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Numerical Competence

Numerical Competence. February 9 th , 2010 Psychology 485. Outline. Introduction Different levels of numerical competence, Why learn? How are numbers learned and processed? What is learned?. History. Clever Hans Oskar Pfungst Showed Clever Hans was responding to subtle cues.

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Numerical Competence

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  1. Numerical Competence February 9th, 2010 Psychology 485

  2. Outline • Introduction • Different levels of numerical competence, • Why learn? • How are numbers learned and processed? • What is learned?

  3. History • Clever Hans • Oskar Pfungst • Showed Clever Hans was responding to subtle cues

  4. What is “numerical competence”? • Different levels of competence • Numerosity Discriminations • Counting • Understanding number as a concept • Arithmetic

  5. Numerosity Discriminations • More or Less • Obvious advantages • The more resources the better OR

  6. Counting • Each item in a set is ‘tagged’ • Final ‘tag’ is cardinal number of the set • Numerons (tags) don’t need to be in any language • Why count? • Keep track of offspring, kin, predators, social hierarchies

  7. Numerical concepts • Abstract concept • e.g. Having a concept of the number 8: • “eightness” is a property of all sets with eight items • Understand the mathematical properties of number • 8 is: • the sum of 7 and 1 • the sum of 5 and 3 • the product of 2 and 4

  8. How are numbers learned and processed?

  9. Subitizingvs Counting • Subitizing • Rapid, accurate and confident judgements of number • Set sizes 1 to 4 • Counting or Estimating • Increased time, or decreased accuracy for set sizes greater than 4 • Amount of time needed increase per item • Demo

  10. Systems of Representation • Object-file system • a separate “file” for each item • Immediate representation of number of “occupied” files • Limited capacity • Good for small sets • Explains subitizing

  11. Systems of Representation • Analog-Magnitude system • Number is represented by a physical magnitude that is proportional to the number of individuals in the set • Accumulator (pulse generator)

  12. Systems of Representation • Analog-Magnitude system • Discriminability is proportional to ratio • Easy to discriminate • 1 vs 2 • 3 vs 8 • Harder to discriminate • 7 vs 8 • 15 vs 16 • Consistent with Weber‘s law

  13. Relationship to Timing Pacemaker (Pulse Generator) Accumulator Reference Memory Working Memory Ratio Comparator Decision or Response Scalar Expectancy Theory

  14. Relationship to Timing • Meck & Church (1983) • Rats trained to: • Press one lever after 2 x 1-second tone pulses • Press another lever after 8 x 1-second pulses • Total duration and number are redundant cues • Test for control by time and number

  15. Meck & Church (1983) • Control by number • Present 2 or 8 pulses over span of 4 seconds • Control by time • Present 4 pulses in 2 or 8 second span

  16. Relationship to Timing • Time and number controlled response equally • Equal responding at geometric mean (not arithmetic) • Time and number processed simultaneously • Cognitive economy/simplicity • Less mechanisms to be “built in”

  17. What do animals learn about numerosity?

  18. Numerosity Discriminations • Many species have been shown to make more/less discriminations • Can be difficult to study • Many confounds (time, surface area, volume, etc)

  19. Counting • Sequential (not simultaneous) numerosity discriminations • Shows animals “keeping track” of values • Capaldi and colleagues • Trained rats with patterns of reward/no reward at end of runway • NRRN or RRN – count to 2 • Rats run fast for reward, slowly for no reward

  20. “Zero” concept • Children don’t usually understand concept of “zero” until 3 or 4 years old • Can be difficult to teach • In animals • Alex, the African Grey Parrot • Ai, chimpanzee

  21. Alex’s zero concept • Was taught the term “none” to compare size • Presented with 2 blocks that are same size • Asked “which block larger?” • Taught to say “none” • Spontaneously transferred “none” to numerosities • Presented with 3 sets: 2, 3, 6 • Asked which set contained 5 blocks • Answered “none” • Further tests showed he applied term to absence of quantity • Shown empty tray, asked “How many?”

  22. Ai’s zero concept • Taught arabic number symbols • Shown numbers 0, 1, 4, 7, 9 • Asked to select the lowest number • Chooses zero • Can match number of dots on screen to arabic numeral • Shown three dots, will select symbol “3” • Shown no dots, will select symbol “0”

  23. Adding & Subtracting ? • Expectancy Violation method • Non-verbal method • Good for children & animals

  24. Expectancy Violation • Method used with dogs, children, monkeys • Look longer at unexpected outcomes • 1 + 1 = 3 or 1 + 1 = 1 • Expected outcomes are “boring”

  25. Video – 8 animals that can count

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