dawS. . . . . . . . . . . . . . . . .
TeacherReported Use of Empirically
Validated and StandardsBased
Instructional Approaches in
Secondary Mathematics
JOSEPH CALVIN GAGNON AND PAULA MACCINI
ABSTRACT
A random sample of 167 secondary special and
general educators who taught math to students with emotional
and behavioral disorders (EBD) and learning disabilities L: . re
sponded to a mail survey. The survey examined teacher percep
tions of (a) definition of math; (b) familiarity with course topics;
(c) effectiveness of methods courses; (d) preparation to use and
frequency of use of effective instructional strategies; and (e) fac
tors . r ,ir;i. ,.i :i to the use of effective instructional strategies. The
number of methods courses taken by teachers uniquely and : ,
icantly contributed to the number of effective instructional ap
proaches that both general and special educators reported using.
Significantly more special educators than general educators re
ported the use of specific, empirically validated instructional prac
tices, Additional results, implications, and suggestions for future
research are provided.
A ACCORDING TO THE THIRD INTERNATIONAL
M ath and Science .ir . i'. I N\ . 1~.\ M iiil.I et al.,  I1jii. U.S.
..ii;Ii * .l.k students scored liitl', higher on mathematics
than the international average, but below the performance
level of 18 other countries (Wang, Coleman, C.,1e . & Phelps,
2003). 1 , most recent data from the I I . I indicated that
U.S. eighth graders were outperformed by students in 9 other
countries (Gonzales et al., :_'I . Also, within the United
.Sur , only one fourth of 8th and I'th. r.,.!e students
achieved at a proficient competency level in math on the Na
tional Assessment of Educational Progress (NAEP; L:.i .' .v I
et al,, .' i11). It is evident that dramatic improvements in stu
dent math achievement are needed at the rid.Jl and high
school levels (..: il in.l Institute on Educational Governance,
Finance, Policymaking, and M.r; .' i.:nti 1998).
A1irl:oi i_ reasons vary for the low math performance of
students in the United States and the poor ji'.f,:,rnuji. .r of
U.S. students versus their international peers, Jt i.ri. ..,, in
instruction may help to Il'l:riin.i. this disparity. For example,
in the United States, less than 1% of math lessons focused on
higher level math i.: ..rii: .." ill , compared to 44% of the
time in Japan (Bernstein, 1997). As such, improving math in
struction via :... .' ,iuri[.hill . for teacher qualifications and re
form of math education is widely advocated (National
Council of Teachers of Mathematics, 1989, 2000; '\i r; et al.,
2i' .). To address teacher i'...lJific;i, . the U.S. C,.;i!: .
,.,iri passed the N.. (C.'l[ Left Behind Act (NCLB) of
2001. The provisions of this legislation include the need for
more highly qualified teachers in core content instructional
areas. Inii,., teachers must have the '.., .Lr._u ..riJ knowledge
and be lil, : l l [ il': to teach a core academic content area.
Also, to increase ;, :...rLuniiji, fir ,il students, including
students with .l1...ill ln, NCLB mandates increased partici
pation on highstakes state assessments.
The N\ ri..il Council of Teachers of Mathematics
(NCTM'i standards are another impetus of current math re
form in American schools (Woodward & M foi t.i, L... 2002).
The N I :. I standards emphasize the need to empower .ll stu
RIMCIAL AND SPECIAL EDUCATION 43
Volume 28, Number i, JantuaavFebruary 2007, Pages 4356
dents to think i, !it: .l: ... 1 ,'. il,,., is accomplished via
teacher facilitation of students' active engagement during
math lessons. Rather than . . ii ii,_ on basic skills instruc
tion in isolation, the instructional focus of the '.i l I\I stan
dards includes teaching higher level 0il ! ir: . reasoning, and
[r 1'..:, 1' .L skills that relate to the real world and that
address conceptual Iu. i..i i :1;id (NCTM, 2 3 1,1 .
However, the focus on more openended problemsolving
tasks may be problematic for students with emotional and
behavioral disorders rFT.n) and learning disabilities (LD).
