Semantic Properties of Quantifiers in Language ...

Semantic Properties of Quantifiers in Language Acquisition and Mathematics Barbara Zurer Pearson University of Massachusetts Amherst Visitor Researcher Colloquium, 27/4/09

Semantic Properties of Quantifiers (such as all, each, every, and not every):

* What added problems can they cause bilingual learners in doing mathematics word problems?

1st: Thanks & Acknowledgments UMass Amherst, Language Acquisition Lab. Especially collaborator Tom Roeper, Professor of Linguistics (Mostly his ideas. My job to make them quantifiable.) University of Miami Bilingualism Study Group (Kim Oller, Vivian Umbel, Virginia Gathercole, Alan Cobo-Lewis, Sylvia Fernandez, Maria Fernandez, Ana Navarro & many others)

Thanks & Acknowledgments - 2 Bilingualism Centre (Margaret, Ginnie, Hans, Rocio, Carmen, Marketa, Ineke, and the faithful lunch-bunch)– for giving me the time and space to prepare a grant submission on this topic. (And you--for being in the select few who are not out-oftown today. Thank you for coming today. Take some birthday cake!)

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-linguistic differences that might create new difficulty in an L-A 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).

1. Interesting Semantic Properties of Quantifiers Exhaustivity Distributivity/ collectivity Displacement

2nd: Definition of Quantifier In plain English, a word that expresses a quantity: 10, many, all In logic and linguistics, a term that binds the variables in a proposition. That is, they help pick out sets of individuals, and they help pick out the relevant sets—and their relationships. Ex. “all” in “all the crayons are broken” tells us about two sets that exhaust the sets of crayons: the set of broken crayons and the set of non-broken crayons. If the statement is true, the set of non-broken crayons must be empty.

Example 2—‖some‖ If “some widgets are wiggly” is true, then we know that neither the set of wiggly widgets nor the set of non-wiggly widgets is empty. There are (some) widgets in each of them.

In this project, we‘re less interested in their abstract, logical properties, and more interested in how children (and adults) from different language backgrounds interpret quantifiers in everyday sentences and situations.

Not going by word, but by property. Distributivity, exhaustivity? What are they, and how are they communicated? What are displacements, and how do they arise? Few real rules; mostly presupposition and implicature. For the learner, nothing obvious.

Exhaustivity – ―don‘t leave any out‖ ―10‖ is always 10. Doesn‘t matter how many there are to choose from. Compare: ―the‖ You have 10 pennies. If I say, ―give me pennies‖ you can give me as many or as few as you want. If I say, ―give me 3 pennies,‖ that‘s pretty clear. If I say, ―give me the pennies,‖ it will be weird for you to give only some of them. To fulfill the sentence, you have to give me all 10 (the whole set). ―a‖ Give me a penny. Don‘t know which, but know how many. (In some languages, the same word as ―one‖)

Exhaustive ―who‖ Who was in the car the night of the murder? That‘s ―who-everyone.‖ If it‘s Tom, Dick, and Harry, and you just say ―Tom,‖ it could be perjury! You have to say everyone. BUT that semantic (& legal) requirement can be ―cancelled‖ in some circumstances/ pragmatics. Who has a pencil I could borrow? You just need one person. (That’s more a fact about indirect requests than quantifiers, but it’s part of the child’s input.)

Sentence syntax enters the picture Tanja Heizmann’s dissertation (in progress)

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.)

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.‖

What is distributivity? ―distributing‖ objects (often in one-to-one relationships)

Distributivity/ collectivity

Compare: ―Every boy is on a horse.‖

(Roeper & de Villiers, 1993)

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.

(From Brooks et al. 2001)

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)

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 does each child have?

On DSLT (Seymour, Roeper & de Villiers, 2000)

Every boy has 3 buckets, and these girls have one bucket.

Is that one bucket for each girl? Or all together??

(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 ―unresolved.‖ She can have it her way.

What are Displacements Floating adverbial quantifiers Plural spreading

Weak quantifier spreading All not  Not all

―Floating‖ a. Is each rabbit eating a carrot? b. Are the rabbits eating each a carrot? c. Each of the rabbits is eating a carrot each.

d. the children (all) have (all) been (all) going home. e. Susan, Mary, and Sally were all here f. *All Susan, Mary, and Sally were here

Note: ―every‖ doesn‘t ―displace‖ It‘s not reversible, it doesn‘t float. It stays where it is. Every girl is riding a bike. • * A girl is riding every bike. • * Every girl is riding every bike.

