Number (Notes and Talking Points)

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; ) "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, )

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

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)

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

ratio performance 1.5

ratio (n1 + n2): n3 performance 1.5

Magnitude of n1 performance 1.4

Magnitude of n1 Performance 1.2

LIFE WITHOUT COUNTING THROWING ON COUNTING AND THROWING

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)

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

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

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)

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!

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

“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

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

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

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

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

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

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

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 (!)

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

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