1ExceptionalChldren.
Vol 72, No. 2, pp. 217234.
�2006 Council fr Exceptional Children.
Mathematics Instructional
Practices and Assessment
Accommodations by Secondary
Special and General Educators
PAULA MACCINI
University ofMaryland, College Park
JOSEPH CALVIN GAGNON
George Mason University
ABSTRACT: A nationwide random sample of 179 secondary general education math and special ed
ucation teachers completed a survey asking them about their use ofspecific instructional practices to
help teach math to students with learning disabilities (LD) and emotional/behavioral disorders
(EB/D) and their use of specific assessment accommodations with these students. Although general
education teachers were generally more knowledgeable about higher level mathematics content
(e.g., algebra) than special education teachers, they were less likely to report that they used specific
instructional practices and testing accommodations. The number of methods courses taken by
teachers and knowledge of course topics both contributed to the number of instructional practices
and accommodations used by teachers.
United States continue to place
increasingly higher demands on
youth with and without disabil
ities. These demands are mea
sured through mandatory district and state
assessments that directly affect whether or not stu
dents graduate (Gagnon & McLaughlin, 2004).
For example, the No Child Left Behind Act
(NCLB, 2001) includes a focus on high standards
and accountability for student learning. The Indi
viduals With Disabilities Education Act (1997) as
sures that students with special needs are also
included in current educational reform via man
dated access to the general education curriculum
to the greatest extent possible and participation in
assessments with accommodations, as needed.
Subsequently, general and special educators are
Exceptional Children
faced with the task of assisting youth with disabili
ties, including those with learning disabilities
(LD) and emotional/behavioral disorders (E/BD),
to achieve maximum benefit from the curriculum
and progress toward academic goals.
Provisions within NCLB (2001) and IDEA
(1997) also increase the expectations of teachers.
To assist students with special needs, teachers
must have the knowledge and training to provide
effective instructional practices and assessment ac
commodations. For example, in the area of math
ematics, teachers are expected to provide effective
instruction on curriculum that addresses higher
level math skills and encompasses openended,
problemsolving tasks (see Maccini & Gagnon,
2000) as set forth by the National Council of
Teachers of Mathematics Standards (NCTM,
2000). Currently, 42 states (Blank & Dalkilic,
1992, as noted in Thurlow, 2000) have developed
state math standards consistent with NCTM.
The use of effective instructional proce
dures and testing accommodations are critical be
cause most youth with LD and E/BD experience
difficulty acquiring and retaining math skills,
such as algebraic reasoning skills (Maccini, Mc
Naughton, & Ruhl, 1999) and basic skills/com
putational skills (Algozzine, O'Shea, Crews, &
Stoddard, 1987). These students also have diffi
culty passing math tests aligned with state stan
dards (Thurlow, Albus, Spicuzza, & Thompson,
1998). For example, Thurlow and colleagues de
termined that 83% of nondisabled eighthgrade
students passed a Basics Standards Math Test, in
comparison to 42% of students with E/BD and
34% of students with LD.
Given the difficulties that most students
with LD and E/BD experience with mathematics,
it is important to identify the instructional prac
tices and assessment accommodations that help
these students succeed in math. In this article, in
structional practices refer to both empirically vali
dated and recommended practices for teaching
math to students with LD and/or E/BD. Empiri
cally validated and recommended instructional
practices and assessment accommodations are de
fined and discussed relative to the literature and
the purpose of the current study.
INSTRUCTIONAL PRACTICES
As mandated by IDEA (1997), students with dis
abilities should have access to the general educa
tion curriculum and are entitled to empirically
validated instructional practices that help them
succeed. Empirically validated practices refer to
researchbased approaches to teaching math skills
(Wilson, Majsterek, & Jones, 2001). Maccini and
Gagnon (2000) determined that special and gen
eral education teachers of secondary students with
LD and E/BD noted using a number of empiri
cally validated approaches for teaching math to
these students. These approaches included the fol
lowing: (a) use of objects for conceptual under
Given the difficulties that most students
with LD and E/BD experience with
mathematics, it is important to identify
the instructional practices and assessment
accommodations that help these students
succeed in math.
standing; (b) peer or crossage tutoring strategies;
and (c) organizational strategies for retention
(e.g., cue cards of strategy steps, graphic organiz
ers, mnemonics, and time for additional practice).
Maccini and Gagnon (2000) also found
that teachers noted using other instructional prac
tices with students with LD and E/BD in math:
(a) use of calculators; (b) assignment modifica
tions (i.e., adjusted workloads); and (c) increased
time for activities and tests. Assignment modifica
tions included reducing classwork problems and
homework problems and adjusting the workload
via color coding or circles for cuing. Although
these practices have yet to be validated with stu
dents with LD and E/BD, they were frequently
noted, perhaps because they are recommended in
special and general education textbooks. For ex
ample, in an introductory text for teaching math
to students with special needs, experts (Bley &
Thornton, 2001) recommend using visual cues
and reduced classwork and homework problems.
These recommended practices presumably help
Winter 2006
math teachers plan and implement instruction
that is beneficial for all students and help teachers
from "spending an inordinate amount of time
with one child to the exclusion of others" (Bley &
Thornton, p. 22). Though cited by teachers in a
statewide investigation (Maccini & Gagnon,
2002), no national studies exist that focus on
teachers' use of empirically validated and recom
mended instructional practices to assist secondary
students with LD and E/BD in math.
ASSESSMENT
ACCOMMODATIONS
The National Council of Teachers of Mathemat
ics (NCLB, 2001) mandates increased participa
tion and accountability for all students, including
students with disabilities, on highstakes assess
ments that are tied to rigorous math standards.
