A Synthesis of Empirical Research on Teaching Mathematics to Low-Achieving Students

Abstract

The purpose of this study was to synthesize research on the effects of interventions to improve the mathematics achievement of students considered low achieving or at risk for failure. Meta-analytic techniques were used to calculate mean effect sizes,for 15 studies that met inclusion criteria. Studies were coded according to 5 categories of mathematics interventions, and effect sizes were examined on a study-by-study basis within each of these categories. Results indicated that different types of interventions led to improvements in the mathematics achievement of students experiencing mathematics difficulty, including the following:.(a) providing teachers and students with data on student performance; (b) using peers as tutors or instructional guides; (c) providing clears specific feedback to parents on their children's mathematics success; and (d) using principles of explicit instruction in teaching math concepts and procedures.

Full text

A Synthesis of Empirical Research on Teaching Mathematics to Low-Achieving Students
Author(s): Scott Baker, Russell Gersten and Dae-Sik Lee
Source:
The Elementary School Journal,
Vol. 103, No. 1 (Sep., 2002), pp. 51-73
Published by: The University of Chicago Press
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A
Synthesis
of
Empirical
Research
on
Teaching
Mathematics
to
Low-Achieving
Students
Scott Baker
Eugene
Research
Institute/University
of
Oregon
Russell
Gersten
Instructional
Research
Group
Long
Beach,
CA
Dae-Sik Lee
Inchon
National
University
of
Education
The
Elementary
School
Journal
Volume
103,
Number 1
?
2002
by
The
University
of
Chicago.
All
rights
reserved.
0013-5984/2003/10301-0003$05.00
Abstract
The
purpose
of this
study
was to
synthesize
re-
search on the effects of interventions
to
improve
the
mathematics achievement of
students con-
sidered
low
achieving
or
at
risk for
failure.
Meta-analytic
techniques
were used to
calculate
mean
effect
sizes
for
15
studies
that
met
inclu-
sion criteria. Studies
were
coded
according
to
5
categories
of mathematics
interventions,
and
ef-
fect sizes were examined
on a
study-by-study
basis within each of these
categories.
Results
in-
dicated that
different
types
of
interventions led
to
improvements
in
the
mathematics
achieve-
ment of
students
experiencing
mathematics
dif-
ficulty, including
the
following:
(a)
providing
teachers and
students
with
data
on student
per-
formance;
(b)
using peers
as
tutors or
instruc-
tional
guides;
(c)
providing
clear,
specific
feed-
back to
parents
on
their
children's
mathematics
success;
and
(d)
using principles
of
explicit
in-
struction
in
teaching
math
concepts
and
proce-
dures.
Recently,
the
National
Research
Council
(Kilpatrick,
Swafford,
&
Findell,
2001)
con-
vened
a
panel
of
experts
to
"synthesize
the
rich and diverse
research"
on
mathematics
learning
in
the
elementary
and
middle
school
years,
to
"provide
research-based
rec-
ommendations
for
teaching
...
and
curric-
ulum
for
improving
student
learning,"
and
to
"identify
areas
where
research is
needed"
(p.
3).
The
panel
examined all
types
of re-
search:
experimental
interventions,
quanti-
tative
studies
linking
observed
classroom
interactions to
growth
in
mathematics
achievement,
qualitative
studies of
class-
room
practice,
comparative
international
studies
of mathematics
achievement,
and
the
vast
array
of
qualitative
studies of
the de-
velopment
of
mathematical
concepts
and
reasoning
in
students.
In
that
formidable re-
port
the
panel attempted
to cut
across the
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52
THE ELEMENTARY
SCHOOL
JOURNAL
wide
array
of
data
sources
and
disciplines
in
order
to
draw
conclusions about
how
math-
ematics instruction in
U.S.
schools
can
be
im-
proved.
Our
goal
in
this
article
is far more
mod-
est. We
synthesize
data
from one
source:
well-controlled
experimental
and
quasi-
experimental
studies that
assess the
effects
of
interventions
designed
to
improve
the
mathematics
achievement of
students con-
sidered
low
achieving
or at risk
for
failure.
All
of
the studies in
this
research
synthesis
meet
standards of
scientific
rigor
similar
to
those
used
in
the
recent
synthesis
of
begin-
ning
reading
research
conducted
by
the Na-
tional
Reading
Panel
(2000).
As
the
National
Research
Council
(Kil-
patrick
et
al.,
2001)
aptly
noted,
"Experi-
mental
rigor
often
requires
narrowing
one's
focus to a
single
feature of
an
instructional
method or
to a
limited
amount
of
mathe-
matical
content"
(p.
25).
The focus of
many
of the
studies
we
reviewed
was
narrow,
limited
to the
effects of
an
assessment
sys-
tem or a
classroom
organizational
structure.
In
some
cases,
fairly
subtle
aspects
of
cur-
riculum
design
were
manipulated
in
order
to
assess
effectiveness. Our
objective
was
not to
create a
new
vision
of
mathematics
instruction for
students with
learning
prob-
lems but
rather to
provide
a
dispassionate,
systematic
look
at what
has been
learned
over the
past
20
years
through
controlled
research in
classroom
settings.
We
are
not
aware
of
previous
quanti-
tative
syntheses
investigating
the
effects
of
instruction on
the
mathematics
achieve-
ment of
students
at risk
of
mathematics
failure.
Swanson,
Hoskyn,
and
Lee
(1999)
used
meta-analysis
to
investigate
the
effects
of
a
variety
of
instructional
interventions on
students
with
diagnosed
learning
disabili-
ties.
Students
with
difficulties in
mathemat-
ics
were
not
included
in
Swanson
et
al.'s
meta-analysis
unless
they
also had
an
iden-
tified
learning
disability.
Swanson et
al.'s
investigation
is
relevant
to our
task,
however,
because it
addressed
instructional
effects
with
a
population
of
students with
significant
achievement
prob-
lems.
Overall,
Swanson et al.
found
that
the
set
of 18
instructional
intervention
studies
that addressed
mathematics
most
specifi-
cally
had
a
mean
effect
size of
.40
on
mea-
sures
of mathematics
performance,
which
is
considered
a
moderate
effect
(Cohen,
1988).
In
the
same
analysis,
effect
sizes on
mea-
sures
of
reading
comprehension,
word
rec-
ognition,
and
writing
were
all
somewhat
higher:
.72,
.57,
and
.63,
respectively.
And,
in
fact,
effect
sizes
in
mathematics
were
among
the
lowest
reported.
Another
inter-
esting
finding
was
that
only
10% of the
in-
tervention
studies
had
a
primary
focus
on
mathematics.
An
important
methodological
feature
of
the
Swanson
et
al.
(1999)
meta-analysis
is
that the
authors
categorized
studies
pri-
marily
on
the basis
of the
types
of
depen-
dent
measures
used
to
determine
effects.
Consequently,
there
is a
rich
source
of
in-
formation
on what
effect
general
aspects
of
instruction
had
on a
range
of
outcomes,
but
there
is
less
information about
the
details
of
the
instructional
interventions that
pro-
duced
those
effects.
The
National
Research
Council
(Kilpa-
trick et
al.,
2001)
recently
summarized
the
knowledge
base
on
helping
students
learn
mathematics.
Although
they
did
not
focus
on
students
experiencing
serious
difficulty
learning
mathematics,
many
of
the
sugges-
tions
they
provide
constitute
sound
in-
structional
recommendations
for
students
struggling
with
mathematics. A
central
recommendation
in
the
report
is that
teach-
ers
should
play
a
more
active
instructional
role
in
helping
their
students
build
mathe-
matical
proficiency
than
they
currently
do.
Active
instruction
is
critical
to
"engaging
students
in
the
mathematical
work,
main-
taining
their
focused
involvement in
it,
and
helping
them
take
advantage
of
instruction
to
learn"
(p.
331).
Use of
multiple
instruc-
tional
methods
to
achieve
this
goal
is
clearly
endorsed.
For
example,
the
report
suggests
that
there
are
times
when
math
content
should
be
constrained in
ways
that
focus
SEPTEMBER
2002
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TEACHING
MATHEMATICS
53
students' attention on
specific
learning
goals,
as well as times
when
problems
should
be
represented
in
multiple
ways
with a
variety
of
strategies
for
solving
them.
Thus,
the
report
argues
for
a
blend
of fo-
cused,
explicit
instruction
with
the
more
open-ended problem-solving approach
ad-
vocated,
for
example,
in
earlier versions
of
the National
Council
of Teachers of
Math-
ematics
(NCTM) (1989)
standards.
Segments
of mathematics
instruction
should
target teaching
students to
generate
explanations
of
math
concepts
in
their
own
words and
to
justify
the methods
they
use
to solve
problems.
Focusing
on
student
er-
rors and
misconceptions
can also be an ef-
fective instructional
method,
especially
when
teachers
anticipate predictable
stu-
dent errors and
prepare
in
advance to use
those errors to
help
students understand
correct solutions.
Knowing
how
to
teach
math well to stu-
dents
with
differing
abilities seems to be
much more
important
than
having
math
teachers
who
possess strong
backgrounds
in
mathematics
(Ball, Lubienski,
&
Mewborn,
2001).
What
is
less
clear from
the
National
Research
Council
Report (Kilpatrick
et
al.,
2001)
is how
teachers
can
balance
different
instructional
approaches
in
a
comprehensive
program.
Also,
unlike the
current research
synthesis,
the council's
report
does
not ad-
dress how
instructional
methods are best ad-
justed
for
students
experiencing
serious
dif-
ficulty
learning
mathematics.
We
discovered a
body
of controlled ex-
perimental
and
quasi-experimental
re-
search on
teaching
mathematics to low-
achieving
students that
includes
reliable,
valid
outcome
measures.
We
believe a
syn-
thesis
of
the
findings
from this
research can
shed
light
on
effective
instructional
ap-
proaches
for
students
with low
mathemat-
ics
achievement.
Although
the number of
studies
is
small,
the
quality
of the
research
is,
in
general,
high.
