JANUARY 2000
FOCUS on
ExceDtional
children
Best Practices for Teaching Mathematics to
Secondary Students with Special Needs
Paula Maccini and Joseph Calvin Gagnon
Having all students achieve in mathematics is considered a national priority, as indi
cated in the Goals 2000: Educate America Act (PL 103227). Mathematics is the gate
keeper to a number of opportunities for occupational and educational advancement (Jetter,
1993). Further, more state and district requirements (e.g., Maryland, Virginia) are includ
ing highschool math assessments that students have to pass to receive diplomas.
Though math is vital to students' future, many students have difficulty with it.
According to the Third International Mathematics and Science Study (TIMSS) (Interna
tional Association for the Evaluation of Educational Achievement, 1996), American
eighthgrade students score significantly below the international average in math and out
perform only seven other nations (Bernstein, 1997). Further, American 12th graders have
an overall math average significantly below the international average.
Discrepancies in the nature of classroom activities across countries may help to
explain student performances. U.S. students spend 96% of their seatwork time practicing
routine procedures, whereas Japanese students engage in this type of exercise only 41% of
their seatwork time (Bernstein, 1997). In addition, Japanese students work on problems
that require the invention of new solutions, proofs, or creative procedures 44% of the time,
compared to U.S. students, who engage in similar activities less than 1%.
In the past, reforms such as back to the basics and individualized instructional pro
grams have attempted to address the issue of how to improve students' mathematics per
formance. More recently, the National Council of Teachers of Mathematics (NCTM, 1989,
1991, 1995) has outlined changes in curricular, assessment, and teaching practices that
emphasize complex math tasks requiring problem solving and mathematical reasoning
skills and deemphasize rote computation and memorization tasks.
Paula Maccini, PhD. And Joseph Calvin Gagnon, M.A. are affiliated with the University of Maryland,
College Park.
� Love Publishing Company, 2000
VOLUME 32
NUMBER 5
FOCUS ON EXCEPTIONAL CHILDREN
This new vision in math education is outlined in the Cur
riculum and Evaluation Standards for School Mathematics
(NCTM, 1989) and includes 1314 focus statements per
grade section (i.e., K4, 58, 912, evaluation). These state
ments are commonly referred to as "Standards" as they rep
resent the philosophy for teaching and assessing mathemat
ics. The three grade sections share four standards for
teaching math concepts or skills:
1. Math as problem solving (e.g., incorporating reallife
math applications and utilizing problemsolving
strategies)
2. Math as communication (i.e., writing, explaining,
discussing math ideas)
3. Math as reasoning (i.e., incorporating logical reason
ing into math areas)
4. Math connections (i.e., relating math concepts to
other math tasks and conceptual understanding, and
to other content areas/reallife situations).
FOCUS On
Exceptional
children
ISSN 001551 IX
FOCUS ON EXCEPTIONAL CHILDREN (USPS 203360) is pub
lished monthly except June, July, and August as a service to teachers,
special educators, curriculum specialists, administrators, and those con
cemed with the special education of exceptional children. This publica
tion is annotated and indexed by the ERIC Clearinghouse on Handi
capped and Gifted children for publication in the monthly Current
Index to Journals in Education (CIJE) and the quarterly index, Excep
tional Children Education Resources (ECER). The full text of Focus on
Exceptional Children is also available in the electronic versions of the
Education Index. It is also available in microfilm from Xerox University
Microfilms, Ann Arbor, MI. Subscription rates: Individual, $30 per
year; institutions, $40 per year. Copyright � 2000, Love Publishing
Company. All rights reserved. Reproduction in whole or part without
written permission is prohibited. Printed in the United States of Amer
ica. Periodicals postage is paid at Denver, Colorado. POSTMASTER:
Send address changes to:
Love Publishing Company
Executive and Editorial Office
P.O. Box 22353
Denver, Colorado 80222
Telephone (303) 2217333
Karen Harris
University of Maryland
Stanley F Love
Publisher
Thomas Skrtic
University of Kansas
James Shriner
University of Illinois
Thomas S. Love
Associate Editor
The five goals or outcomes that encompass the "spirit" of
the Standards include the ability for students to (NCTM,
1989):
* become better problem solvers (i.e., including math
tasks that require problemsolving and related strate
gies)
* learn to reason mathematically (i.e., having students
explain or defend their thinking)
* learn to value mathematics (i.e., understanding why
math is important to realworld situations)
* become more confident in their mathematical ability
(i.e., developing students' selfconfidence with math)
* learn to communicate mathematically (e.g., learning
to read, write, and discuss math ideas)
The suggestions that encompass these goals are guided by
the philosophy of constructivism, in which students' "con
struct" their own knowledge via active engagement in learn
ing activities and assimilating new knowledge with existing
information (Van De Walle, 1994). For the secondary math
classroom, this includes more emphasis on tasks that require
mathematical reasoning and problem solving.
The recommendations for change may be challenging for
teachers and secondary students with learning disabilities
(LD) and emotional/behavioral disorders (ED) because of:
(a) learner characteristics, (b) teacher practices and aware
ness, and (c) clarity and research issues. For example, many
students, particularly students with LD and ED, fall further
behind in their academic performance despite theefforts of
NCTM to reach "all students." These students are atrisk for
school failure as many experience low levels of math per
formance and have social and behavioral issues that impede
their learning. These two disability categories account for
more than 65% of students labeled in special education.
Given this high representation of students with ED and LD,
it is alarming that only 12% of these students take advanced
math courses (e.g., algebra, geometry, calculus, trigonome
try) (Wagner & Blackorby, 1996).
While the results of these two groups of students are not
reported separately within the TIMSS and NAEP, other
research provides data through which their levels of success
may be evaluated. For example, over onefourth of students
labeled LD have been identified specifically due to a dis
crepancy between math aptitude and performance (Brian,
Bay, LopezReyna, & Donahue, 1991). On average, adoles
cents with LD function 2.7 grade levels below their nonla
beled peers (Wagner, 1995). Specifically in math, secondary
teachers have noted that many of their students experience
difficulty in mathematics (McLeod & Armstrong, 1982).
Further, Cawley and Miller (1989) determined that adoles
cents with LD have difficulty with problem application and
JANUARY 2000
generally perform at the 5th grade level. Additionally, many
secondary students with LD experience difficulties with a
range of mathematics tasks, including basic skills
(Algozzine, O'Shea, Crews, & Stoddard, 1987) and higher
level skills/concepts and problem solving (Huntington,
1994; Hutchinson, 1993; Maccini & Hughes, in press; Mac
cini & Ruhl, in press).
Many secondary students with ED also share a common
set of learner characteristics that negatively affect their aca
demic success including: (a) a lack of persistence; (b) anxi
ety; and (c) problems with attention (Bos & Vaughn, 1994).
Although variation in the operational definition of these
characteristics limits generalization, their importance becomes
evident in light of the rigorous goals of the Standards (i.e.,
perseverance with mathematical tasks, confidence in using
math, and appreciation of mathematics application). It is not
surprising, then, that students with ED are typically 1.8
grade levels behind their nonlabeled peers, and of these stu
dents, more than half do not receive a high school diploma
(Wagner, 1995).
In addition to learner characteristics that influence suc
cessful performance in mathematics, Fitzmaurice (1980)
surveyed specialeducation teachers and determined that
adequate instruction in helping students with LD complete
openended problemsolving tasks may be a challenge.
Specifically, these respondents noted more competence with
their own computation skills than conceptual mathematical
tasks. Too, Heshusius (1991) noted that instructional prac
tices common in special education classrooms focus nar
rowly oncomputationaltasks rather than higherorder prob
lemsolving activities that are in line with the goals of the
Standards. Further, it has been suggested that many special
education teachers are unfamiliar with the national Stan
dards for teaching mathematics and receive little support for
increased awareness and training of these Standards
(Goodrich & Stern, 1995).
Researchers in the field of special education also have
equivocal views regarding the Standards and students with
disabilities. For example, Hofmeister (1993), Mercer, Har
ris, and Miller (1993), and Rivera (1993) have noted con
cerns with the "vagueness" of the Standards and the lack of
a clear focus for teaching, as well as a lack of empirical val
idation that addresses the unique characteristics of students
with disabilities (Chard & Kameenui, 1995). Nevertheless,
these researchers and many others in the field of special edu
cation (Mercer, Jordan, & Miller, 1996; Thornton, Langrall,
& Jones, 1997) recognize the importance of embedding
mathematics in problemsolving situations and expanding
the content traditionally underrepresented in math programs
(e.g., statistics, estimation).
The challenge is clear for general and special educators
who struggle to accommodate individual differences in light
of these Standards. This is a result, in part, to the national
trend toward inclusionary practices in which more students
with disabilities are being educated in general education
classes and are exposed to the curricula their nondisabled
peers receive. Given that school districts are redesigning
their math curricula to reflect the NCTM goals (Parmar &
Cawley, 1995), teachers must be informed of validated prac
tices and supports necessary for helping these students in
math relative to the goals of the Standards.
Further, as many students with LD and ED experience
difficulty with skills that relate to the goals of the Standards
(e.g., problem solving, reasoning), teachers must be aware
of instructional interventions to accommodate learner char
acteristics. Thus, the purpose of this review is threefold: (a)
to relate results of a survey that investigates the instructional
supports and adaptations that special and general education
teachers perceive as effective for teaching math to secondary
students with LD and ED in light of the Standards; (b) to
provide examples and support in light of practices that have
been found to be effective with this population, and (c) to
provide recommendations for future practice,
To address the first goal, a representative sample of sec
ondary general and special education teachers in Maryland
were contacted via a mail survey and answered openended
responses to determine their ideas relative to the goals of the
NCTM Standards and students with LD and ED (see Mac
cini & Gagnon, 2000; Gagnon & Maccini, 2000 for a full
description of the study). The openended responses were
coded for major themes and then analyzed relative to the fre
quency ofthe ideas perceived as effective practicesfor sec
ondary students with LD and ED. In addition, the authors
conducted a comprehensive review of the literature of vali
dated teaching practices and related background information
targeting math interventions for secondary students with LD
and ED. The following discussion covers these findings: (a)
advantages of the NCTM Standards; (b) specific instruc
tional approaches, (c) typical adaptations and provisions
based on students' IEP; and (d) teaching methods to improve
successful implementation of the Standards with students
who have LD and ED.
ADVANTAGES OF THE NCTM STANDARDS
General and special education teachers in the current
study identified three specific advantages of implementing
the NCTM standards with students labeled ED and LD.
According to these participants, the Standards:
* promote handson learning (i.e., use of manipulatives
to promote conceptual understanding versus rote
memorization)
* support "equal opportunity" between general and spe
cial education students (i.e., share the same curriculum)
FOCUS ON EXCEPTIONAL CHILDREN
* emphasize a more rigorous mathematics program en
compassing higherorder reasoning and critical think
ing skills.
Teachers also indicated that activities based on the goals of
the Standards address the connection between mathematics
and realworld applications and connections to other subject
areas specifically for students with LD. For example,
respondents noted the importance of incorporating handson
learning activities ("Many students with disabilities have not
internalized concepts at a concrete level, so that part of the
Standards is very helpful"), equal opportunities among gen
eral and special education students ("Implementation of
these Standards to LD students puts them on equal footing
with their peers, and it increases their math abilities and
knowledge"), a more rigorous math program ("Ensures that
they are given the opportunity of experiencing higher level
math beyond functional 'basics'"), and real world connec
tions ("Math concepts are connected and make sense to stu
dents who usually don't see a purpose for high math skills").