These students often have difficulties i irl independent seat
work, particularly when the task r .. ! ir.: extended focus to
solve a.mathematical problem (Carr & I'h,;.:.. 1993; Guntner
& Denny, 1998; Hogan & Prater, 1993). Sm.i'.ii.l. may also
have. I!tilui l ult with l"i..,. r level problem, 1 . ..n.. tasks that
include algebraic ihi;.in ; (Hutchinson, 1993; Maccini, Mc
Naughton, & Ruhl, 1999), rPii :!,.,iin word problems
(Ml.,it. .II'e. Bos, & Doucette, I '.i I. solving for the solution
(Algozzine, O'Shea, Crews, & I' lJ.,i.ir, I' ), and moni
toring their problemsolving plrf .r'ra.,_:. : (Brown & Palinc
sar, :'. 2' Given these learner characteristics, teachers need
to be prepared to teach math to students with special needs
and to accommodate in Ir. . Ilu.i I learner needs.
To :'.f.. i I .: 1.' support secondary students with EBD and
LD in ,.!: teachers are expected to rely on empirically val
idated instructional approaches (NCLB, 2001). However,
teachers' instructional decisions are based on complex per
sonal and situational reasons M. ,_i.;l:lii~i 1998). In a study
r Maryland teachers, researchers i, [... 1 ,i & Gagnon, 'T' 'i '
noted that three factors in:., affect teacher decisions about
what and how to teach students: (a) teacher knowledge of
and Fr'ni;1..'rt with the content, (b) teacher preparation, and
(c) teacher beliefs and orientation. Specifically, Maccini and
Gagnon noted that the number of methods courses taken by
teachers; .i;,. i;,i. contributed to teacher confidence on
ir'lI ri..!'i,.il' the goals of the NCTM standards with students
classified as having EBD and LD. Furthermore, a ininrttI .iT
. I t: t i, I. was found between general and special educators'
confidence in their ability to teach math relative to the goals
of the NCTM standards, with special educators feeling less
prepared. Given current reform t..r ti: and the difficulties that
students with EBD and LD may have .... ?,i." 'ni in math, it is
important to consider teacher factors that .t !. 1 instructional
decision making. In the f.ll.. , in _I sections, the three :,... r.
noted by Maccini and Gagnon are discussed ... ;. rrTi. teacher
:I .' 1i ' I for and actual use of instructional strategies that
are empirically validated and consistent with the NCTM stan
dards. These :'... r. [ served as the basis for the current survey
of secondary .:,:;,,. r l7 and i.,. ,; 1 educators who teach math to
students with EBD and LD.
KNOWLEDGE/FAMILIARITY
Content and pedagogical content knowledge are critical fac
tors in creating and sustaining instruction that promotes stu
dent discourse and conceptual , I iI.; lin. L . i . I .. 1 '1 Ii,
1998). However, Manouchehri noted that math teachers com
monly do not have the necessary mathematical skills and un
1i : '.ci.Lr..' Other researchers i(M. . i:, & i .r . i ;'i:,. also
reported that there was a significant difference between sec
ondary ; :r(Al and special educators, with ,.i,:i I educators
reporting greater [p q i'tr',t;~ to teach all math course topics
except the combined category of 2:i.0. 1.11 math and basic
.1. ill
In addition to ., ,nlit .f .... knowledge, teachers need
to be 'rr:!.:...l ._r and use instructional .i,.. i,i.;  consistent
with the goals of the NCTM standards and .:;n;, r ..I. vali
dated t.:.lc""r. practices to assist students with F il ) and LD
(Maccini & f.. ;r. ir.. 2000). Based on a review of the research,
Maccini & Gagnon (2000) determined that recommended
instructional strategies include (a) instructional strategies
consistent with the ''l :'1 I standards, (b) direct instruction,
(c) graduated instruction, (.: grouping practices, and (e) self
i,.:,ri . ,inr:. Practices i fl..., ' .. . of the '.j I .1 [. !riIIil. ; focus
on understanding mathematical concepts and problem solv
ing via active student participation. Maccini and Gagnon
, I, ":, reported that teachers of students with EBD and LD
Fr:'.. lu.r '1.' reported using instructional approaches consistent
with NCTM, such as calculators and embedding math in real
world tasks. T:i, l '..Ji'. is promising given that activities
embedded in realworld problemsolving tasks positively af
fect both achievement and generalization for students with
i, l:lhli[t.?. ; i. .rt , ... 1999; 1.  ,'i.. & H . . .itI .;', 1993; B. ' :: .