Assessment accommodations are necessary for
some measures to help "level the playing field" for
students with special needs (Thompson, Blount,
& Thurlow, 2002). Experts define accommo
dations as modifying assessment procedures (e.g.,
scheduling, timing, presentation) or materials
(e.g., setting, response) "that enable students with
disabilities to participate in an assessment in a
way that allows abilities to be assessed rather than
disabilities" (Thurlow, Elliott, & Ysseldyke, 1998,
pp. 2829). In a statewide survey, Maccini and
Gagnon (2000) determined that special and gen
eral education teachers of secondary students
with LD and E/BD reported using a number of
accommodations, including (a) presentation
(e.g., color coding); (b) response (e.g., using ref
erence materials such as cue cards or charts of
strategy steps); (c) setting (e.g., behavioral modi
fications such as seating accommodation); (d)
timing/scheduling (e.g., extended time on tests);
and (f) calculators. However, only timing accom
modations (Helwig, RozekTedesco, Tindal,
Heath, & Almond, 1999) and oral presentation
(Calhoon, Fuchs, & Hamlett, 2000) have been
empirically validated in math. The remaining ac
commodations reported included those that are
commonly allowed on local and largescale assess
ments (Menikoff, 2004). To date, no national
studies exist that focus on teachers' use of empiri
callyvalidated and recommended assessment
accommodations to assist secondary students with
LD and/or E/BD in math.
Also lacking in the research on mathematics
instruction is the identification of factors that
may predict the number of instructional practices
and assessment accommodations made for stu
dents with LD and E/BD. No information is
available concerning (a) the contribution of
teacher characteristics (e.g., number of years
teaching); (b) special and general education math
courses taken to teach math to students with LD
and E/BD); (c) knowledge (e.g., familiarity with
subject matter); (d) teacher type (e.g., general vs.
special education teacher); and (e) classroom vari
To date, no national studies exist that
focus on teachers' use of empirically
validated and recommended assessment
accommodations to assist secondary
students with LD and/or E/BD in math.
ables (e.g., number of students per class that need
math assistance) in predicting the number of re
searchbased instructional adaptations and assess
ment accommodations made by teachers.
However, in a related study, Graham, Harris,
Fink, and MacArthur (2003) found that, in addi
tion to the amount of time spent writing, both
teacher variables (e.g., number of years teaching)
and classroom variables (e.g., number of students
needing special education services) predicted the
number of instructional practices teachers used
with students who experienced difficulty with
writing. Teacher type may also predict the num
ber of instructional practices or assessment ac
commodations made for students with LD and/or
E/BD. Researchers (Schumm et al., 1995) found
that general education teachers allow fewer assess
ment accommodations. Also, as the number of
methods courses taken by teachers positively af
fects student learning in secondary math courses
(Monk, 1994), it is possible that the number of
methods courses taken by teachers also influences
the number of instructional practices and assess
ment accommodations.
Exceptional Children
219
TABLE 1
Participant Characteristics
No. (%) Significance
Special General
Education Education
Characteristics Teachers Teachers df X2 p
Gender
Female 69 (39) 53 (30) 1 .124 .725
Male 29 (17) 25 (14)
Age
2029 7 (4) 6 (3) 3 2.570 .463
3039 18(10) 19(11)
4049 42 (24) 24 (14)
50or > 32(18) 28(16)
Educational Level
Undergraduate 36 (21) 31 (18) 1 .317 .574
Graduate 61 (35) 44 (26)
PURPOSE
Provisions within NCLB (2001) and IDEA
(1997) require that students with special needs be
exposed to the same curriculum as their nondis
abled peers, to the greatest extent possible. Also,
as mandated by IDEA, all students are required to
participate in state assessments (with the excep
tion of those that participate in alternate assess
ments) that are linked to the general education
curriculum. The challenge, then, is to help educa
tors provide effective instructional practices and
appropriate assessment accommodations to help
all students achieve within the general education
mathematics curriculum and meet high stan
dards. Using a nationwide sample, the current
study was designed to answer three research ques
tions. First, what specific instructional practices
do special and general education teachers report
edly use for students with LD and E/BD during
instruction on basic math computation skills and
problemsolving tasks? Second, what specific ac
commodations do special and general education
teachers reportedly make for students with LD
and E/BD when assessing basic math skills and
when assessing problemsolving tasks? Third,
what factors predict the number of instructional
practices and assessment accommodations general
and special education teachers reportedly make
for students with LD and E/BD?
METHOD
SURVEYS
A survey of secondary general and special educa
tors teaching math to students with LD and
E/BD was developed across three central topics:
(a) teacher background information, (b) teacher
perceived knowledge of secondary math topics,
and (c) specific instructional practices and accom
modations teachers use for students with LD and
E/BD during instruction and when assessing basic
math/computation skills and problemsolving
tasks (see Tables 1, 2, and 3). For teacher back
ground information, respondents were asked to
provide information on their age, gender, educa
tional level, years teaching students with LD
and/or E/BD, teaching credentials, math methods
courses taken, and type of math courses currently
taught (e.g., prealgebra, algebra, algebra II, alge
bra II/trigonometry, geometry, statistics/probabil
ity, general mathematics or basic skills
mathematics, integrated/unified high school
mathematics, other). In addition, teachers com
pleted a 4point Likerttype scale to indicate their
level of preparation for teaching math to students
with LD and/or E/BD after completing their
methods courses (ranging from very prepared to
very unprepared = 4).
