Slide 1

Number (Notes and Talking Points)

Slide 2

Two Different Views 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)

Slide 3

The Challenge You cant learn what you cant represent. ~ Fodor, 1975 Can humans construct new representational resources? If so, how?

Slide 4

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

Slide 5

Agenda 1. 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. 2. What has changed from initial state to final state? Combinatorial nature of language Bootstrapping

Slide 6

Slide 7

ratio performance 1.5

Slide 8

ratio (n1 + n2): n3 performance 1.5

Slide 9

Magnitude of n1 performance 1.4

Slide 10

Magnitude of n1 Performance 1.2

Slide 11

Slide 12

Slide 13

Slide 14

LIFE WITHOUT COUNTING THROWING ON COUNTING AND THROWING

Slide 15

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

Slide 16

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

Slide 17

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

Slide 18

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)

Slide 19

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, twonine! No counting required!

Slide 20

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

Slide 21

Parallel individuation: toy model Representational Principle: One-to-one correspondence between symbols in mind and objects in world O1O1 O 1, O 2

Slide 22

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

Slide 23

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

Slide 24

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

Slide 25

five, six seven, eight, nine eight, nine, ten, eleven, twelve onetwo three four

Slide 26

Whats On This Card? Whats on this card? Thats right! Its one apple. Whats on this card? Thats right! Its six bears.

Slide 27

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

Slide 28

0.06 -0.010.060.23

Slide 29

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

Slide 30

Slide 31

0.08 Three and four-knowers: one- four only

Slide 32

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

Slide 33

CP-knowers: mappers & non-mappers Non-mappers: slope 6-10 0.3 Mean age: 4;1Mean age: 4;6

Slide 34

A CP non-mapper on Fast Cards age: 4 years 7 months

Slide 35

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

Slide 36

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

Slide 37

one means 1two, three all mean > 1 Singular?Plural?

Slide 38

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

Slide 39

Language affects language: syntactic bootstrapping two, three, = > 1 Because co-occur w/ plural nouns

Slide 40

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

Slide 41

Do 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

Slide 42

Slide 43

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

Slide 44

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

Slide 45

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

Slide 46

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

Slide 47

Which box does the bear want? one vs. eighttwo vs. three six vs. teneight vs. ten

Slide 48

Slide 49

Partial mappers can order eight vs. ten!

Slide 50

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

Slide 51

Learning to count is hard because requires construction of new representational resource (!) Construction process Universal 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 (!)

Slide 52

Counting as a representation of number Initially limited procedure for creating sets Does not support ordinal inferences until some numerals have been integrated with each core system Growth of arithmetic competence beyond counting Depends on integration of numerals with analog magnitudes

Slide 53

Slide 54

118

Slide 55

7212

Slide 56

818

Slide 57

6272