Silver Cai JRME1996

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See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/245280700

An Analysis of Arithmetic Problem Posing byMiddle School Students

ARTICLE in JOURNAL FOR RESEARCH IN MATHEMATICS EDUCATION NOVEMBER 1996

Impact Factor: 1.27 DOI: 10.2307/749846

CITATIONS

140

READS

175

2 AUTHORS:

Edward A. Silver

University of Michigan

95PUBLICATIONS 2,001CITATIONS

SEE PROFILE

Jinfa Cai

University of Delaware

115PUBLICATIONS 1,241CITATIONS

SEE PROFILE

Available from: Edward A. Silver

Retrieved on: 28 February 2016
https://www.researchgate.net/profile/Edward_Silver2?enrichId=rgreq-ed65b765-f1d3-4474-aa7a-9dd1292e1667&enrichSource=Y292ZXJQYWdlOzI0NTI4MDcwMDtBUzoyOTcyMTI5NTAxMzg4OTNAMTQ0Nzg3MjQ5Nzk4NA%3D%3D&el=1_x_7https://www.researchgate.net/institution/University_of_Michigan?enrichId=rgreq-ed65b765-f1d3-4474-aa7a-9dd1292e1667&enrichSource=Y292ZXJQYWdlOzI0NTI4MDcwMDtBUzoyOTcyMTI5NTAxMzg4OTNAMTQ0Nzg3MjQ5Nzk4NA%3D%3D&el=1_x_6https://www.researchgate.net/institution/University_of_Michigan?enrichId=rgreq-ed65b765-f1d3-4474-aa7a-9dd1292e1667&enrichSource=Y292ZXJQYWdlOzI0NTI4MDcwMDtBUzoyOTcyMTI5NTAxMzg4OTNAMTQ0Nzg3MjQ5Nzk4NA%3D%3D&el=1_x_6https://www.researchgate.net/institution/University_of_Michigan?enrichId=rgreq-ed65b765-f1d3-4474-aa7a-9dd1292e1667&enrichSource=Y292ZXJQYWdlOzI0NTI4MDcwMDtBUzoyOTcyMTI5NTAxMzg4OTNAMTQ0Nzg3MjQ5Nzk4NA%3D%3D&el=1_x_6https://www.researchgate.net/institution/University_of_Michigan?enrichId=rgreq-ed65b765-f1d3-4474-aa7a-9dd1292e1667&enrichSource=Y292ZXJQYWdlOzI0NTI4MDcwMDtBUzoyOTcyMTI5NTAxMzg4OTNAMTQ0Nzg3MjQ5Nzk4NA%3D%3D&el=1_x_6https://www.researchgate.net/institution/University_of_Michigan?enrichId=rgreq-ed65b765-f1d3-4474-aa7a-9dd1292e1667&enrichSource=Y292ZXJQYWdlOzI0NTI4MDcwMDtBUzoyOTcyMTI5NTAxMzg4OTNAMTQ0Nzg3MjQ5Nzk4NA%3D%3D&el=1_x_6https://www.researchgate.net/institution/University_of_Michigan?enrichId=rgreq-ed65b765-f1d3-4474-aa7a-9dd1292e1667&enrichSource=Y292ZXJQYWdlOzI0NTI4MDcwMDtBUzoyOTcyMTI5NTAxMzg4OTNAMTQ0Nzg3MjQ5Nzk4NA%3D%3D&el=1_x_6https://www.researchgate.net/institution/University_of_Michigan?enrichId=rgreq-ed65b765-f1d3-4474-aa7a-9dd1292e1667&enrichSource=Y292ZXJQYWdlOzI0NTI4MDcwMDtBUzoyOTcyMTI5NTAxMzg4OTNAMTQ0Nzg3MjQ5Nzk4NA%3D%3D&el=1_x_6https://www.researchgate.net/profile/Edward_Silver2?enrichId=rgreq-ed65b765-f1d3-4474-aa7a-9dd1292e1667&enrichSource=Y292ZXJQYWdlOzI0NTI4MDcwMDtBUzoyOTcyMTI5NTAxMzg4OTNAMTQ0Nzg3MjQ5Nzk4NA%3D%3D&el=1_x_5https://www.researchgate.net/profile/Jinfa_Cai?enrichId=rgreq-ed65b765-f1d3-4474-aa7a-9dd1292e1667&enrichSource=Y292ZXJQYWdlOzI0NTI4MDcwMDtBUzoyOTcyMTI5NTAxMzg4OTNAMTQ0Nzg3MjQ5Nzk4NA%3D%3D&el=1_x_7https://www.researchgate.net/institution/University_of_Delaware?enrichId=rgreq-ed65b765-f1d3-4474-aa7a-9dd1292e1667&enrichSource=Y292ZXJQYWdlOzI0NTI4MDcwMDtBUzoyOTcyMTI5NTAxMzg4OTNAMTQ0Nzg3MjQ5Nzk4NA%3D%3D&el=1_x_6https://www.researchgate.net/institution/University_of_Delaware?enrichId=rgreq-ed65b765-f1d3-4474-aa7a-9dd1292e1667&enrichSource=Y292ZXJQYWdlOzI0NTI4MDcwMDtBUzoyOTcyMTI5NTAxMzg4OTNAMTQ0Nzg3MjQ5Nzk4NA%3D%3D&el=1_x_6https://www.researchgate.net/institution/University_of_Delaware?enrichId=rgreq-ed65b765-f1d3-4474-aa7a-9dd1292e1667&enrichSource=Y292ZXJQYWdlOzI0NTI4MDcwMDtBUzoyOTcyMTI5NTAxMzg4OTNAMTQ0Nzg3MjQ5Nzk4NA%3D%3D&el=1_x_6https://www.researchgate.net/institution/University_of_Delaware?enrichId=rgreq-ed65b765-f1d3-4474-aa7a-9dd1292e1667&enrichSource=Y292ZXJQYWdlOzI0NTI4MDcwMDtBUzoyOTcyMTI5NTAxMzg4OTNAMTQ0Nzg3MjQ5Nzk4NA%3D%3D&el=1_x_6https://www.researchgate.net/institution/University_of_Delaware?enrichId=rgreq-ed65b765-f1d3-4474-aa7a-9dd1292e1667&enrichSource=Y292ZXJQYWdlOzI0NTI4MDcwMDtBUzoyOTcyMTI5NTAxMzg4OTNAMTQ0Nzg3MjQ5Nzk4NA%3D%3D&el=1_x_6https://www.researchgate.net/institution/University_of_Delaware?enrichId=rgreq-ed65b765-f1d3-4474-aa7a-9dd1292e1667&enrichSource=Y292ZXJQYWdlOzI0NTI4MDcwMDtBUzoyOTcyMTI5NTAxMzg4OTNAMTQ0Nzg3MjQ5Nzk4NA%3D%3D&el=1_x_6https://www.researchgate.net/profile/Jinfa_Cai?enrichId=rgreq-ed65b765-f1d3-4474-aa7a-9dd1292e1667&enrichSource=Y292ZXJQYWdlOzI0NTI4MDcwMDtBUzoyOTcyMTI5NTAxMzg4OTNAMTQ0Nzg3MjQ5Nzk4NA%3D%3D&el=1_x_5https://www.researchgate.net/?enrichId=rgreq-ed65b765-f1d3-4474-aa7a-9dd1292e1667&enrichSource=Y292ZXJQYWdlOzI0NTI4MDcwMDtBUzoyOTcyMTI5NTAxMzg4OTNAMTQ0Nzg3MjQ5Nzk4NA%3D%3D&el=1_x_1https://www.researchgate.net/profile/Jinfa_Cai?enrichId=rgreq-ed65b765-f1d3-4474-aa7a-9dd1292e1667&enrichSource=Y292ZXJQYWdlOzI0NTI4MDcwMDtBUzoyOTcyMTI5NTAxMzg4OTNAMTQ0Nzg3MjQ5Nzk4NA%3D%3D&el=1_x_7https://www.researchgate.net/institution/University_of_Delaware?enrichId=rgreq-ed65b765-f1d3-4474-aa7a-9dd1292e1667&enrichSource=Y292ZXJQYWdlOzI0NTI4MDcwMDtBUzoyOTcyMTI5NTAxMzg4OTNAMTQ0Nzg3MjQ5Nzk4NA%3D%3D&el=1_x_6https://www.researchgate.net/profile/Jinfa_Cai?enrichId=rgreq-ed65b765-f1d3-4474-aa7a-9dd1292e1667&enrichSource=Y292ZXJQYWdlOzI0NTI4MDcwMDtBUzoyOTcyMTI5NTAxMzg4OTNAMTQ0Nzg3MjQ5Nzk4NA%3D%3D&el=1_x_5https://www.researchgate.net/profile/Jinfa_Cai?enrichId=rgreq-ed65b765-f1d3-4474-aa7a-9dd1292e1667&enrichSource=Y292ZXJQYWdlOzI0NTI4MDcwMDtBUzoyOTcyMTI5NTAxMzg4OTNAMTQ0Nzg3MjQ5Nzk4NA%3D%3D&el=1_x_4https://www.researchgate.net/profile/Edward_Silver2?enrichId=rgreq-ed65b765-f1d3-4474-aa7a-9dd1292e1667&enrichSource=Y292ZXJQYWdlOzI0NTI4MDcwMDtBUzoyOTcyMTI5NTAxMzg4OTNAMTQ0Nzg3MjQ5Nzk4NA%3D%3D&el=1_x_7https://www.researchgate.net/institution/University_of_Michigan?enrichId=rgreq-ed65b765-f1d3-4474-aa7a-9dd1292e1667&enrichSource=Y292ZXJQYWdlOzI0NTI4MDcwMDtBUzoyOTcyMTI5NTAxMzg4OTNAMTQ0Nzg3MjQ5Nzk4NA%3D%3D&el=1_x_6https://www.researchgate.net/profile/Edward_Silver2?enrichId=rgreq-ed65b765-f1d3-4474-aa7a-9dd1292e1667&enrichSource=Y292ZXJQYWdlOzI0NTI4MDcwMDtBUzoyOTcyMTI5NTAxMzg4OTNAMTQ0Nzg3MjQ5Nzk4NA%3D%3D&el=1_x_5https://www.researchgate.net/profile/Edward_Silver2?enrichId=rgreq-ed65b765-f1d3-4474-aa7a-9dd1292e1667&enrichSource=Y292ZXJQYWdlOzI0NTI4MDcwMDtBUzoyOTcyMTI5NTAxMzg4OTNAMTQ0Nzg3MjQ5Nzk4NA%3D%3D&el=1_x_4https://www.researchgate.net/?enrichId=rgreq-ed65b765-f1d3-4474-aa7a-9dd1292e1667&enrichSource=Y292ZXJQYWdlOzI0NTI4MDcwMDtBUzoyOTcyMTI5NTAxMzg4OTNAMTQ0Nzg3MjQ5Nzk4NA%3D%3D&el=1_x_1https://www.researchgate.net/publication/245280700_An_Analysis_of_Arithmetic_Problem_Posing_by_Middle_School_Students?enrichId=rgreq-ed65b765-f1d3-4474-aa7a-9dd1292e1667&enrichSource=Y292ZXJQYWdlOzI0NTI4MDcwMDtBUzoyOTcyMTI5NTAxMzg4OTNAMTQ0Nzg3MjQ5Nzk4NA%3D%3D&el=1_x_3https://www.researchgate.net/publication/245280700_An_Analysis_of_Arithmetic_Problem_Posing_by_Middle_School_Students?enrichId=rgreq-ed65b765-f1d3-4474-aa7a-9dd1292e1667&enrichSource=Y292ZXJQYWdlOzI0NTI4MDcwMDtBUzoyOTcyMTI5NTAxMzg4OTNAMTQ0Nzg3MjQ5Nzk4NA%3D%3D&el=1_x_2


