Best practices for teaching mathematics to secondary students with special needs: Implications from teacher perceptions ...
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Title: Best practices for teaching mathematics to secondary students with special needs: Implications from teacher perceptions and a review of the literature
Series Title: Maccini, P., & Gagnon, J. C. (2000). Best practices for teaching mathematics to secondary students with special needs: Implications from teacher perceptions and a review of the literature. Focus on Exceptional Children, 32(5), 1-22.
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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 103-227). 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 high-school 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
eighth-grade 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 13-14 focus statements per
grade section (i.e., K-4, 5-8, 9-12, 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 real-life
math applications and utilizing problem-solving
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/real-life situations).


FOCUS On

Exceptional

children
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lished monthly except June, July, and August as a service to teachers,
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cemed with the special education of exceptional children. This publica-
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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 problem-solving 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 real-world situations)
* become more confident in their mathematical ability
(i.e., developing students' self-confidence 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 the-efforts of
NCTM to reach "all students." These students are at-risk 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 one-fourth of students
labeled LD have been identified specifically due to a dis-
crepancy between math aptitude and performance (Brian,
Bay, Lopez-Reyna, & 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 special-education teachers and determined that
adequate instruction in helping students with LD complete
open-ended problem-solving 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 on-computational-tasks rather than higher-order prob-
lem-solving 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 problem-solving 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 open-ended
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 open-ended responses were
coded for major themes and then analyzed relative to the fre-
quency of-the ideas perceived as effective practices-for 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 hands-on 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 higher-order reasoning and critical think-
ing skills.

Teachers also indicated that activities based on the goals of
the Standards address the connection between mathematics
and real-world applications and connections to other subject
areas specifically for students with LD. For example,
respondents noted the importance of incorporating hands-on
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) real-life
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 school-deter-
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 hands-on activities. I make sure that the lesson will be
able to reach the visual and auditory learners"; "Use of
hands-on 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 small-group 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 real-life 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 real-world application to help stu-
dents generalize math skills and concepts, ("Mathematical
connections-I 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 at-risk. 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 student-focused (i.e.,
teacher as facilitator) learning environment than the more
teacher-directed 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 modeling-making 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 teacher-modeling when instructing secondary
students with LD, ("Being very specific-step-by-step 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 at-risk 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 step-by-step 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 step-by-step 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 problem-solving tasks
requiring ratio and proportions. Twenty-nine 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 step-by-step 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 problem-solving 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 technology-based instruction into math-
ematics lessons
2. Utilizing pictorial displays for concept development
3. Focusing on problem-solving 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 step-by-
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 two-colored counters (one-sided painted
lima beans) to express negative and positive integers and
model adding integers using zero pairs"; "I use hands-on
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 teacher-directed 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-
gle-subject 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 stages-concrete, 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 problem-solving 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 problem-solving
strategy, STAR (Maccini, 1998), which involved a general
problem-solving 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 problem-solving
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 teacher-led), and programming for instructional transi-
tions from concrete to symbolic representation.

Real-World 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 real-world context helps
students activate their conceptual knowledge when pre-
sented with a real-life problem-solving 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-
lem-solving situations in a real-life 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 teacher-directed 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 5-day
remediation plan involving a videodisc program was devel-
oped based on the analysis. Students were reassessed to






FOCUS ON EXCEPTIONAL CHILDREN


JANUARY 2000


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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
real-life problem-solving 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 problem-solving
task requiring different math skills than the videodisc prob-
lem-solving 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 real-life settings" (p. 565).
In addition to improving generalization to other math
tasks, providing real-life application improves students' on-
task behaviors. For example, McWhirter and Bloom (1994)
examined the effects of a math curriculum involving a class-
room-based business and assessed on-task behavior. Three
students with ED participated in the single-subject study
within a self-contained 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 attention-to-task
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 self-contained settings, and (c) the percentage
of special educators whose responsibilities are confined to
exclusionary settings (e.g., self-contained, 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 hands-on
activities (16%)
3) Promotes a more rigorous math program (higher-
order thinking, critical thinking skills) (14%)
4) Promotes connections between math and the
real-world (13%)