Well-conceptualized
in-
structional
approaches,
measures
of treat-
ment
fidelity,
students
randomly
assigned
to
treatment and
comparison
conditions,
and outcome measures that
tapped
possible
effects in both the treatment
and
compari-
son conditions
characterized
the
set of
stud-
ies
we reviewed.
Many
of the studies
have
addressed
well-defined
problems
in
teaching
mathe-
matics to students who
struggle
with
learn-
ing
mathematical
concepts
and
procedures.
The
specificity
of
these
problems
and
how
they
were examined
in
controlled
investi-
gations
can
have
great practical utility,
es-
pecially
if
commonalties
among
the
prob-
lems,
as
well
as
the
solutions,
can be
discerned.
One
purpose
of
synthesizing empirical
research is to examine a
given
body
of
stud-
ies,
searching
for
commonalties
and
ways
to
summarize them
accurately
and
suc-
cinctly.
For
example,
in
their
analysis
of 16
studies on
reciprocal
teaching,
Rosenshine
and Meister
(1994)
concluded
that,
across
the
studies
that
used measures
aligned
with
the
intervention
focus,
results "were
gen-
erally
the
same
regardless
of
the
number of
strategies
that were
taught"
(p.
507).
The
authors of the
synthesis
also
began
to
artic-
ulate common features of
effective
reading
comprehension
approaches.
A
second
purpose
of
research
syntheses
can
be to
search for
important
differences
among
studies that share a
common
focus.
Analyzing
differences
typically requires
a
more
detailed
explication
of studies
than
summarizing
commonalties. In
Rosen-
shine and
Meister's
(1994)
synthesis
on
reciprocal
teaching,
for
example,
effects
were
much
greater
when
experimenter-
developed comprehension
measures
were
used than when
standardized
tests
were
used.
These
discrepancies
led
Rosenshine
and Meister to
compare
details
of
the
most
commonly
used
standardized
test
with
the
most
commonly
used
experimenter-devel-
oped
test. Their
analysis
led
to
a
hypothe-
sis
that
the two
types
of
tests
differed
on
six
important
dimensions. The
authors
then
provided
plausible
reasons for
how
students
in
reciprocal
teaching
may
have
benefited
over
comparison
students
on
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54
THE
ELEMENTARY
SCHOOL
JOURNAL
each of
these
dimensions when
completing
experimenter-developed comprehension
mea-
sures,
but not
necessarily
on
standardized
tests.
In
the
current
synthesis,
we used
meta-
analytic
techniques
(Cooper
&
Hedges,
1994)
to
present
mean
effect sizes
for each
study
within
certain
categories
of
mathematics
in-
terventions.
Because the
small
number
of
studies in
each
category
virtually precluded
the
search for
moderator
variables,
we
ex-
amined
variation in
effect
sizes on
a
study-
by-study
basis,
without
applying
homoge-
neity
tests or
other
statistical
techniques.
Using
effect size
as a
common
metric
helps
the
reader
easily
discern
the
relative
effec-
tiveness
of
each
approach.
Rosenshine and
Meister
(1994)
relied
on
meta-analytic
tech-
niques
in
a similar
way
to
examine
the em-
pirical
research
base on
reciprocal
teaching.
Our
objective
was
to
analyze
findings
from
experimental
research
conducted
in
schools
to
improve
the
mathematics
achievement
of
students
struggling
to
learn
math.
Method
All
studies
published
from
1971
to
1999
that
included
specific
instructional
mathematics-
based
intervention
strategies
to
improve
the
mathematics
performance
of
low-achieving
school-age
students
were
included
in
the re-
view.
Our
basic
source
for
identifying
rele-
vant
studies
was a
doctoral
dissertation
by
Lee
(2000),
later
contributing
to a
technical
report
(Lee,
Kame'enui,
&
Gersten,
2000).
The
following
search
procedures
were
used
to
locate
mathematics
intervention
studies
(Lee,
2000).
Computer
searches of
the
ERIC and
PsycINFO
databases
were
conducted
to
locate
studies
published
from
1971
to
1999
that
addressed
mathematics
interventions with
students
who
were
low
in
mathematics
achievement.
The
follow-
ing
combinations of
descriptors
were
used
in
this
search:
mathematics
achievement,
mathematics
education,
mathematics
re-
search,
elementary
education,
secondary
education,
slow
learners,
underachievement,
academically
disadvantaged,
math
anxiety,
low
achieving,
at
risk,
and
learning prob-
lems.
We
examined
bibliographies
of
re-
search
reviews in
the
area of
learning
dis-
abilities
(i.e.,
Maccini
&
Hughes,
1997;
Mastropieri,
Scruggs,
&
Shiah, 1991;
Miller,
Butler,
&
Lee, 1998;
Swanson
&
Hoskyn,
1998;
Swanson
et
al.,
1996)
for
studies
pub-
lished
during
the
same
period
but
that
may
have
been
omitted
from
the
computerized
databases.
Finally,
we
conducted
a
manual
search of
major journals
in
special,
reme-
dial,
and
elementary
education.
This
procedure
resulted
in
the
identifi-
cation
of 599
studies.
Of
this
total,
we
se-
lected 194
studies
for
further
review
based
on
the
title,
key
words,
and
abstracts.
From
these
194
studies,
17
(9%)
met
the
following
criteria
for
inclusion
in the
analysis:
1.
Only
studies
that
provided
math
in-
struction,
or
structured
opportunities
for
students
to
practice
or
apply
class-
room
mathematics
lesson
objectives,
were
included.
Math-related
studies
that
examined
the
effects
of
test-taking
strategies,
or
taught
students
computer
programming,
logic,
or
assessed the
ef-
fects
of
inclusion
and
mainstreaming
on
mathematics
achievement,
for
ex-
ample,
were
excluded.
2.
Math
instruction
must
have
lasted
for
a
minimum
of
90
minutes
during
the
course
of
the
intervention.
3.
Only experimental
or
quasi-experi-
mental
intervention
studies
that
em-
ployed
group-design
methods with
a
control
group
were
included
(i.e.,
no
single-subject
studies
or
case
study
re-
search
reports
were
included).
4.
Quasi-experiments
were
included
as
long
as
one of
three
conditions
was
met:
(a)
posttest
performance
could
be
adjusted
statistically by
factoring
in
pretest
performance
on
relevant
out-
come measures
(Wortman
&
Bryant,
1985),
or
(b)
the
researchers
in
the
original
study
adjusted
posttest
per-
formance
using
appropriate
analysis
of
covariance
(ANCOVA)
techniques.
In
addition,
when
posttest
scores
could
not
be
adjusted
statistically
for
pretest
differences
in
performance,
the
original
study
documented
that
there
SEPTEMBER
2002
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TEACHING MATHEMATICS
55
were
no
significant
differences
be-
tween
groups
at
pretest
on
relevant
measures of mathematics achieve-
ment.
(Table
1
reports
which studies
were
experimental
and
which
were
quasi-experimental.
In
some
cases,
teachers rather than students were ran-
domly
assigned
to condition. These
studies
were considered
quasi-experi-
mental,
and
all were included
because
posttest
performance
could be
adjusted
to
factor
in
pretest
differences.)
5.
Each
study
needed to include at least
one
mathematics
performance
or
achievement measure.
Studies that
measured
computation
skill,
math
problem
solving,
understanding
math-
ematics
concepts,
and other
activities
where students had
to demonstrate
mathematics
proficiency
in
some
way
were
included
in
the
analysis.
When
experimenter-developed
mea-
sures were
used,
reliability
informa-
tion on the
measure needed
to be re-
ported.
Studies that
only
measured
students'
attitude toward
mathemat-
ics or
self-concept
were
excluded
from
analysis.
6.
Studies
must have
reported
means
and
standard
deviations,
or
F-values,
so that
effect sizes
could be
calculated
(Cooper
&
Hedges,
1994).
(Only
one
study
was
excluded
based on
this
cri-
terion.)
We
excluded
studies for
the
following
reasons:
(a)
an
experimental
or
quasi-
experimental
design
was not
used
(33.3%),
(b)
there
was
insufficient
information
doc-
umenting
that
students
were low
achiev-
ing
in
mathematics
(32.2%), (c)
a
mathe-
matics
intervention was
not
implemented
or
the
intervention
was not
described with
enough
clarity
for
coding
(17.0%),
(d)
out-
come data
for
calculating
an
effect size
were
not
reported
(14.1%),
(e)
other
rea-
sons
(3.4%)
(e.g.,
the
intervention was
not
of
sufficient
duration,
only
experimenter-
developed
measures were
used and
the re-
liability
of
those
measures could
not
be
verified).
Definition
of
Low
Achieving
Students
in
these
studies
were
identified
as low
achieving
in
mathematics
on
the ba-
sis of their
performance
on
standardized
or
informal
tests or
by
their
placement
in
re-
medial mathematics
classes.
In
some
stud-
ies students were
receiving
Title
I
services
in
mathematics.
All
studies
provided operational
defi-
nitions of low
achieving. Typically
the
re-
searchers relied on both
teacher
nomination
and a
measure
of math
performance.
For
ex-
ample,
Fuchs et al.
(1997)
asked teachers
to
select
"two
students
whose
mathematics
performance
was at
or near the bottom
of
the class
...
but who had
never been
re-
ferred for
special
education"
(p.
519).
Next,
the
researchers administered
a
pretest,
which
systematically
sampled
math
prob-
lems
from the
Tennessee
mathematics
framework for
grades
1-6. The
student
with the
lower score was
considered
low
achieving.
Woodward and
Baxter
(1997a)
included
all
students who
scored below the
thirty-
fourth
percentile
on a standardized
mathe-
matics test
(i.e.,
the
Iowa Tests
of Basic
Skills,
ITBS).
A
somewhat
broader net
was
applied
in
the research
of
Fantuzzo and
col-
leagues
(Fantuzzo,
Davis,
&
Ginsburg,
1995;
Ginsburg-Block
&
Fantuzzo,
1997;
Heller &
Fantuzzo,
1993).