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Journal

or Research

n

Mathematics

Education

1996,

Vol.

27,

No.

5,

521-539

AN

ANALYSIS

OF

ARITHMETICPROBLEM

POSING

BY MIDDLE SCHOOL

STUDENTS

EDWARD

A.

SILVER,

Universityof

Pittsburgh

JINFA

CAI,

Universityof

Delaware

The mathematical

roblems

enerated y

509

middleschool

students,

who were

given

a briefwrit-

ten

story-problem

escription

nd

asked o

pose

questions

hatcouldbeanswered

sing

the nfor-

mation,were examined orsolvability, inguistic

andmathematical

omplexity,

and

relationships

within the sets

of

posed

problems.

t was found thatstudents

generated

a

large

number

of solv-

able mathematical

roblems,

many

of which were

syntactically

and

semantically omplex,

and

that

nearly

half the students

generated

ets

of related

problems.Subjects

also solved

eight fairly

complexproblems,

nd

he

relationship

etween heir

problem-solving erformance

nd heir

prob-

lem

posing

was examined

to reveal that

good problem

solvers

generated

more mathematical

problems

and more

complex problems

han

poor problem

solvers did. The

multiple-step

data

analysis

cheme

developed

and

usedherein hould

be useful

o

teachers ndother esearchers

nter-

ested

in

evaluating

tudents'

posing

of

arithmetic

tory

problems.

Recentrecommendationsor the reformof school mathematicsuggestanimpor-

tant

ole

or

student-generatedroblem

osing.

For

example,

he

Curriculum

ndEvaluation

Standards

or

School

Mathematics

NCTM,

1989)

explicitly

states that students

should have

some

experience ecognizing

and

formulating

heirown

problems,

an

activity

hat

s

at

theheart f

doing

mathematics

p.

138).

Furthermore,

he

Professional

Standards

or

Teaching

Mathematics

NCTM,

1991)

suggests

the

importance

of

teachers'

providing pportunities

or students o

pose

theirown

problems:

Students

should

be

given

opportunities

o formulate

roblems

rom

given

situations ndcreate

new

problemsby

modifying

he conditions

of a

given problem p.

95).

Thesedocu-

ments

reflect

an

apparent igh

evel of interest

mongmanypractitioners

o make

prob-

lem

posing

a more

prominent

eature f classroom nstruction. videnceof thisinter-

est canalsobe inferredrom herecent

ublication

f a collection f

practitioner-oriented

articles elatedo

problem osing

Brown

&

Walter,

993)

and he

appearance

f numer-

ous

articles

n

popularournals

whose audience

s

primarily

lementary

chool teach-

ers

(e.g.,

Silverman,

Winograd,

&

Strohauer,

992;Maddon,

1994).

n

fact,

atthis

ime,

it

appears

hat

practitioner

nterest s

running

ar aheadof the

development

f

credi-

ble

techniques

or

assessing

mathematical

roblem

posing

andthe

accumulation

of

solid

researchevidence

regarding

ts

nature.

Preparation

f

this

report

was

supported

n

partby

National

ScienceFoundation

rant

MDR-

8850580

and

by

a

grant

from

the Ford Foundation for

the

QUASAR (Quantitative

Understanding:

Amplifying

Student

Achievement and

Reasoning) project.

The

opinions

expressed

arethoseof

the authors nddo not

necessarily

eflect he

views

of

eitherFoundation.

An earlier

versionof this

paper

was

presented

at the

1993 annual

meeting

of the

American

Educational

Research

Association, Atlanta,

GA. The authorsare

grateful

o the

editorand to

several

anonymous

eviewers

who

madevaluable

comments

concerning

an

earlierversionof

this

article,

thereby

contributing

o its

improvement.

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4/21

522

Problem

Posing by

Middle

School Students

Over the

past

several

decades,

considerable

progress

has been made n

studying

many importantaspects of mathematicalproblem solving, including detailed

analyses

of

problem-solvingerformance

nd

studies elated o the

earning

nd

each-

ing

of

problemsolving

(Charles

&

Silver,

1988;

Silver, 1985; Schoenfeld,

1985).

