ANS

• Nonsymbolic number refers to 'number sense' not reflected by Arabic digits
• Core number systems


  • Subitizing system
  • Approximate Number System
• Subitizing system


  • Does not rely on verbal counting skills
  • Mapping of numerosities <4 occurs prior to the acquisition of the counting principles
  • Six months after children acquire the counting principles, they learn to extend their counting range beyond the numbers within the subitizing range
• Subitizing system

  • Task success dependent on absolute number of items presented, not affected by the numerical ratio between the crackers
  • Upper-bound limit of three
• Subitizing system


  • Rhesus Monkeys given a choice between pieces of apple in two opaque containers - succeeded for 1 vs 2, 2 vs 3, 3 vs 4, 3 vs 5 but failed for 4 vs 5, 4 vs 6, 3 vs 8, 4 vs 8
• Subitizing system

  • Rhesus Monkeys have slightly higher capacity than human infants (4 vs. 3)
  • Rhesus monkeys tracking also unaffected by the ratio between the quantities
• Subitizing

  • Predictive of counting skills in 4-6-year-olds
  • Unrelated to symbolic number comparison or number line task performance in 6-7-year-olds
  • Deficits seen in Developmental dyscalculia
• Approximate Number System (ANS)


  • Involved in generating approximate, nonsymbolic representations of quantity
  • Numerons help convert quantities into analogue magnitudes on the mental number line
  • ANS acuity is imprecise - can be measured using Weber fraction (w)
• Nonsymbolic comparison


  • Arguably the most widely used measure of the precision of ANS acuity
  • Involves comparison of two nonsymbolic arrays of items (presented simultaneously or sequentially)
  • Variability of estimates increases with increasing quantity – Scalar variability
  • Underestimation of smaller numerosities and overestimation of larger numerosities
• Meta-analyses demonstrate relationship between ANS and math achievement is relatively weak
• Signatures of symbolic and non-symbolic comparison are unrelated
• Impairments not as strong as in symbolic comparison for children with mathematical difficulties
• Poorer number comparison skills


  Amongst adults without any formal mathematics education
• Direction of relationship between ANS and maths not fully established
• Nonsymbolic comparison task


  • Estimations influenced by non-numerical aspects of the display (dot size, density, perimeter)
  • No single standardized version - tasks differ in visual characteristics, stimuli presentation, duration
• Measurement of ANS acuity


  • No single index of task performance - % accuracy, error rates, reaction times, accuracy/reaction time composite, distance/ratio effect size
  • Weber fraction (W) problematic - heterogenous values, different methods controlling for non-numerical dimensions lead to substantially variable performance
• Tasks with sequential presentation rely on working memory
• Task performance may at least partly depend on participants' ability to ignore visual characteristics and inhibition
• Strength of ratio effect dependent on stimulus presentation and how performance was indexed
• Better test-retest reliability for accuracy vs. W, argued against the use of ratio effect indices as they were found to be unreliable
• Higher reliability for Non-symbolic comparison, compared to Symbolic comparison tasks
• Nonsymbolic addition task


  • Two 'sum' arrays of dots are presented, the sum is compared to a third 'comparison' array
  • Young children can perform above chance and show the same pattern as adults
  • Performance on this task and nonsymbolic comparison task was significantly correlated in children aged 5-11
  • May tap other skills like working memory, inhibition and attentional control
• Performance on nonsymbolic addition task


  Predictive of early maths skills in children who had just begun primary school, even after controlling for intelligence and literacy skills
• Training the ANS can lead to improvements in mathematical ability
• Many issues in the ANS literature, questioning whether the ANS really exists
• It should not be assumed that ANS is numerical in nature (issue of continuous magnitudes)
• The nonsymbolic addition task may be a suitable alternative measure of the ANS