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Psyc 1306 Language and Thought

Psyc 1306 Language and Thought. Number (Notes and Talking Points). The knowledge of mathematical things is almost innate in us … for layman and people who are utterly illiterate know how to count and reckon.” (Roger Bacon;1219-1294) .

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Psyc 1306 Language and Thought

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  1. Psyc 1306 Language and Thought Number (Notes and Talking Points)

  2. The knowledge of mathematical things is almost innate in us … for layman and people who are utterly illiterate know how to count and reckon.” (Roger Bacon;1219-1294) "It must have required many ages to discover that a brace of pheasants and a couple of days were both instances of the number two."(Russell, 1872-1970) Two Different Views

  3. The Challenge “You can’t learn what you can’t represent.” ~ Fodor, 1975 Can humans construct new representational resources? If so, how?

  4. The Challenge • Spelling out stages of concept development New Conceptual Structure & Cognitive Predispositions Initial Conceptual Structure & Cognitive Predispositions Language Acquisition Integer concepts (Exact Numerosity) PI: Parallel Individuation AM: Analog Magnitude

  5. Agenda • Papers: Gordon & Pica • State of development with & without the input Child  Adult • Background Videos • Recap exps. & results • Other views (M.L.’s analogy to “throwing”) & response. • What has changed from initial state to final state? • Combinatorial nature of language • Bootstrapping

  6. 1 performance .5 ratio

  7. 1 performance ratio (n1 + n2): n3 .5

  8. 1 performance .4 Magnitude of n1

  9. 1 Performance Magnitude of n1 .2

  10. LIFE WITHOUT COUNTING THROWING • ON COUNTING AND THROWING

  11. The Challenge • Spelling out stages of concept development New Conceptual Structure & Cognitive Predispositions Initial Conceptual Structure & Cognitive Predispositions Language Acquisition Integer concepts (Exact Numerosity) PI: Parallel Individuation AM: Analog Magnitude

  12. View 1 • Construction of new conceptual primitives • i.e. one, two, three, four, five, six, etc… • Have integers but must learn difficult convention • “next” <--> “+1” • Slowness for “two”, “three” due to task difficulty • Counting principles innate but verbal counting skills initially fragile Remaining slides from Mathieu Le Corre

  13. View 2 • Innate principles : • Subset-knowers on Give a Number, but CP-knowers on easier tasks • Innate integers, but not formated as counting • Construction: • Performance consistent across tasks at two levels: • CP-knower vs. not • Knower-levels

  14. Give a Number • Assessed knower-levels • “N”-knowers give N when asked for “N” but not for any other set size • CP-knowers succeed on all set sizes (up to 6)

  15. Testing robustness of CP-knower vs. non-CP-knower • Ask puppet to put 6, 7, or 8 elephants in opaque trash can • Puppet counts elephants slowly, putting them in can one at a time • Undercounts on 6: one, two, three, four, five! • Correct on 7 • Overcounts on 8: one, two…nine! • No counting required!

  16. How are the counting principles constructed? • The role of core representations of number • Parallel individuation • Analog magnitudes

  17. O1 O1, O2 Parallel individuation: toy model Representational Principle: One-to-one correspondence between symbols in mind and objects in world

  18. Analog magnitudes: toy model Representational Principle Average magnitude is proportional to the size of represented set.

  19. Core representations: extensions • Parallel individuation • Only up to 4 • Analog magnitudes • 1 to ????

  20. Possible construction process One two three four five six seven eight… ?  PI only AM only PI + AM AM only

  21. one two three four seven, eight, nine eight, nine, ten, eleven, twelve five, six

  22. What’s on this card? What’s on this card? That’s right! It’s one apple. What’s on this card? “What’s On This Card?” What’s on this card? That’s right! It’s six bears.

  23. Also map large numerals to magnitudes to construct principles Only map “one”-“four” to construct principles Children on the cusp of acquisition (“four”-knowers)…

  24. -0.01 0.06 0.06 0.23

  25. Fast Cards: set size estimation without counting • Only present sets for 1s • Too quick for counting • Test: 1-4, 6, 8, 10 • First modeled task

  26. 0.08 “Three” and “four”-knowers: “one”- “four” only

  27. Some CP-knowers have not mapped large numerals onto magnitudes!

  28. CP-knowers: mappers & non-mappers Non-mappers: slope 6-10 < 0.3 Mappers: slope 6-10 > 0.3 Mean age: 4;1 Mean age: 4;6

  29. A CP non-mapper on Fast Cardsage: 4 years 7 months

  30. Role of core knowledge in construction of counting principles One two three four five six seven eight… PI only AM only PI + AM AM only

  31. Case study: counting as representation of positive integers • Evidence that counting is a bona fide construction • Role of innate, core representations in construction • Role of numerical morphology in construction • Role of integration of counting with core systems in development of arithmetic competence

  32. “one” means 1 “two”, “three” all mean > 1 Singular? Plural?

  33. How singular/plural could affect numeral learning • Creates new hypotheses (Whorfian) • Linguistic singular/plural morphemes provide symbols for 1/more than 1 distinction

  34. Language affects language: syntactic bootstrapping • “two”, “three”, … = > 1 • Because co-occur w/ plural nouns

  35. Test case: Mandarin (no si/pl on nouns) • “two”, “three”… = > 1. • Not in Mandarin • Leads Mandarin to map numerals to magnitudes?

  36. Do Chinese ever have numerals that mean more than 1? • Tested Chinese children on What’s on This Card • Analyzed with average numeral by set size method

  37. “One”-knower pattern not product of English numerical morphology • No role for large magnitudes • Count list culturally-specific but construction process universal! • How can Mandarin and English have same meanings? • Same core knowledge systems • Both (all?) syntaxes specify “quantifier” category? • Mandarin cue: classifiers

  38. Case study: counting as representation of positive integers • Evidence that counting is a bona fide construction • Role of innate, core representations in construction • Role of numerical morphology in construction • Role of integration of counting with core knowledge (analog magnitudes) in development of arithmetic competence

  39. What inferential powers does counting have on its own? Does acquisition of mapping to magnitudes create new inferential powers?

  40. Core systems & numerical order of numerals • Do children understand how counting represents numerical order when large numerals not mapped to magnitudes?

  41. Which box does the bear want? “one” vs. “eight” “two” vs. “three” “six” vs. “ten” “eight” vs. “ten”

  42. Partial mappers can order “eight” vs. “ten”!

  43. Confirms some CP-knowers have not mapped large numerals to magnitudes • Representations underlying “one” - “four” support ordinal inferences • Counting initially limited procedure for creating sets • Mapping between large numerals and magnitudes necessary to learn their numerical order • Quickly learn how to use counting to make ordinal inferences • Eventually happens: 10,054 vs. 10,055

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