In
one
case,
Heller and
Fantuzzo
(1993)
defined
low
achieving
as
"(a)
scores below
the 50th
per-
centile
on
standardized
mathematics
achievement
scales
(based
on the
School
District of
Philadelphia's
citywide
norms),
and
(b)
poor
performance
in
mathematics
as
rated
by
classroom
teachers"
(p.
519).
In
this
case,
teacher
nominations
were used
to con-
firm
student
performance
data.
Students with
identified
learning
dis-
abilities
were
included in
one-third
of the
studies
in
the
analysis
but in
those
studies
only
constituted a
small
percentage
of the
entire
sample.
Some
researchers
(e.g.,
Fuchs
&
colleagues)
presented
separate
data
for
low-achieving
students and
students with
learning
disabilities.
In
those
cases,
we ex-
cluded
students with
disabilities
from
our
analysis.
In
other
cases,
researchers
did
not
disaggregate
the data
so
it
was
impossible
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56 THE
ELEMENTARY
SCHOOL
JOURNAL
to
separate
the
performance
of
students
with
and without identified
disabilities.
Coding
of
Studies
We
coded studies on a set of standard
variables
common
in
synthesizing
instruc-
tional intervention research. Initial
coding
variables included the
number of students
in
each
condition,
the
procedures
used
to as-
sign
students to
conditions,
and the
length
of the
intervention.
In
the
second
coding
phase,
we
analyzed
studies
according
to the
dimensions of
the
interventions that are
unique
to this
body
of studies.
Describing
the
studies: Phase
1
coding.
A
total of 17
studies met
the criteria for
in-
clusion
in
the
synthesis.
These studies
and
major descriptive
information
are
presented
in
Table
1.
Initial
coding
involved
determin-
ing
the
following
information:
(a)
whether
the
study
randomly assigned
students to
conditions or was a
quasi-experiment,
(b)
the
number
of
students
per
condition,
(c)
grade
level(s)
of
students,
(d)
the
ethnicity
and
in-
come of
students,
(e)
the
length
of
interven-
tion,
and
(f)
how
low
achieving
was
op-
erationally
defined. We
also listed
all
dependent
variables and
noted all
reliabil-
ity
and
validity
data
provided.
Finally,
we
noted
the
pages
in
the
article where
the
in-
terventions were
described
so that we
could
use this
information
in
the
next
phase
of
the
coding process.
Unfortunately,
most
studies did
not re-
port
student
ethnicity.
Of
the
studies
that
did
report
ethnicity,
both
studies
by
Cardelle-
Elawar
(1992,
1995)
involved
primarily
Hispanic
students.
Three
studies
reported
involving
primarily
African-American
stu-
dents
(Fantuzzo
et
al.,
1995).
Our
sense is
that
the
remaining
studies
concerned
pri-
marily
European-American
populations.
Identifying
independent
variable(s):
Phase 2
coding.
The second
phase
of
coding
was our
attempt
to
identify
precisely
the re-
search
question
or
questions
addressed
in
each
study.
The
senior authors
developed
the
coding
scheme
for
the set of
studies over
several months.
The
process
was
iterative.
During
the first
reading
of an
article,
we
coded
features
of the intervention
accord-
ing
to a
developing
set
of
broad
categories
(e.g.,
curriculum
design, providing ongoing
performance
feedback
to
teachers
and
stu-
dents,
using
data to
generate
specific
in-
structional
recommendations,
the
use
of
technology).
We reviewed
these codes
and
our
notes
and reread relevant
sections of
the
study
to
pinpoint
the
precise
research
ques-
tions
being
addressed. This
required
re-
reading major
sections of all the
studies.
Five of the 17
studies included
multiple
in-
tervention
groups
and thus
addressed
mul-
tiple
research
questions.
The
senior
authors
confirmed all
coding
during
the
second
phase
of
the
coding
process.
In
our
final
analysis,
we
settled on
five
major
categories
that
characterized
the
research
questions
and
intervention
ap-
proaches
for the set of
studies. These
were
(a)
providing
data and
ongoing
feedback
to
teachers
and/or
students about
mathemat-
ics
performance,
(b)
peer
tutoring
/
peer-
assisted
mathematics
instruction,
(c)
use of
parents
to
support
classroom
mathematics
instruction,
(d)
the
use of
explicit
or
teacher-
facilitated
instructional
approaches,
and
(e)
computer-assisted
instruction.
We
discuss each
category
in
detail
as we
present
the
findings.
In
the
appendix,
the
studies are
listed
by
category
along
with
the
associated
effect sizes. A
few
studies
are
in-
cluded
in
more than
one
category
because
it
was
possible
for a
study
to
explore
more
than one
research
issue. For
example,
some
studies
examined
the effects
of
peer
tutor-
ing
as well
as
any
"value-added"
effects of
strategies
that
encourage parental
involve-
ment. All
of the
studies
included
in
more
than one
category
involved
three or
more
groups
(e.g.,
two
treatment
groups
and
a
comparison
group).
When
studies
included
more
than two
groups
in
the
overall
anal-
ysis,
we
used
orthogonal
contrasts
(Keppel
&
Zedeck,
1989)
to
calculate effect
sizes.
This was
done
to
ensure
that no
statistical
assumptions
of
independence
were vio-
SEPTEMBER
2002
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58
THE ELEMENTARY SCHOOL
JOURNAL
lated.
Data-analysis procedures
are de-
scribed
in
the next section.
Data
Analysis
Computation of
effect
size. Standard
techniques
for effect-size calculation are
in-
tended for studies
with
one
experimental
group
and one
comparison group.
For stud-
ies
with two
groups,
we used standard
pro-
cedures for
determining
effect sizes
(Coo-
per
&
Hedges,
1994).
The basic index
we
used to calculate an effect size was Cohen's
d,
defined
as the difference between the ex-
perimental
and
comparison
group
means
divided
by
the
pooled
standard deviation
(Cooper
&
Hedges,
1994).
For
calculating
effect sizes when studies were combined for
comparisons
across
categories,
we followed
procedures
of
Shadish and Haddock
(1994),
which
weight
each effect size
by
the number
of students
in
each
study.
For studies that
reported
both
pretest
and
posttest
scores,
we calculated
posttest
effect sizes
adjusting
for
pretest perfor-
mance,
using
the first and the second
equa-
tions
(Wortman
&
Bryant,
1985).
Adjusted
Effect
Size
=
Unadjusted
d
-
Pretest
Correction,
(1)
Pretest Correction
=
(ME[pretest]
-
MC[pretest])
/SDpooled
[pretest]'
(2)
where
ME[pretest]
=
the
mean
of
the
experi-
mental
group
at
pretest,
MC[pretest]
=
the
mean of
the
comparison
group
at
pretest,
and
SDpooled [pretest]
is the
pooled
standard de-
viation at
pretest.
This
technique
is
espe-
cially
useful for
quasi-experimental
studies,
or
studies where the
sample
sizes are
small
and there are
small differences
in
pretest
scores between
samples.
A
somewhat
controversial issue
in
cal-
culating
effect
sizes is how to
determine
the
appropriate
number and
types
of
compari-
sons
to make when
there are
more than two
groups
in
one
study.
Although
standard
procedures
(Cooper
&
Hedges,
1994)
indi-
cate
that
each
study
should
contribute
only
one effect size
for each
relevant
category,
we
did
not
deem this
approach appropriate
for
this set of studies. One-third of the
stud-
ies addressed
research
questions
that
fit
more than
one
category.
Use
of orthogonal
contrasts.
Five
stud-
ies had at least
two
groups receiving
an ex-
perimental
intervention,
as
well
as
a
com-
parison group.
(One
of these studies
had
three intervention
groups
and a
comparison
group.)
The
experimental
interventions
were often subtle variations of one substan-
tive intervention. For
example,
in
one
group
the teachers received data on student
per-
formance,
whereas
in
the other
group
the
teachers received data on student
perfor-
mance as well as ideas
on
curriculum to use
with
particular
students.
In
these cases it
made sense to
compute
one effect size for
each research
question
asked.
If
a
study
had
three
independent
groups
(i.e.,
two
treat-
ment
groups
and a
comparison
group),
one
could
easily
calculate three effect
sizes:
comparing
the first
group
to the
second,
the
first to the
third,
and the
second to the third.
However,
this
would violate
assumptions
related to
independence
of each effect
size.
Conducting
multiple
comparisons
this
way
provides
redundant
and
potentially
misleading
information.
Orthogonal
con-
trasts,
however,
provide
independent
pieces
of information
(Keppel
&
Zedeck,
1989).
They
seemed
the most
appropriate
and
most
elegant
statistical
approach
to
take.
Thus,
we
conducted
only
orthogonal comparisons.
These were
determined
by
an
in-depth
read-
ing
of each
study
to
determine
what we
viewed as the
major
research
questions
the
author
posed.
Effect
size
calculations
for multiple
de-
pendent
variables. To
calculate
effect sizes
for
the
dependent
variables,
we
followed
standard
procedures.
If
a
study
included
more
than one
dependent
measure
of a
similar
construct
(e.g.,
mathematics
com-
putation),
the
average
of
the
measures was
calculated
and entered
in
the
analysis.
We
present
results
centered on each
of
the
major
categories.
We
approach
each as
a
theme,
in
that,
in
each
case,
a
series of re-
SEPTEMBER 2002
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TEACHING
MATHEMATICS
59
searchers
have addressed an
instructionally
important
question.
However,
after
careful
consideration,
we
decided
that one
category
was
not
relevant
for a
contemporary
audi-
ence. After
reviewing
the
computer-assisted
instructional
studies,
we decided
these stud-
ies
addressed
primarily
efficacy
of
dated
software
(e.g.,
Bass, Ries,
&
Sharpe,
1986;
Moore,
1988)
and
should
be excluded. The
types
of
software used 15
to 20
years ago
are too
dated
in
the context
of
contempo-
rary
instruction to
be
of
much use.
These
studies are
represented by
the final
category
in
the
appendix.
However,
one
study
that
used
technology
to
provide
precise
feed-
back on
student
performance
in
mathemat-
ics
(Clairiana
&
Smith,
1989)
was still in-
cluded under the
category
of
provision
of
feedback.