The second portion of the survey focused
on teacher perceived knowledge of secondary
Winter 2006
220
TABLE 2
Participants' Credentials and Current Teaching Responsibilities
No. (%) Significance
Special General
Education Teachers Education Teachers df x2 p ES
Characteristics Yes No Yes No
Secondary math teaching
credential 7 (4) 94 (52) 76 (43) 2 (1) 1 144.966 .000 .90
Elementary math teaching
credential 18 (10) 83 (46) 9 (5) 69 (39) 1 1.357 .244 .09
Special education teaching
credential 97 (54) 4 (2) 5 (3) 73 (41) 1 144.238 .000 .90
Teaches Prealgebra 62 (34) 39 (22) 21 (12) 57 (32) 1 21.020 .000 .34
Teaches Algebra 46(26) 55 (31) 45 (25) 33 (18) 1 2.599 .107 .12
Teaches Geometry 27(15) 74(41) 25(14) 53(30) 1 .604 .437 .06
Teaches General
Mathematics or Basic
Skills (HS) 65(36) 36 (20) 13 (8) 65 (36) 1 40.710 .000 .48
Teaches Algebra II 6 (3) 95 (53) 13 (8) 65(36) 1 5.377 .021 .17
Teaches Algebra II/
Trigonometry 1 (0) 100 (56) 4 (2) 74 (42) 1  .169a .13
Teaches Statistics/
Probability 3 (2) 98 (55) 6 (3) 72(40) 1  .181a .11
Teaches Integrated/Unified
High School Math 6 (3) 95 (53) 7 (4) 71 (40) 1 .601 .438 .06
Note. ES = coefficient phi;  = Insufficient cell size to calculate 2; a = Fisher's exact test value.
math topics. Teachers completed a 4point Likert
type scale (ranging from great deal = 1 to very little
or nothing = 4) to indicate their level of perceived
knowledge to teach the eight math topics. For the
final section of the survey, teachers indicated if
they used specific instructional practices to help
students with LD and/or E/BD with basic
skills/computation and problemsolving tasks
from a list of 14 choices (see Tables 4 and 5).
Teachers also indicated if they used specific assess
ment accommodations to help students with LD
and/or E/BD with basic skills/computation and
for problem solving tasks from a list of eight
choices (see Tables 4 and 5).
One version of the survey was constructed
for general educators and another for special edu
cation teachers to address their unique teaching
situations. The survey questions were identical,
with the exception of three questions that applied
solely to special education teachers (number of
years as a special education teacher, number of
students on caseload, current teaching position)
and one question for general education teachers
(number of years as a math teacher). Finally, it
should be noted that coding for data entry was
conducted for 25% of the responses. Data was
entered accurately as the obtained reliability was
95%.
SAMPLE
Survey Sampling Procedures. The target pop
ulation for the current study consisted of two
groups of secondary teachers: (a) general educa
tion math teachers who teach students with LD
and/or E/BD, and (b) special education teachers
who teach math to students with LD and/or
E/BD and/or collaborate with the general educa
tion math teachers. First, a random sample of 750
special and general educators from all public high
schools in the United States was obtained from
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221
TABLE 3
Years Teaching and Number and Nature ofMethods Courses
Mean Rating ofMethods
Courses on a 4Point Scalea Significance
Special General
Education Education
Survey Questions Teachers Teachers df t p ES
How many years have you taught 12.36 1.15 176 12.433 .000 .68
students with LD and/or E/BD?
How many general education math 1.61 3.16 91 3.719 .000 .36
methods courses have you taken?
How many special education math 1.27 .56 163 4.309 .000 .32
methods courses have you taken that
addressed teaching math to students
with LD and E/BD?
How prepared were you to teach 2.28a 2.75a 123 2.852 .005 .25
math to students with LD and/or
E/BD after completing your math
education methods courses?
How prepared were you to teach 2.34a 1.74a 111 4.171 .000 .37
math to general education students
after completing your math
education methods courses?
How prepared were you to teach 1.94a 2.07 a 88 .868 .388 .09
math to students with LD and/or
E/BD after completing your special
education courses that focused on
teaching math?
Note. LD = learning disabilities; E/BD = emotional/behavioral disorders; ES = Cohen's d.
a4point scale ranging from 1 = very prepared to 4 = very unprepared.
Quality Education Data (QED School Personnel
Data Base, 2000/2001). Approximately 40% (n =
325) of the teachers sampled were general educa
tors listed as having a special education class and
currently teaching (a) algebra, (b) business/
consumer math, (c) general/applied math, (d) ge
ometry, (e) prealgebra, or (f) remedial math. Ap
proximately 60% of the total sample (n = 425)
were special educators. These teachers were listed
as currently teaching students who were emotion
ally challenged, learning challenged, or had spe
cial needs and a special needs resource. The
percentage of special educators sampled was
greater than general educators, as the list of spe
cial educators was not limited to those teachers
who teach only mathematics. Thus, more special
educators needed to be sampled to account for in
eligibility (e.g., special education teachers who
teach students with learning challenges but do
not teach math). Surveys were then mailed to the
targeted teachers.
Response Rate. Of the 750 surveys mailed,
257 surveys were excluded from the analysis be
cause of concerns about (a) ineligibility (teachers
either returned surveys with a note or made con
tact by phone saying they do not teach relevant
courses; n = 250), or (b) incorrect school address
(n = 7). Thus, a total of 493 (278 special educa
tion; 215 general education) teacher surveys met
the criteria for inclusion. Of these, 101 special ed
ucation teachers returned the survey for a response
rate of 36%, and 78 general education teachers re
turned the surveys for a return rate of 36%. The
response rate remained below a common survey
Winter 2006
return rate of 50% (Weisberg, Krosnick, &
Bowen, 1989). This occurred even though there
were two mailings, a followup postcard sent be
tween the mailings, and a phone call to all eligible
participants after the second mailing.
DEMOGRAPHICS
Of the 176 teachers that responded to the ques
tion (3 were missing), 44% (n = 78) were general
education teachers and 56% (n = 98) were special
education teachers who teach math to students
with LD and/or E/BD (see Tables 1 and 2). Fur
ther, 69% (n = 122) of the total respondents were
female, most of whom were special education
teachers (n = 69, 57%). The majority of respon
dents noted being 40 years of age or older (n =
126, 72%) and holding graduate degrees (n =
105, 61%). No significant differences were deter
mined among general and special educators with
regard to gender, age, or educational level.
To determine if there were statistical differ
ences between general and special education
teachers on the variables included in Tables 2 and
3, chi square and ttests were conducted. If statis
tically significant differences were found, the
magnitude of the effect, or effect size (ES), was
computed (Cohen's d for the ttest; coefficient phi
for chi square analysis). Cohen's d values range
from .2 (small), .5 (medium), to .8 (large). For
coefficient phi, the range is 0 (proportions of the
sample are equal to the hypothesized proportions)
to 1 (proportions of the sample are as divergent as
possible to the hypothesized proportions).
