Number development across infancy and childhood

Question 1

what are Symbols and systems?

Answer 1

• Number systems are cultural tools that represent quantities, e.g.:
  
  • Recursive systems
    
    • Roman: I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII … XX … L … C, CI, CII, CIII …
    • Hindu- Arabic: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 … 20 … 50 … 100, 101, 102, 103 …
  • Finite systems
    
    • Oksapmin – a non-recursive, finite system: numbers are represented by body locations.

Question 2

what are Number relations?

Answer 2

• Quantity
  
  • a property of magnitude – how much.
• Cardinality
  
  • being represented by a cardinal number.
  • any set of items with a particular number is equal in quantity to any other set with the same number.
• Ordinality
  
  • numbers come in a serial order of magnitude.
  • transitive inferences are possible

Question 3

how do we Perceive numbers?

Answer 3

• Starkey & Cooper (1980): can 4-month-olds perceive number?
• Method
  
  • Habituation procedure + looking time measure
  • small number vs. large number
• Results
  
  • Infants dishabituate (looked longer) on small number trials
• Conclusions
  
  • Infants discriminate differences between small quantities
  • Infants are born with ability to understand number

Question 4

what did Wynn (1992) study?

Answer 4

• tested 5 month olds’ knowledge of mathematical operations: addition and subtraction
• Method: Violation of expectation study
  
  • E.g., a toy was placed behind a screen then another added to it
  • compared looking time when screen lowered
    
    • for possible outcomes (2 toys behind the screen)
    • or impossible outcomes (1 toy behind the screen)
• Results
  
  • Infants surprised by (looked longer at) impossible outcome
• Conclusions
  
  • Infants understand mathematical operations
  • Infants have innate number structures

Question 5

Is number knowledge innate?

Answer 5

• Alternative explanations: Wynn’s results could be based on change detection
  
  • impossible outcome is the same as starting point – 1 toy – infants’ surprisal could be related to expecting a change
• Mixed success in replications
• Limited to small numbers < 3
  
  • Children do not show understanding of 2 + 2 = 4 until 3-5 years
  • Perception results limited to small numbers too
• Alternative explanation: Infant results could be based on perceptual features
  
  • Number needs to be separated from continuous dimensions of sets: area, contours etc.
  • Infants may perceive difference between continuous quantity not precise number relations

Question 6

what did Xu & Spelke (2000) study?

Answer 6

• tested 6 month old infants’ discrimination of large number
• Method
  
  • habituation procedure – sheets with dots
    8 dots, 12 dots or 16 dots
• Test
  
  • Discrimination of a change in number
    
    • 8 dots  16 dots
    • 8 dots  12 dots
  • Longer looking to new number
• Results
  
  • Infants discriminated large differences – 8 vs 16, NOT 8 vs 12
  • Infants’ perception is based on approximate representations not exact number

Question 7

How do we count?

Answer 7

• Abstract counting vs object counting
  
  • Abstract counting  reciting number sequence
  • Object counting  determining a quantity

Question 8

what are Gelman’s counting principles?

Answer 8

• How to count principles:
  
  • one-to-one: count each item in a set once and only once.
  • stable order: produce the number words in the same set order.
  • last number: the last number counted represents the value of the set.
• Other principles
  
  • order irrelevance: the order in which items are counted makes no difference.
  • abstraction: the number in the set is independent of the qualities of the members.

Question 9

what did Gelman and Gallistel (1978) study?

Answer 9

• 2-5 year olds counted sets of 2 – 19 items, tested:
  
  • counting sequence,
  • one-to-one correspondence
  • last-number significance.
• Children were accurate with small sets
  
  • Understand counting principles
• Children made errors with larger sets
  
  • Attributed to performance errors
• Children recognise counting errors made by a puppet
• Children recognise unusual but correct counting procedures

Question 10

what is Carey’s individuation hypothesis (2004)?

Answer 10

• Children gradually develop number understanding through combination of innate knowledge and experience:
• One learning mechanism is: Parallel individuation: infants recognise and represent small numbers exactly
  
  • Children first recognise ‘one’ - ‘one-knowers’,
  • then recognise ‘one’ vs ‘two’ – ‘two-knowers’,
  • then ‘one’ vs ‘two’ vs ‘three’ – ‘three-knowers’
• Enriched parallel individuation: children learn larger numbers, bootstrapping from counting
  
  • Learning the count list teaches them that quantities extend beyond ‘three’ and helps children discriminate larger numbers

Question 11

what are Comparing sets?

Answer 11

• Number words imply relationship between sets
• Cardinality: any set of items with a particular number is equal in quantity to any other set with the same number.
• According to Piaget, understanding cardinality is key to understanding number
• Greco (1962): Conservation task.
  
  • Test 4-8 year olds on three tasks:
    
    • classic conservation
    • classic conservation + counting 1 set
    • classic conservation + counting both sets

Question 12

what are Counting systems?

Answer 12

• Number systems are cultural tools that represent quantities, e.g.:
  
  • Decimal system: base of 10
    
    • Numbers repeat after 10: 35 = three 10s + 5; 40 = four 10s; 60 = six 10s
  • Number words used reflect multiplicative and additive nature of numbers:
    
    • multiplicative: two hundred, three hundred
    • additive: twenty-one, thirty-three
    • exceptions in English: eleven, twelve, thirteen, fourteen, nineteen, twenty, thirty
    • exceptions in French: onze, douze, treize, quatorze, quinze, seize, vingt, quatre-vingts
    • exceptions in German: elf, zwolf, zwanzig

Question 13

what are Cross-linguistic differences?

Answer 13

• Some languages have more transparent number words – English vs Chinese (Miller and Stigler (1987)):
• Is this important for number development?

Question 14

what did Miller and Stigler (1987) study?

Answer 14

• tested 4, 5 and 6 year old English- and Chinese-speaking children
  
  • abstract counting
  • object counting
• Chinese-speaking children count ‘better’ than English-speaking children.
• Alternative possible explanations:
  
  • Education systems differ
  • Parental expectations differ
• Or, language characteristics influence the representation of number due to the greater transparency of number names.

Question 15

what did Miura et al., (1994) study?

Answer 15

• Comparisons of Children’s Cognitive Representation of Number.
• In Chinese languages:
  
  • Additive: “ten–two” = 12
  • Multiplicative: “two tens” = 20
• In European languages:
  
  • Additive: “twelve” = 12
  • Multiplicative: “twenty” = 20
• Tested 7-year-old children learning Chinese, Korean, Japanese, English, French and Swedish.
• Use number blocks of 10 and 1 to represent different numbers in different ways.
  
  • 11, 13, 28, 30, 42
• Asian-language learners used 10 blocks first more and used more ways of representing number.
• European language learners used unit blocks first more.
• English speakers were least likely to produce canonical (base 10 + units) representations.