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Number: Pt 1.

Number: Pt 1. Early Years Lecture 12. Numerical sensitivity. When are children: - first able to represent number? - able to reason about number? - able to recognize a problem as a numerical problem? Over next 2 lectures, we’ll address the following question:

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Number: Pt 1.

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  1. Number: Pt 1. Early Years Lecture 12

  2. Numerical sensitivity • When are children: - first able to represent number? - able to reason about number? - able to recognize a problem as a numerical problem? Over next 2 lectures, we’ll address the following question: Q. Is number a ‘privileged domain’?

  3. Numerical sensitivity • In this lecture, we will: • assess the evidence for infant representations of number. • In the next lecture we will: - assess the evidence that preschoolers recognize the numerical significance of counting.

  4. First.. ...just what is Number? Write down a definition of ‘Number’

  5. Number? Natural numbers are... “classes such that any two inside the set are similar to each other, and none outside the set are similar to any inside the set.... .... the number of a class corresponds to a certain manyness” (Bertrand Russell, 1919)

  6. Do infants represent number? • Problem of assessment: 2 methods; 1. Preference/Habituation (see earlier lectures) used primarily for number discrimination

  7. Do infants represent number? • Problem of assessment: 2 methods; 1. Preference/Habituation (see earlier lectures) used primarily for number discrimination 2. ‘Violation-of-expectancy’ technique used for numerical reasoning

  8. Do infants represent number? Preference/Habituation studies • used to assess infants’ ability to discriminate between sets on the basis of number

  9. Do infants represent number? Preference/Habituation studies • used to assess infants’ ability to discriminate between sets on the basis of number Visual Preference? • infants look longer at large sets (128 items) than smaller sets (32 items) (Fantz & Fagan, 1975).

  10. Do infants represent number? Starkey & Cooper (1980) • 4-month-olds • habituate to 2 or 3 items? 2 > 3 3 > 2 H1 H2 PH

  11. Do infants represent number? Results? • habituate to 2/3 items • renewed looking at 3/2 items • no effect for 4 vs 6 items (not until 3-4 years)

  12. Do infants represent number? Results? • habituate to 2/3 items • renewed looking at 3/2 items • no effect for 4 vs 6 items (not until 3-4 years) Conclusion. infants do not perceive absolute numbers above 3 (subitazable range - see later). Antell & Keating (1983) - replicated Starkey & Cooper’s results .... but with neonates (21-144 hours old)

  13. Limits to discrimination? • Xu & Spelke (2000). - 6-month-olds - ratio limit - 8 vs 16 = ok - 8 vs 12 ≠ ok - innate mechanism for representing approximate - but not exact - numerosity. Evidence for exact number representation?

  14. Can infants ‘match’ number? Starkey, Spelke & Gelman (1990) - 7-month-olds • pair of displays - 1 x 2; 1 x 3 • drumbeats (2 or 3) - looked longer at matching display Cross-modal matching? Infants look longer at the unexpected display (Mix et al, 1997). results are...mixed...

  15. Can infants ‘match’ number? Wynn (1996). Habituate to a puppet jumping (e.g., 2 jumps) Present [a] 2 or [b] 3 jumps in 2nd set • look longer at different number! • integrate temporal and numerical info.

  16. Discriminate number or something else? • Number often confounded with other features (e.g., area, contour, etc.). • ‘amount of stuff’ hypothesis. • Clearfield & Mix (1999). OR [A] [B] Habituate

  17. Discriminate number or something else? • Result [A] = same contour/diff. number. [B] = diff. contour/same number. Dishabituate to different contour (B), not number (A)! OR [A] [B]

  18. So far... • Infants discriminate small sets of 2-3 items

  19. So far... • Infants discriminate small sets of 2-3 items • Sensitive to cross-modal matching

  20. So far... • Infants discriminate small sets of 2-3 items • Sensitive to cross-modal matching • Limited to large ratios (e.g., 1:2 not 4:5)

  21. So far... • Infants discriminate small sets of 2-3 items • Sensitive to cross-modal matching • Limited to large ratios (e.g., 1:2 not 4:5) Q. Might children be responding to change in global amount (e.g., by contour) rather than number?

  22. So far... • Infants discriminate small sets of 2-3 items • Sensitive to cross-modal matching • Limited to large ratios (e.g., 1:2 not 4:5) Q. Might children be responding to change in global amount (e.g., by contour) rather than number? Q. Are infants responding to number...or simply ‘change’? Do they understand nature of change? Need more compelling evidence...

