10/5/2009
What is your quantifier quotient
(QQ)?
Barbara Zurer Pearson
bpearson@research.umass.edu
University of Massachusetts Amherst
Language Acquisition Colloquium
Background
•This work is part of the preliminary studies
for an October 1 submission to Institute of
Education Sciences (IES) with Tom Roeper,
PI, John Francisco (UMass School of Ed), coPI, Peggy Speas, Michael Lavine (Math and
Statistics), and Bev Woolf (Computer Science).
•Consultants: Jill de Villiers (Smith College),
Carolyn Maher (Rutgers), Ana Perez-Leroux
(Toronto), & Janice Jackson (U of West
Georgia).
The Hypothesis of the IES
Grant
Quantifiers represent a learnability problem for
children in general, but most especially for nonnative speakers (L1 Spanish), and potentially
speakers of AAE as a first dialect.
1.
2.
Implicit interpretations—targets not clear
Shifting interpretations—
•
3.
Need sensitivity to syntax and pragmatics
Little awareness of difficulties, so there is no teaching
to them.
The Plan for the IES Grant
1. Discussion of which properties of quantifiers we’re
focusing on as difficulties.
2. How we know these pose a problem for children in
typical L1 acquisition.
3. What are cross-linguistics differences that might
make them cause difficulty in an L2.
4. Where might each have some effect on math;
perhaps doubly hard for bilinguals and L2 learners
(or where might there be facilitation).
1st: How can we operationalize
“difficulty w/ quantifiers”?
• Need a measure of quantifier comprehension /
interpretation to permit comparisons
Between adults of the same language group
Between adults and children of the same language group
Between adults of different language groups
Between children of one language group and adults of the
same and different language group
• Between children of different language groups
•
•
•
•
What does success with quantifiers look like?!
What is the target?
The plan for today
1. Which interesting properties of quantifiers we’re
focusing on (very briefly).
2. Some examples of ambiguities the semantic
properties of quantifiers can cause in math word
problems.
______________________________________________
_
How can we characterize an individual’s knowledge
of the semantic properties of quantifiers?
•With a quantifier quotient (QQ)!
•A play on words with IQ
•Making “norms” for quantifier interpretation
With respect to distributivity,
exhaustivity, & displacements
• How consistent are interpretations?
• Within an individual
• Within a group
• Native speaking adults
• Non-native-speaking adults
• Children
• What is “difference”? What is development?
Why?
Issues to Address
• Learnability (a moving target?)
• How powerful is syntax in constraining interp’s?
• Does pragmatics trump syntax?
• How different are native speakers from each other?
• How different are non-native speakers (of different
language backgrounds)?
How?
The Norming Instrument
• On-line Quantifier Intuitions Survey
• http://www.kwiksurveys.com/onlinesurvey.php?surveyID=OIHKG_7f21b1b
The Norming Sample
• 42 native speakers of English (ages 18-71, mean ~32)
• 18 non-native speakers of English (mean ~32 yrs)
First an overview
• Of the properties we’re looking at
• Displacements
• Exhaustivity
• Distributivity/ Collectivity
• Why we think there’s some math issues
What are Displacements
• Plural spreading
• Quantifier spreading (e.g. with every)
• Floating adverbial quantifiers
• All not  Not all
Simple example: plurals move
•
(from Roeper, 2006)
• Is the corner of the boxes bent?
How did the plural get from boxes to corner?
Piaget: Are all the circles blue?
(1952)
Not this one!
? = Are all the circles all of the
blue things?
“Every”
•What does “every”
mean?
•Consider this common
error:
•Is every girl holding a
lollypop?
•Not this lollypop.
•Cf. Piaget…
Roeper, Strauss & Pearson: (bunny spreading)
Is every dog eating a bone?
Copyright The Psychological Corporation 2000
“Floating”
• a. Is each rabbit eating a carrot?
•
b. Are the rabbits eating each a carrot?
•
c. Each of the rabbits is eating two carrots each.
• d. the children (all) have (all) been (all) going home.
• e. (Both) boys (both) left early.
Diary examples
• Each glove is on each hand.
• (A hand is in each glove and a glove is on each hand.)
• (3-year-old grandson)
• _____________________________________
• Both rabbits are on both sides. (for one on each side)
• (from B. Partee)
• _____________________________________
• Everybody isn’t here. (when no one was there.)