In the discussion that follows, the advantages will be
explored within the context of the remaining questions.
INSTRUCTIONAL APPROACHES AND METHODS
Teacher responses regarding specific instructional ap
proaches or methods used to implement the goals of the
NCTM Standards were categorized into 16 areas according
to similarity in content. Responses were evaluated for com
monalties within and between teacher categories (e.g., spe
cial education teacher, general education teachers) and stu
dent categories (e.g., LD, ED). The most prevalent
responses by general and special education teachers be
tween and across categories included: (a) effective instruc
tional techniques; (b) use of manipulatives; and (c) reallife
application.
Specifically, teachers indicated that use of manipulatives
is important for implementing the goals of the NCTM Stan
dards with students who have LD and ED (general educa
tion, 19%; special education, 12%). For example, teacher
responses included general statements ("I use a lot of hands
on activities using dice, pattern blocks, regular blocks, spin
ners, and other manipulatives"); and specific recommenda
tions for geometry or algebra, respectively ("Geometry
skills and concepts are taught through manipulatives. We
measure perimeter and area of our classroom, desk space,
etc. Measure off a garden area in front of the schooldeter
mine space the plants need to grow before planting"; Show
ing simple integer problems using different color chips on
overhead"). Some special education teachers noted the
importance of frequency for the use of manipulative ("A lot
of handson activities. I make sure that the lesson will be
able to reach the visual and auditory learners"; "Use of
handson materials as often as possible (i.e., use of actual
menus to determine sales tax, tip, trips to restaurants, etc.").
Special education teachers' most prevalent response
involved effective instruction techniques for students with
LD (18%) and ED (19%), whereas general education teach
ers indicated either manipulatives for students with LD
(19%) or cooperative grouping arrangements (16%).
Responses categorized under "effective instructional tech
niques" involved components found to be effective for
teaching students with mild disabilities, including teacher
directed instruction, pacing, and smallgroup instruction.
For example, one teacher noted, "I model everything con
stantly. I show over and over one correct way, but challenge
them to develop their own way to become problem solvers."
In addition, use of reallife applications was noted as the
second most prevalent instructional consideration (special
education, 15%) for students with ED and the fourth instruc
tional consideration for students with LD (special education,
11%) or ED (general education, 11%). Teachers noted the
importance of providing realworld application to help stu
dents generalize math skills and concepts, ("Mathematical
connectionsI have students bring in grocery ads from at
least two different stores. We then shop from the ads to pre
pare three meals for three days and the cost...").
In the following discussion, teacher responses are dis
cussed further and examples from the literature are high
lighted to support the three most prevalent recommendations
(effective instruction techniques, manipulatives, and real
life application) for implementing the goals of the NCTM
Standards. In addition, guidelines for the effective imple
mentation of the three recommendations are included.
Effective Instruction
Research on "effective instruction" supports instructional
and curricular methods that help students with disabilities
and others who are considered academically atrisk. The
studies involve specific curricular design variables and
teaching techniques (i.e., Direct Instruction) and general
teaching methodologies (i.e., direct instruction). Specifi
cally, Direct Instruction (DI) refers to a method of instruc
tion that addresses both curricular design (i.e., "what" to
teach) and teaching methodologies (i.e., "how" to teach) and
includes six components:
1. Explicit strategy instruction (i.e., teaching a strategy
that can be generalized to many examples or problems)
2. Mastery learning (i.e., having students reach a crite
rion before advancing to a new step)
3. Error corrections (i.e., immediate teacher correction
if a student errs via prompting or repeating the fact)
JANUARY 2000
4. Fading teacher involvement as the student assumes
more responsibility for learning the material
5. Wide range of examples and nonexamples to enhance
generalization
6. Cumulative reviews of previously learned skills.
Certain teaching practices are also recommended, such as
providing immediate feedback, using signals for individ
ual/group responses, and monitoring student performance
(Tarver, 1992). The general direct instruction method
includes similar teaching methodologies that can be applied
across curricular areas to promote student learning (Rosen
shine & Stevens, 1986):
1. Reviewing previously learned skills
2. Teaching content (teacher demonstration, guided
practice, and independent practice)
3. Providing student feedback and monitoring student
performance
4. Providing corrective feedback and review or reteach
ing as necessary
5. Independent student practice
6. Cumulative reviews, monthly and weekly.
Although instructional approaches advocated by the
goals of the Standards support a more studentfocused (i.e.,
teacher as facilitator) learning environment than the more
teacherdirected approaches, respondents in the current
study noted ways in which the two approaches complement
each other. For example, one teacher noted, "I model every
thing constantly. I show over and over one correct way, but
also challenge them to develop their own way to become
problem solvers." Similarly, general and special education
teachers noted the importance of teacher modeling and pac
ing for teaching students with LD and ED ("I always list the
objectives and NCTM Standards on the board. I always give
students the rationale for studying a specific concept"; "The
pace at which I teach is much slower. I use a variety of activ
ities to teach one concept. . . . I also reteach before moving
on if I see a problem with the majority when I have taught a
concept").
Teachers in the current study also supported the use of
several components of effective instruction ("Scaffolding
and modelingmaking a task easier for the student by mod
eling a similar problem first or providing the framework for
the students and having them fill in the missing details"; "DI
Instruction techniques: repeat and drill and pacing; teaching
until mastery; structure"; "Repetition of skills/directions").
General and special education teachers both stated the need
for explicit teachermodeling when instructing secondary
students with LD, ("Being very specificstepbystep in
structions"; "Giving many examples with the steps to help
them solve the problems on their own").
When asked to provide specific approaches for teaching
students with ED, teachers noted the importance of organiz
ing a structured learning environment to help these students
succeed, ("structured environment"; "maintain a structured
(to a point) environment at all times"; "Give a lot of praise
and quick correctional feedback"; Providing structure for
the course and each class is the only special thing I know to
do with these children"). Although a wealth of research val
idates components of effective instruction (Tarver, 1992),
two studies in particular exemplify the effectiveness of com
bining principles of effective instruction with goals of the
NCTM Standards.
In the first study, researchers (Kelly, Gersten, & Carnine,
1990) investigated the effects of a math curriculum involv
ing features of instructional design versus a basal curricu
lum for teaching fraction concepts to students with LD and
other academically atrisk secondary students within a gen
eral education setting. The instructional design curriculum
involved videodisc instruction for teaching fractional num
ber concepts (e.g., numerator, denominator, basic fractional
operations). The intervention included the following instruc
tional design variables:
1. Discriminating among problem types (i.e., adding or
multiplying fractions) and terms (i.e., numerator and
denominator)
2. Separating potentially confusing math tasks, such as
introducing the terms numerator and denominator
during different lessons
3. Incorporating a range of problems (e.g., fractions
greater than 1, unknowns on either side of the equa
tions).
Further, the stepbystep method for teaching fractions
entailed learning to translate pictorial displays of fractions
into numeric equations and then receiving immediate feed
back via the computer. Students then learned the rule for
adding fractions involving stepbystep directions ("When
you add or subtract fractions with the same denominator,
first copy the denominator then work on top") (Kelly, Ger
sten, & Carnine, 1990, p. 26). Conversely, students in the
control group received basal instruction that did not include
the instructional design variables. Rather, students learned
skills in isolation or one skill per lesson. Potentially confus
ing terms were not separated (e.g., students learned the
terms numerator and denominator during the same lesson),
and a narrow range of examples was presented (e.g., frac
tions less than 1, unknown on the right side of the equation
only). As a control measure, instruction in both groups
included methods of effective instruction, such as corrective
feedback, guided and independent practice, and classroom
management techniques.
FOCUS ON EXCEPTIONAL CHILDREN
The researchers determined that students in both groups
improved their performance as compared to pretest mea
sures. Students in the control group, however, made more
errors than students in the treatment group in discriminating
between the rules for addition and multiplication of frac
tions, and they confused the terms numerator and denomi
nator. Further, more than 80% of the students in the control
group had difficulty analyzing fractions greater than 1
(39.10%), whereas students in the treatment successfully
analyzed a range of example problems (93.80%). Kelley et
al. (1990) stated, "Specific error patterns can arise as a direct
result of aspects of the curriculum used. The range and
sequence of examples used in a curriculum can have a pow
erful effect on student performance. When a curriculum ade
quately conveys a particular concept or skill, transfer to
related tasks occurs more readily" (p. 28).
Moore and Carnine (1989) also studied the effect of cer
tain curriculum design variables versus basal instruction on
students' performance involving problemsolving tasks
requiring ratio and proportions. Twentynine students,
including six students with math disabilities, were assigned
randomly to either the treatment group, curriculum design
variables (ATCD), or the control group, basal design cur
riculum (ATBC). Treatment in the ATCD group included a
videodisc program involving the following curriculum
design features: (a) explicit strategy instruction (i.e., stu
dents were taught one strategy generalizable to various
problems in a stepbystep sequence that required mastery at
each step); and (b) exposure to a wide range of examples
and nonexamples to help students with discrimination (e.g.,
mixed sample of problems during cumulative reviews).
For example, to solve the problem, "Nine pencils cost 69
cents. How much do 21 pencils cost?" students were taught
to: (a) identify the correct units of measure, (b) write the
label identifying the units of measure per column, and (c)
align the number quantities with the similar units of measure
per column (Moore & Carnine, 1990, p. 35). For example:
pencils
_9
21
cents
= 69
Conversely, students in the ATBC group were exposed to
basal instruction that did not include these variables. Rather,
students learned different strategies for solving word prob
lems, including three procedures for determining the percent
of a number during one lesson. Further, students did not
learn the strategies in steps, and they practiced skills in iso
lation from other skills learned previously. Instruction in
both groups also included effective teaching components
(e.g., model/demonstration, monitoring student performance
via questioning, guided practice, independent practice, fast
pacing, mastery learning, immediate feedback) as a control
measure. As a result of the treatment, it was determined that
students in both groups improved their problemsolving per
formance as compared to pretest measures. Students in the
treatment group, however, outperformed control group stu
dents significantly on a posttest assessment and maintained
higher scores on unit tests.
The goals of the NCTM Standards, evident in these stud
ies, include:
1. Incorporating technologybased instruction into math
ematics lessons
2. Utilizing pictorial displays for concept development
3. Focusing on problemsolving tasks
4. Applying strategies to problem solving tasks.
Researchers in both studies, as well as what is known about
effective instruction, demonstrate the need for effective
math programs to also include the following curriculum
design variables when designing instruction for secondary
students with disabilities:
1. Teach explicit strategy instruction within a stepby
step approach, and require mastery at each step.
2. Include effective teaching components (e.g., teacher
modeling, guided practice, independent practice,
corrective and positive immediate feedback, monitor
student responses).
3. Use a wide range of examples and nonexamples to
promote generalization.
4. Separate potentially confusing terms/skills to reduce
student errors.
5. Promote discrimination practice such as cumulative
reviews involving new and previously mastered
material.
Manipulatives and Conceptual Knowledge
One avenue for student exploration is the use of manipu
lative aids in mathematics. Manipulatives are concrete ob
jects that students can physically arrange or group to repre
sent an array of mathematical relationships (e.g., coins, base
10 blocks, counters, toothpicks). General education and spe
cial education teachers alike indicated using specific manip
ulative materials when teaching secondary students with LD
and ED ("I use twocolored counters (onesided painted
lima beans) to express negative and positive integers and
model adding integers using zero pairs"; "I use handson
equations where each student has a balance mat and number
cubes to manipulate and demonstrate the basic rules for
solving equations").