Researchhas described

many

of

the

cognitive processes

associatedwith

the

solu-

tion

of

mathematical

problems

that are

posed by

a source

outside

the solver.

Although

urrentnterest

n

mathematical

roblemposing

can be seen as

representing

a new facet of a

longstanding

nterest

n

mathematical

roblem

solving

(Stanic

&

Kilpatrick,

1988),

far less is known

about

the

cognitive

processes

involved when

solvers

generate

their own

problems (Kilpatrick,

1987),

or about instructional

strategies hatcaneffectively promoteproductiveproblemposing, althoughmore

progress

has been made on the latter rontthan on the former.

Therehave been several

reports

of

instructional

pproaches

used to

incorporate

problemposing

into the

mathematicsnstruction f studentsat

a

wide

range

of edu-

cational evels

in

the U.S.

(e.

g., Healy,

1993;

Keil, 1964/1965;Perez, 1985/1986;

Winograd,

1990)

and

abroad

e.g.,

Hashimoto,

1987;

van

den

Brink,

1987).

Some

reports

have also

includedan examinationof the

impact

on

students

of

experience

orformal nstruction

mphasizing

mathematical

roblem osing

e.g.,

Keil, 1964/1965;

Perez, 1985/1986;

Scott,

1977).

In

general,

hese studieshave

found hat

having

stu-

dentsengagein some kind of generativeactivityrelated o problemposing--often

something

as

simple

as

rewriting iven

storyproblems-has

a

positive

nfluence

on

their

word-problem-solving

chievement

Hashimoto,

987;Keil,

1964/1965;Perez,

1985/1986; cott,

1977)

or heir ttitudeoward

mathematics

Perez,

985/1986;

Winograd,

1990/1991).

Thus,

he evidenceaccumulated

n

these

studies

suggests

hat

even

very

simpleexperiences

withmathematical

roblem

posing

can have a

positive mpact

on

students.

Nevertheless,

espite

hefact hat omeaccounts

f

instructionalnterventions

emphasizing

mathematical

roblem osing

have

shown he

positive

ffectsof the

nter-

ventions

on

student

achievement

nd

attitude,

hese

studieshave

not

directly

exam-

ined mathematical roblemposingitself. Thispriorresearchhasthereforeprovided

relatively

ittle

nformationbout

ither he

processes

used

by

students

n

problem

en-

erationor the

products

of students'

problem-posing

ctivity.

A few researchers

have

examined

the

mathematics

problems

posed by

children

(e.g.,

Ellerton,

1986;

Silverman

et

al.,

1992),

by prospective

elementary

chool

or

secondary

school teachers

e.g., Leung,

1993;

Silver,

Mamona-Downs,

Leung,

&

Kenney,

1996),

or

by

in-service

middle

school

teachers

e.g.,

Silver

et

al.,

1996).

Thus

far,

research

on children's

problemposing

has

tended o focus

on small

num-

bers of

subjects

and

to

provide only

a

fairly superficial

analysis

of

the

posed

problems, f anyanalysisat all. Forexample,Ellerton 1986) compared he math-

ematical

problems

generated

by

eight high-ability

young

children

with those

gen-

erated

by eight

low-ability young

children

by asking

each

to

pose

a

mathematical

problem

hatwould be

difficult

or a

friend

o

solve.

Ellerton

eported

hat

he

more

able

students

posed problems

hatwere

more

complex

than

those

posed

by

the

less

able

students,

but

her criteria

or

determiningproblem

complexity

were not well

specified.

In another

nvestigation,

Silverman

et

al.

(1992)

reported

hat

a class of



5/21

Edward

A.

Silver and

Jinfa

Cai

523

fifth-grade

tudentswas able to

generate

toryproblems

hat

exceeded

in

difficulty,

novelty,andinterest he wordproblemexercises found n their extbooks,buttheir

criteria or

judging

these

qualities

were

likewise

underspecified.

f

progress

s

to

be made

in

understanding

he

natureof

mathematical

problem

posing,

or if

rigor-

ous

attempts

re

to

be madeto

study

he instructional

mpact

of

interventions

elated

to mathematical

roblem

posing,

thenbetter

analytic

echniques

mustbe

developed

to

study problemposing by

elementary

chool and middle

school

students.

Some

guidance

or the

development

f schemes

o

analyze

hildren's

mathematical

problem

posing

is

provided

by

the

approach

used

in

those studiesof

adult

problem

posing

thathave ncluded

moreextensive

and

rigorous

nalyses.

For

example,Leung

(1993)successfully sedavariety f cognitiveanalysis ools,suchas GeneralProblem

Solver

(GPS)

graphs

(Newell

&

Simon,

1972)

and arithmetic

story problem

schema

nalysis

Marshall,

995),

o

examine he

problem-posingroducts

nd

processes

of

about50

prospective

elementary

school teachers

who

posed

written

arithmetic

story problems

n

response

to written

prompts.

n

another

nvestigation

nvolving

adult

subjects,

Silver et

al.

(1996)

developed

a

differentkind of

scheme to

analyze

the

written-problem-posingproducts

of

about 80

preservice

secondary

school

teachersand

in-service

middle school

teachers

who

posed

mathematical

roblems

related o a

complex

task

environment

nvolving

the

hypothesizedpath

of

a

billiard

ball ontablesof various izes andshapes.Theiranalytic chemeattendedo thenature

of

posedproblems

n

relation o the

information

iven

in

the task

environment.

hey

also

examined the

relationship

between

subjects'

problem

posing

and their

solu-

tion of a

specified

mathematical

roblem

n

the

same task

environment.