Heinrichs, h al & Serlin, 2001; F:.' _I' Heinrichs, Mehta,
& Hung, 11',
In addition to studentcentered instructional strategies,
the teachers surveyed by . .a.. i.i and Gagnon I :''I'II also
recommended more 'i.:.,i ..r directed" .ipi'r T I' . reflective
of direct instruction principles (e.g,, tc. 1 .': l 1 ', teacher
feedback, review; see Rosenshine & M'.,. I, 1986). Inter
ventions that include .., Liir illi...[.'il approaches are effec
tive for ".a..1J,ii' math concepts to secondary students with
highincidence .i . 'l iiri i ._ I I Gersten, & Carnine, 1990;
Moore & Carnine, 1989).
Yii, , additional instructional .'r~: i:.. have been vali
dated with students with L L.1 and LD in math, I ir : gradu
ated instruction refers to a threephase i;''r.: 1 to math that
includes (a) a concrete phase ('  . representing concepts via
cI':;.L t; (b) a semiconcrete phase (e.g., drawing pictures of
objects), and (c) an abstract phase (using numerical represen
tations). Use of a graduated .aiiT I..... to instruction is. *:. 
tive for hlpI :l'i students with LD learn higher level math
(e.g., relational word problems, ;rir. r TTliri.i,. 1994;
'1.lu. ,n & Hughes, '1 i' Maccini & Ruhl, 2000).
Second, the use of grouping ; ,..'. . . such as coopera
tive learning activities, can promote academic success for
secondary students with EBD and LD. For example, class
wide peer tui.:.. *~n, was effective in '...:.h ir,; algebra problem
solving skills to 14 to 15yearold students (All "i. , 1997).
As ir,. n students with EBD and LD may take algebra in high
school, this approach is i':";lli.'i,. for Li; i school algebra
44 P gMEDlAL AND SPECIAL EDUCATION
Volume 28. Numbher I. JanwiayFebriary 2007
classes and allows the teacher to conduct smallgroup in
struction and provide assistance while other students work in
cooperative  :..11
The final validated instructional , ii:  of interest is
selfmonitoring. Use of this strategy helps students to moni
tor their problemsolving behaviors via selfquestions (e.g.,
what do I know? what do I need to find?) while _: r:1 . up and
solving a problem ?.,Iii .1i.. & i:.. 1 1''I;. Use of self
monitoring strategies is a necessary component of successful
algebra performance (Maccini, MT. V.iu._ *... & Ruhl, 1999).
TEACHER PREPARATION
Teacher preparation, which i1 I: affect teacher use of instruc
tional strategies, includes trJir iin programs for preservice
special education teachers and j. .! . !. comprehensive pro
fessional de l 1r'r:i.i opportunities (i.' workshops, pro
grams, courses) f. r r .1. !..::. teachers (Culatta, Tompkiis, &
Werts, 2003). The importance of the preservice and inservice
training, as noted in the Individuals with Disabilities Educa
tion Act (IDEA) of .y,"'i, is supported by research. For exam
ple, ,. iriL (I *' 1. noted a link between student achievement
and the number of math courses taken by teachers. 'Siin.r r;.,
Tooke (1993) found that the achievement level of students in
prealgebra increased as the skill level in math of the teachers
increased. A positive . tt' . r was determined between content
course completion by teachers and the skill level of students,
As Koency and Swanson ..: 'I, stated, ir.: hcr who pos
sess a deep and broad ir:.;, I;,ii _.. of fundamental math
provide more rI. .1i. , instruction for their students, which in
turn leads to higher student achievement in math" (p. 3).