Use of these concrete aids has been determined to be an
effective medium for students across grade and developmen
tal levels, including students with disabilities (Huntington,
JANUARY 2000
1994; Maccini & Hughes, in press; Maccini & Ruhl, in
press). For example, Huntington (1994) investigated the
effects of using manipulatives and teacherdirected instruc
tion on the algebra performance of three secondary students
with LD. Students were taught relational statements requir
ing problem representation and problem solution (e.g., "My
brother and I drove to Chicago. It was 900 miles. He drove
three times as far as I did. How far did I drive?"). In this sin
glesubject design study, students were taught via teacher
direction (i.e., teacher modeling, guided feedback, and inde
pendent practice) how to solve algebraic word problems
involving relational statements and the use of the Algebra
Lab Gear (Picciotto, 1993). The Algebra Lab Gear involved
colored plastic tiles to represent both numeric and variable
amounts during problem representation and solution.
Specifically, students advanced through three instruc
tional stagesconcrete, semiconcrete, and abstract (CSA)
instruction. At the concrete phase, students were taught to
manipulate the algebra tiles to represent relational state
ments. Once students reached a criterion of 100% over three
consecutive trials, they learned to represent relational state
ments via pictorial representations at the semiconcrete level.
After reaching criterion, students advanced to the abstract
level and learned to write mathematical equations and cal
culate the solutions. It was determined that students im
proved their problemsolving performance significantly as
compared to their baseline measures on both representing
and solving relational algebraic word problems.
Moreover, two students transferred concrete representa
tions to semiconcrete representations. Subjects not only met
criterion, but also generalized to other persons, settings, and
tasks. For example, Huntington (1994) noted, "Subject 3 vol
unteered that one participant volunteered she had not really
understood the meaning of the terms difference, twice, and
consecutive. She reported that it had not been enough when
teachers had her memorize definitions of the terms. She
stated that only after working with the Algebra Lab Gear did
she really understand what these words meant" (p. 113).
Similar to the previous study, use of the Algebra Lab Gear
was effective for teaching integer numbers and related word
problems to students with mild disabilities (Maccini &
Hughes, in press; Maccini & Ruhl, in press). Students were
taught to represent integer operations via a problemsolving
strategy, STAR (Maccini, 1998), which involved a general
problemsolving strategy for problem representation and
solution (Search the word problem, Represent the problem,
Answer the problem, Review the answer). Students advanced
through three levels of instruction: (a) concrete application
(i.e., using the algebra tiles to represent integer problems), (b)
semiconcrete application (drawing pictorial representations of
the mathematics problems), and (c) abstract application (writ
ing mathematical symbols to represent and solve problems).
For instance, during concrete instruction, students manip
ulated the tiles to represent word problems involving integer
numbers (see Figure 1). Participants in both studies im
proved their percent strategy use over the instructional
phases and significantly improved their problemsolving
skills involving integer numbers as compared to baseline
measures.
Though use of concrete aids was determined to be effec
tive with students with disabilities, simply using manipula
tives when teaching a math concept will not guarantee
acquisition of a concept. Marzola (1987) summarized
important guidelines when using math manipulatives with
students who have disabilities (see Table 2). These include
selecting manipulatives that are connected to the concept
and to students' developmental level, as well as incorporat
ing a variety of manipulatives, verbal explanations (student
and teacherled), and programming for instructional transi
tions from concrete to symbolic representation.
RealWorld Application and Problem Solving
In relation to teaching and activating conceptual knowl
edge, some authorities have suggested contextualizing infor
mation in a "real world" context. Embedding the problem
solving information within a realworld context helps
students activate their conceptual knowledge when pre
sented with a reallife problemsolving situation (Gagne,
Yekovich, & Yekovich, 1993) and improves student motiva
tion, participation, and generalization (Polloway & Patton,
1997). As Mercer, Jordan, and Miller (1994) stated, "If
mathematical content is to be relevant to learning, it is
imperative that it be presented in a real world context. For
example, if the instructional content fails to relate 6y + 2y +
6 = 48 to a pragmatic word problem, students are memoriz
ing meaningless procedures for obtaining answers" (p. 300).
Teacher quotes from the present study also reflect the impor
tance of contextualized learning ("I incorporate fun activi
ties such as timing a wave, weighing bananas, and counting
chips in a cookie to acquire data. Students are included in
groups and usually have successful experiences with others
as they do the activities").
Anchored instruction is one example of embedding prob
lemsolving situations in a reallife situation via interactive
videodisc instruction. For example, Bottge and Hasselbring
(1993) researched the effects of teaching contextualized
word problems (CP) via videodisc instruction versus teach
ing word problems via teacherdirected instruction (WP)
with secondary students with behavioral or learning diffi
culties from two remedial math courses. Prior to the inter
vention, students were assessed on their fraction computa
tion skills to determine patterns of error analysis. A 5day
remediation plan involving a videodisc program was devel
oped based on the analysis. Students were reassessed to
FOCUS ON EXCEPTIONAL CHILDREN
JANUARY 2000
0)
a2 a,
E 2, 5
c, .
E E * 5 
o 0) 0,.
a) a)g
QEE l
c9 c. Q
o
a, cv a, 2v c
0 U
s s � o w
I . I
'4 " nS
2 2 0
o g I a 2
(D
IL
20
co
a)
a.E
E
a)
0
0 )
2 c
0 c
V)
� ca
Is 
I 0)
c
d)>
CLLa
2V
SE l
CU 
8 s
a a
(0
c v
(Ud U)
< U
cQ
o V3
0 V
CU
~CL
a)v
,c C>
03 i
QIS
Eli
ExwE
Saa
"a,
Q(C
Scv
co E
E
a
0 E
a o
0 2
* E
(0D
0
a) Lo
E
0
a)
L 0
 .
a= "
co
ca
a('
as
SS
P "�
Ej i'
ET
"0
Er
c:C
ca
o 
So
CO .
c c
C: N
: 0
1'
0 o
Wa)
"8 
cvac
'V
4 :2
0~
0 3p
C (
cc
u1 cu
c .
CD
(cc
06
Un
c'E
j
0
4"
w
cc
cc
0)
Q.'
C
0
0
co
.0)o
u c
as
0
a
E
(U
determine their level of improvement and preparation for
the intervention. Students then were matched by test scores
and assigned to either the CP condition or the WP condition.
The CP condition included a videodisc program depicting a
reallife problemsolving task involving the application of
fraction skills and other math tasks to building a cage for a
pet.
The researchers determined that students in both groups
significantly improved their math performance on a contex
tualized posttest measure; however, students in the CP con
dition generalized to another videodisc problemsolving
task requiring different math skills than the videodisc prob
lemsolving task presented during instruction. As Bottge
and Hasselbring (1993) stated, "CP students who had suc
cessfully solved the transfer problem did not narrow their
focus only to fractions and measurement, but they seemed to
view the problem as a more global representation of how
problems naturally occur in reallife settings" (p. 565).
In addition to improving generalization to other math
tasks, providing reallife application improves students' on
task behaviors. For example, McWhirter and Bloom (1994)
examined the effects of a math curriculum involving a class
roombased business and assessed ontask behavior. Three
students with ED participated in the singlesubject study
within a selfcontained classroom. Students learned how to
run a business, making and selling wood baskets for holi
days. The math curriculum included application of basic
math facts for determining overall costs, time cards,
receipts, and payroll. As compared to baseline measures, the
researchers found that students' overall attentiontotask
improved when the intervention was in effect.
ADAPTATIONS AND PROVISIONS BASED ON
INDIVIDUALIZED EDUCATION PLANS
With the inception of the Education for All Handicapped
Children Act of 1975, the importance of individualized edu
cational programming for students with disabilities was
realized. Since 1975, the prevalent view relates teacher
adaptations to student academic success (Corno & Snow,
1986). The nature of instructional adaptations involves
teacher judgment, via formal and informal assessment,
resulting in variations and adjustments in teaching strategies
and goals (Fuchs, Fuchs, Phillips, & Simmons, 1993). The
continued necessity of adaptations for students with LD and
ED in light of the current popularity of the NCTM Standards
cannot be overemphasized. A common concern with the
Standards is the lack of attention given to students with spe
cial and diverse needs (Hofmeister, 1993; Hutchinson, 1993;
Mercer, Harris, & Miller, 1993). Given the lack of specific
guidelines for students with special needs, it is important
to identify ways in which teachers currently adapt the
Standards within the context of IEPs for students with LD
and ED.
Teacher responses to the question addressing typical
adaptations and provisions based on IEPs for students with
LD and ED were grouped into 18 categories. The most fre
quent adaptations included: (a) calculator use, (b) assign
ment modification, (c) behavior management, and (d) extra
time on assignments and tests (see Table 1). Overall, the
most popular adaptation was the utilization of calculators
for students with LD (27% of special educators, 15% of gen
eral educators).
Special educators noted calculators as a tool for students
with ED more often than general educators. One possible
explanation could be the large percentage of general educa
tors who listed behavior management (22%) as a useful
adaptation for students with ED, whereas, fewer special edu
cators noted behavior management (10%). Factors affecting
this difference include: (a) the number of students with ED
served in the general education setting without special edu
cation support, (b) the relatively large number of students
per class in the general education setting, as opposed to
resource and selfcontained settings, and (c) the percentage
of special educators whose responsibilities are confined to
exclusionary settings (e.g., selfcontained, full time
resource) (Gagnon & Maccini, 2000).
The second most common adaptation recorded for stu
dents with LD was modification of assignments (11% of
special educators, 15% of general educators). Similar to use
of calculators, general educators identified assignment mod
ification less often (7%) for students with ED, whereas spe
cial educators mentioned it 15% of the time.
Another noticeable trend was the notation of extra time
for assignments and tests as a strategy that general educators
(12% for students with LD, 13% for students with ED) and
special educators (9% and 8%, respectively) use.
The following discussion addresses the four most com
mon responses to adaptations based on student IEP goals by
general and special educators as they apply the NCTM
Standards to the individual needs of students with LD and
ED. Examples of specific teacher statements and an evalua
tion of these statements through a consideration of current
literature will follow. In addition, general guidelines for the
effective use of the adaptations are included.