Aspects

of

the

analytic

approaches

sed

by

Leung

(1993)

and

by

Silver

et al.

(1996)

were

ncor-

porated

nto the

current

tudy.

A

major

goal

of

the

study

reported

erewas to

develop

and

use

an

analytic

cheme

to

examinethe

problem

posing

of

middle

school

students. n

particular,

he

scheme

developed

and

used in

this

study

employed

semantic

category analysis

and

other

analytic tools borrowed and adaptedfrom researchon mathematicalproblem

solving.

This

analytic

scheme

provided

he

basis for an

examinationof

the nature

and

complexity

of the

arithmetic

story

problems

posed by

middle

school

stu-

dents.Another

goal

of this

study

was to

examine he

relationship

etween

students'

problem

posing

and

their

problem

olving.

In

this

study,

his

goal

was

accomplished

by

probing

he

differences

between the

problem

posing

of

studentswho

were suc-

cessful

problem

solvers and

that of

studentswho

were

less

successful.

Silver

(1994)

has

noted

that the term

problem

posing

is

generally

applied

to

three

quite

distinct

forms of

mathematical

ognitive

activity:

(a)

presolution

pos-

ing, in which one generatesoriginalproblems romapresented timulussituation;

(b)

within-solution

osing,

n

whichone

reformulates

problem

s

it is

being

solved;

and

c)

postsolution

osing,

n

which

one

modifies he

goals

or

conditions f

an

already

solved

problem

to

generate

new

problems.

It is

a form of

presolution

posing

that

is

examined

in

this

investigation.

The

decision

to focus on

arithmetic

tory prob-

lems

was based on

the

availability

of an

extensive research

base on

which

to

build,

on the

appropriateness

f

this

type

of

activity

or

middle

school

students,

and



6/21

524

Problem

Posing by

Middle

School Students

on the fact that

prior

research

has shown the

efficacy

for

elementary

and

commu-

nity college students of problem-posing experiences related to the writing or

rewriting

of arithmetic

tory

problems.

A

greater

understanding

f this

particular

type

of mathematical

roblemposing

can havebotha

practical

nda theoretical

ay-

off.

In

particular,

scheme to

analyze

the

complexity

of arithmetic

tory

problems

generated

by

students ould be useful both

to

teachers,

who

might

wish to use such

a scheme o

evaluate heeffectiveness f their

nstructionr to measure tudent

rogress,

and

to

researchers,

who

might

use it and/or he results

obtained rom ts use

to

help

them

understand t least

one form of a

cognitive

activity

called

problem

posing.

METHOD

Subjects

Subjects

were

509

sixth-and

seventh-grade

iddle choolstudents

ttending

chools

in four different ow-income

communities

n

urban ocations

in

the United

States.

The students

attended our

middle schools

thatwere

part

of

the

QUASAR

project

during

the

1990-91

school

year.

QUASAR

was intended

to foster

innovative

mathematics

nstruction

n middleschools

serving

economicallydisadvantaged

om-

munities

Silver

&

Stein,1996).Except

ortheir nterest

n

participating

n the

QUASAR

project,

he

schools were

typical

of

urbanmiddle

schools

in

the U.S.

The students

in

the schools

were also

typical:

An

ethnically

and

linguistically

diverse

popula-

tion

(about

50% of

the students

were

African

American,

about 20%

were

White,

about20%

Latino,

andabout10%

Asian

American)

who

performed

enerally

below

average

on standardized

achievement

tests.

The

sample

was divided

approxi-

mately

equally

between

boys

and

girls.

Tasks

and

Administration

Eachsubjectcompletedaproblem-posing ask andeight problem-solving asks

in

a

single

class

period

of

approximately

5

minutes.

The tasks

were administered

by

the students'

eachers

during

mathematics

lass,

as

part

of

the

biannual

fall

and

spring)

project

testing

at

QUASAR

schools.

The

eight open-ended

problems,

together

with

the

posing

task,

comprised

one of four

forms

in the

QUASAR

Cognitive

Assessment

nstrument

QCAI)

or

grades

6 and

7

(Lane,

1993).

The

QCAI

is an assessment

nstrument

developed

by

the

project

to

measurestudents'

math-

ematical

hinking,

easoning,

and

understanding

Silver

&

Lane,

1993).

QCAI

tasks

have

undergone

xtensive

crutiny

o ensure

heir

quality

and airness

Lane

&

Silver,

1995).Theproblem-posing askand theproblem-solving asksin this form of the

QCAI

had been

pilot

tested

several

times to

ensure

hatthe tasks

assessed

the

cog-

nitive

processes

andcontent

areas

hey

were

designed

o assess

(Magone,

Cai,

Silver,

&

Wang,

1994).

The

eight problem-solving

asks

were

constructed-response

asks,

nvolving

var-

ious

mathematics

ontent

areas:

ractions,

geometry/measurement,

umber

heory,

patterns

nd

relationships, atio/proportion,

ndstatistics.

Forthese

problem-solving



7/21

Edward

A.

Silver

and

Jinfa

Cai

525

tasks,

students

were

required

not

only

to

produce

answers

but also to

justify

their

solutionsor to explaintheirsolutionprocesses.Theproblem-posingask,whichis

shown

in

Figure

1,

asked

students o

pose

three

questions

that could be

answered

on the basis of some

given

information.