• (Esme, age 3 (bl)
Different constraints for
“floating”
•
Floating quantifiers (in French)
•
French has “Quantifiers at a distance”
• QAD: Les enfants ont beaucoup recu de ballons
• ‘the children have a-lot received of balloons’
• (beaucoup quantifies over balloons)
•
But from the same position, , “floating quantifiers”
• FQ: Les enfants ont chacun recu un ballon.
• ‘the children have each received a balloon’
• (chacun quantifies over the children).
•
Not surprisingly, there’s a time when children will float both ways for
beaucoup (beaucoup des enfants).
•
Labelle & Valois (2001)
All…not  Not all
•
Panic:
• (sign on train: All doors do not open.)
•
Relax: It means NOT ALL doors open.
•
Most common interpretation:
• Here are extra handouts, in case everyone doesn’t have one.
(i.e. only some have them).
• All is not lost. Etc.
Exhaustivity
test with who:
Who is wearing a hat?
Double wh—adds a new requirement
This dad and
his baby were
eating lunch
together.
Who ate what?
On DELV-NR (Seymour, Roeper & de Villiers, 2003, 2005)
Who ate what?
• Have to answer the “who” --exhaustively.
• Have to answer the “what” --exhaustively.
• But can’t just say, “baby, father, apple, banana.”
• Have to pair them.
• “Baby ate banana, father ate the apple.”
Sentence syntax enters the
picture (cleft)
•
Tanja Heizmann’s dissertation
• We see Grover eat a sandwich, a banana, and a cookie.
• A puppet says: “Oh I see. It’s the cookie that he ate.”
• Tanja asks the child: Did the puppet say it right?
• Could be fine: I see, he ate the cookie. (no need to be
“exhaustive”), but with the cleft, now you have to be.
What is distributivity?
From 6th grade MCAS:
“He wrote each name on one card” [and then picked
one out of a hat.]
Wait a minute. Isn’t there only one card?
Did the speaker distribute names to separate cards, or
collect them all on one card?
Distributivity/ collectivity
Compare: “Every boy is on a horse.”
Test presupposition by
“cancellation”
Pick up all the rocks, but don’t do it one by one.
*Pick up each rock, but don’t do it one by one.
?Pick up every rock, but don’t do it one by one.
•
Depends on the predicate:
•
Lift all these books (together) vs
•
Paint all these chairs (one by one).
Add in syntax constraints
• A nurse cares for every patient.
• (one nurse for all/ or one-for-each)
• A nurse wanted to care for every patient.
• (only one)
• Every boy is on a horse.
• (one horse for all/ one for each)
• There’s a horse that every boy is on.
• (only one)
(current example)
• With my coauthor, “The children took the test
individually in a quiet room at their school.”
• For me, it distributes.
• For her, she insists since there are many different schools,
it’s “in quiet rooms in their schools.”
• For me, that makes it more than one room in each
school—but since I really know it’s “under-determined,”
she can have it her way.
4. Potential Pitfalls in
Mathematics Story Problems
Exhaustivity
Distributivity/ collectivity
Displacement
Exhaustivity
• Jalal has ten pockets and forty-four pennies. He
wants to put his pennies in his pockets in such a way
so each pocket contains a different number of
pennies. Can he do it? Explain your answer.
• The adult bilinguals in our friend’s elementary math
class did not immediately grasp that “each” directed
them to make 1 group per pocket. Nor did they
understand without explicit explanation that they
were to use all 44 pennies.
Ambiguous Conjunction/
ambiguous collectivity?
•“How many sets of 10 and how
many ones are in the picture?”
•Answer key, 3 and 4. My answer was potentially 34.
•Child very carefully counted two sets of 10 to confirm
that there were 10 in each of them, and then said “3 tens
and 1 ones”
—as if the question had an elliptical “sets”: “how many
sets of ones were there.” (PS said 34 ones and 3 sets of 10
From the Amherst Elementary School Math Placement Test: a girl aged 6;9
MCAS test asks for “one more number that goes in each
of the four spaces in the diagram.”
(could be impossible)
5
Multiples
of 3
Multiples
of 4
24
8
9
1
From 6th grade Math CAS, p. 129
No “every”—lots of
“each”
• Each child voted for one game.