This
study
addressed
questions
that remain
relevant
for
contemporary
math-
ematics
instruction.
Effect
size
calculations
when the
class is
the unit
of analysis.
Two
studies
(Cardelle-
Elawar, 1995;
Fuchs et
al.,
1997)
used the
class or
a
subset of
students
in
the
class as
the unit
of
analysis.
Although
this
is a le-
gitimate
means
of
data
analysis,
the
stan-
dard
deviations
presented
in
these
studies
are much
smaller
than
studies
where the
student
is
the unit
of
analysis.
Left
uncor-
rected,
this would
tend
to
inflate
effect
sizes
substantially.
As a
statistical
correction,
we
multiplied
the
standard
deviation
of the
unit
of
analysis presented
by
the
square
root of
the
class or unit
size
(Hopkins,
1982;
Peckham, Glass,
&
Hopkins,
1969).
This re-
sulted in
standard
deviations that
closely
approximated
what
they
would
have been
if
students had
been
the
unit of
analysis.
Coding of dependent
variables.
We
coded
each
dependent
measure
as
either a
computation
measure or a
general
mathe-
matics
achievement
measure.
The
purpose
of
computation
measures
was to
determine
how
accurately-and
in
many
cases,
how
quickly-students
could
add,
subtract,
multiply,
and
divide
numbers
(including
numbers
with
decimals).
These
measures
invariably
were
closely
aligned
with
the
fo-
cus of the instructional
interventions.
Some
of the
computation
measures included
the
Math
Operations
Test
(Fuchs
et
al., 1997),
the Curriculum-Based
Computation
Rate
(Ginsburg-Block
&
Fantuzzo, 1997),
and
the
Math Skill
Test
(Schunk,
1982).
The
assessments
coded as
general
math-
ematics
achievement
measures included
an
array
of
mathematical
topics.
All
included
some
computation,
but
they
also
included
word
problems
and items
assessing
under-
standing
of
mathematical
concepts
such
as
equivalence
of
fractions.
Published
general
math
achievement
measures
included
the
ITBS total
score
in
mathematics
(Cardelle-
Elawar,
1992;
Woodward
&
Baxter,
1997b).
General
mathematics
achievement
mea-
sures
developed
by
researchers
connected
to the
target study
included
the
Comprehen-
sive Math
Test
(Fuchs
et
al.,
1997).
These
measures
were not
completely aligned
with
the focus of
the intervention.
Rather,
they
as-
sessed
whether
the
intervention
improved
general
mathematical
competence.
We calculated
separate
effect
sizes
for
computation
and
general
achievement.
Fol-
lowing
standard
procedures,
however,
if
a
study
included
two
computation
measures,
we
calculated
the
mean of
the
two
measures
in
determining
the
computation
effect
size.
Results
and
Discussion
In
this
section
we
discuss
the
most
impor-
tant
findings
in
each
of
the four
major
study
categories.
Our
goal
is to
pinpoint
what
each
set of
studies
says
about
means
to
im-
prove
the
mathematical
performance
of
low-achieving
students.
In
particular,
we
look at
how
consistently
the
practice
en-
hanced
performance
and
the
overall
mag-
nitude of
the
effect.
The
number of
studies
meeting
our
ini-
tial
criteria for
inclusion
in
the
synthesis
was
17.
With
the
exclusion
of
the two
com-
puter
software
studies
by
Bass
et
al.
(1986)
and
Moore
(1988),
the total
number
of
studies
in
the
analysis
was
15.
These 15
studies
were
coded
into four
major
cate-
gories
and
generated
39
independent
effect
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60
THE
ELEMENTARY
SCHOOL
JOURNAL
sizes.
These
effect
sizes
ranged
from
-.59
to
1.49
(see
App.).
In
quantitative
syntheses
using
effect
sizes,
it
is common
to
present
the
overall
mean
effect size for all
the intervention
studies in
the
analysis.
We
decided not
to
do this
because
the mean effect size
would
have
little
meaning
in
the
context of the
range
of
questions
being
addressed
in
the set of
studies. It
would
merely
present
the
aggregate
of an
array
of
interventions
ranging
from
curriculum
redesign,
to tu-
toring,
or
to
improving
the
information
teachers
receive
on
student
progress.
Inter-
pretation
would
be difficult.
Providing
Data or
Recommendations
to
Teachers and
Students
In
four
studies,
students
and/or
their
teachers
were
provided
with
specific
data
on
student
performance.
In
some of
these
studies,
the
computer
also
generated
rec-
ommendations
about
what
types
of
prob-
lems
to
work or
how
many
problems
to
work on
a
given topic.
The
comparison
group
in
these
four
studies
either was
pro-
vided
with
no
performance
feedback or
with
such
limited
feedback
that a
relevant
contrast
between
the
experimental
and
comparison
group
was
meaningful.
Six
comparisons
were
conducted
across
these four
studies.
All of
these
comparisons
were
orthogonal-that
is,
they
contributed
independent
sources
of
information
(Kep-
pel
&
Zedeck,
1989)
(see
Table
2).
In
five
comparisons
(involving
all
four
studies)
students
received
information on
their
ef-
fort
or
performance
in
solving
mathematics
problems
or
received
recommendations
from the
teacher
or
computer
regarding
the
number
of
problems
they
should
work in
a
given
time.
In
many
cases,
a
computer
pro-
vided
this
feedback
to
teachers as
well as
students.
The
overall effect
size for
these
five
comparisons
was .57
(unweighted,
.71),
and
the
confidence
interval
indicated
that
the
overall
effect
was
significantly
different
from
zero. This
is a
moderate effect
and the
second
largest
mean effect
size we found
in
this
synthesis.
One
study
in
this
category
was
con-
ducted
by
Fuchs,
Fuchs, Hamlett,
Phillips,
and Bentz
(1994),
and
we describe
it
in
more
depth
than
the others.
It
is,
in
our
view,
the
most
complex
study
in
the set but
also
the
richest.
Fuchs et al.
included
two
experi-
mental
groups
and
one
comparison
group.
On a
computer,
students in
the
two
exper-
imental
groups
took
weekly
tests on
items
that reflected state
content standards.
The
software
created
individualized
graphs
de-
picting
each
student's
performance
over
time.
Performance
graphs
were
given
to
both
teachers and
students.
Teachers
also
received
a
performance
summary
of
all
stu-
dents
in
the class. In
the
comparison
con-
dition,
teachers used
their own
techniques
for
monitoring
student
progress.
Thus,
the
major
difference
between the
experimental
groups
and
the
comparison
group
was
pro-
viding
teachers and
students with
weekly
information
on
student
progress
in
mathe-
matics.
One
of
the
unique
features
of
this
study
was the
difference
between
the two
exper-
imental
groups.
In
the more
complex
ex-
perimental
group,
teachers
also
received
computer-generated
recommendations
on
what
content to
teach
the
full
class
in
up-
coming
lessons
based on
the
aggregate per-
formance
data.
Recommendations
regard-
ing
which
students
to
group
together
for
small-group
instruction
were
provided
(based
on
their
individual
Curriculum-
Based
Measurement
math
tests).
Teachers
also
received a
listing
of
computer
lessons
to
use with
individual
students and
sugges-
tions on how
to use
peer
tutoring
(Green-
wood,
Delquadri,
&
Hall,
1989)
to
provide
students
with
practice
and
reinforcement
on
concepts
and skills with
which
they
were
struggling.
We
conducted two
orthogonal
contrasts
from this
study.
The
first
investigated
the
effect of
providing
weekly
progress
infor-
mation to
students
and their
teachers.
Here,
we
contrasted the
average
performance
of
SEPTEMBER
2002
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TEACHING
MATHEMATICS
61
the two
experimental groups
with
the
per-
formance
of the
comparison
group.
This
ef-
fect size was
.29,
indicating
a small
effect
on
mathematics
achievement
when
teachers
systematically
monitored the
progress
of
their
low-achieving
students
and
graphed
this
information for
themselves
and
their
students.
In
the
second contrast
we
investigated
the
effect of
providing
teachers
with
in-
structional
recommendations
based on
the
progress-monitoring
data. For
this
question
we
examined
the difference
between
the
two
experimental
groups.
The
effect size
for
this
comparison
was
.51,
which
is consid-
ered
moderate.
The two
effect
sizes
suggest
that
perhaps
merely
providing
teachers
with
data on
student
performance
may
not
be
as beneficial
as the
combination of
pro-
viding
data and
then
making specific
in-
structional
recommendations
to
address
problem
areas
identified in
current
student
performance.
In
summary,
the
small
number of
stud-
ies and
comparisons
supported
the
practice
of
providing
feedback
to
students and
rec-
ommendations
to them
on
what
problems
to
work.
Advantages
and
limitations of
pro-
viding
teachers with
feedback
versus
pro-
viding
them with
feedback
accompanied
by
instructional
recommendations
are
specu-
lative
and
need
further
investigation.
Com-
puters
seem
to be
valuable
in
generating
mathematics
problems
for
students
to
work,
being
able
to
target
areas
where
more
practice
is
needed,
and
providing perfor-
mance
feedback
to
students
and
teachers
along
with
specific
recommendations.
Peer-Assisted
Learning
A
major
focus of
research on
mathemat-
ics
instruction
for
low-achieving
students
has been
on
development
and
evaluation
of
strategies
and
structures
that
enable stu-
dents
to
provide
each
other
with
feedback
and
support.
Six
studies
addressed
this
topic.
The
overall
aggregate
effect
size
for
this
category
appears
in
Table
2.
We
believe
there
are
several
reasons
for
the
relatively
heavy
emphasis
on
studying
how
students
can work with
each
other
to
learn
mathematics.
While
working
on
math-
ematics
problems
independently,
students
who
are uncertain
about
problem
solving
of-
ten
want to
ask
questions
about
what
to
do.
A
teacher cannot
be
available
to each
indi-
vidual
student
to address
questions
and
un-
certainty.
Oftentimes
peers
can
provide
the
answer,
or
(if
taught
to do
so)
provide
sug-
gestions
that
help
students
solve
the
prob-
lem
themselves.