Simple example: plurals move (from Roeper, 2006)

Is the corner of the boxes bent?

How did the plural get from boxes to corner? (distribution!)

not following surface scope (weak) quantifiers are reversible. Many girls are riding bikes Girls are riding many bikes. Many French are winners of the Tour de France. Many winners of the Tour de France are French. (from Drozd, 2001)

All…not  Not all Panic:

(sign on train: All doors do not open.) Relax: It means NOT ALL doors open—but some do. 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.

Summary The properties are ―real,‖ but underrepresented in the speech signal. There is not full agreement among adult speakers. Not so far-out to contemplate interpreting the quantifier in a site other than the one where it appears.

2. Children‘s Documented Problems with these Properties of Quantifiers Exhaustivity Distributivity/ collectivity Displacement

Have to learn which and when. Learning we don‘t take into account. May be cognitive, but we claim it‘s both a mapping problem (knowing which words have which properties in which circumstances), and also a syntactic issue.

Note: Production by children only part of the story. They use in contexts where they understand. Only rarely wrong. But they don‘t necessarily have full understanding, and may not produce correctly when pushed (Altreuter & de Villiers, 2005).

Piaget: Are all the circles blue? (1952, 1964)

Not this one!

? = Are all the circles all of the blue things?

Also see with ―some‖ & ―most‖ Matthei experiments (1974).

Roeper & Matthei: choose the picture where ―some of the boxes are black.‖

?= some boxes are some black

Exhaustive who: Who is wearing a hat?

Who-someone? Or who-everyone?

Through age 6, most prevalent answer was to point to one. Not random performance strategy: as one child said, ―I don‘t know which one to tell you.‖ Roeper, Pearson, Penner, & Schulz, 2002

Double –wh: Who ate what? Most 4-year-olds got it right. BUT, many typical errors: • • • •

a. Give just one pair b. Give a singleton answer. c. Give just the what, or just the who.

(c the most common)

―Every‖ What does ―every‖ mean? Like ―all‖, that the set of non-x is empty. But consider this common error: Is every girl holding a lollypop? Not this lollypop. (Philip, 1995. Cf. Piaget…)

Roeper, Strauss & Pearson: (bunny spreading) Is every dog eating a bone?

Copyright The Psychological Corporation 2000

What can ―every‖ mean to that child? Adverb? Like always: ―Are girls always holding lollipops?‖ Concord: Is every girl holding every lollipop? (Takahashi, 1991)

Displacement: Every ―moves‖ to lollipop? Roeper, Strauss, & Pearson, 2005 – syntactic argument involving focus nodes in the tree as ―every‖ moves into the determiner phrase.

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 here.) (Esme, age 3 (bl)

De Villiers & Roeper (‘93):

There‘s a horse that every boy is on.

Every….not (Musolino & Lidz, 2006)

Every horse did not jump the fence.

1 or 2 out of 3 jumped, adults will accept. (Kids will reject unless no horse jumps the fence.)

Every….not: Change pragmatics Every horse jumped the log, but every horse did not jump the fence. More kids got it.

But still not adult like in L&M L&M next did it with All x’s not doing the action (both sub & ob). The strong man didn‘t lift every elephant. (when none did it.) Adults don’t like this as a description (when it could have been “didn’t lift any of the elephants.”) Contrasted description with the phrase in a prediction: “I bet the strong man won’t lift every elephant.” (kids not sensitive to the difference.)

3. Cross-linguistic Differences wrt Properties of Quantifiers Exhaustivity Distributivity/ collectivity Displacement

Compare with other languages (according to Nishigauchi, 1990)

Japanese: Distinguishes ―who-specific one‖ Southern American English: Who-all (on model of ―youall‖) Few languages have ―every‖—all and each, but not every (Bach et al., 1995)

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)

Tolerance for ―floating‖ What is the effect of having Negative concord/ multiple negation

as in Spanish and African American English?

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.)