Maccini and GC  Li.' I,. ',' also noted that the number of
methods courses  i; ti: .. :1 differed between general and
special educators. ii r :i:i ..' . they reported that the number
of methods courses predicted teachers' ..... t,.i:i. in their
. ,lii, to teach math relative to the NCTM standards.
TEACHER BELIEFS
Teacher instructional practices may be manifestations of their
consciously held notions, I :I;.. r: and preferences (A. G.
I i.. .,,". , . 1984). According to Ernst (1988), there are three
primary views of math: (a) a 'I 1l. imi,,  . i i view (i.e., a dy
namic and problembased approach, wherein math is a dy
namic process of inquiry); (I. a platonist view (i.e., math is a
static and unified body of knowledge); and (c) an instrumen
talist view (i.e., math is a collection of tools consisting of
facts, rules, and skills used to achieve an external i in.,, In one
study of 11 secondary math education majors, the most com
mon I.. . i .t i. .r:. . . t. ni.lT i " i included a tool, language,
adventure, exercise for the mind, exploration, and problem
solving (Pachnowski, 1997).
If teachers' definition of math is, in fact, related to their
use of instructional ir.,tc'e. , when e it iin. math, then at
tempts to improve math instruction must begin with an un
.1kiT 1i' h. of that r. l.L. 1. i,: (A. G. Thompson, i".. i. For
example, if teachers define math as a set of calculations, ..
may invariably neglect the use of instructional strategies con
sistent with Ni T . I If teachers ,*r' .', I. math as a tool, there
is an indication that it would link math to problems set in a
realworld context.
FACTORS AFFECTING INSTRUCTIONAL
STRATEGY USE
Given the importance of researchbased instructional strate
:, it is important to identify variables that affect teachers'
use of these strategies. Limited evidence has I:. . .:.: that
two variables uniquely predict the number of researchbased
and recommended instructional approaches made by special
and general education teachers: (a) teachers' perceived knowl
edge of math topics (for special education teachers) and
(b) the number of methods courses taken (by ,rr.,:i.,T edu
cation teachers; Maccini & G0 1ii,. 1l',  Moreover, the
number of years of ' .. ,it � experience may affect teachers'
use of instructional strategies. l>.'; teachers (i.e., 5 years or
less) are less effective than experienced teachers (D. rli; ..
Hammond, 2000; Rivkin, Hanushek, & Kain, 2., 'P 1.. As Gra
ham, Harris, Fink, and MacArthur (2003) noted in a study
with primarygrade teachers, this may be due to teachers with
more years of experience being more knowledgeable about
the modifications necessary to help students who are aca
demically at risk. However, further research is needed to
i. rLt:'  if teachers' perceived knowledge of math t1 .r . . the
number of methods courses taken, and years of experience
predict teacher use of researchbased instructional practices.
PURPOSE OF STUDY
Teacher mi i: . and use of.. ipr...i.i,' validated instruc
tional strategies and strategies consistent with the NCTM
standards are critical to prepare students with i;.i'iiili.. to
pass state assessments !.:c.;r: & Gagnon, .'I 1,'1 However,
to date, no national iF ,. ,i,., rii exists that examines the fac
tors that contribute to teachers' use and frequency of use of
these instructional strategies. Also, there is limited informa
tion available that .ii..i.. r. a relationship between teachers'
t f :iiii i'. of math and their approach to math 1 NI i X 11 I ,
A. G. Thompson, ': i. Therefore, the current survey was
designed to answer the following research questions concern
ing special and general education teachers:
1. How do teachers ,... linr math, and does their
orientation contribute to their use of instruc
tional strategies?
2. How Ir, i I .i are teachers with secondary
course topics (e.g., prealgebra, algebra,
_':",,el. l ". Ir  '
REMEDIAL AND SPECIAL EDUCATION 45
Vouime 28, Niumhber 1. JantarylFebruamy 2007
3. Do teachers perceive special and .  �. ,
methods courses as effective in preparing them
to teach math to students with special needs
and general education students?