• Each child’s name was written on one card, [and then were
picked out a hat]
• What is the cost of having to reanalyze? (cf. Frazier on gardenpathing into collective….)
Drawing by an 11 year old boy doing our production task
For “Every boy is on a box”
AND
“There is a box that every boy is on.”
Cf. de Villiers & Roeper 1993
—6-year-olds get the relative clause barrier.
Drawing by boy 6;9 (and similar one by 11-year-old) for “There is
a box every boy is on.”
An apparent violation of the presupposition for “every” that there
needs to be more than one—and maybe more than two—for
“every.”
Quantifiers do mysterious
things
• How do we learn them?
• What they mean lexically
• What they entail semantically
• How they derive or change meaning according to the
pragmatics of the situation.
• Set the acquisition questions aside
Back to the QQ!
• By property
Displacement Examples
•Sign in a railway car: All
doors do not open.
•Are we stuck (in the railway car)?
•Native English: 24 change to “not-all doors” [not
stuck]; 13 said “none do” [we’re stuck]
•(but 6 of the 24, answered based on the situation, not the
sentence. “people who write train signs aren’t very careful,”
“how did we get in?” )
Displacement Example-con’t
•Non-native: 8 no [not stuck], 9 yes [“All-not”/ stuck]
(vs. almost 2:1 not-stuck to stuck, among native speakers)
More displacement: Example 2
“Every cat does not have a cookie? Is that right? Show me.”
Answer Yes (displaces) i.e. = (not every cat) has a cookie—Will
point to the cats with no cookie
Answers No i.e. = every cat (does not have a cookie) = every
cat has no cookie –> point to cats with cookies
All the cats don’t have a piece of
watermelon.
How consistent were people?
• Native speakers: displace on all 3 or not displace on
all 3 = 25 (15 : 10)
• Non-native: 7:5 displace to non-displace
Especially for the train-door example, we saw them
abandon the sentence and just give answer based on
reality.
Displacement Example-Consistency
•(Somewhat surprised at the 13 “stuck” responses among
native speakers.) How many who gave an “All-not”
interpretation here did so on other similar items?
For example, my niece and my sister-in-law both did the
questionnaire: the niece (age 28) displaced (all not  not
all) on all three such items; the mother , 50+ didn’t displace
any of them.
Of the 13 (native-speaker) “No-doors-open” people,
10 were consistent on all 3, 3 did the 2 cat examples “notall”
Collectively or distributively
Do the boys have two hands or four hands?
(Understand with an implicit “each”, does each boy have…”)
What makes collectivity?
1. (How many hands do the
children have?)
2. How many books do the
children have?
3. How many books do the
boy and the girl have?
4. How many books do the
boy and girl have?
On DSLT (Seymour, Roeper & de Villiers,
2000)
Distributive or Collective bias
(or tendency)?
• 20 flat out collective—hands 4, books 2 (except “the
boy and girl” tended to get “1,” most of these 20.
plus some others = 32 “boy and girl” with 1)
• 3 didn’t trust anyone to distribute or collect and said
explicitly each time: “one each or two together” (or
2 and 4)
• Of 15 who said “2 each” for hands, 8 carried “each”
through on the books; 7 just said “2 books”—
nothing to make it explicit).
Distributivity continued
Nonnative
• German and Hebrew, all collective (n = 6)
• Romance, split
Distributivity interaction with syntax
Consider a hospital with 100 patients
a) A nurse cared for every patient
b) A nurse wanted to care for every patient
How many nurses do you think there are?
a) Could be either distributed (1 nurse per patient) or
collective (1 nurse for everyone)
b) But b) can only be 1 nurse
Native speaker near-unanimity
People didn’t like the idea of one nurse for 100, but
typically said a) could be any number of nurses, but
b) had to be one
34 of 41 restricted distributivity when embedded; 4
didn’t make a distinction between the 2 sentences;
only one said “it doesn’t say”
Non-native speaker, no clue
6 of 16 got the distinction; 7 of 16 got restriction on
embedding, but one had the other sentence collective as
well.
9 made no distinction between the sentences
3 said “it doesn’t say”
3 (!) said “at least one” for the last sentence.
(What does “at least one” mean--when the answer is 1?)
Every boy has 3 buckets, and these girls
have one bucket.