Calculators
The use of calculators was the most prevalent adaptation
based on the IEPs of students with LD and ED. Although
many of the respondents simply listed "calculators," others
developed their explanation to incorporate ways in which
this technology was used. Teacher responses involved two
primary categories consistent with Etlinger and Ogletree's
(1982) view that calculator use has two main functions in
FOCUS ON EXCEPTIONAL CHILDREN
Table 1
Percent of Teacher Perceptions
Advantages of Implementing the Standards
Percent of General Education Percent of Special Education
Category Teachers' Responses Teachers' Responses
LD 1) Positive effect on student learning and positive
student reaction (19%)
2) Promotes conceptual learning via handson
activities (16%)
3) Promotes a more rigorous math program (higher
order thinking, critical thinking skills) (14%)
4) Promotes connections between math and the
realworld (13%)
ED 1) Promotes connections between math and the
realworld (17%)
2) Positive effect on student learning and positive
student reaction (15%)
3) Promotes conceptual learning and handson
activities (12%)
1) Preparation for school/state requirements and
life beyond school (16%)
2) Promotes conceptual learning and handson
activities (11 %)
3) Equal opportunity among general and special
education students (11%)
1) Preparation for school/state requirements and
life beyond school (26%)
2) Promotes a more rigorous math program
(higherorder thinking, critical thinking skills)
(13%)
3) Promotes conceptual learning/handson
activities (11 %)
4) Equal opportunity among general and special
education students (11%)
Specific Instructional Approaches/Methods Teachers use to Implement the Goals of the NCTM Standards:
Manipulatives (19%)
Cooperative Groups (17%)
Effective Instruction (16%)
Cooperative Groups (16%)
Effective Instruction (15%)
Manipulatives (13%)
1) Effective Instruction (18%)
2) Calculators/Computers (15%)
3) Manipulatives (12%)
4) Reallife Application (11%)
1) Effective Instruction
2) Reallife Application (15%)
3) Calculators/Computers (13%)
Typical Adaptations and Provisions Based on Students' IEP:
LD 1) Calculators (15%)
2) Assignment Modification (15%)
3) Extra Time for Tests and Activities (12%)
ED 1) Behavior Management (22%)
2) Extra Time for Tests and Activities (13%)
3) Grouping Practices (13%)
1) Calculators (27%)
2) Assignment Modification (11%)
3) Strategies or Charts (11%)
1) Calculators (23%)
2) Assignment Modification (15%)
3) Testing (10%)
4) Behavior Management (10%)
Specific Teaching Methods to Improve Successful Implementation of the Goals of the NCTM Standards:
LD 1) Effective Instruction (21%)
2) Cooperative Groups (11 %)
ED 1) Behavior Management Strategies (27%)
2) Cooperative Groups (14%)
3) Effective Instruction (10%)
1) Effective Instruction (21%)
2) Manipulatives (18%)
3) Cooperative Groups (9%)
1) Effective Instruction (18%)
2) Strategies (15%)
3) Manipulatives (13%)
LD 1)
2)
3)
ED 1)
2)
3)
JANUARY 2000
Table 2
Guidelines for Using Manipulatives with Students with Disabilities
Suggestions for Addressing Guidelines
Select manipulatives that are connected to the
concept and to students' developmental level
Incorporate a variety of manipulatives for
concept exploration and attainment
Provide verbal explanations and
questions with demonstrations
Provide opportunities for student
interaction and explanations
Encourage the use of manipulatives
and strategies across settings
Program for transitioning from
concrete to symbolic representation
* Choose manipulatives that clearly illustrate the concept during initial
concept exploration to help students connect the objects to the
concept and then advance to more abstract representations.
* Illustrate the concept via multiple representations to aid student
generalization. For example, after students are familiar with adding
integer numbers via colorcoded chips, introduce the concept using
algebra tiles.
* Incorporate teacherdirected verbal explanations while presenting a
concept via manipulatives. Also include selfquestions and verbal
explanations to improve students' selfmonitoring strategies.
* Encourage student participation (opportunities to use
manipulatives).
* Monitor student performance as students manipulate objects,
explain, and write down their problemsolving steps.
Provide corrective and positive feedback as needed.
* Organize a box of materials (e.g., strategy or cue cards, calculators,
algebra tiles) that students can use across settings, such as
resource and general math class).
* Incorporate a graduated instructional sequence when teaching a
concept: (1) concrete application (i.e., students manipulate objects,
such as algebra tiles, to illustrate positive and negative quantities);
(2) semiconcrete application (i.e., use pictorial displays to represent
the concept, such as drawing pictures of the algebra tiles to
illustrate integer numbers); and (3) abstract application
(i.e., incorporate numerical representations, such as 3 + 4 =).
* Use "portable packets" involving a box or carton of manipulative
materials (e.g., cue cards of strategy steps or examples of problems
to serve as models, calculators, tiles) that students can carry to
different classes and use as needed.
* Incorporate teacherdirected instruction: (1) model examples) of the
target concept and thinkaloud while demonstrating the concept via
manipulatives. Write the numerical notation for each example
presented; (2) monitor student performance as students think aloud
while solving a problems) via manipulatives, and have them record
numeric notations. Students can use the manipulatives to check
their problemsolving steps; (3) fade teacherassistance gradually
until students are able to independently think aloud each step of the
process while writing each step and checking for problem solution
using manipulatives.
Guidelines
FOCUS ON EXCEPTIONAL CHILDREN
the classroom. First, the "practical" function refers to the
use of calculators to complete tedious calculations, save
time, increase student motivation, and decrease math anxi
ety. Second, the "pedagogical" function relates to similari
ties between calculators, textbooks, and manipulatives in
that each enhances students' understanding and competence
in mathematics. These classifications are consistent with the
five primary functions of calculators as stated by the
NCTM.
Within the practical classification, NCTM (1986) identi
fied the use of calculators to:
* perform tedious computations that arise when work
ing with real data in problemsolving situations
* concentrate on the problemsolving process rather
than calculations associated with problems
* gain access to mathematics beyond their level of com
putational skill.
In addition, the pedagogical function coincides with two
other uses identified by NCTM (1986):
* to explore, develop, and reinforce concepts including
estimation, computation, approximation, and proper
ties
* to experiment with math ideas and discover patterns.
Clearly, these general headings are not completely distinct.
However, they do provide a useful framework with which to
consider the teachers' responses to adaptations required
within student IEPs.
The most common statement by teachers was simply,
"Use of calculators." The remainder of teacher responses
varied across either pedagogical or practical domains with
pedagogical uses employed more often. Further, statements
on the need for training students to use calculators were
common, as was calculator use "Under all circumstances."
Teachers considered the affective benefit as one key practi
cal use: "Daily use of calculators] to eliminate arithmetic
phobia." Use of calculators to increase motivation also has
been noted by researchers (Deshler, Ellis, & Lenz, 1996).
Another practical use was saving time through "Use of the
calculator to correct work."
Pedagogical functions of calculators centered on their
use as an aid to solve problems, ("[Students] use calculators
only after attempting to solve problems" and "[Student] use
of calculators to help solve problems, but he/she still must
show an understanding by listing their steps"). In addition,
calculators were perceived as a tool to facilitate learning
("Use of calculators for students who have not acquired
basic math facts"; "I allow the use of calculators, but also
encourage mental math and paper/pencil math"). Another
teacher response, "[Student] use of a calculator or graphing
calculator for Algebra II," is consistent with NCTM (1998)
and researchers' (Milou, Gambler, & Moyer, 1997; Demana
& Waits, 1990) beliefs in the use of calculators to enhance
learning by helping students to visualize connections
between symbolic and graphic solutions.
The other important category of teacher responses is
related to teaching students to use calculators: ("Generally if
students have a learning disability in mathematics opera
tions, they are given the opportunity to be trained on a cal
culator for the Maryland Functional Math Test and MSPAP
(Maryland School Performance Assessment Program)"; "I
do extensive work with students on how to use the calcula
tor. I use an overhead calculator to assist with VAKT [i.e.,
visual, auditory, kinesthetic, and tactile learning")]. Simi
larly, Salend and Hofstetter (1996) asserted the importance
of training students to use calculators and the effectiveness
of overhead projectors for teaching this skill. The authors
described the significance of locating and describing the
function of each key to students, as well as providing exam
ples of calculator use. Further, Salend and Hofstetter recom
mended that students be provided opportunities to practice
calculations, including estimation skills and reviewing
answers obtained through calculator use.
In addition to training students to use calculators effec
tively, Advani (1972) determined the positive effects of
calculator usage on the achievement and attitudes of ado
lescents with LD and ED. Students were taught to use cal
culators to: (a) check answers to computational problems,
(b) solve problemsolving tasks, and (c) check grocery
receipts. The researchers used a pre and posttest design
and a significant positive effect was observed in student
achievement. In addition, the attitude survey indicated a
substantial increase in student interest and attitude toward
math. Though some concerns exist within the study
related to replicability of procedures, the positive trends
provide initial support for the use of calculators in the sec
ondary classroom with students identified as having LD
and ED.
Based on teacher responses, the literature, and NCTM
position statements (1998), the following recommendations
for teachers are noted:
1. Model calculator application.
2. Use calculators in computation, problem solving,
concept development, pattern recognition, data analy
sis, and graphing.
3. Integrate calculator use in assessment and evaluation.
4. Remain current with stateoftheart technology.
5. Explore and develop new ways to use calculators to
support instruction and assessment.
JANUARY 2000
Behavior Management
Considering the very nature of the ED label, it is logical
that many behavioral management adaptations are available
for these students. Interestingly, 22% of generaleducation
teachers and 10% of special educators noted behavior man
agement as an adaptation based on student IEPs. A review of
preliminary quantitative data from teacher responses
(Gagnon & Maccini, 2000) reveals three issues that account
for the differing importance of behavior management
between general and special education teachers:
1. More than 50% of general educators and 33% of spe
cial educators acknowledged that they had students
labeled ED in their class or on their caseload who
were included in the general education environment
with no special education supports.
2. Of the general educators, 43% reported having
2635 students per class and 25% noted 36 or more
students per class.
3. Of the special educators, 55% noted that they teach
either in a selfcontained classroom or a fulltime
resource room, typically situations that have a much
lower studentteacher ratio.
Given that a relatively large percentage of general education
teachers reported having students with ED who are not
receiving special education services and the large general
education class sizes, it is reasonable to conclude that the
combination of these factors may lead to more behavioral
issues for general educators and the subsequent necessity of
using behavioral management techniques.
Timeout
Within the category of behavior management, the use of
timeout, seating accommodations, and specific behavior
plans (e.g., token economy, behavioral contracting) were the
three central adaptations based on student IEPs. Timeout
from reinforcement, the most frequent of these responses, is
defined as the contingent withdrawal of reinforcing stimuli
that maintain the targeted inappropriate behavior (Johnston,
1972). Researchers (Ruhl, 1985; Gast & Nelson, 1977) have
affirmed the effectiveness of timeout procedures for reduc
ing inappropriate behavior of students across a range of ages
and settings.
The popularity of timeout is evident from a review of
respondents' comments. Teachers noted the usefulness of
"Letting them studentsl take a timeout when needed" and,
"When conflicts arise, students are not cornered or pres
sured. Students are allowed selfesteem and left alone."
Another teacher wrote, "Procedures are in place when a stu
dent needs to leave suddenly to stay in control or get back
into control." The importance of precise standards and pro
cedures for the effective implementation of timeout has been
well documented (Alberto & Troutman, 1999; Gast & Nel
son, 1977; Nelson & Rutherford, 1983). In addition, Emmer
(1981) stated that "effective management classes" have rules
and procedures with clearly established expectations.
An example of effective use of timeout is a study by
Webster (1976). The participant was a 13yearold male in
sixthgrade public school. He had difficulties with violent
actingout and a history of psychiatric hospitalization. Phys
ical aggression, operationalized as hitting with hands or
objects, kicking, pushing, and biting, were the criterion for
placement in timeout. A steady decline in aggressive behav
ior was observed during the intervention which used time
out, and the behavior was extinguished after 7 weeks. Dur
ing an 8week follow up, no incidences transpired that
required the use of timeout.
In addition to decreasing aggressive behavior, timeout
provides an opportunity for students and teachers to disen
gage from power struggles. The likelihood of these instances
and student defiance of authority increase with adolescents
(Mercer & Mercer, 1998) and are common with many stu
dents with ED. Effective use of timeout, however, requires
that the classroom be sufficiently positive and reinforcing
for the student. If the student views the classroom environ
ment negatively, timeout may serve as a reinforcer (Plum
mer, Baer, & LeBlanc, 1977).
Sprick (1985) identified seven criterion for establishing
effective use of timeout within the classroom:
1. Set up a place to isolate a student within the class
room.