As can be seen from Tables 2 and 3, there
were a number of differences between general and
special education teachers. Special education
teachers were almost 11 times more likely to teach
students with LD and/or E/BD, than general ed
ucation teachers. In addition, special education
teachers were almost 20 times more likely to have
special education credentials than general educa
tion participants, but more than 10 times less
likely to hold secondary math credentials. As ex
pected, no significant difference was determined
among general and special educators with regard
to holding an elementary math credential.
Statistical significance (t= 7.001, df= 89,
p = .000; ES = .60) was determined across teacher
type and the average number of students with LD
and/or E/BD who are provided assistance (e.g., in
class support, resource room), with special educa
tion teachers noting a greater average of students
who are provided assistance (M= 13.73, SD =
10.89) than general education teachers (M'I
3.08,. = 3.51). Concerning current teaching
position, 60% (n = 61) of special educators taught
full time in either a resource room or segregated
classroom setting. Only 12% of special educators
(n = 12) reported being a fulltime consultant or
team teaching on a fulltime basis.
Teachers also reported the type of math
courses they currently teach to students with LD
and/or E/BD. Special education teachers were five
times more likely to teach general math or basic
skills classes and three times more likely to teach
prealgebra classes to students with LD and/or
E/BD than were general education participants.
There was insufficient cell size to calculate chi
square statistics for the number of teachers teach
ing algebra II/trigonometry classes and
statistics/probability classes. However, general ed
ucation teachers were twice as likely to teach alge
bra II to students with LD and/or E/BD than
were special education teachers. No statistically
significant differences were found across teacher
type for algebra, geometry, and integrated/unified
mathematics.
Teachers were asked to indicate the number
of general and special education methods courses
taken that addressed teaching matlV to students
with LD and E/BD and their level of preparation
for teaching math to these students. GCeneral edu
cation teachers were almost twice as likely to take
general education riethods courses than were spe
cial education participants. Special education
teachers were more than twice as likely to take
special education methods courses than were geni
eral education participants.
For teachers who noted taking at least'one
special or general education methods course (n
187), respondents also reported their level qf
preparation for teaching math to students with
LD and/or E/BD as a result of the metho.ds
courses they had completed. For general educa
tion methods courses completed, special eduica
tion teachers indicated feeling more prepared to
teach math to students with LD and/or E/BD (M
= 2.28, SD = .95) than were general education
teachers (M = 2.75, SD = .92); whereas general
Exceptional Children
223
education teachers felt more prepared to teach
math to general education students (M = 1.74,
SD = .69) than were special education teachers
(M= 2.34, SD = .93). There were no statistically
significant differences in teachers' perceptions to
teach math to students with LD and E/BD as a
result of taking special education methods courses
(M =2.07, SD = .72 for general educators; M =
1.94, SD = .67 for special educators).
RESULTS
RESEARCH QUESTION 1:
INSTRUCTIONAL PRACTICES
Teachers were queried on their use of 14 instruc
tional practices with students with LD and/or
E/BD during instruction of basic math
skills/computational tasks (see Table 4) and multi
step problemsolving tasks (see Table 5). Overall,
the mean number of instructional practices on
basic math skills/computational tasks they re
ported using was 9.13 (SD = 2.86) for special ed
ucators and 6.17 (SD = 2.89) for general
education teachers. The mean number of instruc
tional practices they reported using on multistep
problemsolving tasks was 8.46 (SD = 3.08) for
special educators and 9.09 (SD = 2.57) for gen
eral education teachers. A statistically significant
difference existed 1Letween the groups on the aver
age number of instructional practices used with
basic math facts (t = 6.860, df= 177, p = .000;
ES = .46), with special educators reporting using
more instructional practices than general educa
tors. No statistically significant differences were
noted between special and general educators for
the overall use of these procedures with problem
solving tasks.
Basic Skills/Computational Tasks. As can be
seen in Table 4, special educators were more likely
to report that they used the following instruc
tional procedures with students with LD and/or
E/BD when teaching basic skills/computational
tasks: individualized instruction by teacher, addi
tional practice, reduced classwork problems, ex
tended time on assignments, problems read to
students, cue cards of strategy steps, individual
ized attention given to student by classroom aide,
and graphic organizers. Across special education
teachers, the four most reported applied instruc
tional practices during basic skills/computational
tasks included: calculators, individualized instruc
tion by teacher, extended time on assignments,
and problems read to students. General education
teachers reported using the following most fre
quent instructional practices: calculators, ex
tended time on assignment, individualized
instruction by teacher, and peer or crossage tu
toring.
ProblemSolving Tasks. As shown in Table 5,
special educators were more likely to report that
they used the following instructional procedures
with students with LD and/or E/BD when teach
ing multistep problemsolving tasks: individual
ized instruction by teacher, additional practice,
reduced classwork problems, extended time on as
signments, reduced homework problems, prob
lems read to students, cue cards of strategy steps,
use of concrete objects, and graphic organizers.
Across special education teachers, the four most
reported applied instructional practices during
multistep problemsolving tasks included: prob
lems read to students, individualized instruction
by teacher, extended time on assignments, and
calculators. General education teachers reported
most frequent use of the following instructional
practices: calculators, individualized instruction
by teacher, extended time on assignment, and
peer or crossage tutoring.