To minimize he effect of task

order,

QCAI

askswere

systematically

ariedacross

problem

booklets.

In

particular,

herewere threedistinct

arrangements

f

the

nine

tasks

within the test

booklet,

so that about one thirdof the

sample completed

the

problem-posing

askas the secondtask

n

the

booklet,

one third

ompleted

he

prob-

lem-posing

task as the fifth

task

in

the

booklet;

and

the otherone third

completed

the

problem-posing

task as the

eighth

task in the booklet. About half of

the

responsesconsidered n this studywereobtained n fall 1990 and the otherhalf in

spring

1991.

Write hree

different

uestions

hat

can be answered rom he information

below.

Jerome,

Elliot,

nd Arturoook

turns

driving

ome

froma

trip.

Arturo rove80

miles more

than

Elliot.Elliot rove wice

as

many

milesas Jerome.Jerome

drove50

miles.

Question

#1

Question

#2

Question

#3

Note:

In

he task

booklet,

students were

given

more

space

in

which o

write heir

responses.

Figure

1.

Problem-posing

ask.

Data

Coding

A

summary

f the

coding

cheme

developed

nd

used n

this

nvestigation

s

provided

in

Figure

2.

Each

step

n

the

coding

process

s

explained

more

fully

in

this section.

Students'

problem-posing

responses

were first

categorized

as

mathematical

questions,

nonmathematical

uestions,

or

statements.

Those

responses

given

in

the



8/21

526

Problem

Posing by

Middle

School Students

form of mathematical

questions,

when

taken

together

with

the information

given

in thetaskcore, can be considered o constitutea mathematical roblem.Thus,it

was

possible

to consider he

student-generateduestions

o be

problems

andto

ana-

lyze

them as such. The

next

step

involved

categorizing

he

mathematical

roblems

as

solvableor not solvable.Problems

were

considered

o be not solvable f

they

acked

sufficient nformation

r

if

they posed

a

goal

thatwas

incompatible

with the

given

information. or

example,

he

response,

DidArturo

drive

aster

han

Jerome? was

considered

to

represent

a

problem

that was not solvable because

information

regarding

elative

drivingspeeds

or times was

neither

given

in

the tasknor

supplied

by

the student.

An

example

of an

impossible roblem-one

in which

the

goal

is

incompatible ith he conditions-is theresponse, HowmanymilesmoredidJerome

drive thanElliot?

e s p o n s e s

Nonmath Math

questions

Statements

questions

Solvable

Nonsolvable

Semantic

Linguistic

analysis

syntactic

analysis

Figure

2.

Summary

of

multiple-step

data

coding

scheme.

The last

step

n the

codingprocess

nvolved

examining

he

complexity

of the

posed

problems.

One

type

of

complexity

was related

o

the

linguistic

or

syntactic

struc-

tures

embedded

n the

posed

problems.

n

some

prior

research

e.g.,

Mayer,

Lewis,

&

Hegarty,

1992)

the

linguistic

structure

f mathematics

toryproblems

has

been

examined

by

focusing

on

the

presence

of

assignment,

relational,

and

conditional

propositions

n

problem

statements.

An

assignment

proposition

s

a

question

such



9/21

EdwardA. Silver

and

Jinfa

Cai

527

as

How

many

miles did

they

drive n all? A relational

proposition

s a

statement

suchas Howmanymoremiles didArturodrive hanJerome? A conditional ropo-

sition is

a

question

such as

If

Arturodrove 80 miles more than

Elliot,

how

many

miles

did

Arturo

drive?

Mayer

et al.

(1992)

found that

problem-solving

difficulty

appeared

o

be relatedto

linguistic

complexity,

in

that

problems

with

conditional

and

relational

ropositions

ended o be

moredifficult

or students

o solve than

hose

containingonly assignmentpropositions.

Thus,

the

presence

of conditional

or

rela-

tional

propositions

can be taken as an indication of

problem

complexity.

It

is

important

o

note thatthe task core

(i.e.,

the information

presented

o the students

from which

they

were to

generateproblems)

ontained

wo

assignment ropositions

( Jerome, lliot,andArturo ook turnsdrivinghome fromatrip and Jerome rove

50

miles )

and

two relational

ropositions

Arturo

rove

80 miles more hanElliot

and

Elliotdrove wice as

many

miles as

Jerome ).

The taskcore

thereforewas itself

quite

complex

from a

linguistic

perspective.

Our

analysis

ocused

on

the additional

complexity

contributed

by

the

questions

posed

by

the students.Because this

type

of

complexity

analysis

was feasible for nonsolvable

mathematics

problems

as

well as for those

that

were

solvable,

both

types

of

responses

were considered

n

the

analysis reported

here.

Another

ype

of

complexity

related o the

mathematical tructures ound

in

the

posedproblems.Because theposedproblemscouldbe solvedusing some combi-

nation

of

arithmetic

operations,

one

plausible

measureof

mathematical

omplex-

ity

would

be

the number f

operations,

r

the

number

f

computational

teps,

required

for

solution.

Leung

(1993)

successfullyanalyzed

he

complexity

of arithmetic

rob-

lems

posed by

preservice lementary

chool teachers

by

using

GPS

graphs

o deter-

mine

he

number

f

operators

sed n

solving

he

posedproblems.

lthough

his

approach

was shown

by

Leung

to be

very

useful and

powerful,

t is

not the

only

reasonable

way

to

measure mathematical

complexity.

