thelanguageofnatural number -...
TRANSCRIPT
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The Language of Natural
Number
Rochel Gelman
Rutgers University
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Learning the meaning of number words
• TWO CLASSES OF THEORIES
–1. Discontinuous
•Animals & infants abstractnumerical information fromdisplays & do something likearithmetic
Learning language of natural N rests on thewidely held view:
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Discontinuity ContinuedDIFFERENT SYSTEMS & LEARNING MECHANISMS FOR
1 to 3(4) VS. Larger values
Possible accounts
1. Small n´s treated as canonical sets of object files;
larger values as quantities with their accompanyingvariability. (Carey; Feigenson?, Spelke)
2. (a) Direct percept of oneness, twoness & threeness
like cowness, shoeness & treeness – that are labelled ; (b) Counting, which comes later for larger N´s (Klahr)
3? Distributional occcurrence in particular syntacticenvironments about words for small N´s in sentences & then an induction. e.g., there are 3 birds
Example of Tasks
Wynn´s “Give n``
Large number of heterogenous items in a bag or on the floor
Child asked to give adult (or puppet) exactly x
1. Robust finding –
2. - on average children younger than 3 ½ yrs
fail when asked for N > even for 2 or 3
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Example of Tasks Continued
2. Discrimination – which is “3” (4),etc.
There are variations on this kind of task
Result: Children either go with quantity or are random
Continuity
A nonverbal skeletal structure helps children
identify the relevant data and use rules
The principles underlying verbal counting are
isomorphic to the non-verbal ones. Once the
data and their use conditions are identified, a
child can proceed to learn the verbal count list
and where number words can be used.
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Continuity Cont´d
In this view, the learning of the verbally mediated arithmetic reasoning is founded on nonverbal counting and nonverbal arithmetic
The domain-specific nonverbal system of quantitativeestimation and reasonings is what makes thelearning possible.
The domain must possess a notion of exact equality(Leslie, Gelman & Gallistel)
Continuity Account
• Evidence
• Infant work Intermodal, Cordes
• WOC
• MAGIC
• IDIOSYNCRATIC LIST
• VERY NOISY, STILL STRONG TENDENCY FOR LARGER N´S TO BE USED WITH LARGER NUMBERS
• FELICIA
• WORK WITH SYRETT AND JULIAN
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Counting Does Not Stand Alone
Counting principles are embedded in a
structure organized by principles of
arithmetic.
• COUNTING PRINCIPLES
HOW TO’S
– 1. One-one
– 2. Stable order
– 3. Cardinal
PERMISSIONS
– 4. Order Irrelevance
– 5. Item Kind Irrelevance
Counting and Arithmetic
Principles: A Specific Domain
Arithmetic reasoningPrinciples
Numerons stand for cardinal value (or
quantity) give count of discrete
entities.
Following: Gelman & Gallistel 1978
Irrelevant changes =
+ - x ÷ ≠ < >
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Relating Nonverbal and Verbal Systems
There are key structural properties that map
between the nonverbal system and verbal use rules
for counting and arithmetic
1. Once and only once
2. Order
3. Final ‘tag’ represents value of counted set
4. Unique nexts, critical for a discrete system
5. (N + a) > N
6. (N - a) < N
Infants Reconsidered
McKrink and Wynn - 5 + 5 Study in very young children
Wynn, Bloom and Chiang 2002 – infants discriminate
between sets of 2 & 4 groups of moving dots
Cordes and Brannon – infants discriminate between 2 & 8
as well as 1 & 4 but not 2 & 4 and 3 & 6
Why? Perhaps it takes a larger difference for babies to
pay attention to N vs items in the small N range.
A possible hypothesis
-----------------------------------------------------------------------------------------------
Cordes and Brannon (2008) provide an excellent review in the Annual
Review of Psychology.
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Ambiguity
Label vs Number word
Black Circles
Five Circles
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E: See this card? (N=1)
What's on this card?C: A heart.
E: That's right, there is one heart on the card.
EXAMPLE OF START OF ONE PROBLEM SET OF “WHAT’S ON THE CARD” EXPERIMENT
First Trial
E: See this card? (N=1)
What 's o n this c ard?C: A heart .
E: Th at's right , t here is one hear t on th e c ard.
E: See this card? (N=2)
Wh at's on t his card?
C. 2 heart s.
E. Sh ow me.
c. 1, 2.
E. So, what kin d of a car d is th at?
C. 2
E. O.K. What’ s on thi s card? (N=3)
(W hat kin d of a ca rd is th is?)