RESEARCH QUESTION 2:
ASSESSMENT ACCOMMODATIONS
Teachers were queried on their use of eight assess
ment accommodations with students with LD
and/or E/BD while assessing student knowledge
of basic math skills/computational tasks (see Table
4) and multistep problemsolving tasks (see Table
5). Overall, the mean number of assessment ac
commodations on basic math skills/computa
tional tasks they reported using was 4.52 (SD =
1.75) for special educators and 3.14 (SD = 1.36)
for general education teachers. The mean number
of assessment accommodations they reported
using on multistep problemsolving tasks was
4.54 (SD = 2.05) for special educators and 3.18
(SD = 1.40) for general education teachers. Statis
tically significant differences existed between the
groups for the average number of assessment
accommodations used with basic math facts
Winter 2006
TABLE 4
Instructional Practices and Assessment Accommodations Used During Instruction on Basic Math
Skills/Computational Tasks
No. (%) Significance
Special General
Education Teachers Education Teachers df X2 p ES
Characteristics Yes No Yes No
Type of Instructional Practices
Individualized
instruction by teacher 93 (52) 8 (4) 57 (32) 21 (12)
Additional practice 83 (47) 18 (10) 31 (17) 47 (26)
Reduced classwork
problems 74 (41) 27 (15) 33 (19) 45 (25)
Extended time on
assignments
Reduced homework
problems
Color coding
Peer or crossage
tutoring
Problems read to
students
93 (52) 8 (4) 61 (34) 17 (10)
58 (33) 43 (24) 31 (17) 47 (26)
23 (13) 78 (44) 9 (5) 69 (38)
64 (36) 37 (20) 48 (27) 30 (17)
93 (52) 8 (5) 40 (22) 38 (21)
Cue cards of strategy
steps 44 (25) 57 (32) 8 (4) 70 (39)
Use of concrete objects 66 (37) 35 (20) 38 (21) 40 (22)
Individualized attention
by class aide
Calculators
Graphic organizers
Mnemonics
55 (30) 46 (26) 21(12) 57 (32)
98(54) 3 (2) 71 (40) 7 (4)
36 (20) 65 (36) 13 (8) 65 (36)
42(24) 59(33) 20(11) 58(32)
1 11.706 .001 .26
1 34.268 .000 .44
1 17.545 .000 .31
1 7.050 .008 .20
1 5.504 .019 .18
1 3.783 .052 .15
1 .063 .802 .02
1 38.364 .000 .46
1 23.690 .000 .36
1 4.999 .025 .17
1 13.655 .000 .28
1  .106a .13
1 7.972 .005 .21
1 4.941 .026 .17
Type of Assessment Accommodations
Reduced problems on
tests 60 (34) 41 (23) 18 (10) 60 (33)
Extended time on tests 94 (53) 7 (4) 67 (37) 11 (6)
Calculators 93 (52) 8 (4) 69 (39) 9 (5)
Cue cards of strategy
steps 28 (15) 73 (41) 7 (4) 71 (40)
Color coding 13 (7) 88 (50) 4 (2) 74 (41)
Use of concrete objects 39 (22) 62 (35) 20 (11) 58 (32)
Problems read to
students
Individualized attention
given by class aide
86(48) 15 (8) 37(21) 41(23)
44(24) 57(32) 23 (13) 55 (31)
1 23.624 .000 .36
1 2.503 .114 .12
1 .670 .413 .06
1 9.835 .002 .23
1 3.070 .080 .13
1 3.352 .067 .14
1 29.117 .000 .40
1 3.724 .054 .14
Note. ES = coefficient phi;  = Insufficient cell size to calculate x2; = Fisher's exact test value.
Exceptional Children
225
TABLE 5
Instructional Practices Used During Instruction on Multistep ProblemSolving Tasks
No. (%) Significance
Special General
Education Teachers Education Teachers df x2 p ES
Characteristics Yes No Yes No
Type of Instructional Practices
Individualized
instruction by teacher 90 (51) 11 (6) 56 (31) 22 (12)
Additional practice 77 (43) 24 (13) 34 (19) 44 (25)
Reduced classwork
problems
Extended time on
assignments
Reduced homework
problems
Color coding
Peer or crossage
tutoring
Problems read to
students
Cue cards of strategy
steps
Use of concrete objec
Individualized attention
by class aide
Calculators
Graphic organizers
Mnemonics
62 (35) 39 (22) 29 (16) 49 (27)
87 (49) 14 (8) 53 (29) 25 (14)
58 (33) 43 (24) 22 (12) 56 (31)
17 (10) 84 (47) 5 (3) 73 (41)
57 (32) 44 (25) 46 (25) 32 (18)
91 (51) 10 (6) 45 (25) 33 (18)
48 (27) 53 (30) 13 (7) 65(36)
ts 63 (35) 38 (21) 28 (15) 50 (28)
50 (28) 51 (29) 25(14) 53 (29)
87 (49) 14 (8) 66 (36) 12 (7)
34 (19) 67 (37) 9 (5) 69 (38)
33 (18) 68 (38) 16 (9) 62 (35)
1 8.774 .003 .22
1 19.913 .000 .33
1 10.318 .001 .24
1 8.545 .003 .22
1 15.203 .000 .29
1 4.434 .035 .16
1 .116 .733 .03
1 25.324 .000 .38
1 18.655 .000 .32
1 12.346 .000 .26
1 5.507 .019 .18
1 .082 .774 .02
1 11.804 .001 .26
1 3.274 .070 .14
Type of Assessment Accommodations
Reduced problems on
tests 66 (37) 35 (20) 22 (12) 56 (31)
Extended time on tests 90 (50) 11 (6) 63 (35) 15 (9)
Calculators 89 (50) 12 (7) 67 (37) 11 (6)
Cue cards of strategy
steps 30 (17) 71 (40) 8 (4) 70 (39)
Color coding 13 (8) 88 (49) 6 (3) 72 (40)
Use of concrete objects 49 (28) 52 (29) 17 (9) 61 (34)
Problems read to
students
Individualized attention
by class aide
82(46) 19 (11) 41(23) 37(20)
40(22) 61(34) 24 (14) 54 (30)
1 24.292 .000 .37
1 2.465 .116 .12
1 .194 .660 .03
1 9.953 .002 .24
1 1.244 .265 .08
1 13.500 .000 .28
1 16.774 .000 .31
1 1.495 .221 .09
Note. ES = coefficient phi.