2. Specify any behavior that will result in the class
room isolation.
3. Determine the length of time the student will stay in
isolation once he/she is in control.
4. Establish procedures with the student prior to
implementation.
5. Discuss the procedures to use if the student refuses
to go to the isolation area.
6. Reinforce the student for appropriate behavior
[although not while in timeout].
7. Evaluate the effectiveness of your consequence and
reinforcement procedures. (pp. 107109)
In addition to these guidelines, current laws and ethical con
siderations should be observed.
Teachers also might consider arranging for a quiet place
for the student to take a timeout with another teacher,
librarian, or other school personnel (British Columbia Min
istry of Education, 1996). Similarly, specific criterion and an
agreedupon time frame are essential. In addition, if students
begin to misuse this option, a provision can be added for the
student to make up work that is missed.
FOCUS ON EXCEPTIONAL CHILDREN
Preferential Seating
Teachers identified use of preferential seating as another
adaptation based on students' IEPs. Classroom seating has
been shown to be an important variable in the behavior of
students with ED (Wheldall & Lam, 1987). Further, Walker
(1981) asserted the physical organization of a class can
enhance or inhibit teacher attempts to implement each stu
dent's IEP Respondents to the current study wrote general
comments ("seating arrangement"), as well as identifying
the specific use of isolated seating ("I may have them sit iso
lated from the rest of the group so they can focus," "sitting
in the front of the class," "I try to place those students in the
front with me to keep [them] focused").
Placing students in the front of the class may serve dual
purposes: (a) maintaining proximity between the teacher
and students with ED, and (b) facilitating attention to task.
Proximity between teacher and student, defined as a dis
tance of 3 feet or less (Etscheidt, Stainback, & Stainback,
1984; Van Houten, Nau, MacKenzieKeating, Sameoto, &
Colavecchia, 1982), may result in the discontinuation of
inappropriate behavior and aid in the reduction of student
anxiety and frustration (Walker & Shea, 1999). A study by
Fifer (1986) exemplifies the positive effects of teacher prox
imity in a secondary mathematics classroom. Reduction of
inappropriate behavior throughout the class was observed in
classrooms where teachers circulated around the room. In
contrast, a high incidence of inappropriate behaviors was
observed when teachers circulated the least (Fifer, 1986).
Further, Weinstein (1979) determined that students seated in
the front of the room had a more positive attitude and par
ticipated more in class activities than students seated in the
back of the room.
One form of isolated seating is the use of study carrels.
These may function as private work areas and serve as a
physical boundary, reducing stimulation for easily dis
tracted students (Gallagher, 1995). In addition, this
approach to seating may provide greater personal space that
aggressive adolescents may require (Newman & Pollack,
1973).
Although decisions relating to the physical arrangement
of the classroom and student seating depend upon the
instructional goals and extent of interaction desired, general
recommendations do exist. Alberto and Troutman (1999)
synthesized information from researchers (Gallagher, 1995;
Haring & Phillips, 1962: Hewett & Taylor, 1980; Stephens,
Hartman, & Lucas, 1978), and the following may be applic
able to secondary mathematics classes:
1. Provision for easy teacher observation of all students
2. Sufficient physical separation of students to mini
mize inappropriate behaviors
3. Availability of study carrels. (p. 462)
Behavioral Plans
The other significant behavioral adaptation noted by
respondents was the use of positively focused behavioral
plans. Teachers cited the use of "Behavior check sheets and
a token economy system" and, "Behavior contracts are
used." The use of a token system accompanied by teacher
praise has been shown to be effective (Drege & Beare,
1991), and a procedure utilized commonly by effective
teachers (Stallings & Kaskowitz, 1974).
Systems that specify behavioral criteria and secondary
reinforcement (e.g., praise, tokens, or points that can be
exchanged for another reinforcer) (Alberto & Troutman,
1999) can be used to sustain appropriate student behavior,
guide students from materialistic to social reinforcers, and
provide a practical method of reinforcing appropriate behav
iors. An important consideration for the reinforcement of
students within the secondary math class is recognition for
completion of correct mathematical processes regardless of
outcome (Lock, 1996). This approach may promote student
effort and concentration on the underlying mathematical
concepts and procedures and is a goal of the NCTM Stan
dards. In addition, Blackham and Silberman (1975) identi
fied several guidelines that facilitate the effective imple
mentation of a token system:
1. The target behaviors that earn tokens should be spec
ified clearly.
2. The reinforcers that the tokens are exchanged for
must be appealing and available only within the
token system.
3. The number of tokens earned must match the effort
required for performing the target behavior.
4. The teacher should keep a record of the number of
tokens each student and the group earn. This could
provide an additional incentive.
5. If response cost (token fines) is used, the conditions
under which the tokens are earned and lost must be
specified clearly.
6. A scheduled token exchange at the end of the day
usually works best.
7. The system should be designed to encourage self
competition rather than competition between stu
dents.
8. Gradually withdraw material reinforcers and stress
activities and events.
9. The system should be simple, functional, and not dis
tracting. (pp. 145146)
In addition, teachers should plan for eventual fading of
the program through increased expectations, increased cost
of reinforcers, or increased time between token reinforce
ment.
JANUARY 2000
Assignment Modification
In addition to the use of calculators and behavior man
agement strategies, teachers identified assignment modifica
tion as an important adaptation based on students' IEPs.
Assignment modification is an essential component of math
instruction for students with special needs (Salend, 1994).
An increasing number of these students are being served in
the general education environment (Fuchs, Fuchs, &
Bishop, 1992), and many are functioning below grade level
in mathematics. As student IEPs commonly include assign
ment modifications, teachers are challenged to integrate
modifications with instructional practices relative to the
goals of the NCTM Standards.
For example, in grades 58, the NCTM Standards (1998)
state the necessity of using reading and listening to interpret
and evaluate mathematical ideas. Modifications (e.g., read
ing for students, using visuals or manipulatives) might be
necessary for students whose disabilities affect these skills.
The teachers in the current study noted three assignment
modifications most commonly: (a) adjusted workload, (b)
reading information for students, and (c) providing written
information/notes or allowing verbal answers.
Adjusted Workload
Respondents to the present survey noted adjusted work
load as a common adaptation for students with LD and ED.
Teachers mentioned "adjusted workload," "small number of
problems to solve," and "shorter assignments." Salend
(1990) supports the adaptation of assignments through a
decrease in the number of problems assigned and suggests
(Bley & Thornton, 1981):
1. Reviewing previously mastered skills
2. Dividing a task or worksheet into smaller tasks or
sections
3. Using a similar assignment/worksheet format (e.g.,
standardized spacing, color coding, use of boxes and
circles for cueing, listing of procedural steps).
Relatedly, Dunlap et al., (1993) determined that inappropri
ate student behavior decreases when students are presented
with a sequence of shortened assignments versus one long
assignment.
Reading
The teacher's reading of word problems to students was
another common adaptation. This type of assignment modi
fication is not surprising given learner characteristics of
students with ED and LD. Specifically, secondary students
with ED function an average 3.5 grade equivalent units
behind their nonlabeled peers in reading (Coutinho, 1986),
and students with LD function an average of 3.1 grade levels
below nonlabeled students (Wagner, 1995). Given the
emphasis on problem solving in the high school math cur
riculum (Dossey, Mullis, Lindquist, and Chambers, 1988) in
light of the Standards (NCTM, 1989) and the widespread
use of textbooks that present concepts in an incomprehensi
ble manner (Elliot & Wiles, 1980), these students have dif
ficulties reading and solving mathematical word problems.
To support the connection between the two subjects, Cul
lyer (1988) devised a comparison of structural similarities.
Specifically, student success in reading requires proficiency
with basic sight words and meaning of vocabulary words.
Similarly, success in math necessitates a command of content
specific sight words (e.g., sum, difference, product) and an
understanding of the meaning of these words. Cullyer (1988)
also notes the need for a specialized vocabulary for both
subjects. This would include abbreviations for reading and
comparable abbreviations in math related to measurement.
Respondents to the current study also noted the relation
ship between reading and math as evidenced by their com
ments: "I have to read to them the application problems
because they get confused and frustrated easily and lose
their concentration;" "Reading of word problems, directions
etc., on all tests and assignments."
In an attempt to gauge the readability of a commonly
used eighthgrade math text, Elliot and Wiles (1980)
assessed 91 certified middle/junior high mathematics teach
ers. The majority held advanced degrees with a mean of 12
years teaching experience. Researchers determined that
more than 25% of the teachers were unable to comprehend
a well known mathematical concept (percent) with relative
ease within the text. The authors noted that these results
could not be accounted for by differences in the participants'
sex, degree earned, or experience. The issue of math text
readability, then, compounds the math difficulties for stu
dents with LD and ED.
Writing
Another challenge for teachers as they attempt to modify
assignments for secondary math students with LD and ED
is the difficulty that these students experience with writing
and notetaking. Given the large amount of information pre
sented to students in a lecture format and through textbook
explanations (Lazarus, 1993) beginning in the middle
school years, proficiency with notetaking is essential for
students' success (Robinson, Braxdale, & Colson, 1988). A
review of available studies (Suritsky & Hughes, 1991),
however, indicates that students often neglect the essential
ideas of a lecture within their notes. In addition, the read
ability of math texts could compound the difficulties with
extracting key ideas.
Although respondents to the current study did not
delineate the exact writing difficulties that affect student
FOCUS ON EXCEPTIONAL CHILDREN
performance in the mathematics class, research (Graham,
Harris, MacArthur, & Schwartz, 1991) on students with LD
includes factors supporting assignment modification for stu
dents with special needs in this academic environment.
Three factors are especially relevant and may interfere with
students' text production:
1. The physical demands of writing
2. Utilization of ineffective strategies and processes
3. An extensive focus on lowerlevel skills (e.g.,
spelling, punctuation).
Given these known factors, it is comprehensible that teach
ers identified the importance of modifying written tasks.
Adaptations that respondents of the current study made
include: "[I] have students verbally recite their answers to a
question while I document word for word their response.
This helps them express themselves in an openended ques
tion"; "Expectations in the written goal are lowered some
what"; "I sometimes allow oral instead of written answers."
In addition to verbal answers, Polloway and Patton
(1993) suggested accepting other modes of response based
on student individual needs and IEP's, such as:
1. Use of physical materials (manipulatives)
2. Physical identification of an object or answer
3. Written response
Respondents to the current study also stated their use of
teachermade notes in support of students who have diffi
culties with written expression and notetaking: "Typed
notes/notesheets" and "Teacher made notecards." Researchers
Mohr (1995) and Meese (1994) support providing notes to
students and reducing the amount of copying expected.
Although the exact nature of the notes that the respondents
supply to students is not specifically known, guided note
taking has been shown to aid in the accuracy of student
notes and retention of material (Heward, 1994). In this pro
cedure, students are given a handout in outline form, on
which they provide the missing key words based on the
teacher's lecture (Olson & Platt, 2000). Despite the need for
more research involving guided notetaking for secondary
mathematics classrooms with students labeled LD and ED,
it is a potentially effective adaptation that teachers currently
utilize.
Within the topic of assignment modification, discussion
has centered on adjusted workload, reading to students,
allowing verbal responses, and providing notes to students.
These modifications are among those that Meese (1994)
noted as effective for students with disabilities:
1. Divide assignments into chunks and have timelines
for each chunk.
