The Relationship between
Quantifiers & Math: A Proposal
Barbara Zurer Pearson (w/ Tom Roeper) in
preparation for a visit to Univ of WalesBangor in Spring ‘09
UMass-Amherst Language Acquisition
Colloquium 9/22/08
Evolution of the project:
• Submitted NIH R03 in 2005
• Submitted NSF Role (w/ Bev Woolf of CS)
• Submitted NIH R21 (exploratory study) 2006
• (some of the NIH critique on Handout p. 2….Guess who
was on the panel?!)
• Small funding from ESRC in Bangor, Spring
2009 (need to pilot more before then). Hence
our meeting today to get started.
Organization
 1. Some observations of non-adult interpretations of
quantifiers (not new)
 2. Linguistic (syntactic/ semantic/ pragmatic) aspects of
the misinterpretations (not new)
 3. Prediction of effect of quantifier understanding on
math performance (new)
 4. Potential cross-linguistic/ cross dialect effects of
quantifier interpretation, with corollary effects on math
understanding (new—needs more development)
 4b. Potential 2-way relationship Quantifiers <=> Math
(also thin)
 5. How could one study the hypothesized relationships?
Historical Examples
•
•
•
•
•
Piaget (1964)
Roeper & Matthei (1974)
De Villiers & Roeper (1993)
Philip 1995 (classic spreading)
Roeper, Pearson & Strauss (2004-6) (add
in bunny-spreading)
• Negative displacement: (many, Lidz &
Musolino 2002?)
Piaget: Are all the circles blue?
Not this one!
? = Are all the circles all of the
blue things?
Roeper & Matthei: choose the picture where
“some of the boxes are black.”
?= the boxes are some black
De Villiers & Roeper (’93):
There’s a horse that every boy is on.
Bill Philip: (“classic” spreading):
Is every girl riding a bike?
Copyright The Psychological Corporation 2000
Roeper, Strauss & Pearson: (bunny spreading)
Is every dog eating a bone?
Copyright The Psychological Corporation 2000
Negative displacement
• Every student can’t afford a car.
• Adults prefer: = “not every” = “some”
Only some students can afford a car
• Children prefer: = no students can afford
cars, i.e. no students have cars
One of Tom’s favorites:
Do the boys have two hands or four hands?
Collectively or distributively—i.e. with an implicit “each”
Our proposal:
• These are linguistic (potentially syntactic,
semantic, or pragmatic)—but not necessarily
mathematical--obstacles.
• They may have an impact on children’s
understanding of math concepts—and
especially their ability to do “word problems.”
• Problems even greater for those speaking a
non-mainstream dialect –or with a
background in a different language.
It’s an empirical question, i.e. testable.
• Test children’s knowledge of quantifiers
• Test children’s knowledge of math (concepts,
computation, and word problems)
• Include participants from mainstream Am English,
Hispanic, and African American English background.
• Do think-alouds for the comprehension and production
tasks to probe children’s thinking about the meanings of
the quantified expressions in the story problems.
• Use a regression analysis to see whether quantifier
knowledge contributes to word problem success over
and above computational skill (and especially for
children of different linguistic backgrounds)
Testing children’s knowledge of quantifiers
• 4 quantifiers: all, every, each, some, most
• 3 language groups: MAE, AAE, Hispanic
BL
• 7 ages (4, 6, 8, 10, 12, 14, adult) (or 4)
• 10 participants per cell
• 4 semantic properties
–
–
–
–
Exhaustivity,
Distributivity/ collectivity,
concord/ scope,
displacement
And testing math--Materials/ Tasks
• Comprehension w/ abstract stimuli
• Comprehension through production
(drawing)
• Quantifier probes in text context
• Math probes (ARPS placement tests;
Robbie Case tests of math concepts;
Berkeley people??)
• Plus DELV-Screener LVS or BESA
First steps
1. Find out where this is likely to actually be an
issue for children (i.e. examine math materials
for potential ambiguities)
2. Establish that children’s quantifier
understanding is engaged when they’re doing
math (i.e. pilot more kids)
3. Hone the tests for quantifier understanding to
make sure we can operationalize it adequately.
4. Find what we hypothesize won’t differ across
languages (to motivate the crucial role of
language in the quantifier parts).
Step 1: Find Quantifier issues in real math
tests and classes
From www.mcasmentor.com (Sorry, can’t find the picture)
• Liu is separating the figures below according to their
properties. There are at least three figures in each
group. So far, he has made two different groups. List at
least 3 figures that could go into each group. Explain
what all the figures in each group have in common.
• I assume it’s something like this:
Potential ambiguities, especially if you’re not sure
about the meaning of “each” in:
“some figures could go into each group”
• Is that 1 group per figure?
• Or can one figure go into both groups?
• Do all the figures have something in
common regardless of group, or only by
group?
From an elementary math class for
adult bilinguals:
• Example from Vanessa Hill
• Given this problem:
• 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 students (adult bilinguals) 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 the Amherst Elementary School Math
Placement Test: a girl aged 6;9
• “How many sets of 10
and how many ones are
in the picture?”
•She 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.”
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
Teenagers/ dialect issues
• Eleanor Orr (1987), Twice as less: Black
English and the performance of Black children in
mathematics and science.