Related
to
the
idea of
peers
working
together
is
the
finding
that
success
in
mathematics
requires
considerable
task
persistence
(Kolligian
&
Sternberg,
1987).
Researchers
investigating peer
tutors
have
stressed
that
peer
tutoring
is
likely
to
en-
courage
low
achievers
to
persist
in
their
work.
Peer-assisted
learning
interventions
in-
variably
led
to
positive
effects
on
student
achievement.
The
average
effect
size
was
.62,
with
a
median
of .51.
Effect sizes
ranged
from
.34
to
1.26.
With
the
exception
of
the
one
outlier
(Heller
&
Fantuzzo,
1993),
which
had
the
weakest
comparison
group,
the
effects
were
reasonably
consistent
and
in
the low
to
moderate
range.
The
majority
of
the
studies
(five
of
the
six)
examined
ef-
fects on
computation;
only
two
studies
in-
cluded
a
measure
of
general
math
achieve-
ment.
The
magnitude
of
effect
sizes
was
greater
on
computation
than
general
math
ability.
The
average
effect
size on
compu-
tation
problems
was
.62
(weighted),
which
was
significantly greater
than
zero.
On
gen-
eral
math
achievement,
the
two effect
sizes
were .06
and
.40,
producing
a
weighted
mean
of .29
that
was
not
significantly
differ-
ent
from
0.
It is
safest
to
conclude
that
the
peer-assisted
learning
approaches
demon-
strated
a
consistent,
moderately
strong
posi-
tive
effect
on
the
computation
abilities
of
low
achievers.
To
date,
it
is
unclear
how
helpful
peer
tutoring
might
be
in
other
areas of
mathematics.
In
discussing
the
individual
studies,
it
is
helpful
to
disentangle
the
two
major
streams of
research
that
have
been
con-
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TEACHING
MATHEMATICS
63
ducted
on this
topic. Lynn
Fuchs and her
colleagues
have conducted
the first.
Fan-
tuzzo and
colleagues
conducted
the second.
There are
many
similarities
between
the
two
approaches
as
well as some subtle dif-
ferences. Both
rely
on students
working
in
pairs (dyads)
as
opposed
to the
groups
of
four to
six that
typify
cooperative
learning.
In
each
approach,
curriculum-based assess-
ment data
are used to
pair
students and
to
determine the content of
the
tutoring
ses-
sions.
Both
approaches
are
reciprocal
in
na-
ture.
In
other
words,
students alternate
be-
tween the role of tutor and tutee.
(The
one
exception
is the
study by
Fuchs, Fuchs,
Phil-
lips,
Hamlett,
and Karns
[1995],
which used
a more
traditional tutor-tutee
model.)
Role
reciprocity
became a critical innovation as
Fuchs et al.'s
peer tutoring approach
evolved
and
may
be a central feature
in
its success.
Both Fuchs et
al.
and
Fantuzzo et al. use a
tightly
structured
format for the
tutoring
sessions. Students are
carefully
trained
in
tu-
toring procedures.
Both
approaches
employ
prompt
cards and
procedures
for
tutors to
use as
they
help
fellow students work
through
a
series of
problems.
There are two
major
differences be-
tween the
approaches.
The research of Fan-
tuzzo and
colleagues
focused
only
on com-
putation,
whereas the
work of Fuchs and
colleagues
included
a broader
range
of
mathematics
topics
and
involved a more
complex
feedback
system
for the
peer
tutors
to
use.
In
the
Fuchs et al.
system,
each tutor
gives
the
partner
feedback on each
step
in
the math
problem
attempted.
Although
tu-
tors are
provided
with
a
specific
step-by-
step strategy
for
approaching
each
type
of
problem,
when
the tutees
encounter
diffi-
culty,
tutors are
encouraged
to
construct ex-
planations
in
their own
words.
Once
students master the
basics
of
the
peer
tutoring procedures
(typically
by
the
third
week
of the
program), they
are
taught
several
teaching strategies
to
use
as
they
work with their
partners.
These
strategies
were
adapted
from
the research
of Hiebert
and
Wearne
(1993).
The
goal
of the
tutoring
strategies
is to
help
students contextualize
problems.
Tutors are
encouraged
to
repre-
sent abstract
mathematical
quantities
with
visuals or
manipulatives.
Tutors
are also
en-
couraged
to discuss
solution
strategies
with
their
partner.
In
summary,
the use of
peers
to
provide
feedback and
support
is
consistently sup-
ported by
research
as a means to
improve
computational
abilities and
is a
promising
means to enhance
problem-solving
abilities.
Explicit
Teacher-Led
and
Contextualized Teacher-Facilitated
Approaches
Seven
studies
investigated
the effects of
instructional
practices
on math achieve-
ment of low achievers. Effect sizes for
these
studies are
presented
in
Table
2.
Although
this is the
largest
set of
studies,
it is still an
extremely
small number of
research studies
on the
broad
topic
of effective
mathematics
instruction
for
low-achieving
students. This
paucity
reflects the
general
lack of
experi-
mental field
research
in
mathematics
(Kil-
patrick
et
al.,
2001).
The
seven studies fell
into two
general
categories:
those
involving explicit
instruc-
tion
in
mathematics,
and
those that
stressed
contextualized
approaches.
Three
studies,
each
contributing
one
comparison
to the
overall effect
size,
investigated
an
approach
involving explicit
instruction
in
mathemat-
ics.
In
these
studies,
the
manner
in
which
concepts
and
problem
solving
were
taught
to
students
was far more
explicit
than is
typical.
Three
studies
investigated
the
effects of
math
instruction that
emphasized
the
con-
text of
the
mathematics
problems
in
which
teaching
occurs and
stressed
conceptual
un-
derstanding
over
procedural
compliance
and
accuracy.
Following
this
approach,
teaching
emphasized
real-world
applica-
tions of
mathematical
principles.
One
study
(Woodward,
Baxter,
&
Robinson,
1999)
had
both an
explicit
instruction
group
and
a
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64
THE
ELEMENTARY
SCHOOL
JOURNAL
contextualized
instruction
group
and
con-
sequently
was
included
in
both
contrasts.
Explicit
instruction.
Two of the inter-
ventions followed
principles
articulated
in
Engelmann
and
Carnine
(1982),
which are
often
referred to as
direct
instruction.
Di-
rect
instruction
involves
teaching
rules,
concepts,
principles,
and
problem-solving
strategies
in
an
explicit
fashion.
This
in-
cludes
providing
a
wide
range
of
examples
of the
principle
or
concept
and
providing
extensive review
and
discriminative
prac-
tice.
Two other
studies
in
this
category,
both
by
Cardelle-Elawar
(1992,
1995),
used an
approach
that
was also
explicit
but
focused
on
teaching
generic
problem-solving
strat-
egies
using
a
specific
set of
problems.
The
method of
problem
solving
used
was de-
rived from
the
cognitive
research
of
Mayer
(1987).
The
aggregate
weighted
effect
size
in
the
explicit
instruction
category
was .58
(un-
weighted
=
.65).
Individual effect
sizes
ranged
from
.32
(Cardelle-Elawar,
1995)
to
1.1
(Moore
&
Carnine,
1989).
The
95% con-
fidence
interval was .40
to
.77,
indicating
that the
effect was
statistically significant.
This result
indicates
that,
overall,
the
ap-
proaches
that used
explicit
instruction
had
a
positive,
moderately strong
effect on
the
mathematics
achievement of
at-risk
stu-
dents.
Cardelle-Elawar
(1992,
1995)
investi-
gated
the
effects
of the
Mayer problem-
solving
approach
on a
general
measure
of
mathematics
achievement.
The
overall
ef-
fect
for
the
problem-solving
approach
was
.55
(weighted),
which
was
statistically sig-
nificant.
Woodward et
al.
(1999)
examined
effects
of the
intervention
on a
measure of
computation
involving
decimals.
Moore
and
Carnine
(1989)
assessed
students'
pro-
ficiency
and
understanding
of
elementary
problems
involving
decimals
and
propor-
tions
(e.g.,
What
is 25%
of
22?).
Both
the
Woodward
et
al.
and
Moore and
Carnine
studies
involved
curricula
designed
ac-
cording
to
Engelmann
and
Carnine's
(1982)
principles
of
direct
instruction.
The
weighted
effect
size
was
.80,
also
statisti-
cally significant.
Cardelle-Elawar's
instructional
approach
focused
on
strategy
instruction.
Specifically,
teachers
extensively
modeled
how
students
should
ask themselves
a series of
questions
when faced
with
mathematics
problems.
In-
struction
emphasized
story problems.
Deci-
phering
the
vocabulary
used
in
problems
was
stressed.
In
part,
this was
because
the
students in
these
studies
were
primarily
En-
glish
language
learners.
After
working
to
understand
the
vocabulary
used,
students
learned
to
determine
if
the
necessary
infor-
mation
was
available to
solve the
problem,
and if
so,
how
to
organize
the
problem.
Then
they
proceeded
step by
step
through
the
calculations
phase
to
arrive
at
the
cor-
rect answer.
After
extensive
teacher
modeling
of
all
components,
students
worked
individu-
ally
on
similar
problems,
under
close
su-
pervision
and
monitoring
by
the
teacher.
Teachers
provided
feedback
to
students
that
closely
followed
the
question-asking
strategy
students
learned
from
the
teacher
modeling.
At
the end of
each
lesson,
stu-
dents were
required
to
formulate
in
their
own
words
what
they
learned
that
day
as
a
means
of
processing
the
key
principles
and
strategies
presented
in
the
lesson.
In
the
explicit
instruction
approaches
in
the
Moore and
Carnine
(1989)
and
Wood-
ward
et
al.
(1999)
studies,
concepts
and
operations
involving
ratios and
proportions
were
taught
following
the
principles
of
in-
structional
design
articulated
by
Engel-
mann
and
Carnine
(1982).