Counting

the number of

steps

in

a

hypothesized

olutionhas

the

advantage

f

assessing

complexity

n

a

straightforward

andreasonablemanner,but it also has thedisadvantage f makingsomerelatively

simple

arithmetic

toryproblems

appear

o be

fairly

complex.

For

example,

f

a

prob-

lem'

s

solution nvolved

the additionof

five

numbers,

hen it would

be counted

as

requiring

our

operation

teps

for its solution.

If

one

determines

problem

complex-

ity by

counting

operators,

hen this

problem

would be

seen as more

complex

than

another

problem

hat

required

a

multiplication

ollowed

by

a

subtraction,

ecause

this latter

problem

equired

nly

two

operation

teps

for

solution.On the

other

hand,

if one

determines

problem

complexity

by

enumerating

istinct

semantic

relations,

then

he atter

roblem

wouldbe

seen o be

more

complex

han he

first

problem,

ecause

the latterproblemembodiestwo distinctsemanticrelationsandthe firstembodies

only

one. It is this

second

approach

hat was taken n

the

study

reported

here;

that

is,

the

numberof

semanticrelations

rather han

the

numberof

operators

was used

to determine he

mathematical

omplexity

of

the

posed

problems.

All

mathematically

solvable

problems

were

subjected

to semantic

category

analysis

using

a

classification

scheme of

arithmeticword

problems

developed by

Marshall

1995).

The

posed

arithmetic

word

problems

were

classified

on

the

basis



10/21

528

Problem

Posing by

Middle

School Students

of their

underlying

semanticstructural elations

using

Marshall's ive

categories:

Change,Group,Compare,Restate,Vary.A mathematicalproblem hatwas clas-

sified

as

involving

N semanticstructural elations

n

the classification

scheme

was

designated

an

N-relation

problem.

If

a

mathematical

problem

could

be answered

directly

rom

the

given

information,

t was

designated

a

zero-relation

roblem,

and

in

this

analysis

it

would be said to involve zero semantic relations. Problems

involving

a

greater

numberof

semanticrelations

areconsidered o be

semantically

more

complex

thanthose

involving

fewer relations.

To examine interrater

eliability,

one

person

classified all students'

problem-

posing

responses,

fterwhicha second

person

andomly

elected 0 students'

esponses

andindependentlyclassified them. Interrater greement or the basic classifica-

tion

(mathematical

question,

nonmathematical

uestion,

or

statement)

was

93%.

Rates

of

agreement

n

the

classifications

f

linguistic

andmathematical

omplexity

of students'

posed

mathematical

problems

were

also

highly acceptable,

93% for

linguistic complexity

and 89%

for mathematical

complexity.

Because there was

substantial

agreement

between

raters,

the

first

person's

classifications of

all

students'

problem-posing

responses

were used

in

the

subsequentanalyses.

The

problem-solving esponses

were evaluated

using

a

focused,

holistic

scoring

method

Silver

&

Lane,

1993).

A

generalized

coring

rubric

with

three nterrelated

components (mathematical,conceptual, and proceduralknowledge; strategic

knowledge;

and

communication)

pecified

criteria or each

of

five score

evels

(0-4)

and

guided

the

development

of

a

specific

rubric or each task

(Lane, 1993).

Each

of the students'

responses

was scored

ndependently

by

two middle school

teach-

ers,

who were trained

o

use

the

scoring

rubric.

Interrater

greements

or each

of

the

eight

tasks

ranged

rom 75%

to

89%,

which was

judged

to be

acceptable.

RESULTS

The results

are

presented

n

two

sections.

The first section

provides

a

summary

of students'

problem-posing

esponses,

ncluding

the

analyses

of

complexity

and

relatedness;

nd

he second

presents

n

analysis

of

the

relationship

etweenstudents'

problemposing

and their

problem

solving.

In the

analyses

reported

here,

the

sam-

ple

is

treated

as a whole

rather hanexamined

by

grade

evel

(6

or

7)

or

by

testing

occasion

(fall

or

spring).

Data

for

this

study

were collected

during

he

first

year

of

QUASAR

project ctivity

at

each of the

sample

chools.

During

hat

year,

substantial

attention

as devoted

o the

design

of innovative

nstructional

rograms

nd o enhanc-

ing

teachers'

knowledge

of

contentand

pedagogy;

ess

change

was

actually

mple-

mented

n classroom nstruction

n

a

day-to-day

basis.

Thus,

for

the data

rom this

year,thereappearedobe no compellingreason o separatehesampleby response

occasion or

by

grade.

Problem-Posing

Responses

Subjects

provided

a total

of 1465

responses.

More

than70%of the

responses

were

classified

as mathematical

questions,

about

20%

were

statements,

and 10%

were



11/21

Edward

A.

Silver and

Jinfa

Cai

529

nonmathematical

questions.

Approximately

the same distribution

of

response

types was evident foreach of the threeresponsescalled for in the task.Although

many

combinations f

response

ypes

were

theoreticallypossible,

the

data

ndicate

that

students

tended to be

consistent with

respect

to

response

types,

because

approximately

5%

of

the

students

generated

hree

mathematical

questions,

three

nonmathematical

uestions,

or three

statements.

Nearly

80%

of

the

students

generated

at

least one

mathematical

uestion.