4. How do teachers perceive their preparation to
use instructional strategies consistent with
NCTM standards, .1fllr.i..,; ,II, direct
instruction,, .i__.111..t,1 instruction, and student
groupings, and how often do ~;, use instruc
tional strategies consistent with these
approaches?
5. What factors contribute to teachers' use of
instructional strategies?
METHOD
Sample
The i.; . t population consisted of a random sample of two
':. i .u, of teachers from all public high schools in the United
States: (a) secondary general education math teachers who
teach students with LD and EBD in their math classes; and
(b) special education teachers who either teach math to
students with LD and EBD or collaborate with general edu
cation math teachers. lii sample was obtained from the
Quality Education Data (,.:L. database (....' :., Education
Data School Personnel Database, 20002001). QED is a
comprehensive database of U.S. schools and school person
nel and has been used in national research (Carlson, Lee,
Schroll, Klein, & lii_.., 2002). The sample included .s
general educators who were listed as teaching math under
several course subcodes (i.e., algebra, business/consumer
math, general/applied math, .'.. r.:tr,, . l:,i;.r.i remedial
math, and il,.,h and teaching students served in special edu
cation. Also included in the sample were 425 special educa
tors who were listed under subcodes if currently teaching
students with emotional ,. h.Il nmc.. learning challenges, spe
cial needs resource assistance, or special needs. No special
education subcode :'.'.i'in: ully I .inr!i'k,. those educators
teaching rn ,l, ini.r.., r. , the .1, ,,dil :,' that some partici
pants were ineligible for participation (e.g., special education
teachers who teach students with learning 5. 1lnii. , but do
not teach math). iin; , a l.,r_ : number of rcii.il educators
than general educators was sampled. Following the survey
administration, 101 ,.. i:r.l education teachers were found to
be in1i 2il!: (i.e., 99 teachers returned surveys with a note or
contacted researchers by phone saying they did not teach rel
evant courses; 2 school addresses were incorrect). Further
more, 172 special educators were found to be ir.,:ii.hl.: for
the survey (i.e., 171 teachers returned :i: . with a note or
contacted researchers by phone saying :lii;, did not teach
relevant courses; one school address was incorrect), Ilr , the
total sample size was 253 for special educators and 224 for
general educators.
Surveys
Separate surveys for .i' I.2 educators and special educators
were 'l: :1, '.*: based on an extension of previous research
(see Maccini & G. i.n ', 1'." i 2002), r . !.. l, I ,rr a focus
..I I ,, of teachers, consultants (e.g., professors in special ed
ucation and math .e'i,..ti,,':. feedback from graduate stu
dents and pt ':.' .i in math education, and consultants from
the Survey Research Center. The current 1'.,i on teacher
orientation included closedended and ordinal questions re
garding five topics: (a) teacher and student information;
(b) teacher ., ii ..J.n.., (c) teacher preparation and use of in
structional strategies consistent with NCTM, :!.'f '! i , , ii 
direct instruction, graduated instruction, and student group
ings; and (d) the frequency with which students :n_ .'t.:.d in
math tasks consistent with the ' , i ..I standards. One .,i.:.
tional topic involved an openended response to the f. 11 ii, ,
question: "The way teachers view math may influence their
orientation or instructional focus. In your own words, please
define what math means to you." General and special educa
tor , ir't.:. questions were identical, with the exception of
three questions concerning teacher and student information.
I,'r example, special education teachers were asked how
many years Ihc;. had been a special education teacher, whereas
general educators were asked how many years they had been
a math teacher.
Reliability and Validity
I' 1, i!tr. was addressed via three approaches. I ,rI stan
dardized directions were incorporated into the i� ' instru
ments. Second, reliability for data entry was tested on  .. of
the responses and was determined to be 9' i.i I., ,.:l.i.1
ity checks were conducted on 11' . of openended responses
to assess the F..J;l' of .,1;n ;l .,r, ..  into categories
(Fink, 1995). The agreement :.r openended responses was
1 1'' A.2r,:;:nnt was calculated as the number of agree
ments IJi i' by the number of ..:r,.';. t and disagree
ments multiplied 1. 100.