POSSIBLE AN SW ER A :
C: 1, 2, 3 (He arts)
E: So, w hat kin d of a ca rd is it?
C: 3 (or a 3-h ea rt card or a 3-card).
POSSIBLE AN SW ER B :
C: 3
E: Show me.
C: 1-2- 3.
Example of Procedure and Answers for WOC Task
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a heart (label)
“thing”
“one thing”
Adding and Subtracting
Stan mentioned some results
MP1
Diapositiva 20
MP1 Marcela Pena, 3/14/2011
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The Magic Show:Phase 1
From Gleitman’s Psychology
Phase 2
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Phase 3
:
Results Across Experiments
• If we added or subtracted children
– Were surprised
– Made a relevant comment
– Searched and asked how an item came in (+) or wasgone (-).
If we make number-irrelevant changes, children say thatthese do not matter or do not seem to notice thechange. True even if we change items or color. Are surprised but say they still ¨win¨ because of N, oftencount, as if to prove this.
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Fallout of Magic Show with Beginning Language Learner:
Idiosyncratic Count Lists
Child: 2 – 6 – 10
Me - Huh?
Child 2 – 6 – 10!
Me - Huh??
Child 2 – 6 – 10!
Me - Huh??
.
.
.
.
.
Magic Tasks that Vary Changes in N
• If add or subtract 1, children of all ages from
2- 5 yrs know how to ¨fix¨the game
• If more than 1, then use a counting strategy:
– A solvability principle (G&G, 1978).
Important – attuned to exactly 1
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Another possible ambiguity
Count words and quantifiers
Task: place sticker on correct picture
The alligator took four of the cookies
took none
all/four
two/some
distractor
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Task: place sticker on correct pictureThe alligator took two of the cookies
took none
all/four
two/some
distractor
Task: place sticker on correct pictureThe alligator took all of the cookies
took none
all/four
two/some
distractor
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Task: place sticker on correct pictureThe alligator took some of the cookies
took none
all/four
two/some
distractor
Experimental Details
– 12 items, blocked for condition, within Ss– (counterbalanced: location, block order, etc.)
– quantifiers and numbers were run in different
sessions (different days)
– age 3:0-4;0, evenly distributed (mean = 3;6)
– N=20
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Adult controls at ceilingApply scalar implicature: some ≠ all; 2 ≠4
The alligator took
some/two of the cookies
The alligator took
all/four of the cookies
n=10
Percentage Correct 3;0-4;0 (3;6)
0
10
20
30
40
50
60
70
80
90
100
N=20, Conditions Blocked
All SomeFour Two
exact
reading
at least
reading
Alligator Took
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Adding & Subtracting in the Bakery
B&W Cnt´d
• For example, their analysis showed that
– Number words and many can both be used with
the partitive frame
• Both adults and their children do this.
• But – not with mass nouns, after adjectives, or
be modified by to or very.
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However When Ask Where Else Transcripts Contain of
• Table: ‘Quantity-denoting’ lexical items immediately before of
Type of item Examples
amount terms (including quantifiers)
– all, any, bit, both, a couple, each, half, (a) lot, many, most,much, none, oodles, pair, plenty, (the) rest, some
segment terms
– back, beginning, bits, bottom, edge, end, front, part, piece,side, top; body parts (e.g., head, neck)
units of measurement
standard: -foot, hour, inch, minute, pint, pound, quart, week,year
non-standard:
-bite, bottle, bowl, box, bucket, bunch, can, chunk, cup, drink,glass, pail, plate, reel, taste
Predicting and Checking the Effects of Addition and Subtraction
Have child check their prediction, countingallowed and used
Add or subtract 1, 2, or 3 items-array covered
Give child N(3-15) to construct initial array for a
problem ask How Many
Have child predict result,
without countingrepeat
Zur & Gelman 2001
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More About the Task
Inverse Principle
a+b= x First Problem
X-b = a Second Problem
Control
a+b= x First Problem
X-b = a -/+ 1 Second Problem
Is there a difference on the second problem.
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Young Children’s Implicit Use of the Arithmetic
Inverse Principle as Indicated by Second in Pair of
Inverse Problems (Zur & Gelman)
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Bloom and Wynn
Asked whether the distributional cues for 3
sets of lexical items:
-1. the number words two through ten,
-some quantifiers,
-and some adjectives,
-2. Are both present in caregiver speech in a
way that is informative about number-word
meaning, and
&
-3) whether these frequencies are also
reflected in children’s own productions
Developmental Risk Groups
Downs
Autism
Dyscalculia – Reasoning without
counting and number facts
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Hartnett Dissertation
• Puppets in a bag and then pops up and asksthe kid to count on …..
