10/5/2009

What is your quantifier quotient (QQ)?

Barbara Zurer Pearsonbpearson@research.umass.edu

University of Massachusetts AmherstLanguage 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), co-PI, 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 non-native speakers (L1 Spanish), and potentially speakers of AAE as a first dialect.

1. Implicit interpretations—targets not clear

2. Shifting interpretations—• Need sensitivity to syntax and pragmatics

3. 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/online-survey.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.

Exhaustivitytest 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

ProblemsExhaustivity

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 10From 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)

Multiples of 3

Multiples of 4

9

24

8

5

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 garden-pathing 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?” )

•Non-native: 8 no [not stuck], 9 yes [“All-not”/ stuck]

(vs. almost 2:1 not-stuck to stuck, among native speakers)

Displacement Example-con’t

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.

•(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 “not-all”

Displacement Example-Consistency

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 exampleFour 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

Split cookies?

Split flowers?

N Y N Y

All 14 40 28 26

Non-native

3 12 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.

• (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 allYes’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)

Example (not from our 3 properties)

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 vases1. 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 MeetingUmass AmherstOctober 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).