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.‖

We‘re testing displacement cross-ling

―Every cat does not have a cookie? Is that right? Show me.‖

Answer Yes 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

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 a 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. (from V. Hill, STCC)

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.‖ 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.

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.‖

References (not exhaustive) Altreuter, E. & de Villiers, G. (2005). Does every child produce every correctly? In T. Heizmann (Ed.), Papers in Language Acquisition, University of Massachusetts Occasional Papers UMOP, 34, Amherst, MA: GLSA. Bach, E., E. Jelinek, A. Kratzer and B.H. Partee, eds., (1995). Quantification in Natural Languages, Dordrecht: Kluwer. Brooks, P. J., Braine, M. D. S., Jia, G. & Dias, M. G. (2001). Early representations for all, each, and their counterparts in Mandarin Chinese and Portuguese. In M. Bowerman & S. C. Levinson (Eds.), Language acquisition and conceptual development (pp. 316-339). Cambridge: Cambridge University Press. DellaCarpini, M. (2003). “Developmental stages in the semantic acquisition of quantification by adult L2 learners of English: A pilot study”. In J.M. Liceras, H. Zobl and H. Goodluck (eds.), Proceedings of the 6th Generative Approaches to Second Language Acquisition Conference (GASLA 2002): L2 Links. Somerville, MA: Cascadilla Press. (Donaldson, M. & Lloyd, P. (1974). Sentences and situations: children’s judgments of match and mismatch. In Bresson, F. (Ed.). Current Problems in Psycholinguistics. Paris: Centre National de la Recherche Scientifique.) Drozd, K.F. (2001). “Children’s weak interpretation of universally quantified sentences”. In Bowerman, M. and S. Levinson (eds.) Language Acquisition and Conceptual Development, p. 340-376. Cambridge: Cambridge University Press. Inhelder, B. & Piaget, J. (1964). The Early Growth of Logic in the Child. London: Routledge, Kegan and Paul.

References (con‘t) Labelle, M. & Valois, D. (2001). “Functional categories and the acquisition of distance quantification.” Ms. Universite du Quebec a Montreal. Musolino, J.& Lidz, J. (2006). Why Children aren't universally successful with Quantification. Linguistics, 44(4), 817-852. Nishigauchi, T. (1990). Quantification in the theory of grammar. Dordrecht: Kluwer. Philip, W. 1995. Event quantification in the acquisition of universal quantification, Doctoral dissertation, UMass Amherst. Roeper, T. (2007, 2009). Prism of grammar. Cambridge MA: MIT Press. Roeper, T. & de Villiers, J. G. (1993). The emergence of bound variable structures. In E. Reuland and W. Abraham (eds.), Knowledge and Language: From Orwell's Problem to Plato's Problem. Dordrecht: Kluwer. Roeper, T. & Matthei, E. (1974). On the acquisition of some and all. Presented at the Sixth Child Language Research Forum, Stanford University, April 1974. Appeared in Papers and reports on child language development (l975), Stanford University, 63-74. Roeper, T., Pearson, B. Z., Schulz, P., & Penner, Z. (July 2002). The Emergence of Wh-Variables: Crosslinguistic explorations. Paper presented at the Joint Meeting of the IXth International Congress for the Study of Child Language and the Symposium for Research in Child Language Disorders, University of Wisconsin, Madison, WI. (Also presented at the UConn-UMass Colloquium, December 2001).

References - 3 Roeper, T., Strauss, U., & Pearson, B. Z. (2005). The acquisition path of the determiner quantifier every: Two kinds of spreading. In T. Heizmann (Ed.), Papers in Language Acquisition, University of Massachusetts Occasional Papers UMOP, 34, (pp. 97-128). Amherst, MA: GLSA. Seymour, H., Roeper, T., & de Villiers, J. G., with contributions by P.A. de Villiers. 2005. Diagnostic Evaluation of Language Variation-Norm Referenced (DELV-NR). San Antonio, TX: The Psychological Corporation, Harcourt Assessments, Inc. (Also, Dialect-sensitive Language Test, pilot version, 2000). Takahashi, M. (1991). “Children’s interpretation of sentences containing every”. In T.L. Maxfield and B. Plunkett (eds.), Papers in the Acquisition of WH. Amherst, MA: GLSA Publications.