  23. Do infants operate on number? Use ‘violation of expectancy’ technique. Wynn (1992). • one puppet placed in case. • screen hides puppet. • second puppet added. • hand removed empty. • screen drops down to reveal 2 puppets (expected outcome) or 1 puppet (unexpected/impossible outcome)

  24. Do infants operate on number? Wynn (1992) - results? • infants (5 months) understand numerical transformations. • ...but problem. Numerically invalid display = impossible display. Q. Are infants sensitive to [a] number, or [b] physical impossibility?

  25. Do infants represent number as a constant? Simon, Hespos, & Rochat (1995). • extension of Wynn (1992) • 2 (number) x 2 (identity/possibility), i.e., [a] correct number x correct identity [b] correct number x impossible identity [c] wrong number x correct identity [d] wrong number x impossible identity

  26. Do infants represent number as a constant? Simon, Hespos, & Rochat (1995). • extension of Wynn (1992) • 2 (number) x 2 (identity/possibility), i.e., [a] correct number x correct identity [b] correct number x impossible identity [c] wrong number x correct identity [d] wrong number x impossible identity • [a] and [c] = same as Wynn (1992).

  27. Do infants represent number as a constant? Simon, Hespos, & Rochat (1995). • extension of Wynn (1992) • 2 (number) x 2 (identity/possibility), i.e., [a] correct number x correct identity [b] correct number x impossible identity [c] wrong number x correct identity [d] wrong number x impossible identity ...... looked longer at [d] than [b] ....infants discriminate on the basis of quantitative difference, not physical impossibility.

  28. Do infants represent number as a constant? Fiegenson & Spelke (1998) Number of puppets vs ‘amount’ of puppets One small puppet > Screen + One small puppet (remove empty hand) At test, either.. [a] two larger puppets (correct no. – diff contour) or [b] one large puppet (wrong no. but same contour) ..looked longer at [a] ..support for Clearfield & Mix

  29. Summary of infant capability • 2 types of numerical discrimination: [1] absolute (i.e., exact) [2] estimated (approx. order of magnitude). Newborns = absolute with 2 & 3 items Older infants = abstract large sets. ...but ‘more stuff’ hypothesis?

  30. Models of number representation How might infants have nonverbal representation of number? 2 Dominant Models 1. Subitizing. 2. Accumulator.

  31. Subitizing model • Enumeration-without-counting • Automatic; Rapid; Fewer than 5 items. 1000 Trick & Pylyshyn (1994) 800 600 Different systems? 400 RT (ms) No. of items

  32. Subitizing model What mechanism explains subitizing? - canonical representations (Mandler & Shebo, 1982). • but why an increase over 4 items? • also, linear arrays (e.g., 3) = triangular arrays (Trick, 1987)

  33. Accumulator model Meck & Church (1983). - timing/counting behaviour in rats. - ‘pacemaker’ operates a ‘gate’ into ‘container’. if timing - gate ‘pulses’ at constant rate. if counting – gate pulses one at a time - magnitude = contents of container.

  34. Accumulator model Addition = Accumulator ‘A’ + Accumulator ‘B’ A C B

  35. Accumulator model • like verbal counting = precursor of verbal enumeration (Gallistel & Gelman, 1992).

  36. Accumulator model • like verbal counting = precursor of verbal enumeration (Gallistel & Gelman, 1992). • obeys counting principles (Gelman & Gallistel, 1978 - see next lecture).

  37. Accumulator model • like verbal counting = precursor of verbal enumeration (Gallistel & Gelman, 1992). • obeys counting principles (Gelman & Gallistel, 1978 - see next lecture). • accumulator model feeds nativist account of number development including counting.... ...but learning to count, and recognizing what counting achieves, is a lengthy journey. Do preverbal discriminations really imply number sense?

  38. Next lecture • Infants appear to equate number with amount. • Accumulator = isomorphic with counting • Learning to count > easy? To be continued......

  39. Tutorial Week 9/10 Reading in ‘Tutorial Solutions’ in Library Cole, M. (1992). Context, modularity and the cultural constitution of development.

  40. Reading Essential Wynn, K. (1995). In Slater & Muir (1999), The Blackwell reader in developmental psychology, Chapter 13. Also For Wynn, 1992. Dehaene, S. (1999). The number sense. London:Penguin. • Siegler & Alibali (2005). p. 292 - 297.

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