Symbolic

• Symbolic Representation


  The process of mentally represent objects and experiences through the use of symbols
• No matter what languages we speak, we share the same universal mathematical symbolic system
• Triple Code Model (Dehaene, 1992)


  According to this model, children develop these codes over time, starting with a nonverbal understanding of quantity and gradually learning verbal names and symbols for numbers
• Verbal Number Names


  • Rote counting – reciting the number words in sequence (procedural)
  • Initially meaningless (lack of conceptual understanding)
  • Cultural differences in the names of number words
• Counting principles (Gallistel, 1978)


  • One-to-one principle
  • Stable order principle
  • Abstraction principle
  • Order irrelevance principle
  • Cardinality Principle
• Cardinality Principle


  The last number in the count sequence describes how many objects there are in total in the set
• Give-N-Task (Wynn, 1990)


  • Pre-number-knower (non-knowers)
  • One/two/three/four-knowers
  • Cardinal principle (CP)-knower
• Mapping among spoken number words, Arabic digits and the quantity they stand for

  1. Quantity to number-word

  2. Number-word to quantity

  3. Number-word to digit

  4. Digit to number-word

  5. Quantity to digit

  6. Digit to Quantity
• Numerical Stroop task


  Participants see two single digits presented in various physical size and have to choose either the numerically or physically larger digit of the pair
• Size Congruity Effect


  • Faster response time on congruent trials compared to neutral ones (facilitation effect)
  • Slower response time on incongruent trials compared to neutral ones (interference effect)
• Automaticity of Numerical Magnitude


  • It is fundamental to success in many areas of more advanced mathematics
  • It is important for everyday tasks
• Problem Size Effect


  The larger the two numbers are, the longer it takes us to process them
• Unit-decade compatibility effect


  Participants automatically process the irrelevant unit digits
• Meta-Analysis: Symbolic number processing

  • It is reliably associated with mathematical competence over the lifespan
  • The strength of the correlation was .30
  • The correlation decreased slightly with age
  • Higher correlation for mathematical competencies that rely more heavily on the processing of magnitude
• Percent absolute error (PAE)

  • Where does number X go?
  • PAE = (|Child's estimate—Target Number)/scale of the estimate|) x 100%
• Foundation for Further Development

  • The strength of the correlation was .44
  • The correlation increased with age
  • The correlation was not moderated by the type of number line estimation measure used