As noted, survey validity was addressed via teacher
focus groups. Teachers responded to issues including survey
objectives, clarity and appropriateness of .. .: i,.n . and over
,.t format. Furthermore, consultants (i.e., professors in the
field in special education and consultants from the ,r."
Research Center) reviewed the surveys to address construct
validity and study i n,:..'l ..:. The researchers i":...1t !.: 1 the
survey and rn.. Ii, .1.:1 .: based on the F.....l..i. received.
Survey Administration and Response Rate
1 1 i i! !,.iilili, to secondary general and special educators
included a cover letter rin, . n. the purpose of the study and
assuring confidentiality, a survey, and a business reply re
turn envelop. A week later, all teachers were sent a card as
a reminder or thank you. Approximately 6 weeks after the
first :i.iilir :. a second mailing was sent to nonrespondents.
46 REMEDIAL AND SPECIAL EDUCATION
Volume 28. Number 1. January/February 2007
Followup phone calls to nonrespondents were conducted
3 weeks it. i, the second mailing.
Despite the two II... 1 a reminder postcard, and
fi ...  1' phone .:i . i the r:p..r, .. rate remained low. A total
of 35.97% (n = 91) 'p. i ' educators and 33.'  (n = 76)
general educators responded. This rate is below the common
survey return rate of :.'.i ('",..i I":; Krosnick, & Bowen,
) ': .. and is a limitation of the current study. One other no
table limitation of the study was the lack of information on
nonrespondents. Because the survey was mailed by a re
search organization, and complete confidentiality was main
tained for respondents and nonrespondents, the researchers
had no . .''r t.ni:it', to obtain data on nonrespondents.
Data Analysis
Data analysis 'r. '..::Jrl . ,r the survey included descriptive
statistics and additional statistical methods, such as t tests to
compare r.u pI means and .'r. rT i...rr. Effect sizes were cal
culated for each t test (Cohen's d), and if the Levene's test for
equality of variances p value was less than .01, equal vari
ances were not assumed. Furthermore, nonparametric statis
tics were conducted, i; 1.lin.1 ; chisquare tests to compare
proportions and i .r, i, ..n analysis. For each chisquare
analysis, effect sizes (coefficient phi) were also calculated.
Prior to .r, inl..,. i:;. regression analyses, . ' r t 1. c,: ni alpha was
calculated to ensure internal reliability and consistency of
questions within each of the four criterion variables (i.e.,
NCTM, direct iii:rin.l.iiri. graduated instruction, zr....:in"
To maintain a balance between il;  I and II errors, the alpha
level was set at .01. It should be noted that the n for individ
ual questions varied, as some respondents did not answer cer
tainm tl:.' i .
For the openended response question (i.e., in your
own words, please I..!n. what math means to you."), a six
stage process was used for ...' ,1;11 responses. First, the open
ended responses were coded for major themes (Maccini &
G. r.i *n. 2000). Nc i, the first author identified the frequency
of each theme. ili r J the second author independently coded
the teacher responses uL.ii _ the i.1., ;r themes. Fourth, the au
thors discussed each teacher response and the I;l'.:'.ri.l'
ness of the themes and adjusted themes as necessary. The
authors then ;.ki.,iri'ritl', recorded the responses with the
clarified themes, and a final r !i.,lilii. check was conducted.
Ten themes or categories of openended responses were iden
tified (i.e., necessary tool, basic arithmetic, think 1 .:.11..
process, puzzle, academic discipline, language, philosophy,
consisting of properties, other).
