longer at 2 than 1 puppet

Addition and subtraction study - Exp 3

Wynn (1992)

1. Infants compute precise results of simple additions/subtractions

2. Infants expect arithmetical operation to result in numerical change (no expectation of size/ or direction of change)

• Findings:

  • 1+1 = 2 OR 3

  • Infants preferred 3 in the trials, but not pre-test trials

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

• Replicated with larger sets (ex: 10 v. 5; McCrink & Wynn, 2005)

Criticism of Wynn (1992)

Wakeley et al (2000)

3 Experiments

• Replications of exps 1 & 2

• Subtraction counterpart to Exp 3

  • 3-1 = 1 or 2

  • Controls for possibility that preferred answer is always greater number of items

Results: No systematic preference for “incorrect” versus “correct”

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 attentiveness of infants

• 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

Numerical abilities in animals

Santos et al (2005)

• Can discriminate amounts (sounds, arrays, food)

Core knowledge hypothesis

Humans have a small number of core functions that help them deal with the environment. This system is:

• Domain-specific, task-specific, and encapsulated. 

• Abstract and can support numerical operations like addition, subtraction, and comparison. 

• Present in human infants and nonhuman primates. 

• A building block for developing new cognitive abilities, like symbolic arithmetic. 

Empiricist views on numeracy

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)