Winter 2006
226
(t = 5.753, df= 177, p = .000; ES = .40) and
problemsolving tasks (t = 5.044, df= 177, p =
.000; ES = .35), with special educators reporting
using more assessment accommodations than
general educators.
Basic Skills/Computational Tasks. As can be
seen in Table 4, special educators reported that
they more likely used the following accommoda
tions with students with LD and/or E/BD when
assessing knowledge on basic skills/computational
tasks: reduced problems on tests, cue cards of
strategy steps, and problems read to students.
Across special education teachers, the four most
reportedly applied accommodations provided
when assessing knowledge of basic math
skills/computational tasks included: extended
time on tests, calculators, problems read to stu
dents, and reduced problems on tests. General ed
ucation teachers reported using the following
accommodations most frequently: calculators, ex
tended time on assignment, problems read to stu
dents, and individualized attention given to
student by classroom aide.
ProblemSolving Tasks. As shown in Table 5,
special educators reported that they were more
likely to use the following accommodations with
LD and/or E/BD when assessing knowledge of
multistep problemsolving tasks: reduced prob
lems on tests, cue cards of strategy steps, use of
concrete objects, and problems read to students.
Across special education teachers, the four most
reported applied accommodations when assessing
knowledge of multistep problemsolving tasks in
cluded: extended time on tests, calculators, prob
lems read to students, and reduced problems on
tests. General education teachers reported using
most frequently the following accommodations:
calculators, extended time on tests, problems read
to students, and individualized attention given to
student by classroom aide.
RESEARCH QUESTION 3: FACTORS
AFFECTING TEACHER RESPONSE
Four nonhierarchical regression analyses were
conducted to examine the contribution of three
predictor variables on the number of instructional
practices and assessment accommodations re
ported by general and special education teachers.
This was done separately for general and special
education teachers for the following two indices:
total number of reported instructional practices
and total number of reported accommodations.
First, each predictor variable was entered into the
first position of the linear equation to determine
if it was significantly related to the outcome vari
able (e.g., total number of reported instructional
practices). In addition, each variable was entered
into the last position to determine if it con
tributed to the prediction of the outcome variable
(e.g., total number of reported accommodations)
beyond the contribution of the other two vari
ables. The predictor variables were (a) years teach
ing students with LD and/or E/BD; (b) number
of methods courses completed; and (c) knowledge
of math topics (i.e., prealgebra, algebra, geometry,
general or basic skills math, algebra II, and alge
bra II/trigonometry). Data for knowledge of math
topics is included in Table 6. It is interesting to
note that general education teachers were more
familiar than were special education teachers with
the topics of prealgebra, algebra, geometry, alge
bra II, algebra II/trigonometry, statistics/probabil
ity, and integrated/unified high school math.
INSTRUCTIONAL PRACTICES
Special Education Teachers. The three pre
dictor variables accounted for 11.7% of the vari
ance in the total number of instructional practices
special education teachers noted using with stu
dents with LD and/or E/BD on basic math/com
putational skills and problemsolving tasks (see
Table 7). The ttest for Beta weights was statisti
cally significant for two predictor variables
(knowledge of math topics and number of meth
ods courses) when entered into the first position
of the regression model. When entered into the
last position, one variable (knowledge of math
topics) contributed significantly to the prediction
of instructional practices above and beyond the
other predictor variables. Knowledge of math top
ics accounted for 6.1% of the total variance after
the other variables were controlled.
General Education Teachers. The three pre
dictor variables accounted for 12.5% of the vari
ance in the total number of instructional practices
general teachers reported using with students with
LD and/or E/BD on basic math/computational
skills and problemsolving tasks (see Table 7). The
ttest for Beta weights was statistically significant
for the first predictor variable (number of meth
Exceptional Children
TABLE 6
Teacher Perceived Knowledge of Secondary Math Topics
Significance
Special General
Education Education
Teacher Perceptions Teachers Teachers df t p ES
How much do you feel you know
about each of the math topics listed
below?
Prealgebra 1.53 (.70) 1.06 (.30) 134 5.497 .000 .43
Algebra 2.07 (.84) 1.12 (.40) 143 9.153 .000 .61
Geometry 2.54 (.86) 1.49 (.70) 169 8.590 .000 .55
General Mathematics or Basic
Skills (HS) 1.14 (.38) 1.08 (.32) 167 1.117 .265 .09
Algebra II 2.92 (.82) 1.35 (.60) 166 13.958 .000 .74
Algebra II/Trigonometry 3.37 (.80) 1.62 (.76) 164 14.464 .000 .75
Statistics/Probability 3.07 (.80) 2.01 (.84) 157 8.275 .000 .55
Integrated/Unified High School
Math 2.98 (1.03) 2.14 (1.06) 152 5.095 .000 .38
Note. ES = Cohen's d. Ratings based on a 4point scale with 1 = a great deal; 2 = quite a lot; 3 = some;
4 = very little or nothing.
ods courses) when it was entered into the first and
last position of the regression model. For general
education teachers, the number of methods
courses taken accounted for 9.6% of the variance
after the other variables were controlled.
ASSESSMENT ACCOMMODATIONS
Special Education Teachers. The three pre
dictor variables (years teaching students with LD
and/or E/BD, knowledge of math topics, and
number of methods courses) accounted for 4.7%
of the variance in the number of assessment ac
commodations used by special educators (see
Table 8). The ttests for Beta weights were not
statistically significant for any of the predictor
variables when they were entered into the first po
sition or the last position of the regression model.
General Education Teachers. The three pre
dictor variables accounted for 12.5% of the
variance in the number of assessment accommo
dations made by general educators (see Table 8).
The ttest for Beta weights was statistically signifi
cant for one predictor variable (total number of
methods courses) when entered into both the first
and last position. For general education teachers,
the number of methods courses taken accounted
for 11.1% of the total variance above and beyond
the other variables.