The
curriculum
stressed
presenting
a
wide
range
of
exam-
ples
to
demonstrate each
concept,
extensive
practice,
and
cumulative
review
of
previ-
ously
taught
material. The
curriculum
used
in
the
Moore and
Carnine
(1989)
study
ex-
plicitly
taught
students
strategies
for
dis-
cerning
relevant
from
irrelevant
material in
the
problems.
Students
also worked
on
problems
independently,
errors
were
cor-
rected
quickly
by
the
teacher,
and
students
reviewed
the
strategies
taught
or
the
rele-
SEPTEMBER
2002
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TEACHING
MATHEMATICS
65
vant
mathematical
principle.
There was
vir-
tually
no
emphasis
on students
being
able
to verbalize
their
problem-solving
strate-
gies
in
this
approach,
unlike the
approach
Cardelle-Elawar
used.
In Woodward et al.
(1999),
students
in
one of
the two
groups
were
taught
concepts involving
decimals
using
the identical theoretical
framework as
the Moore
and
Carnine
study.
Contextualized
instruction and
prac-
tice. Four recent studies
(all
conducted
in
the
1990s)
investigated
what
we have la-
beled contextualized
teaching.
A
character-
istic
of this
approach
was for some or all of
the instruction
in
the
experimental
group
to
stress
real-world
applications
and,
at least
to some
extent,
to focus on
understanding
underlying concepts
of authentic
problems.
The treatments
in
studies
in
this
category
sought
to teach students about mathemati-
cal
thinking, arguing
that a more
vigorous
emphasis
on
concept development
was
critical to mathematics success and would
lead not
only
to a
deeper understanding
of
math
but
to
computational
proficiency
as
well. These studies were
all influenced
by
the framework of
mathematics instruction
developed by
the NCTM
in
1989 and re-
cently
revised
drastically
(NCTM,
2000).
One
study
(Woodward
et
al.,
1999)
investi-
gated explicit
instruction
versus the NCTM
framework
and thus fell into both
the ex-
plicit
instruction and
contextualized instruc-
tion
categories.
The
overall effect size of
studies
in
con-
textualized
instruction was
.01,
essentially
zero. In
other
words,
students
in
the com-
parison
groups
scored
as well as
students
in
the
experimental
groups.
In
two of the
four
studies,
the
overall
effect size favored
the
experimental
group,
and
in
two of the
four it
favored the
comparison
group.
The
effect size
in
the
study by
Henderson and
Landesman
(1992)
was
.18,
indicating
that
the effect of
contextualized
instruction was
small.
Henderson and
Landesman
admin-
istered
both
a
general
achievement
measure
(concepts
and
applications)
and
a
compu-
tation
measure.
Both
measures
produced
small,
positive
effects
(.22
and
.14,
respec-
tively).
The
study by
Bottge
and
Hasselbring
(1993)
produced
the
largest
positive
effect
for this
group
of studies
(effect
size
=
.48).
This was one of
the most
interesting
and
thoughtful
studies
in
the
synthesis
because
of the creative nature
of the
design
and
in-
struction
and the
way
the authors
tried
to
assess
the effect of the intervention
on a
range
of
dependent
measures.
Bottge
and
Hasselbring
did
two
things
that are worth
discussing.
First,
during
a
5-day
baseline
phase they taught
both
experimental
and
comparison
students
mathematics skills that
would
help
the re-
searchers better understand the
subsequent
effects
of the
intervention. The intervention
compared
math
learning
via contextualized
instruction-the
presentation
of an auth-
entic
problem
delivered via
videodisc-
versus
learning
from instruction
delivered
through
a more traditional focus on
word
problems.
The
second
noteworthy
feature
was that
dependent
measures were
carefully
con-
structed to assess
learning
and transfer.
Two
dependent
measures were
closely aligned
to
each of the
teaching approaches-that
is,
one measure
was
aligned
to
instruction the
experimental
group experienced,
and one
measure
was
closely aligned
to
instruction
the
comparison
group experienced.
Two
measures
assessed
transfer,
one of the
most
troublesome areas of
special
education re-
search
and
practice.
Students
who received
contextualized
instruction
scored
higher
on
a
contextualized word
problem
test and on
transfer
measures
than students
in
the con-
trol
group.
The two studies
by
Woodward and
his
colleagues
(Woodward
&
Baxter,
1997a;
Woodward et
al.,
1999)
were the
closest to
Bottge
and
Hasselbring
in
trying
to
inves-
tigate
how
contextualized
instruction
af-
fects
mathematics
performance.
The
overall
effect
sizes for both
of these
studies
were
negative.
Woodward
and
Baxter
(1997a)
compared
students
receiving
instruction
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66 THE ELEMENTARY
SCHOOL
JOURNAL
based
on
the
Everyday
Mathematics
(Bell,
Bell,
&
Hartfield,
1993)
program developed
at the
University
of
Chicago
to students re-
ceiving
traditional instruction
following
a
basal mathematics
program.
This
program
de-emphasized computation
and included
many
problems
with
relevance to students'
everyday experiences.
Problems were se-
lected
so
that
students could not
capitalize
on
"key
words" as
shortcuts
to
understand-
ing
the
concepts.
"Students
are
encouraged
to use or
develop
a
variety
of
number mod-
els which
display
relevant
quantities (e.g.,
total and
parts;
start-change-end;
quantity-
quantity-difference)
to
be
manipulated
in
solving
these
problems"
(Woodward
&
Baxter, 1997a,
p.
376).
Estimation
is
actively
encouraged.
An
array
of mathematical
games
is
an
integral
part
of the curriculum.
Unlike
the
approach
taken
in
the Moore and
Carnine
(1989)
study,
where
relatively
brief
problems
were
taught
first and students
gradually
built
to
complex problems,
stu-
dents
in
the Woodward
and
Baxter
(1997a)
study
were confronted
early
on with com-
plex
multi-step problems.
The
goal
was to
develop in-depth
conceptual
understand-
ing.
The overall effect
size,
based on
perfor-
mance on a
general
measure of math
achievement
(total
mathematics score on the
ITBS),
was
-.24,
meaning
that
students
in
the
basal
group
(the
comparison
condition)
performed
better than
students
in
the
NCTM
group
(the
experimental
condition).
A
strength
of the
Woodward and Baxter
(1997a)
study
was
the
use of a
standardized
measure of
mathematics
achievement. The
study
also
included
a
student
interview to
assess
conceptual
understanding,
but
this
was
only
administered to a small
subsam-
ple,
precluding
statistical
analyses.
In
the
subsequent
study
in
1999,
Wood-
ward
and his
colleagues
used
a
stronger
comparison
group
-explicit
instruction-
and
investigated
the
effects of
instruction
using
two
experimenter-developed
mea-
sures of
computation
involving
decimals.
Students solved
computation
problems
with
and without
the use
of a
calculator.
The
effect sizes
were
nearly
identical,
-.59
and
-.58,
indicating
that
students in
the
comparison group
did
better than
students
in
the
experimental
group,
and
the
effect
in
both cases was moderate.
This
finding
is not
too
surprising given
that the
computational
focus
could be
expected
to
favor the
explicit
instruction
group.
Woodward
et al.
(1999)
also
presented
the
results
of an
individual student
inter-
view
scored
quantitatively,
which
signifi-
cantly
favored students
in the
NCTM
group.
The authors indicated that this
mea-
sure showed
how
NCTM instruction bene-
fits
students'
conceptual
knowledge
of
mathematics
compared
to
explicit
instruc-
tion. We
did
not include
this interview
in
our
synthesis,
however,
because
it
was
not
clear how the measure was
scored,
and its
reliability
was uncertain.
Also,
it
was not
clear whether the
interviewers were
blind
as
to
which
group
students were
in
when
they
were interviewed.
Instruction for the
conceptual,
or contex-
tualized,
condition
in
the
Woodward et al.
(1999)
study emphasized
the
development
of
conceptual
understanding through
the
use of
visual
representations
and
physical
manipulatives
(e.g.,
wood
block
rectangles).
Lessons from Mathematics in
the Mind's
Eye
(Bennett,
Maier,
&
Nelson,
1988)
served as a
basis for
daily
instruction.
Students
were
taught
to
develop
visual
representations
of
problems
using pie
chart
diagrams
and
wood
block
rectangles, squares,
or
cubes.
According
to Woodward et al.
(1999,
pp.
17-
18),
"Instruction
was
intended to
provide
much
greater
depth
in
initial
decimal con-
cepts
than these
students had
received in
the
past."
Links between
fractions
and
decimals
were
stressed.
In
summary,
the
studies
involving
con-
textualized
mathematics
instruction
present
a
complex
puzzle
of
findings, open
to mul-
tiple
interpretations.
These
studies
have
furthered the
understanding
of
how
in-
struction
focusing
on
concept
development
compares
to other
approaches,
such
as ex-
SEPTEMBER
2002
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TEACHING
MATHEMATICS
67
plicit
instruction
in
fundamentals
and
prob-
lem
solving.
Further studies
in
this
area
should be
conducted,
using,
as
Woodward
et al.
(1999)
did,
techniques
such as
high-
quality
control
groups,
and
employing,
as
Bottge
and
Hasselbring
(1993)
did,
a
range
of
dependent
measures of
learning
and
transfer. Researchers should also
attempt
to
combine the
best
aspects
of contextualized
instruction
with
other
approaches
such as
explicit
instruction
with
the same
group
of
students.
Regarding explicit
instruction
in
math-
ematics,
effects were
consistently positive.
Approaches ranged
from
very explicit
in-
struction
in
mathematics
operations,
with
extensive and
carefully
crafted
practice,
to
approaches
that focused on the
explicit
teaching
of
strategies
students
needed
in
or-
der to
understand the content of
story prob-
lems.
Providing
Parents with
Information
about Student
Successes
Fantuzzo and
colleagues
(Fantuzzo
et
al.,
1995;
Heller &
Fantuzzo,
1993)
con-
ducted
studies
in
this
category.
In
both
cases,
providing
information
to
parents
was
assessed
as an "add on"
to a
reciprocal
peer-tutoring
program.
Both
studies fo-
cused on
improving
computational
skills
and
used
measures of math
computation.
The
studies
produced
identical effect sizes
of
.42.
