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Psyc 1306 Language and Thought Number (Notes and Talking Points)
Psyc 1306 Language and Thought
Psyc 1306 Language and ThoughtNumber (Notes and Talking Points)1Two Different ViewsThe 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)
2The ChallengeYou cant learn what you cant represent. ~ Fodor, 1975Can humans construct new representational resources? If so, how?3The ChallengeSpelling out stages of concept developmentInitial Conceptual Structure & Cognitive PredispositionsLanguageAcquisition New Conceptual Structure & Cognitive PredispositionsPI: Parallel IndividuationAM: Analog MagnitudeInteger concepts(Exact Numerosity)4AgendaPapers: Gordon & PicaState of development with & without the inputChild AdultBackground VideosRecap exps. & resultsOther views (M.L.s analogy to throwing) & response.What has changed from initial state to final state?Combinatorial nature of languageBootstrapping
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ratioperformance1.57
ratio (n1 + n2): n3performance1.58
Magnitude of n1performance1.49
Magnitude of n1Performance1
.210
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13LIFE WITHOUT COUNTING THROWINGON COUNTING AND THROWING14The ChallengeSpelling out stages of concept developmentInitial Conceptual Structure & Cognitive PredispositionsLanguageAcquisition New Conceptual Structure & Cognitive PredispositionsPI: Parallel IndividuationAM: Analog MagnitudeInteger concepts(Exact Numerosity)15View 1Construction of new conceptual primitivesi.e. one, two, three, four, five, six, etcHave integers but must learn difficult conventionnext +1Slowness for two, three due to task difficultyCounting principles innate but verbal counting skills initially fragileRemaining slides from Mathieu Le Corre16View 2Innate principles : Subset-knowers on Give a Number, but CP-knowers on easier tasksInnate integers, but not formated as countingConstruction:Performance consistent across tasks at two levels:CP-knower vs. notKnower-levels
17Give a NumberAssessed knower-levelsN-knowers give N when asked for N but not for any other set sizeCP-knowers succeed on all set sizes (up to 6)
18Testing robustness of CP-knower vs. non-CP-knowerAsk puppet to put 6, 7, or 8 elephants in opaque trash canPuppet counts elephants slowly, putting them in can one at a timeUndercounts on 6: one, two, three, four, five!Correct on 7Overcounts on 8: one, twonine!No counting required!1912 1+2, 12 3+4, 13 CP.15 children got 6,7,8 then 8,7,6How are the counting principles constructed?The role of core representations of numberParallel individuationAnalog magnitudes20Parallel individuation: toy modelRepresentational Principle: One-to-one correspondence between symbols in mind and objects in worldO1O1, O221Analog magnitudes: toy modelRepresentational PrincipleAverage magnitude is proportional to the size of represented set.22Core representations: extensionsParallel individuationOnly up to 4Analog magnitudes1 to ????