2. Extend time for completing assignments.
3. Encourage the use of calculators and computers.
4. Allow groups to complete some written assignments.
5. Reduce the amount of copying needed throughout
the assignment (e.g., from board, notetaking).
6. Require students to paraphrase an assignment's
tasks. (pp.350351)
In addition to agreement with points I and 2 above, Salend
(1994) includes reduction in the number of problems
assigned to students as another effective modification for
students with special needs.
Increased Time for Activities and Tests
The final adaptation based on student IEPs that the
teacher respondents noted was increased time for students
with LD and ED to complete assignments and tests. Because
students with ED tend to become anxious, stressed, and
unable to focus within strict time constraints (Gallagher,
1995) and students with LD often require a great deal of
time and energy to complete higherlevel math problems
(Deshler, Ellis, & Lenz, 1996), it is understandable that
teachers would cite this adaptation. Responses included spe
cific references toward assessment ("Allow extra time on
tests") and general statements related to assignments ("My
LD students always need extra time to complete work";
"Extended time with supervision"; "Unlimited time").
Researchers (Chalmers, 1991; Salend, 1994; Mohr, 1995)
also acknowledge the importance of (a) allowing an increase
in time to complete assignments for students with special
needs and (b) providing additional opportunities for practic
ing new skills (Peacock Hill Working Group, 1991).
In addition to respondents in the current study, Fuchs,
Fuchs, Phillips, and Simmons, (1993) determined that
teachers, when asked to generate their own adaptations for
fictitious students, favor an increase in practice time.
Although the Fuchs et al. (1993) study focused on teachers
at both the elementary and middle levels (grades 16), it
provides an example of the value that teachers place on
increased time as an adaptation. Similarly, respondents and
researchers recognize the potential benefits of adapting time
allowed for students during assessments. The purposes of
assessment within the context of the NCTM Standards are
noted by Rivera, Taylor, and Bryant (19941995):
1. Determining mathematical achievement in compari
son to one's peer group
2. Gathering diagnostic information
3. Obtaining information to provide instructional feed
back and grading
4. Evaluating program effectiveness. (p. 144)
Clearly, the information garnered through assessment is
critical. In light of research (Alley, Deshler, & Warner,
JANUARY 2000
1979) indicating that as many as 85% of students with LD
have difficulties taking tests, adaptations become crucial in
obtaining the most accurate picture possible of students with
special needs. An increase in time to complete assessments
is one such provision that researchers (Deshler, Ellis, &
Lenz, 1996) and teacher respondents support.
Although an increase in time allotted to students to com
plete assignments and tests supports individualization of
instruction, an increase in time alone may not provide suffi
cient adaptation for students with LD and ED. To effectively
utilize additional time on a topic, several factors have to be
considered. As described previously and synthesized by
Deshler, Ellis, and Lenz (1996), five variables are essential
for the modification of math curricula leading to enhanced
skill acquisition and procedural competence:
1. Provide many examples and nonexamples.
2. Provide practice in discrimination.
3. Provide explicit instruction.
4. Separate confusing elements.
5. Consider parsimony. (p. 330)
Consideration of these five variables allows teachers to
focus productively and utilize the increase in time provided
to students. For example, Edgarton (1992) found that teach
ers who implemented the instructional practices set forth in
the NCTM Standards required less student seatwork and
completion of routine tasks. The authors noted a simultane
ous increase in systematically applying previously learned
material within newly presented concepts and skills to aid
generalization.
TEACHING METHODS TO IMPROVE FUTURE
NCTM STANDARDS IMPLEMENTATION
In addition to specific recommendations for adaptations
necessary for implementing the goals of the NCTM Stan
dards and student IEPs, teachers were queried about specific
teaching methods they perceive as necessary for improving
successful implementation of the Standards with students
who have LD and ED. Whereas the intent of the question
was to determine practices that teachers perceive as neces
sary for future implementation of the Standards. rather than
recommendations for specific instructional approaches or
methods they use currently to implement goals of the
NCTM Standards, teacher responses were somewhat similar
to the two questions.
For example, teachers perceived "effective teaching"
practices as necessary for both current and future practice.
Utilizing cooperative or group activities, however, was
determined to be the second most frequent response for
improving future implementation of the Standards, rather
than use of manipulatives or reallife applications as indi
cated in the previous question. As one teacher noted, "For
instruction to be successful in implementing the NCTM
Standards, various teaching methods are necessary. Cooper
ative learning paired with teacherdirected lessons (on con
cept building) are very important."
Specifically, special education teachers indicated effec
tive teaching components as the most prevalent response for
students with LD (21%) and ED (18%) for improving
implementation of NCTM Standards in the future (see Table
1). Further, general education teachers noted certain compo
nents of effective instruction for students with LD (21%).
The majority of general education teacher responses (27%),
however, addressed behavior management strategies (e.g.,
proximity control, ignoring inappropriate behavior, being
patient, and reducing "down time") for students with ED.
Further, the second most prevalent response by general
education teachers involved cooperative learning or group
ing practices for students with LD (11%) and ED (14%) and
the third most prevalent response by special education
teachers for students with LD (9%). The two most prevalent
responses across teachers (general and special educators)
and disability categories (ED and LD) were effective teach
ing practices and cooperative learning/grouping activities.
These are discussed below within the context of the goals of
the NCTM Standards for teaching students with special
needs.
Effective Instruction
Predominant themes in teacher responses addressed spe
cific teaching methods to improve future implementation of
the Standards with students having LD and ED. The
responses included components found to be effective for
teaching math to students with disabilities, such as (a) repe
tition and review, (b) small group instruction, and (c) mastery
teaching. For example, programming for frequent reviews is
important for students with LD as many experience memory
deficits because of distractibility and problems concentrat
ing during instruction (Bezuk & Cegelka, 1995).
As teachers in the current study noted for students with
LD: "Repetition of concepts throughout the year is essential.
Students forget so quickly that activities must be repeated
periodically to help them retain methods and concepts" and,
"Continual and varied review and practice of the skills being
taught." To help students retain mathematical skills and con
cepts, Bezuk and Cegelka (1995) recommend: (a) teaching
for understanding via conceptual lessons, and (b) program
ming frequent reviews. As discussed earlier, one method for
introducing conceptually oriented lessons involves the
graduated instructional sequenceconcretesemiconcrete
abstract (CSA) continuum. This ensures problem under
standing prior to problem solution, which aids in concept or
FOCUS ON EXCEPTIONAL CHILDREN
skill acquisition and retention (Huntington, 1994; Maccini
& Huges, in press; Maccini & Ruhl, in press; Miller, 1996).
In addition, to address frequent reviews, it is recom
mended that lessons include daily, weekly, and monthly
reviews of new information and previously mastered skills
or concepts (Good & Grouws, 1979). For example, Good
and Grouws (1979) recommend a: (a) daily review consist
ing of homework problems and/or mental computation skills
for approximately 8 minutes at the beginning of class, four
times per week; (b) weekly review focusing on math tasks
introduced the previous week for approximately 20 minutes
every Monday; and (c) monthly review focusing on math
tasks introduced the past month.
In addition to frequent and varied reviews, teachers rec
ommended smallgroup instruction and mastery learning as
effective teaching practices to improve successful imple
mentation of the Standards with students having LD and
ED. As Archer, Gleason, and Isaacson (1995) stated,
"Smallgroup instruction translates into more instructional
time and significantly more active learning opportunities for
all students, typically resulting in increased achievement for
each individual" (p. 166). For example, teachers could divide
students into heterogeneous groups (small groups consisting
of a variety of ability levels) and have students work to
gether on problemsolving tasks, classwork, and/or reviews.
Too, requiring a criterion for student mastery before
advancing to a new skill or concept (e.g., 80% or higher dur
ing initial learning and 90% or higher during independent
practice prior to independent work) is effective for students
who have special needs. Teachers in the present study noted
the importance of student mastery: "Students must be given
time to achieve mastery of concepts before being pushed
through the curriculum or socially promoted year after
year"; and "Teach only one concept per class periodthen
use time to master concept."
Cooperative Grouping Activities
Another instructional method that teachers in the current
study noted as improving implementation of the Standards
with students who have LD and ED involved cooperative or
group activities. Teacher responses included: (a) general
statements addressing cooperative learning (e.g., "Coopera
tive teaching and learning"), (b) expert groups (e.g.,
"Another teaching method that is necessary to improve suc
cess is the use of expert groups where students learn from
each other"); and (c) peertutors or rolemodels (e.g., "Small
group (lab) activitiespair LD student with "good" math
student," "Have these students in class or groups with good
role models").
Cooperative learning experiences improve students' on
task behaviors, academic skills, social skills (help one
another and receive assistance), and experience in working
toward group goals or rewards. Slavin (1983) determined
that for cooperative learning experiences to be effective, stu
dents must be held accountable within the groups (i.e., indi
vidual accountability) and there must be a group goal for the
students to work toward. This can be particularly beneficial
with students who experience academic and social deficits,
such as students with LD and ED. For example, in a review
of peertutoring interventions for students with BD,
researchers concluded that, overall, tutees and tutors benefit
socially and academically from these interactions (Scruggs,
Mastropieri, & Richter, 1985).
In one investigation, Franca, Kerr, Reitz, and Lambert
(1990) determined that a peertutoring intervention im
proved the academic and social skills of eight middleschool
students with EBD. Students worked in dyads tuteee and
tutor) during 15minute tutoring sessions on fraction con
cepts and skills. Prior to teaching, tutors were trained to
solve the problems, provide stepbystep procedures for
problem representation, correct errors, and provide positive
reinforcement. Researchers determined that the tutor and
tutees improved their fraction skills (decreased error rates,
increased rate of correct responses), attitude toward mathe
matics, and social skills within the dyads.
Another effective grouping approach in mathematics
involves working with peer partners (Archer, Gleason,
Englert, & Isaacson, 1995). As one teacher in the current
study noted, "Peer tutoring, coaching, having student explain
methods to other students." For example, students work
together in pairs on an assignment/worksheet and provide
peer assistance via the following steps (Archer et al., 1995):
1. Students solve the first problem independently.
2. Students check their respective answers with a key.
3. If one student errs, the other student illustrates how
to solve the problem.
4. Students ask the instructor if they both erred.
These approaches are essential given the rigor of the NCTM
Standards and the difficulty that students with LD and ED
have with higherorder math tasks.
CONCLUSIONS AND IMPLICATIONS
FOR FUTURE PRACTICE
Having researched the instructional supports and adapta
tions that teachers perceive as effective for teaching math to
secondary students with LD and ED in light of the Standards
and the literature, the second author was reminded of his
eighth grade son's (Walter's) experience in prealgebra and
algebra within a progressive school district in the Northeast:
Despite the high reputation of the school district,
I was dismayed by the lack of recognition and
JANUARY 2000
implementation of the goals of the NCTM Stan
dards in my son's math class. All too often, Walter
came home with assignments that lacked real world
connection, such as solving for an unknown vari
able or creating Z tables, and his homework was
either too simple or reviewed skills that he had pre
viously mastered. Walter also acknowledged that
during an accelerated summer program, in prepara
tion for Algebra 1, no new skills were introduced
and this affected his motivation to stay ontask. Fur
ther, Walter and I reviewed his present Algebra text
book and determined that he could easily pass the
first three chapter tests (24% of the text) based
solely on his previous year's work in prealgebra.