Although
on the
surface
this seems
like
a moderate
effect,
it
is not
statistically
significant.
The
"low cost"
of the
interven-
tion is still
impressive,
however,
and war-
rants
a closer
look at the
studies and
further
empirical
investigations.
The
parent
support
intervention
was
"designed
to
enhance the
parent's
role as
supporter
and
motivator of
students'
aca-
demic
effort and
success"
(Fantuzzo
et
al.,
1995,
p.
274).
It
involved
regular
home-
school
contacts
(by
note or
telephone)
that
described
examples
of
students'
efforts
and
successes
in
mathematics.
All
exam-
ples
were
positive,
focusing
on
what the
student had
learned or
accomplished
or on
activities
on which
the student
had
worked
particularly
hard.
Messages
focused on in-
stances of students
showing
academic
ini-
tiative and task
persistence.
The
purpose
of
these
contacts
was to
encourage parental
celebrations of
students' successes
in
math-
ematics.
It
is
important
to
emphasize
that in
these
interventions the
parent's
role
was
not
that of a math teacher
but rather of
a
knowl-
edgeable supporter
to
her/his
children
in
their efforts to work hard on
learning
math-
ematics.
These studies
suggest
that
providing
parents
with
information on their
chil-
dren's mathematics
accomplishments
and
encouraging parents
to celebrate those
ac-
complishments
with
their children can
result
in
improved
student
achievement.
Moreover,
this
parent-support
technique
is
remarkably easy
to
implement,
and the
positive
effect has been
replicated.
The
ap-
proach
has,
at
times,
worked
in
conjunc-
tion
with
other instructional
efforts such as
peer tutoring.
Summary
and
Conclusions
The
set of 15 studies
provides
some
ideas
about
ways
to
improve
the
performance
of
low-achieving
students
in
mathematics.
Al-
though
this is not a
large
body
of
research,
four
findings
are consistent
enough
to be
considered
components
of
best
practice.
Other
findings
are more
tentative,
based on
only
a few
studies.
One consistent
finding
is
that
providing
teachers and
students with
specific
infor-
mation on how
each
student is
performing
seems to
enhance
mathematics
achievement
consistently.
This
practice
has
been
recom-
mended for
many years,
yet
the
extent to
which it
is
implemented
is
unclear.
The ef-
fect
of such
practice
is
substantial,
raising
scores,
on
average,
by
.68
SD units.
A
second consistent
finding represents
an
important
strand
in
contemporary
re-
search.
Using peers
as tutors
or
guides
en-
hances
achievement.
Research
shows that
the
use
of
peers
to
provide
feedback and
support improves
low
achievers'
compu-
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68
THE ELEMENTARY
SCHOOL
JOURNAL
tational
abilities
and holds
promise
as a
means
to
enhance
problem-solving
abili-
ties.
If
nothing
else,
having
a
partner
avail-
able to
provide
immediate
feedback
is
likely
to
be of
great
benefit
to a low achiever
struggling
with
a
problem.
A
crucial
feature
of this
approach
is that
the
topics
being
cov-
ered are
ones on
which
curriculum-based
measurement
data
suggest
areas
where
a
student needs
extra
practice
and
support.
Third,
providing
clear,
specific
feedback
to
parents
of low achievers
on
their chil-
dren's successes
in
mathematics
seems
to
have
the
potential
to enhance
achievement,
although perhaps
only modestly.
More
re-
search
needs to
be conducted
before
firm
conclusions
are drawn.
Advantages
of
this
approach
are
that
it
is
relatively
easy
to
im-
plement
and can
lead
to
other
long-range
benefits
in school-home
communication.
The
two
relevant
studies
suggest
that the
feedback
should
(a)
be
specific, objective,
and honest
and
(b)
detail successes
(or
rela-
tive
successes)
as
opposed
to failures
or
dif-
ficulties.
Fourth,
in
terms
of
curricula,
a
small
body
of research
suggests
that
principles
of
direct
or
explicit
instruction
can
be
useful
in
teaching
mathematical
concepts
and
pro-
cedures.
This includes
both the
use
of
strat-
egies
derived
from
cognitive psychology
to
develop generic
problem-solving
strategies
and
more classic
direct instruction
ap-
proaches
where students
are
taught
one
way
to
solve a
problem
and are
provided
with
extensive
practice.
With the
latter
ap-
proach,
concepts involving
fractions,
ratios,
or decimals
are
presented
using
a
wide
range
of
examples.
There is less
clarity
about
the benefits
of contextualized
approaches,
where
the
teacher serves
primarily
as
a facilitator as
students
work
through
real-world
exam-
ples
of mathematical
concepts
and
discuss
alternative
solutions to
problems
with
their
peers
or
teachers.
A
small
positive
ef-
fect was found when students worked out
complex,
real-world
problems
only
after
they
had been
clearly
and
explicitly
taught
the
underlying
foundational
mathematical
concepts
(e.g.,
Bottge
&
Hasselbring,
1993).
Thus,
low
achievers
seem
not
to
do well
at
authentic
problem
solving
and
discussion
of
mathematical
concepts
without solid
preparation
in
the
underlying
mathematical
foundations.
At this
point,
however,
it is
premature
to
draw
strong
inferences
on the effective-
ness of this
recently
developed approach.
For one
thing,
the mean
effect size of
the
set
of
studies is
essentially
zero,
indicating
that
there
is no clear
trend
in the
findings.
In
addition,
the
concept
of
contextualized
in-
struction is
an
emerging
one,
and
we
did
not
find
strong
coherence
in the
approaches
used
in
the four
studies.
There
are other
plausible
explanations
for
the overall
ineffectiveness
of
the four
studies
that
used a
contextualized
in-
struction
approach.
All
four
were
quasi-
experiments,
so
differences between
the
experimental
and
comparison groups
un-
related
to
the intervention
may
have
influ-
enced
the outcome.
Also,
three
of the four
studies
involved
older
students
(grades
7-
9),
which was a far
larger percentage
than
we
found
in
the
other
categories.
In addi-
tion,
only
one of
the four studies included
a measure
of
implementation
fidelity.
Not
only
is this a
lower
percentage
of measured
fidelity
than
in
the other
categories,
but be-
cause
contextualized
instruction is
not
yet
a well-defined
approach,
lack
of
assess-
ment
of
what
actually
occurred
during
les-
sons
may
have led to erratic
implementa-
tion
and
thus to fewer
effects on student
learning
than would have been
obtained
with
expert
implementation.
Examined as
a
group,
the
instructional
studies
seem to
support
the
position
taken
by
the
National Research Council
(Kilpa-
trick
et
al.,
2001),
which
argues
for a
mix
of
explicit
instruction
in
procedures
and
am-
ple opportunity
to
apply
procedures
to
open-ended
problems
with
real-world rele-
vance. Manzo
(2001,
p.
1)
commented
that
the
report emphasized
that,
"while both
SEPTEMBER
2002
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TEACHING MATHEMATICS
69
computational
skills and a
deep
under-
standing
of
math
concepts
are essential
parts
of
a
complete
mathematics education
...
other
elements,
including
problem-
solving
and
reasoning
abilities,
as well
as
an
awareness of the relevance
of math
in
everyday
life,
are also
necessary
for
math-
ematical
proficiency."
Furthermore,
the
panel
also
suggested
that
algebraic princi-
ples
should
be
built
into the
curriculum be-
ginning
in
the
early
grades.
Earlier research
(e.g.,
Gersten &
Carnine,
1984)
suggests
that
this is a wise course of
action for
at-risk stu-
dents.
It
is unfortunate
that such
a limited
number of
controlled
research
studies ad-
dress means for
improving
the
mathematics
knowledge
of
students
who
are considered
low
achievers.
By
limiting
our
synthesis
to
studies that
were well
controlled,
we did
not
address
some
of
the
subtle,
intricate,
and
profound
issues in
the
teaching
of
mathematics
raised,
for
example,
by
Ball
(1990, 1993,
1995),
Griffin, Case,
and
Siegler
(1994),
and
Hiebert
and
Wearne
(1993).
Yet
we find
in
these
studies a
burgeoning
sense
of
the
concept
of
mathematical
proficiency,
that
is,
"the
integrated
attainment of con-
ceptual
understanding, procedural
fluency,
strategic
competence,
adaptive
reasoning,
and
productive
disposition"
(Kilpatrick
et
al.,
2001,
p.
313).
Each
of
the
researchers
cited in
the
in-
structional/curricula
strand
grappled
with
this
issue
(Bottge
&
Hasselbring,
1993;
Cardelle-Elawar,
1992,
1995;
Henderson
&
Landesman, 1995;
Woodward
&
Baxter,
1997a),
albeit
using
different
language
and
coming
at it
from
different
traditions.
There
are too
few
studies on
curriculum and in-
struction to
allow
inferences to
be
drawn;
there is a
pressing
need
for
well-designed
research on
this
topic.
Even
the
National
Research
Council
report
(Kilpatrick
et
al.,
2001)
is
elusive,
at
best,
as to
how to
design
and
orchestrate
instruction
for
students
with
chronic
problems
in
mathematics.
But
the
report's
admonition that
"sound
re-
search can
help guide
the
design
of
effective
mathematics instruction"
(p.
24)
reinforces
the
important point
that
strong
research
studies
are not
only
the
best
way
to
answer
questions
about
the effects
of
specific
ap-
proaches,
but
they
should
also
play
a
sub-
stantive
role
in
shaping
better
methods
to
teach
mathematics.
The
National
Research
Council
(Kilpa-
trick et
al., 2001,
p.
26)
also
suggests
that
"high-quality
research
should
play
a
central
role
in
any
effort
to
improve
mathematics
learning.
That
research
can
never
provide
prescriptions,
but it
can
be used
to
help
guide
skilled teachers in
crafting
methods
that
will
work
in
their
particular
circum-
stances."
Designing
instruction
so that
even
students with
chronic
problems
in
mathe-
matics
can
succeed and
develop
a
solid
con-
ceptual
understanding
is
a
formidable
chal-
lenge.