• To our surprise, this had no effect comparedto children who could count on themselves in the likelihood of their passing our successortask
• As if needed to be cued that there are natural numbers for which they do not have the termsand that addition creates these.
Math into Science
(insert photo of book)
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Date stamps
Stamp used as decoration
- Sept.
Stamp used as science
tool - March
Learning about Units of Measurement
Measuring the
length of a 37
foot snake
using the
length of their
own bodies as
units of
measurement
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Using Measurement Instruments
Growing a Sunflower House
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No se puede mostrar la imagen. Puede que su equipo no tenga suficiente memoria para abrir la imagen o que ésta esté dañada. Reinicie el equipo y, a continuación, abra el archivo de nuevo. Si sigue apareciendo la x roja, puede que tenga que borrar la imagen e insertarla de nuevo.
Making
observations by
counting and
recording
numbers so
they can be
compared
Summary
The model of number-word learning we favor
combines
the Gelman & Gallistel & one that has the
child attending to syntax and its interface
with semantics
The language learner identifies the relevant
linguistic environments in a given language as
to where number words can be inserted.
In a way the natural number domain is
parasatic on language. Consider the stable
order principle - nothing in natural language
structures that gives rise to this.
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Collaborators
• Kimberly Brenneman, PhD - colleague
• Sara Cordes PhD - former graduate student
• Jenny Cooper PhD - former graduate student
• Dana Chesney PhD - former graduate student
• Felicia Hurewitz PhD - former postdoctoral fellow
• C,R, Gallistel PhD -colleague
• Lila Gleitman, PhD -colleague
• Christine Massey PhD - colleague
• Jaimie Liberty - Research Assistant
• Anna Papafragou PhD - postdoctoral fellow
• Earl Williams PhD - former graduate student
• Osnat Zur PhD - - former graduate student
• Lots and lots of undergraduatesLots and lots of undergraduatesLots and lots of undergraduatesLots and lots of undergraduates
• Funding Sources-- NSF(LIS,ROLE): NSF & NIMH ; Predocs; Spencer Postdoc; Rutgers, UCLA.
.
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Fraction Placement Task
CCChhhiiilllddd (((888 yyyrrrsss))) ppplllaaaccceeesss ttthhhrrreeeeee
cccaaarrrdddsss aaasss ssshhhooowww iiinnn 111...
111... 111///222 444 222---111///222
NNNeeexxxttt CCCaaarrrddd PPPuuuttt BBBeeelllooowww 444
222... 111///222 444 222---111///222
2 / 2
NNNeeexxxttt cccaaarrrddd iiisss pppuuuttt aaattt eeennnddd ooofff rrrooowww iiinnn 222
333... 111///222 444 222---111///222 2 - 2 / 3
222///222
NNNeeexxxttt CCCaaarrrddd PPPuuuttt BBBeeetttwwweeeeeennn 222---111///222 &&& 222---222///333
444... 111///222 444 222 111///222 2 / 4 222 222///333
222///222
R. Gelman
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Table 1: Examples of Ways Primary School Children Ordered Numerical
Representations. (Each number was on a separate card and children
were allowed to self-correct and knew they could place more than one
card at the same rank order.)
Example 1:
0 1 12 13 14 1
12 1
34 2
22
24 2
12 2
13 2
23 3
33
34 3
12 4
--0 &1 before unit fractions with ascending denominater, then clusters of ascending
whole numbers with “related” mixed fractions -- x/x’s NOT treated as equals
Example 2:
0 1 12
22
13
33
14
24
34 1 1
12 1
14 1
34 2 2
12 2
13 2
23 3 3
12 4
-- 0 before symbols that look like fractions; then whole number and “their
fractions”;-- ordering rule for fractions – rank order denominators and within
each cluster rank the numerator-- x/x’s not treated as equals
From Hartnett & Gelman, 1998
Numerate individuals less likely to use small sample information
over more normative information (Obrecht, Chapman & Gelman)
50
55
60
65
70
75
80
High Low
Numeracy
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Range Matters
College students do much better on
%, decimals, etc., problems with 100
vs 1000
Numeracy questions (numeracy score = #
correct):
1. .07 is the same as ____ out of 100.
2. 1 out of 1000 is the same as ____ %.
3. If a fair coin is tossed 1,000 times, how many
times
would you expect it to come up tails? ____ .
4. 2 out of 100 is ____%.
5. 12% is ____ out of 100.
6. .2 is the same as ____ out of 100.
7. 27 out of 1000 is the same as ____%.
8. 3% is the same as ____ out of 1000.