More

thanhalf

(about57%)

of

the

students

enerated

hree

mathematical

uestions.

n

fact,

the

studentswho

generated

hree

mathematical

uestions

accounted or

over

80%

of

the

mathematical

uestions

generated

y

all

students. n other

words,

he

remain-

ing 40% of the studentsgeneratedess than20% of themathematical uestions.

The mathematical

uestions

posed by

the

students

were of

particular

nterest,

and

they

were

subjected

o

further

analyses

of

mathematical

olvability

and

inguistic

and

mathematical

complexity.

The

results of

these

analyses

are

discussed next.

Mathematical

Solvability

More

than

90%

of

the

mathematical

roblems

generated

i.e.,

the

questions

posed

by

students

nd he

given

nformationn

the

task

core)

were

udged

o be

mathematically

solvable.

Although

he

solution

of

some

solvable

problems

might

have

required

nfor-

mationbeyondthatgiven in the taskcore and theposedquestion,the majorityof

the

solvable

problems

could be

answeredon

the

basis of

information

given

in

the

task core.

Twelve

studentseach

generated

one

mathematically

olvable

problem

that

could be

answeredon

the

basis

of the

given

information

and new

information

supplied

by

the

student n

the

posed

question.

An

example

of

this

kind of

hypoth-

esis-based

mathematical

question

s

the

following:

How

many

times

would

they

have

to

get gas

if

they

got

160 miles

each

fill-up?

Linguistic

Complexity

Thelinguisticorsyntacticcomplexityof theposedproblemswas determined

y

examining

all

posed

mathematical

questions

for the

presence

of

assignment,

rela-

tional,

and

conditional

propositions.

As

mentioned

earlier,

he

presence

of

condi-

tional or

relational

propositions

n

the

posed

question

s taken

to

be an

indication

of

problem

complexity.

In

the

responses

obtained n

this

study,

nearly

60% of

the

mathematical

uestions

nvolved

only

assignment

ropositions,

bout

35%

nvolved

relational

propositions,

and

only

5%

involved

conditional

propositions.

Almost all

students

80%)

generated

at

least

one

mathematical

question

nvolving

an

assign-

ment

proposition.

Although

relational

propositions

were

found

in

only

about

one

thirdof theresponses,aboutone half of the studentsgeneratedat leastone math-

ematical

question

nvolving

a

relational

proposition.

About

10%

of

students

posed

at

least

one

mathematical

question

with

a

conditional

proposition.

Mathematical

Complexity

All

mathematically

olvable

problems

were

examined

or

the

presence

of

the

five

fundamental

emantic

tructural

relations--Change,

roup,Compare,

Restate,

Vary--



12/21

530

Problem

Posing

by

Middle School Students

or combinationsof these

relations.

Using

this

approach,

ver

90%

of the solvable

mathematical roblemscould be classifiedwithrespect o semanticcomplexity,or

the

number

of relations

required

or solution.The number

of relations

n

the

posed

problems

ranged

rom

0

to

5.

Examples

are

provided

n Table 1.

Table

1

Examples

of

Mathematical

Problems

and the

Corresponding

Number

of

Semantic

Relations

Numberof relations

Examples

Zero

Did

Arturo

drive 80 miles more than

Elliot?

[None]

One

How

many

miles did

Elliot drive?

[Restate]

Two

How

many

more miles did

Elliot drive

than

Jerome?

[Compare/restate]

Three

How

many

miles

did the three

boys

drive

altogether?

[Group/restate/restate]

Four

How

many

times would

they

have

to

get

gas

if

they got

60

miles

each fill

up?

[Vary/group/restate/restate]

Five

Did Arturodrive

a

longer

time thanJeromeand

Elliot drove

altogether

n the

regular

way?

[Compare/restate/group/restate/vary]

Most

of

the

problems

posed

were

semantically

complex.

In

fact,

about

60%

of

the

solvable

mathematical

roblems

nvolved

two or

more

relations,

and these are

hereafter eferred

o as multirelation

mathematical

roblems.

Slightly

more han20%

of

the

problems

nvolved

one

relation,

andabout

16% nvolved

zero

relations.

The

generation

f

semantically

omplex problems

was

fairly

well distributed

cross

he

sample.

n

fact,nearly

70%

of the students

enerated

t least

one

multirelation

math-

ematical

problem,

and

a little

less than

half of the

students

generated

at least

two

multirelation

roblems.

There

was a

tendency

or multirelation

roblems

o

appear

later

rather hanearlier

n the

response sequence.

Only

34%

of

the

first

responses

were

multirelation

roblems,

whereas

41% of the

second

responses

and

49% of

the

third

responses

were

multirelation

problems.

The later

responses

of students

ended

to be somewhat

more

complex

semanti-

cally

thanwere

the earlier

esponses.

Table

2 shows

the

percentages

f

students

who

shifted

or

did

not

shift

the

complexity

levels of

their

posed

mathematically

olv-

ableproblemsrom he first o the secondresponses nd hefirst o thethird esponses.

In

particular,

bout

half of those

students

ave

a more

complex

problem

as their

sec-

ond

response

than

their

first;

whereas,

the

responses

of about

one

third

of the

stu-

dents

moved

in the

opposite

direction.

A

matched-pair

Wilcoxon

test indicated

hat

students

had

significantly

more

complex

mathematically

olvable

problems

n

the

second

responses

han

n

their

irst

responses

z

=

2.71,

p

Silver Cai JRME1996

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Transcript of Silver Cai JRME1996