The
authors
of the
National
Research
Council
report
note "it
can be
challenging
to
draw
scientifically
sound
conclusions
from
a
selected
set of
observations. In
con-
trast,
experimental
methods ...
establish
stronger
bases for
drawing
conclusions,
al-
though
even
these
conclusions
have
impor-
tant
limitations
and
qualifications"
(Kilpa-
trick
et
al.,
2001,
p.
25).
Some of the
qualifications
cited in
the
National
Research Council
report
seem
valid.
Others do
not
apply
to the set of
stud-
ies
we
reviewed.
For
example,
the
report
suggests
that
effects
found in
controlled
studies
may
not
apply
to
the real
world
of
classrooms. Yet
most of
the
studies in
our
synthesis
were
conducted in
school
class-
rooms
or
tutoring
settings
that exist in
Title
I
programs.
The
report's
authors
note that
"most
published
studies in
education
con-
firm
the
predictions
made
by
the
investi-
gators"
(Kilpatrick
et
al.,
2001,
p.
26).
This
was
not
always
the
case
in
the
studies
we
reviewed.
We
believe our
synthesis
pro-
vides
suggestions
that can
serve
as
initial
steps
in
the
improvement,
and
perhaps
the
ultimate
transformation,
of
the
teaching
of
mathematics
for
low
achievers.
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70
THE
ELEMENTARY
SCHOOL
JOURNAL
Appendix
Studies
Included
in
Meta-Analysis,
by
Category
Category
1:
How
Effective Is
Providing
Data
and
Feedback to
Teachers
and
Students?
Effectiveness of Providing
Students with
Information
and/or
Data
Clairiana,
R.
B.,
&
Smith,
L.
J.
(1989).
Progress
reports improve
students'
course
completion
rate
and achievement in math
computer-assisted
in-
struction.
(ERIC
Document
Reproduction
Service
No. ED
317
170) (Effect
size
=
.39)
Fuchs,
L.
S.,
Fuchs, D.,
Hamlett,
C.
L.,
Phillips,
N.
B.,
&
Bentz,
J.
(1994).
Classwide curriculum-
based
measurement:
Helping
general
edu-
cators meet the
challenge
of student diver-
sity.
Exceptional
Children,
60,
518-537.
(Effect
size
=
.29)
Fuchs,
L.
S., Fuchs, D.,
Karns, K., Hamlett,
C.
L.,
Katzaroff, M.,
&
Dutka,
S.
(1997).
Effects of
task-focused
goals
on
low-achieving
stu-
dents
with
and without
learning
disabilities.
American Educational
Research
Journal,
34,
513-543.
(Effect
size
=
.25)
Schunk,
D.
H.
(1982).
Efficacy
and skill
development
through
social
comparison
and
goal
setting.
(ERIC
Document
Reproduction
Service
No.
ED 222
279)
(Effect
size
=
1.31)
Effectiveness of
Providing
Instructional
Recommen-
dations to Teachers
Fuchs,
L.
S.,
Fuchs, D.,
Hamlett,
C.
L.,
Phillips,
N.
B.,
&
Bentz,
J.
(1994).
Classwide curriculum-
based measurement:
Helping general
edu-
cators meet the
challenge
of student diver-
sity.
Exceptional
Children,
60,
518-537.
(Effect
size
=
.51)
Category
2:
How Effective
Are
Peer-Assisted
Learning
Formats?
Fantuzzo,
J.
W., Davis,
G.
Y.,
&
Ginsburg,
M. D.
(1995).
Effects of
parent
involvement in iso-
lation
or in
combination with
peer
tutoring
on
student
self-concept
and
mathematics
achievement.
Journal
of
Educational
Psychol-
ogy,
87(2),
272-281.
(Effect
size
=
.47)
Fuchs,
L.
S.,
Fuchs,
D., Hamlett,
C.
L.,
Phillips,
N.
B.,
&
Bentz,
J.
(1994).
Classwide curriculum-
based
measurement:
Helping general
edu-
cators meet
the
challenge
of
student diver-
sity. Exceptional
Children,
60,
518-537.
(Effect
size
=
.51)
Fuchs,
L.
S.,
Fuchs, D.,
Karns, K.,
Hamlett,
C.
L.,
Katzaroff,
M.,
&
Dutka,
S.
(1997).
Effects
of
task-focused
goals
on
low-achieving
stu-
dents
with
and without
learning
disabilities.
American
Educational
Research
Journal,
34,
513-543.
(Effect
size
=
.40)
Fuchs,
L.
S., Fuchs,
D.,
Phillips,
N.
B.,
Hamlett,
C.
L.,
&
Karns,
K.
(1995).
Acquisition
and
transfer effects
of
classwide
peer-assisted
learning strategies
in
mathematics
for
stu-
dents
with
varying learning
histories.
School
Psychology
Review,
24,
604-620.
(Effect
size
=
.34)
Ginsburg-Block,
M.,
&
Fantuzzo,
J.
W.
(1997).
Reciprocal peer
tutoring:
An
analysis
of
"teacher" and
"student" interactions
as
a
function
of
training
and
experience.
School
Psychology Quarterly,
12(2),
134-149.
(Effect
size
=
.69)
Heller,
L.
R.,
&
Fantuzzo,
J.
W.
(1993).
Reciprocal
peer
tutoring
and
parent
partnership:
Does
parent
involvement
make a
difference?
School
Psychology
Review,
22,517-534.
(Effect
size
=
1.49)
Category
3: How
Effective Are
Explicit
Teacher-
Led and
Contextualized
Teacher-Facilitated
Approaches?
Effectiveness of
Explicit
Teacher-Led
Instruction
Cardelle-Elawar,
M.
(1992).
Effects of
teaching
metacognitive
skills to
students
with
low
mathematics
ability. Teaching
and Teacher
Education,
8(2),
109-121.
(Effect
size
=
.32)
Cardelle-Elawar,
M.
(1995).
Effects of
metacog-
nitive
instruction on low
achievers
in
math-
ematics
problems. Teaching
and Teacher
Edu-
cation,
11,
81-95.
(Effect
size
=
.61)
Moore,
L.
J.,
&
Carnine,
D. W.
(1989).
A
com-
parison
of two
approaches
to
teaching
ratio
and
proportions
to remedial
and
learning
disabled students:
Active
teaching
with
ei-
ther
basal or
empirically
validated curricu-
lum
design
material. Remedial and
Special
Education,
10(4),
28-37.
(Effect
size
=
1.1)
Woodward,
J.,
Baxter,
J.,
&
Robinson,
R.
(1999).
Rules and
reasons:
Decimal
instruction
for
academically
low
achieving
students. Learn-
ing
Disabilities Research and
Practice,
14,
15-
24.
(Effect
size
=
.59)
Effectiveness of
Contextualized
Teacher-Facilitated
Approach
Bottge,
B.,
&
Hasselbring,
T.
S.
(1993).
A
com-
parison
of
two
approaches
for
teaching
complex,
authentic
mathematics
problems
to
adolescents
in
remedial
math
classes. Ex-
ceptional
Children,
59,
556-566.
(Effect
size
=
.48)
Henderson,
R.
W.,
&
Landesman, E.
M.
(1995).
Effect
of
thematically integrated
mathemat-
ics
instruction on
students of
Mexican
de-
scent.
Journal
of
Educational
Research,
88,
290-300.
(Effect
size
=
.18)
Woodward,
J.,
&
Baxter,
J.
(1997a).
The effects of
an
innovative
approach
to
mathematics on
SEPTEMBER
2002
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All use subject to JSTOR Terms and Conditions

TEACHING
MATHEMATICS
71
academically
low-achieving
students
in
mainstreamed
settings.
Exceptional
Children,
63(3),
373-388.
(Effect
size
=
-.24)
Woodward,
J.,
Baxter,
J.,
&
Robinson,
R.
(1999).
Rules and
reasons: Decimal instruction for
academically
low-achieving
students.
Learn-
ing
Disabilities Research
and
Practice,
14,
15-
24.
(Effect
size
=
-.59)
Category
4:
Can
Parents Be
Used
to
Enhance
the
Math
Achievement
of
Their Children?
Fantuzzo,
J.
W.,
Davis,
G.
Y.,
&
Ginsburg,
M.
D.
(1995).
Effects
of
parent
involvement
in
iso-
lation
or
in
combination with
peer tutoring
on
student
self-concept
and
mathematics
achievement.
Journal
of
Educational
Psychol-
ogy,
87(2),
272-281.
(Effect
size
=
.44)
Heller,
L.
R.,
&
Fantuzzo,
J.
W.
(1993).
Reciprocal
peer
tutoring
and
parent
partnership:
Does
parent
involvement
make a
difference?
School
Psychology
Review,
22,517-534.
(Effect
size
=
.42)
Category
5: How
Effective
Is
Computer
Instruc-
tion
Using
Software
from
the 1980s?
Bass, G.,
Ries, R.,
&
Sharpe,
W.
(1986).
Teaching
basic skills
through
microcomputer
assisted
instruction.
Journal
of
Educational
Computing
Research,
2(2),
207-219.
(Effect
size
=
.15)
Moore,
B. M.
(1988).
Achievement
in
basic math
skills for
low-performing
students:
A
study
of
teachers'
affect and CAI.
Journal
of
Exper-
imental
Education,
57(1),
38-44.
(Effect
size
=
.21)
Note
This
research
was
supported
in
part by
a
grant
from
the
Texas
Education
Agency
and in
part
from
a contract
from
the
Research to
Practice
Division
of the
Office of
Special
Education Pro-
grams,
U.S.
Department
of
Education,
awarded to
American
Institutes for
Research. We
wish to ac-
knowledge
the
assistance
in
data
analysis
and
coding
provided
by
Janet
Otterstedt,
Jonathan
Flojo,
and
Joyce
Smith-Johnson.
Jennifer
Palmer
and
Janet
Otterstedt
provided
valuable assis-
tance
in
preparing
the
manuscript.
Finally,
we
would
like to
thank
Doug
Carnine
and Bob
Slavin
for
their
helpful
feedback on
an
earlier
draft of
this
article.
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72
THE
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JOURNAL
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