RESULTS
Teacher and Class Characteristics
i,.ru:h.r and class characteristics were analyzed across
teacher ' 'r: (.":n r.,l or special educator). l;iiilli..l i; itir i
ences existed for teacher .:. in . with more female (n = 65;
39'' 1 special educators than male general educators (n =
r '; 2 " , X 12.887,p = '111. Although more special ed
ucators (n = 57; 35.6%) held graduate J1. _i.:,. than did gen
eral educators (n = 44; 27..' ., the difference in proportions
was not significant. >. nil i;' no significant differences were
found based on teacher age. The mean teacher age for special
educators was 44 years, and the mean for .,:Ji r I1 educators
was I years.
C.t. .irri., teacher credentials and experience, several
variables were ti i,:. l! ';,I .'i .i .nt when c.;ril. i:irl,. gen
eral and special educators. As expected, more general edu
cators (n = 71; 43.6%) held secondary math ;....'ili.
credentials than special educators (n = 2; I ' . y2 =
1',.. ', . .. p < .01. More special educators (n = 83; 50.9%)
held special education teaching credentials than general edu
cators (n = 0; 0 .. X2 = 147.730, p < .01. When ... :,r..;:i
the mean number of general education math methods courses
taken, general educators i' =2 '.. SD = 3.66) reported
taking more general education math methods courses than
special educators (M = 1.92, SD = 3.10). However, this li,
Ir r.n, .' was not .i'.iit .. 'I. C r,... iri._ the number of spe
cial education courses taken that addressed ',..hiri math,
special educators i, = 1.44, SD = I _Ir had taken signifi
cantly more courses than ,.:ni r.l1 educators (M = 0.29, SD =
0.63), t(98.72) = 5.046, p = .000, ES = .440. Of the re
i..,, ,:.J ,, ; 1,',,...l and special educators (n = 76, n = 82, re
i"'.t . , 60 general educators and 26 :...:'1 educators
had taken no special education courses that addressed teach
ing math. Also, of the teachers r r:r nnirn to the number of
i: 1r i1 education courses taken that addressed math (general
educators, n ='., special educators, n = 82), 8 general edu
cators and 28 special educators had taken no courses.
To answer certain questions .i i .: n.i r. class character
istics, special education teachers were ri r i asked about their
current:' Ir. ii , position. There was no .ni1. t ii . i: .ri.. ..
for years of . ,.!lirt_ experience with students labeled f:P' [
and LD between general (M = 11.89, SD = . .' 1i, and special
(M = 12.63, SD = 7 ,"i'i educators. Most special educators re
ported teaching in a resource room (n = 42; 1. ' o. , or a . if
contained class (n = 18; 20.7'~ whereas r:.irl ' five teachers
reported team teaching with a general education teacher.
Only teachers who were team teaching with a general educa
tion teacher completed two followup questions concerning
students with EBD and LD in those inclusionary classes.
Within the ',...,li I u: ih: math courses, there were no : . r i
cant hil !, _ .. between special educators (M = 13.78, SD =
,32.70) and general educators (M = 1.48, SD = 3.60) for the
number of students with i. i or LD who were not provided
with resource or inclass math assistance. The average num
ber of students with EBD or LD that were provided with in
class assistance in math was not significant for the ttest
analysis.
As shown in Table 1, special educators were almost
twice as likely to teach pre.l:.,r.i and almost eight times
more likely to teach general math basic skills to students
with EBD or LD than general educators were. General edu
I 2MEQIAL AND SPECIAL eDUCATION 47
Volume 28. NwInber L Jamiar/Februay 2007
TABLE 1. lypes of Math Courses Teachers Are Teaching to Students with LD and EBD
Course
Prealgebra
. .,.I r :
Geometry
General ni..[h i l: . I1.