It is unfortunate that many of Walter's experiences are
similar to those of both authors in algebra more than 20
years ago. We hope that some of the noted guidelines out
lined in the present review will provide a framework for
integrating what is known to be effective for teaching stu
dents with disabilities relative to teacher perceptions and the
goals of the NCTM Standards. We hope that future mathe
matics instruction for secondary students with LD and ED
extend beyond strictly "drill and practice" approaches to
incorporate the following teacher recommendations:
1. When designing and implementing lessons for stu
dents with disabilities, incorporate elements of effec
tive instruction, such as teaching explicit strategy
instruction, teacher modeling, guided and indepen
dent practice, monitoring of student performance,
using a wide range of examples and nonexamples,
separating potentially confusing terms, and cumula
tive reviews.
2. Select manipulatives related to the target concept or
skill and students' level of functioning. Incorporate a
variety of manipulatives with student and teacherled
verbal explanations to illustrate and explain mathe
matical concepts. In addition, program for transitions
from use of concrete manipulatives to abstract repre
sentations to promote student generalization.
3. Provide lessons and activities that embed mathemat
ics in realworld situations to foster student under
standing of mathematics and promote generalization
beyond the classroom.
4. Integrate calculators within instruction and assess
ment activities via teacherdirected and more discov
erybased approaches. Keep current on stateofthe
art technological advances and their classroom
application.
5. Individualize mathematics instruction through ad
justed workload and modifications based on students'
reading (e.g., reading to students) and writing ( e.g.,
guided notetaking, oral responses to questions) skills.
6. Implement positive, proactive, and consistent be
havioral management strategies including timeout,
token economies and contracts, and preferential seat
ing to provide specific guidelines for motivating stu
dents and encouraging appropriate behavior within
mathematics class.
7. Provide additional time for students to complete
mathematics assignments and assessments. Consider
instructional design variables to provide enhanced
skill acquisition and procedural competence.
8. Provide opportunities for students to work in cooper
ative and group activities to promote positive social
skills and concept/skill acquisition and retention.
Teachers have a continuing challenge to help students to
become better problem solvers, reason mathematically,
value math, become more confident in their ability to do
mathematics, and communicate mathematically. This review
highlights teacher perceptions of strategies and modifica
tions determined to be effective when teaching math to sec
ondary students with LD and ED in light of the goals of the
NCTM Standards. The specific recommendations developed
based on these findings provide an initial framework
through which the goals may be realized. The importance of
student learning extends beyond the classroom and includes
affective benefits. As one teacher noted regarding the advan
tages of implementing the Standards with students who have
special needs: "They feel successful in math, something
many of these students have not felt in many years."
REFERENCES
Advani, K. (1972). The effect of the use of desk calculators on achievement
and attitude of children with learning and behavior problems (research
report). Paper presented at 14th annual conference of Ontario Educa
tional Research Council, Toronto.
Alberto, P. A., & Troutman, A. C. (1999). Applied behavior analysis for
teachers (pp. 462484). Upper Saddle, NJ: Merrill.
Algozzine, B., O'Shea, D. J., Crews, W. B., & Stoddard K. (1987). Analy
sis of mathematics competence of learning disabled adolescents. Jour
nal of Special Education, 21. 97107.
Alley, G. R., Deshler, D. D., & Warner, M. M. (1979). Identification of
learning disabled adolescents: A Bayesian approach. Learning Dis
ability Quarterly, 2(2), 7683.
Archer, A. L, Gleason, M. M., & Isaacson, S. (1995). Effective instructional
delivery. In P. T. Cegelka & W. H. Berdine, Effective instruction for
students with learning difficulties (pp. 161194). Boston: Allyn &
Bacon.
Archer, A. L., Gleason, M. M., Englert, C. S., & Isaacson, S. (19950. Meet
ing individual instructional needs. In P. T. Cegelka & W. H. Berdin,
Effective instruction for students with learning difficulties (pp.
195225). Boston: Allyn and Bacon.
Bernstein. B. (1997). Message and meaning: The third international math
and science study. Educational Horizons, 76(1), 2327.
FOCUS ON EXCEPTIONAL CHILDREN
Bezuk, N. S., & Cegelka, P. T. (1995). Effective mathematics instruction for
all students. In P. T. Cegelka & W. H. Berdine, Effective instruction for
students with learning difficulties (pp. 345384). Boston: Allyn &
Bacon.
Blackham, G. J., & Silberman, A. (1975). Modifications of child and ado
lescent behavior (2nd ed., pp. 145146). Belmont, CA: Wadsworth
Publishing Company.
Bley, N. S., & Thornton, C. A. (1981). Teaching mathematics to the learn
ing disabled (pp. 134). Rockville, MD: Aspen Systems.
Bos, C. S., & Vaughn, S. (1994). Strategies for teaching students with learn
ing and behavior problems (3rd ed.). Boston, MA: Allyn & Bacon.
Bottge, B., & Hasselbring, T. S. (1993). A comparison of two approaches
for teaching complex, authentic mathematics problems to adolescents
in remedial math classes. Exceptional Children, 59, 556566.
Brian, T., Bay, M., LopezReyna, N., Donahue, M. (1991). Characteristics
of students with learning disabilities: A summary of the extant data
base and its implications for educational programs. In J. W. Lloyd, N.
Nirbhay. & A. C. Repp (Eds.), The regular education initiative: Alter
native perspectives on concepts, issues, and models (pp. 113131).
Sycamore, IL: Sycamore.
British Columbia Ministry of Education, Special Education Branch. (1996).
Teaching students with learning and behavioral differences: A
resource guide for teachers (pp. 142). (ERIC Document Reproduc
tion Service No. ED 414 712).
Cawley, J. R, & Miller, J. H. (1989). Crosssectional comparisons of the
mathematical performance of children with learning disabilities: Are
we on the right track toward comprehensive programming? Journal of
Learning Disabilities, 22, 250259.
Cawley, J.F., & Webster, R.E. (1981). Reading and behavior disorders. In
G. Brown, R. L. McDowell, & J. Smith (Eds.), Educating adolescents
with behavior disorders. Columbus, OH: Merrill.
Chalmers, L. (1991). Classroom modifications for the mainstreamed stu
dent with mild handicaps. Intervention in School and Clinic, 27,
4042, 51.
Chard, D. J, & Kameenui, E. J. (1995). Mathematics instruction for stu
dents with diverse learning needs: Heeding the message of the cheshire
cat. Focus on Learning Problems in Mathematics, 17, 2438.
Corno, L., & Snow, R. E. (1986). Adapting teaching to differences among
individual learners. In M. Wittrock (Ed.), Third handbook of research
on teaching (pp. 605629). New York: Macmillan.
Coutinho, M. J. (1986). Reading achievement of students identified behav
iorally disordered at the secondary level. Behavioral Disorders, 11,
200207.
Cullyer, R. C. (1988). Reading and math go hand in hand. Reading Im
provement, 25, 189195.
Demana, F. & Waits, B. K. (1990). Enhancing mathematics teaching and
learning through technology. In T. J. Cooney & R. Hirsch (Eds.),
Teaching and learning mathematics in the 1990's (pp. 212222).
Reston, VA: National Council Teachers of Mathematics.
Deshler, D. D., Ellis, E. S., & Lenz, B. K. (1996). Teaching adolescents
with learning disabilities: Strategies and methods (2d ed., pp.
315367). Denver: Love Publishing.
Dossey, J., Mullis, I., Lindquist, M., & Chambers, D. (1988). The mathe
matics report card: Are we measuring up? Princeton, NJ: ETS.
Drege, P., & Beare, P. L. (1991). The effect of a token reinforcement sys
tem with a timeout backup consequence on the classroom behavior
of E/BD students. British Columbia Journal of Special Education,
15(1), 3946.
Dunlap, G., Kern, L., dePerczel, M., Clarke, S., Wilson, D., Childs, K. E.,
White, R, & Falk, G. D. (1993). Functional analysis of classroom vari
ables for students with emotional and behavioral disorders. Behavioral
Disorders, 18, 275291.
Edgarton, R. T. (1992). A description of the assessment practices of teach
ers who have begun to implement the instructional practices suggested
in the NCTM standards. Paper presented at annual meeting of Ameri
can Educational Research Association, San Francisco.
Elliot, P. G., Wiles, C. A. (1980). The print is part of the problem. School
Science & Math, 80, 3742.
Emmer, T. (1981). Effective management in junior high mathematics class
rooms. (ERIC Document Reproduction Service No. ED 206 448).
Etlinger, L. E., & Ogeltree, E. J. (1982). Using calculators and microcom
puters with exceptional children (ERIC Document Reproduction Ser
vice No. ED 215 884).
Etscheidt, S., Stainback, S., & Stainback, W. (1984). The effectiveness of
teacher proximity as an initial technique of helping pupils control their
behavior. Pointer, 28, 3335.
Fifer, F. L., Jr. (1986). Effective classroom management. Academic Ther
apy, 21, 401410.
Fitzmaurice, A. M. (1980). LD teachers' selfratings on mathematics edu
cation competencies. Learning Disability Quarterly, 3, 9095.
Franca, V. M., Kerr, M. M., Reitz, A. L., & Lambert, D. (1990). Peer tutoring
among behaviorally disordered students: Academic and social benefits
to tutor and tutee. Education & Treatment of Children, 13, 109128.
Fuchs, L. S., Fuchs, D., & Bishop, N. (1992). Teacher planning for students
with learning disabilities: Differences between general and special
educators. Learning Disabilities Research & Practice, 7, 120129.
Fuchs, L.S., Fuchs, D., Phillips, N.B., & Simmons, D. (1993). Contextual
variables affecting instructional adaptations for difficulttoteach stu
dents. School Psychology Review, 22, (4), 725743.
Gagne, E. D..Yekovich, C. W., &Yekovich, E R. (1993). The cognitive psy
chology of school learning (2d ed.). New York: HarperCollins College
Publishers.
Gagnon. J. C., & Maccini, P. (2000). General and special education teach
ers' perceptions: Teaching mathematics to secondary students with
emotional disturbances. Manuscript submitted for publication.
Gallagher, P. A. (1995). Teaching students with behavior disorders: Tech
niques and activities for classroom instruction (2d ed.) . Denver: Love
Publishing.
Gast, D. L., & Nelson, C. M. (1977). Time out in the classroom: Implica
tions for special education. Exceptional Children, 43(7). 461464.
Good, T. L., & Grouws, D. A. (1979). The Missouri mathematics effective
ness project: An experimental study in fourthgrade classrooms. Jour
nal of Educational Psychology, 71, 355362.
Goodrich, B., & Stern, V. W. (1995). Teaching science and mathematics to
students with learning disabilities: Challenges and resources.
NSF/AAS Invitational Conference on Learning Disabilities and the
Teaching of Science and Mathematics, Washington, DC.
Graham, S., Harris, K.R., MacArthur, C.A., & Schwartz, S. (1991). Writing
and writing instruction for students with learning disabilities: Review of
a research program. Learning Disability Quarterly, 14, 89114.
Haring, N. G., & Phillips, E. L. (1962). Educating emotionally disturbed
children. New York: McGraw Hill.
Heshusius, L. (1991). Curriculumbased assessment and direct instruction:
Critical reflections on fundamental assumptions. Exceptional Chil
dren, 57, 315328.
Heward, W. L., (1994). Three "lowtech" strategies for increasing the fre
quency of active student response during group instruction. In R. Gard
ner III., D. M. Sainato, J. 0. Cooper, T. E. Heron, W. L. Heward, J. Esh
leman. & T. A. Grossi (Eds.), Behavior analysis in education: Focus
on measurably superior instruction (pp. 283320). Monterey, CA:
Brooks/Cole Publishing Company.