• Orr worked with AA teenagers at a private school
in DC. They did a lot of talking and writing in
working their math problems, and she
uncovered lots of conceptual errors.
• For example: Twice as less—how much is that?!
– How much is twice as less as (100?)
– (less than what?)
Steps 2 & 3: Examples of our probes (with
pre-pilot results)—
ABSTRACT STIMULI
Distributivity/ Collectivity
f. Do the boxes have three triangle tops=> collective
(yes)
g. Do all the boxes have three triangle tops => collective
(yes)/ distrib (no)
h. Do most of the boxes have three triangle tops => no
i. Does every box have three triangle tops => no
Abstract stimuli for concord/ scope
Do all the boxes have a circle => yes
Do all the boxes have all circles => if yes =
concord/ if no, non-concord.
To test displacement
“Every boy does not have a hat? Is that right? Show me.”
Answer Yes i.e. = (not every boy) has a hat—Will point to
the boys with no hats
Answers No i.e. = every boy (does not have a hat) = every
boy has no hat –> point to boys with hats
Production Questions: “Draw what the
sentences say.”
• . Draw a picture with lots of circles
– Where all the circles are black.
– Where all the circles are all black.
– Where the circles are some black.
– Where every circle has no black.
• Draw a picture with 3 boys:
– Where every boy is on a box.
– Where there is a box that every boy is on
Drawing distributivity (or not)
• Draw me some flowers and vases, like
this:
– The flowers are all in a vase.
– All of the flowers are in vases.
– Each flower is in a vase.
– Each flower is in vases.
– The flowers are each in a vase.
– Every flower is in every vase.
• Word problems on a separate handout.
• Examples from “pre-pilot”—kids engaging
their quantifier knowledge (whatever it is)
in doing math….
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-yearold) 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.”
Other curious interpretations
• As one 6-year-old told us when asked
“who was wearing a hat” (from an array
with several boys wearing hats and others
without), “I don’t know which one to tell
you”
(But, on the way back to the classroom, he told
me about his baseball team (“Who is on your
team?”)—exhaustively. Is it a pragmatic issue?)
11-year-old boy: “Do all the boxes have
every circle?”
No, (he wanted all
the boxes completely
filled with circles—he
colored the boxes in
with tons of circles.)
Same interpretation for “Does each box have each
circle?” AND “Does every box have every circle?” Letting
aside what his interpretation was, all of the quantifiers
had the same properties.
Control sentence: “Do boxes have circles
inside?”
Several children said “no” (requiring exhaustivity
where it’s not warranted)
Math to Quantifiers
• How much does one have to know about
comparing inequalities to interpret “most”? (Had
a 4-year-old who couldn’t do one-to-one
counting very well and couldn’t compare near
quantities. Couldn’t do “most” at all.)
• One-to-one counting allows what??
)
have some speculation in NSF??!
(note to self: do we
Cross-dialect dimensions: AAE
• AAE & bunny spreading: AAE kids didn’t use
more Classic spreading, but more children did BOTH CS
and BS)
• Multiple negation—may predispose to
spreading
Cross-linguistic dimensions:
Spanish-influenced English
• Multiple negation
• (Ana P’s) interpretation of generics (the lion over
there vs “the lion” as a species)
• Much/ many (no grammatical count-non-count
distinction). See Gathercole, 2002
• Each/every (no word for every) – Use each and
all
• What will Spanish children prefer in distinctions
between collective and distributive readings?
Welsh predisposition to collective
over individual reading
• What's in a noun? Welsh-, English-, and Spanishspeaking children see it differently Mueller Gathercole et
al. First Language.2000; 20: 055-90
Welsh children, whose language is not as clearly marked
for singular and plural—and there are many plurals
which are unmarked and they take a suffix for the
singular--Children were more likely to interpret a novel
noun as a collection than were English children. (As
background, they list some experiments with English
children, in support of innate biases, that collection
nouns (“forest”) are less likely guesses for English
children till older. Not so in Welsh.)
We’re going to look at some strange things that I have—
that I bet you’ve never seen. The bear always wants just
what I have.
• .
A
B
Which of these is the bear’s blicket?
Welsh children chose group significantly more often
than English or Spanish children.
• .
Other choices didn’t show
cross-linguistic
differences.
(Did another more
complicated experiment
to make sure they weren’t
just “matching.”
Previous studies
• Not on the radar for math teachers, even
those who write about math and language
• Jose Mestre in late 80s early 90s. Studied
how college students translated word
problems into equations (like Orr).
• Orr Twice as Less (examples in NSF??)
• Helen Stickney on “most” (like “mostly”)
Consensus
• There’s much too much for one study,
even one set of studies (and Tom gets
even more ideas to broaden it whenever
we talk about it (like today).
• Should work more on the actual math
problems to find the nature of the
ambiguities. (Get Peggy to look at them
with me??)
Next studies for us??
• Need to narrow question to something do-able
for this year, including the time in Wales.
• I would be happy to have an age-stratified
sample do a set of drawings of the
expressions—for collective/ distributive (like the
flowers in vases).
• Also want to make a methodology for using a
tablet PC for stimuli and recording responses
and even a light pad for the children’s drawings.