ED 1) Promotes connections between math and the
real-world (17%)
2) Positive effect on student learning and positive
student reaction (15%)
3) Promotes conceptual learning and hands-on
activities (12%)


1) Preparation for school/state requirements and
life beyond school (16%)
2) Promotes conceptual learning and hands-on
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
(higher-order thinking, critical thinking skills)
(13%)
3) Promotes conceptual learning/hands-on
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) Real-life Application (11%)

1) Effective Instruction
2) Real-life 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 color-coded chips, introduce the concept using
algebra tiles.

* Incorporate teacher-directed verbal explanations while presenting a
concept via manipulatives. Also include self-questions and verbal
explanations to improve students' self-monitoring strategies.

* Encourage student participation (opportunities to use
manipulatives).
* Monitor student performance as students manipulate objects,
explain, and write down their problem-solving 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 teacher-directed instruction: (1) model examples) of the
target concept and think-aloud 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 problem-solving steps; (3) fade teacher-assistance 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 problem-solving situations
* concentrate on the problem-solving 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 problem-solving tasks, and (c) check grocery
receipts. The researchers used a pre- and post-test 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 state-of-the-art 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 general-education
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
26-35 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 self-contained classroom or a full-time
resource room, typically situations that have a much
lower student-teacher 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.
Time-out
Within the category of behavior management, the use of
time-out, seating accommodations, and specific behavior
plans (e.g., token economy, behavioral contracting) were the
three central adaptations based on student IEPs. Time-out
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 time-out procedures for reduc-
ing inappropriate behavior of students across a range of ages
and settings.
The popularity of time-out is evident from a review of
respondents' comments. Teachers noted the usefulness of
"Letting them studentsl take a time-out when needed" and,
"When conflicts arise, students are not cornered or pres-
sured. Students are allowed self-esteem 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 time-out 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 time-out is a study by
Webster (1976). The participant was a 13-year-old male in
sixth-grade public school. He had difficulties with violent
acting-out 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 time-out. 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 8-week follow up, no incidences transpired that
required the use of time-out.
In addition to decreasing aggressive behavior, time-out
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 time-out, however, requires
that the classroom be sufficiently positive and reinforcing
for the student. If the student views the classroom environ-
ment negatively, time-out may serve as a reinforcer (Plum-
mer, Baer, & LeBlanc, 1977).
Sprick (1985) identified seven criterion for establishing
effective use of time-out 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 time-out].
7. Evaluate the effectiveness of your consequence and
reinforcement procedures. (pp. 107-109)


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 time-out with another teacher,
librarian, or other school personnel (British Columbia Min-
istry of Education, 1996). Similarly, specific criterion and an
agreed-upon 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, MacKenzie-Keating, 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. 145-146)

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 5-8, 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 eighth-grade 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 lower-level 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 open-ended 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
teacher-made 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.350-351)

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 higher-level 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 1-6), 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 (1994-1995):
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 real-life 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 teacher-directed 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 sequence-concrete-semiconcrete-
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 small-group 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,
"Small-group 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 problem-solving 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 period-then
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) peer-tutors or role-models (e.g., "Small
group (lab) activities-pair 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 peer-tutoring 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 peer-tutoring intervention im-
proved the academic and social skills of eight middle-school
students with EBD. Students worked in dyads tuteee and
tutor) during 15-minute tutoring sessions on fraction con-
cepts and skills. Prior to teaching, tutors were trained to
solve the problems, provide step-by-step 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 higher-order 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 pre-algebra 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 on-task. 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 pre-algebra.

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 teacher-led
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 real-world situations to foster student under-
standing of mathematics and promote generalization
beyond the classroom.
4. Integrate calculators within instruction and assess-
ment activities via teacher-directed and more discov-
ery-based approaches. Keep current on state-of-the-
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, pro-active, and consistent be-
havioral management strategies including time-out,
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."

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