DISCUSSION
This study provides a national picture of sec
ondary general and special educator views con
cerning their reported level of preparation to
teach math and their perceived knowledge of
math. We also examined teacher use of specific
instructional practices and assessment accommo
dations. In addition, we examined the contribu
tion of experience, knowledge, and preparation in
predicting teachers' reported use of instructional
practices and testing accommodations.
We found that special education teachers
indicated they were less familiar with secondary
math knowledge than were general educators (the
only exception involved general mathematics or
basic skills). Furthermore, both general and spe
cial education teachers take few methods courses
that focus on teaching math to students with LD
and/or E/BD. Only one of every two general
Winter 2006
228
TABLE 7
Nonhierarchical Regression Analysis for General Education and Special Education
Teachers' Instructional Approaches
Construct Entered
Initial Entry of the Construct in Last Position
Variables Simple R R2 pvalue R2 Increment pvalue
General Education Teachers
Mean perceived knowledge .090 .008 .444 .009 .488
Total number of methods courses .368 .136 .006 .096 .025
Total years teaching LD and/or E/BD .124 .015 .279 .007 .533
Special Education Teachers
Mean perceived knowledge .234 .055 .031 .063 .030
Total number of methods courses .247 .061 .023 .031 .124
Total years teaching LD and/or E/BD .055 .003 .587 .026 .157
education teachers took such a course, whereas educators for computational and problemsolving
special educators averaged only 1.27 courses of tasks; (b) special education teachers' familiarity
this nature. It is possible that the relatively small with course topics significantly and uniquely con
numbers of methods courses taken by these teach tribute to the prediction of the number of in
ers influenced their general perceptions that they structional practices they provided to secondary
were not adequately prepared to teach math to students with LD and/or E/BD in math; and (c)
these students. These findings are consistent with the number of methods courses taken by general
previous research (Maccini & Gagnon, 2002). education teachers contributed to the prediction
SPECIFIC INSTRUCTIONAL of the number of instructional practices they made
for students with LD and/or E/BD in math.
PRACTICES AND FACTORS
A statistically significant difference existed
In the present study, it was determined that (a) between special educators and general educators
special educators used significantly more recom on the average number of instructional practices
mended instructional practices than did general used with basic math facts, with special educators
TABLE 8
Nonhierarchical Regression Analysis for General Education and Special Education
Teachers'Assessment Accommodations
Construct Entered
Initial Entry of the Construct in Last Position
Variables Simple R R2 pvalue R2 Increment pvalue
General Education Teachers
Mean perceived knowledge .047 .002 .688 .002 .744
Total number of methods courses .368 .135 .006 .111 .016
Total years teaching LD and/or E/BD .063 .004 .583 .009 .493
Special Education Teachers
Mean perceived knowledge .080 .006 .465 .012 .345
Total number of methods courses .207 .043 .057 .028 .161
Total years teaching LD and/or E/BD .043 .002 .670 .007 .473
Exceptional Children
reporting using more instructional practices than
general educators. In fact, for each instructional
procedure, all but three instructional practices
(i.e., color coding, peer or crossage tutoring, and
calculators) were statistically significantly differ
ent, favoring special educators. No statistically
... both general and special education
teachers take few methods courses that
focus on teaching math to students with
LD and/or E/BD.
significant differences were noted between special
and general educators for the overall use of the 14
instructional procedures with problemsolving
tasks. However, for each instructional procedure,
special educators favored all but three (peer or
crossage tutoring, calculators, and mnemonics).
Certain contextual information may help to
explain teacher differences. First, these two groups
of teachers differ greatly in terms of years of expe
rience teaching students with LD and/or E/BD
(M= 12.36 years for special educators; M= 1.15
years for general educators). Also, general educa
tion math teachers may have only a few students
with disabilities in their classes; whereas special
educators may be working almost entirely with
students with special needs.
Despite differences between general and
special educators in the number of recommended
instructional practices, there were several similari
ties among the most frequent instructional prac
tices reported. Specifically, during basic
skills/computational tasks, special and general ed
ucation teachers both reported using the recom
mended practices of individualized instruction by
teacher, extended time on assignments, and use of
calculators. The use of calculators was also a com
mon response across teachertype during prob
lemsolving tasks. These results are encouraging
given the support from the literature for the criti
cal need of providing enough time for students
with LD to learn information, establishing high
success rates, and developing strategies for helping
students to learn and retain information (Mercer,
1997). Although no studies currently exist that
examine the efficacy of standard and/or graphing
calculators on the math performance of secondary
students with LD, calculators hold great promise
for students with special needs. Maccini and
Gagnon (2005) noted the positive results of cal
culator use with nondisabled students relative to
motivational factors, concept development, and
student achievement.
Although the fourth most frequent re
sponse across teacher types differed, the variation
may be a function of resource availability. For in
stance, special education teachers reported using
"problem read to students" as their fourth most
frequent instructional procedure, whereas general
education teachers noted the use of "peer or cross
age tutoring" for helping students with special
needs on problemsolving and basic skills/compu
tational tasks. This is not surprising given the het
erogeneous nature of general classrooms with
varied skill levels and the need to cover the cur
riculum. Use of peer tutoring has proven benefi
cial during independent practice for secondary
students studying introductory algebra skills (All
sopp, 1997).
SPECIFIC ASSESSMENT
ACCOMMODATIONS AND FACTORS
We found that special educators used more assess
ment accommodations than did general educators
for both computational and problemsolving
tasks, and the total number of methods courses
taken predicted teacher use of assessment accom
modations. More specifically, we found that spe
cial educators were more likely than general
We found that special educators used
more assessment accommodations
than did general educators for both
computational and problemsolving
tasks...
educators to reduce the number of problems on
tests, use cue cards of strategy steps, and problems
read to students for both basic math skills and
problemsolving. They were also more likely to
use concrete objects as an accommodation for
problemsolving.