23Possible construction processOne two three fourPI onlyAM onlyPI + AMfive six seven eightAM only?24five, sixseven, eight,nineeight, nine, ten, eleven, twelveonetwothreefour25Whats On This Card?Whats on this card?Whats on this card?Thats right! Its one apple.Whats on this card?Whats on this card?Thats right! Its six bears.26Children on the cusp of acquisition (four-knowers)Only map one-fourto construct principles
Also map large numerals to magnitudes to construct principles
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0.06-0.010.060.2328Fast Cards: set size estimation without countingOnly present sets for 1sToo quick for countingTest: 1-4, 6, 8, 10 First modeled task2930
0.08Three and four-knowers: one- four only31ANOVA 3&4: 2 (level) x 7 (set size)No effect of level (p = .49), no set size X level interaction (p = .49)5 month-olds only need 2s to encode magnitude values with enough precision for 2:1
Some CP-knowers have not mapped large numerals onto magnitudes!32Kolmogorov-Smirnov w/ Lilliefors correction, D = 0.11, p < .05.CP-knowers: mappers & non-mappersNon-mappers: slope 6-10 < 0.3Mappers: slope 6-10 > 0.3
Mean age: 4;1Mean age: 4;633A CP non-mapper on Fast Cardsage: 4 years 7 months34Role of core knowledge in construction of counting principlesOne two three fourPI onlyAM onlyPI + AMfive six seven eightAM only35Case study: counting as representation of positive integersEvidence that counting is a bona fide constructionRole of innate, core representations in constructionRole of numerical morphology in constructionRole of integration of counting with core systems in development of arithmetic competence36
one means 1two, three all mean > 1Singular?Plural?37How singular/plural could affect numeral learningCreates new hypotheses (Whorfian)Linguistic singular/plural morphemes provide symbols for 1/more than 1 distinction
38Language affects language: syntactic bootstrappingtwo, three, = > 1Because co-occur w/ plural nouns39Test case: Mandarin (no si/pl on nouns)two, three = > 1.Not in MandarinLeads Mandarin to map numerals to magnitudes?40Do Chinese ever have numerals that mean more than 1?Tested Chinese children on Whats on This Card Analyzed with average numeral by set size method
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42Chinese: 11 x one-k, 16 x two-k, 7 x 3-kAm: 24 x 1-ks, 17 x two-ks, 9 x 3-ksKlevel x lang X set size: None of effects w/ language significant.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 systemsBoth (all?) syntaxes specify quantifier category?Mandarin cue: classifiers
43Case study: counting as representation of positive integersEvidence that counting is a bona fide constructionRole of innate, core representations in constructionRole of numerical morphology in constructionRole of integration of counting with core knowledge (analog magnitudes) in development of arithmetic competence
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What inferential powers does counting have on its own?Does acquisition of mapping to magnitudes create new inferential powers?45Core systems & numerical order of numeralsDo children understand how counting represents numerical order when large numerals not mapped to magnitudes?46Which box does the bear want?
one vs. eighttwo vs. three
six vs. teneight vs. ten47
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Partial mappers can order eight vs. ten!49Confirms some CP-knowers have not mapped large numerals to magnitudesRepresentations underlying one - four support ordinal inferencesCounting initially limited procedure for creating setsMapping between large numerals and magnitudes necessary to learn their numerical orderQuickly learn how to use counting to make ordinal inferencesEventually happens: 10,054 vs. 10,05550Learning to count is hard because requires construction of new representational resource (!)Construction processUniversal structure (?)Use syntax to pick out class of quantifiers (?)Map one through four onto representations of sets provided by parallel individuation (?)two, three = more than 1 not particular to languages with singular plural (!)Do not map large numerals to large magnitudes (!)
51Counting as a representation of numberInitially limited procedure for creating setsDoes not support ordinal inferences until some numerals have been integrated with each core systemGrowth of arithmetic competence beyond countingDepends on integration of numerals with analog magnitudes52531185472125581856627257Chart11.030303030310.03030303030.0303030303002.28787878792.346250.21241862270.21241862270.15616247550.15616247552.43939393942.62916666670.23572660850.23572660850.34690930770.34690930772.9752.7870370370.35966103570.35966103570.32812453330.32812453332.40740740742.78245614040.24887513490.24887513490.31399943260.31399943262.718752.53333333330.3977156770.3977156770.24147249710.24147249712.65151515152.89705882350.33063647330.33063647330.28226993280.2822699328
ChineseAmerican"One"-knowers
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ChineseAmerican"One"-knowers
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ChineseAmerican"Two"-knowers
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ChineseAmerican"Three"-knowers
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ChineseAmerican"Two"-knowers
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ChineseAmerican"One"-knowers
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ChineseAmerican"Two"-knowers
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ChineseAmerican"Three"-knowers
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ChineseAmerican"Three"-knowers
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ChineseAmerican"One"-knowers
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ChineseAmerican"Two"-knowers
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ChineseAmerican"Three"-knowers