Hewett, F.M., & Taylor, FD. (1980). The emotionally disturbed child in the
classroom: The orchestration of success (pp. 214236). Boston: Allyn
and Bacon.
JANUARY 2000
Hofmeister, A.M. (1993). Elitism and reform in school mathematics. Reme
dial & Special Education, 14, (6), 813.
Huntington, D. J. (1994). Instruction in concrete, semiconcrete, and abstract
representation as an aid to the solution of relational problems by adoles
cents with learning disabilities (Doctoral dissertation, University of
Georgia, 1994). Dissertation Abstracts International, 56/02, 512.
Hutchinson, N. L, (1993). Students with disabilities and mathematics edu
cation reformlet the dialogue begin. Remedial & Special Education,
14, (6), 2027.
Hutchinson, N. L. (1993). Effects of cognitive strategy instruction on alge
bra problems solving of adolescents with learning disabilities. Learn
ing Disability Quarterly, 16, 3463.
International Association for the Evaluation of Educational Achievement.
(1996). Mathematics achievement in the middle school years: LEA's
third international mathematics and science study (TIMSS). Chestnut
Hill, MA: TIMSS International Study Center, Boston College.
Jetter, A. (1993, February 21). "Mississippi learning." The New York Times
Magazine, pp. 28835, 50, 51, 64, 72,
Jones, R. (1998). Solving problems in math and science education. The
American School Board Journal, 185, (7), 1619.
Johnston, J. M. (1972). Punishment of human behavior. American Psychol
ogist, 27, 10331054.
Kelly, B., & Carnine, D. (1996), Teaching problem solving strategies for
word problems to students with learning disabilities. LD Forum, 21,
59.
Kelly, B., Gersten, R. & Carnine, D. (1990). Student error patterns as a
function of curriculum design: Teaching fractions to remedial high
school students and high school students with learning disabilities.
Journal of Learning Disabilities, 1, 2329.
Lazarus, B. D. (1993). Guided notes: Effects with secondary and post sec
ondary students with mild disabilities. Education & Treatment of Chil
dren, 16(3), 272289.
Lock, R.H. (1996). Adapting mathematics instruction in the general educa
tion classroom for students with mathematics disabilities. LD Forum,
21(2), 1923.
Maccini, P., & Gagnon, J. C. (2000). General and special education teach
ers' perceptions: Teaching mathematics to secondary students with
learning disabilities. Manuscript submitted for publication.
Maccini, P., (1998). Effects of an instructional strategy incorporating con
crete problem representation of the introductory algebra performance
of secondary students with learning disabilities. Unpublished doctoral
dissertation, The Pennsylvania State University, University Park.
Maccini, P.. & Hughes, C. A. (1997). Mathematics interventions for ado
lescents with learning disabilities. Learning Disabilities Research &
Practice, 12, 165176.
Maccini, P. & Hughes, C. A. (in press). Effects of a Problemsolving Strat
egy on the Introductory Algebra Performance of Secondary Students
with Learning Disabilities. Learning Disabilities Research & Practice,
Maccini, P., & Ruhl, K. L. (in press). Effects of a graduated instructional
sequence on the algebraic subtraction of integers by secondary stu
dents with learning disabilities. Education & Treatment of Children,
Marzola. E. S. (1987). Using manipulatives in math instruction. Reading,
Writing, & Learning Disabilities, 3, 920.
McLeod, T. M., & Armstrong, S. W. (1982). Learning disabilities in math
ematics skill deficits and remedial approaches at the intermediate and
secondary level. Learning Disability Quarterly, 5, 305311.
McWhirter, C. C. & Bloom, L. A. (1994). The effects of a studentoperated
business curriculum on the ontask behavior of students with behav
ioral disorders. Behavioral Disorders, 19, 136141.
Meese, L. (1994). Teaching learners with mild disabilities: Integrating
research and practice (pp. 279370). Pacific Grove, CA: Brooks/Cole
Publishing Company.
Mercer, C. D., Harris, C., & Miller, S. P. (1993). Reforming reforms in
mathematics. Remedial & Special Education, 14, 1419.
Mercer, C. D., Jordan, L., & Miller, S. P. (1994). Implications of construc
tivism for teaching math to students with moderate to mild disabilities.
Journal of Special Education, 28, 290306.
Mercer, C.D., & Mercer, A.R. (1998). Teaching students with learning
problems (5th ed., pp. 171225). Upper Saddle, NJ: Merrill.
Miller, S. P. (1996). Perspectives on mathematics instruction. In D. D.
Deshler, E. S. Ellis, & B. K. Lenz, Teaching adolescents with learning
disabilities: Strategies and methods (2d ed., pp. 313367). Denver:
Love Publishing.
Milou, E., Gambler, E. D., & Moyer, T. 0. (1997). Using the graphing cal
culator in the classroom: Helping students solve the "unsolvable".
(ERIC Document Reproduction Service No. ED 223 111).
Mohr, L. L. (1995). Teaching diverse learners in inclusive settings: Steps
for adapting instruction. Paper presented at annual international con
vention of Council for Exceptional Children, Indianapolis, IN.
Moore. L. J., & Carnine, D. (1989). Evaluating curriculum design in the
context of active teaching. Remedial & Special Education, 10, 2837.
National Center for Educational Statistics (1997). NAEP 1996 mathemat
ics report card for the nation and states: Findings from the national
assessment of educational progress (Online], Available:
http://nces.ed.gov/pubsearch/pubsiinfo.asp?pubid=97488.
National Council of Teachers of Mathematics (NCTM) (1986). Position
statement of calculator usage. Reston, VA. National Council of Teach
ers of mathematics.
National Council of Teachers of Mathematics (NCTM) (1989). Curriculum
and evaluation standards for school mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics (NCTM) (1991). Profes
sional standards for teaching mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics (NCTM) (1995). Assessment
standards for school mathematics. Reston, VA: Author,
National Council of Teachers of Mathematics (NCTM) (1998). NCTM
position statement [Online]. Available: www.nctm.org,
Nelson, C. M., Rutherford, R. B. (1983). Timeout revisited: Guidelines for
its use in special education. Exceptional Children Quarterly, 3, (4),
5667.
Newman. R., & Pollack, D. (1973). Proxemics in deviant adolescents. Jour
nal of Consulting & Clinical Psychology, 40, 68.
Olson, J. L., & Platt, J. M. (1996). Teaching children and adolescents with
special needs (3d ed.). Upper Saddle, NJ: PrenticeHall.
Parmar, R. S,, & Cawley, J. F. (1995). Mathematics curricula framework:
Goals for general and special education. Focus on Learning Problems
in Mathematics, 17, 5066.
Peacock Hill Working Group (1991). Problems and promises in special
education and related services for children and youth with emotional
or behavioral disorders. Behavioral Disorders, 16, (4), 299313.
Picciotto, H. (1990). The algebra lab. Sunnyvale, CA: Creative Publica
tions.
Plummer, S. Baer, D. M., & Blance, J. M. (1977). Functional considera
tions in the use of procedural timeout and an effective alternative.
Journal of Applied Behavior Analysis, 10. 689705.
Polloway. E. A., & Patton, J. R. (1993). Strategies for teaching learners
with special needs (5th ed.). New York: Merrill.
Rivera, D. M. (1993). Examining mathematics reform and the implications
for students with mathematics disabilities. Remedial and Special Edu
cation, 14, 2427.
Rivera, D.M., Bryant, B.R. (1992). Mathematics instruction for students
with special needs. Intervention in School and Clinic, 28, (2), 7186.
Robinson. S. M., Braxdale, C. T, & Colson, S. E. (1988). Preparing dys
functional learners to enter junior high school: A transitional curricu
lum. In E. L. Meyen, G. A. Vergason, & R. J., Whelan (Eds.), Effective
FOCUS ON EXCEPTIONAL CHILDREN
JANUARY 2000
instructional strategies for exceptional children (pp. 243258). Den
ver, CO: Love Publishing Company.
Rosenshine, B., & Stevens, R. (1986). Teaching functions. In M. C. Wit
trock (Ed.), Handbook of research on teaching (3d ed., pp. 376391).
New York: Macmillan.
Ruhl, K. L. (1985). Handling aggression: Fourteen methods teachers use.
Pointer 29, 3033.
Salend, S. J. (1990). Effective mainstreaming (pp. 249281). New York:
Macmillan.
Salend, S. J. (1994). Effective mainstreaming: Creating inclusive class
rooms (2d ed., pp. 316381). New York: Macmillan.
Salend, S. J., & Hoffstetter, E. (1996). Adapting a problemsolving
approach to teaching mathematics to students with mild disabilities.
Intervention in School & Clinic, 31, (4), 209217.
Scruggs, T. E., Mastropieri, M. A., & Richter, L. (1985). Peer tutoring with
behaviorally disordered students: Social and academic benefits.
Behavioral Disorders, 1O, 283294.
Slavin, R. E. (1983). When does cooperative learning increase student
achievement? Psychological Bulletin, 94, 429445.
Sprick, R. S. (1985). Discipline in the secondary classroom: A problemby
problem survival guide (pp. 66110). West Nyack, NY: The Center for
Applied Research in Education.
Stallings, J. A., & Kaskowitz, D. (1974). Follow through classroom obser
vation evaluation 19721973. Menlo Park, CA: Stanford Research
Institute.
Stephens, T.M.. Hartman, A.C., Lucas, V.H. (1978). Teaching children
basic skills: A curriculum handbook. Columbus. OH: Merrill.
Suritsky, S. K., & Hughes, C. A. (1991). Benefits of notetaking: Implica
tions for secondary and postsecondary students with learning disabili
ties. Learning Disability Quarterly, 14, 718.
Tarver, S. G. (1996). Direct instruction. In Stainback, W. & Stainback, S.,
Controversial issues confronting special education: Divergent per
spectives (2nd ed., pp. 143165). Boston: Allyn & Bacon.
Thornton. C. A., Langrall, C. W., & Jones, G. A. (1997). Mathematics
instruction for elementary students with learning disabilities. Journal
of Learning Disabilities, 30, 142150.
Van De Walle. J. (1994). Elementary school mathematics (2d ed.). NY:
Longman.
Van Houten, R., Nau, P.A., MacKenzieKeating, S.E., Sameoto, D., &
Colavecchia (1982). An analysis of some variables influencing the
effectiveness of reprimands. Journal ofApplied Behavior Analysis, 15,
6583.
Wagner, M. (1995). Outcomes for youth with serious emotional disturbance
in secondary school and early adulthood. Future of Children, 5(2),
90113.
Wagner. M.M., & Blackorby, J. (1996). Transition from high school to
work or college: How special education students fare. Future of Chil
dren, 6(1), 103120.
Walker, S. (1981). Structuring the learning environment for minority hand
icapped students. Washington, DC: National Alliance of Black School
Educators, Special Education Programs.
Walker, J. E., & Shea. T. M. (1999). Behavior management: A practical
approach for educators (pp. 217254). Upper Saddle, NJ: Merrill.
Webster, R. E. (1976). A timeout procedure in a public school setting. Psy
chology in the Schools, 13, 7276.
Weinstein, C.S. (1979). The physical environment of the school: A review
of the research. Review of Educational Research, 49, (4), 577610.
Wheldall, K., & Lam, Y.Y (1987). Rows versus tables. II. The effect of two
classroom seating arrangements on classroom disruptive rate, ontask
behavior and teacher behavior in three special school classes. Educa
tional Psychology, 7, (4), 303312.