Winter 2006
Despite differences among general and spe
cial educators in the number of assessment accom
modations, there were several similarities among
the three most frequent assessment accommoda
tions reported by both groups. These included (a)
extended time on tests, (b) calculators, and (c)
problems read to students. These results are posi
tive given the documented research. Specifically,
Thompson et al. (2002) summarized and critically
evaluated empirical research studies from 1999 to
2001 that examined the effects of specific assess
ment accommodations. The researchers catego
rized the types of accommodations across
common formats (presentation, response, setting,
timing/scheduling, multiple accommodations,
other) and determined that the use of two formats
(extended time and oral presentation) had gener
ally positive results on students' test scores. For ex
ample, Calhoon et al. (2000) noted positive results
on a math performance assessment for secondary
students with LD with a teacher read aloud ac
commodation, rather than standard paper/pencil
administration. Helwig et al. (1999) reported sim
ilar results for students experiencing math difficul
ties, regardless of their reading ability level. With
extended time, Fuchs, Fuchs, Eaton, Hamlett, and
Karns (2000) determined positive results for stu
dents with LD in math on a curriculumbased
math measurement.
It should also be noted that there were
some inconsistencies between teachers' use of spe
cific procedures for instruction and for assess
ment. For example, the use of calculators was a
frequent response for special education teachers as
an assessment accommodation during problem
solving tasks, but was not among the most fre
quent responses as an instructional procedure. It
is important that students have training and expe
rience with available assessment accommodations
during instruction (Thurlow, 2000).
LIMITATIONS AND FUTURE
RESEARCH
Five important limitations existed within the cur
rent study. First, generalization of findings may be
at risk because of the small sample size and the
low response rate. The response rate remained low
despite multiple mailings and repeated reminders.
Also, comparisons between the respondents and
nonrespondents were not available. This was due
to the restrictions concerning participant confi
dentiality from the survey company that managed
the mailings and returns. Future research needs to
include a larger sample and comparisons across
respondents and nonrespondents to increase the
validity of findings. Also, it is not known whether
the general educators sampled had some students
with special needs in their class or if they taught a
class completely comprised of students with spe
cial needs. Variations in students within a class
may have an impact on instructional approaches
and assessment strategies and should be studied
further. Furthermore, it may be of concern that
the same choices existed for questions concerning
basic computational and problemsolving tasks.
Future research should include a more compre
hensive list of instructional practices.
In the current study the number of meth
ods courses uniquely and significantly con
tributed to the prediction of the number of
assessment accommodations used by general edu
cators. However, two issues remain unanswered:
(a) details concerning the methods courses taken
(e.g., focus of the courses, when courses were
taken, how course activities related to field experi
ence); and (b) the extent to which the reported
assessment accommodations were appropriate in
type and number for students and whether they
were aligned with state policies (Pitoniak &
Royer, 2001). Future research should examine the
effects of types of methods courses to the number
of instructional practices and assessment accom
modations used. Also, it is important to note that
the total amount of variance accounted for in the
regression analysis was relatively low. Future re
search needs to expand the list of possible predic
tors used to assess reported instructional practices
and accommodations.
INSTRUCTIONAL IMPLICATIONS
This study raises several questions about who
should teach math to secondary students with LD
and/or E/BD. On the one hand, general educa
tion teachers appear to be more knowledgeable
about secondary math. On the other hand, spe
cial education teachers have more formal course
Exceptional Children
231
work on how to teach math to students with LD
and/or E/BD, and they are more likely to use the
recommended instructional practices and accom
modations that were included in this study. In
stead of worrying about who is best equipped to
teach these students, it may be more productive
to examine how general and special educators can
work together to facilitate these students' progress
in math (Thompson, Lazarus, & Thurlow, 2003).
As Kozleski, Mainzer, and Deshler (2000) stated,
"Special and general educators must work to
gether to ensure that the highest possible number
of students with exceptionalities successfully ac
cesses the important concepts and skills in the
general education curriculum" (p. 11).
The findings from this study indicate that
special education teachers need more preparation
in content of secondary mathematics, and general
educators would benefit from additional prepara
tion in instructional practices and accommoda
tions for learners with special needs. It is
unfortunate that some states do not have math re
quirements for special education preparation pro
grams (Graham & Fennell, 2001), as teacher
content knowledge and course completion have a
positive effect on student achievement (Good &
Grouws, 1987; Tooke, 1993).
One approach to increasing special educa
tors' knowledge of secondary mathematics is to in
tegrate such information into existing preservice
methods courses. Methods courses that addressed
both mathematics pedagogy and domain knowl
edge had a significant impact on preservice teach
ers' attitude toward math and significantly
increased their domainspecific knowledge of
math (Quinn, 1997). Jenkins, Pateman, and Black
(2002) also agreed that dual certification programs
can help secondary special education teachers ob
tain contentspecific knowledge and general edu
cators sharpen their instructional practices.
Another approach for enhancing special ed
ucators' knowledge and general educators' in
structional practice is the use of ongoing,
comprehensive staff development opportunities
(Koency & Swanson, 2000; Whittington, 2002).
Whittington reported that 55% of all mathemat
ics teachers surveyed in the 2000 National Survey
of Science and Mathematics Education indicated
a need for more professional development oppor
tunities that focus on teaching students with spe
cial needs. Obviously, such support can be offered
in conjunction with university coursework (Time
is Key, 2002 in Thompson et al., 2003).
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ABOUT THE AUTHORS
PAULA MACCINI (CEC MD Federation), Associ
ate Professor, Department of Special Education,
University of Maryland, College Park. JOSEPH
CALVIN GAGNON (CEC VA Federation), Assis
tant Professor, Graduate School of Education,
George Mason University, Fairfax, Virginia.
Correspondence concerning this article may be
sent to Paula Maccini, University of Maryland,
Special Education, 1308 Benjamin Building, Col
lege Park, MD 20742. (email: maccini@
umd.edu)
This research was supported by Grant
#H324N990029, CFDA #84.324N Office of
Special Education Programs, U.S. Department of
Education, Office of Special Education and Reha
bilitative Services. The views expressed herein do
not necessarily represent those of USDE.
Manuscript received November 2003; accepted July
2005.
Winter 2006
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