•Is that one bucket for each girl? Or all together??
Do we see the biases from the
other items?
• 13 collective-people, said it was collective
• 3 distrib with “each” said collective
• And several said, I need “each” to distribute, and
one said, it would say “together” if it meant
together. But most said “it says the girls are a
group”
How use syntactic cues in
ambiguous sentence?
Could go both ways:
Use parallel—carry clear distribution from one part of
sentence to the next (7)
Use “contrast” of each versus “group” (3)
1 used number agreement on verb to disambiguate: “these
girls have” means a group
Non-native—more used parallel; native tended to say “it
says the girls are a group.”
Pragmatics example
Four people bought a dozen cookies for $12.
Did each person spend $12?
Four people bought a dozen roses for $3.
Did each person spend $3.
Same grammar,
different decisions
All
Split cookies?
N
Y
14
40
Non- 3
native
12
Split flowers?
N
Y
28
26
6
9
Only one person didn’t split the cookies but split the flowers.
Twice as many people would not split the cookies but would
split the flowers. Reasons based on normal cost of the objects.
Example (not from our 3 properties)
• (picture of 4 ducks with apples) Q: “Some ducks have
apples.” Is that right?
i.e. Is “all” ok for “some”?
•No = 14
•Yes = 16 (of 39)
•Plus 9 “yeses” who made reference to other ducks not in the picture. So
for them “all” is not “some.”
•Reasons given:
•No’s, basically should say “all” if it’s all
Yes’s, some includes all/ indef: “There are some ducks. They have
apples.” (some as plural of “a”)
•Non-native, also split, not-ok / ok: (9-7)
Girls carrying
cakes
1. All of the girls are carrying a
cake.
2. Each girl is carrying a cake.
3. The girls are carrying a cake.
(From Brooks et al. 2001)
All/ each
• Which is it? All together? All carrying one cake? Is
it different from Each of the girls is carrying a cake?
• The girls are carrying a cake??
Flowers in vases
1. All the flowers are in a
vase.
2. All the flowers are in one
vase.
3. Each flower is in a vase.
4. Each flower is in one vase.
5. Every flower is in a vase.
6. One flower is in one vase.
•
(From Brooks et al. 2001)
Thank you : )
Syntactic aids?
• Who is wearing a hat, German versus English
• In English: does homophony with relative pronoun “who”
create a problem? “the boy who is x, versus the boys who are x.”
Might expect that “who” is singular because of the agreement
pattern.
• Until age 5 in Germany, age 6 in the states, children would
point to just one.
• In Spanish (and Dutch), interrogatives also have agreement.
Will they understand exhaustivity sooner?
DelliCarpini Result with Adult
L2
(2003, Gasla Proceedings)
• Low Proficiency Group—
• similar errors to children i.e. “adverbial”
• (presumably not cognitive)
• No (discernible) effect of L1
• High Profiency Group—
• Better performance overall
• L1 effects? All Chinese, Japanese, and Korean learners
rejected perfectly grammatical distributive readings for
“Every girl is on a tractor” (w/ 3 girls on 3 tractors): because
there was “more than one tractor.” (Also “a girl is feeding
every rabbit” because there was more than one girl.)
Marsden Result with Adult L2
(2009, Lang Acq article)
• Shows an effect of L1 in??
African American English features
• AAE existential “it”:
• Does it make a difference for “there’s a horse that every
boy is on” versus “it’s a horse that every boy is on.”
• For GAE speakers, there’s a kind of focus reading that
has nothing to do with distributivity.
• What about AAE background?
•3 levels
1. The range of different answer types
within the set of answers from a single
individual: consistent “profiles”?
2. The range of response patterns or profiles
within a native speaker group?
3. The range of the same response patterns
among groups with different language
background.
•What is your quantifier quotient (QQ)?
•Semantic Intuitions by Language Background
•Barbara Zurer Pearson (& Tom Roeper)
Language Acquisition Meeting
Umass Amherst
October 5, 2009
The Plan
1. Which interesting properties of quantifiers we’re
focusing on.
2. How do we know these pose a problem for children
in typical L1 acquisition.
3. What are cross-linguistics differences that might
make them cause difficulty in an L-alpha or L2.
4. Where might each have some effect on math;
perhaps doubly hard for bilinguals and L2 learners
(or where might there be facilitation).