Algebra 'i ii .. .i 
Ti. '.'.', ti";: , ;, .J high school matha
Other
Special education
teachers
Yes No
n % n %
General education
teachers
Yes
n %
46 28.2 41 25.2 25
No
n %
15.3 51 31.3
35 21.5 52 31.9 33 20.2 43 26.4
22 13,5 65 39.9 24 14.7 52 31.9
63 38.7 24 14.7
8 4.9 68 41.7
10 6.1 77 47.2 20 12.3 56 34.4
6 3.7 81 49.7
10 6.1 77 47.2
18 11.0 69 42.3 11
2 1.2 74 .1:.4
3 1.8 73 44.8
6.7 65 39.9
2 p ES
6.586 .010 .20
0.170 .680 .03
0.793 .373 .07
63.195 .000 .62
5.934 .015 .19
 .286 .10
 .089 .14
1.072 .301 .08
Note. df= I for all chisquare statistics. LD = learning disabilities; EBD = emotional and behavioral disorders; ES = effect size.
*Insufficient cell size to calculate chisquare; p values derived from Fisher's exact test.
cators were twice as !i. 1. to teach il.'r.i II and/or alge
bra IT'trr;,,'i i:. n.ri; to students with EBD or LD than special
educators.
Additional survey questions applied either to general
or special educators. For example, special educators were
queried on the number of years experience 1.,.. had as spe
cial education teachers. The average number of years was
15.52 (n = 87). Similarly, general educators noted the number
of: :.a. experience as a math teacher. The average number of
years was 18.33 (n = ' ,. Special education teachers also
noted the number of students on their caseload. The aver
age number on a caseload was 25 students, with a range of 5
to 80.
Teacher Definition of Math
Teachers' responses to an openended question concerning
their views of math were used to address a twopart research
question: (a) how do teachers .J. l lin math, and (b) does their
dit:n: ili contribute to their use of instructional strategies
consistent with ".I I N I. lir..l instruction, :r.Dtl;.r ... instruc
tion, and student r.i:lup.i, ' Most general (n = 32; 47.1 .;
and special (n = 4, 59 " education teachers defined nu'.
as a necessary tool. F..r example, teacher responses included,
'.[alh is an essential component for . W. .:r' .', l.." and
"There is no question that without math . 111 you would
have a .iit;i.:ilr time .:�r i i: in life." In the second most fre
quent response, for both :;'nciil. (n = 9; 13.2%) and special
(n = 7; 9 �. I educators, teachers defined math as a ... ..L..,..
For example, teachers reported, ' ]..i, is just a language that
uses numbers instead of words," and ' \ L; is a common lan
guage by which individuals may communicate." The third
most common response by , r;c .r l (n = 5; 7.4%) and special
(n = 4; 5.6%) educators defined math as the ability to think
.. ..".. For example, "I believe that learning math develops
areas of the brain that are used in logic and problem , I * in; "
The remaining responses were evenly distributed over several
categories, ,i..lIii.r J rFil;. math as a process (n = 3),
a puzzle (n = 2), an academic li .c:lir,:. (n = 5), a philoso
phy (n = 1), .. ., i li ._: of properties (n = 1), basic arithmetic
(n = 4), and other (n = 4). No ]c;,!t;.:.nrt differences existed
when :..r" i .rrir2 p,r. r'".'rIp r of general and special education
teachers on their views toward math. It was anticipated that
overall teacher views toward math would ;.r.',i. their use of
instructional strategies consistent with NCTM, direct instruc
tion, _iru . I ii, instruction, and student groupings. However,
no :i.i:nli.. vin correlations existed for these variables. Thus,
:, . r,.. n analyses were not conducted.
Teacher Familiarity with Course Topics
Teachers responded to questions on their t.ar,,i rI; with sec
ondary course topics (i,e., prealgebra, algebra, ''. .r tr., gen
eral math or basic skills, algebra II, algebra i1 i ,iiri ;n: tr;,
statistics/probability, ;it.';:.si:.'lJnif.i high school math).
Teachers rated their famril; i ' with the topics on a 4point
scale, where 1 = a great deal, and 4 = very *'. or nothing.
Special educators reported greatest familiarity with pre
.;l,::ht..i l = 1.52, SD = 0.71) and _.irL r 1 ..;ith. 1h.i skills
(M = 1.07, SD = 0.26) and least l.,n,:ii.r,. with algebra II/
trigonometry (M = 3.44, SD = 0.69) and statistics/probability
(M = 3.23, SD = (I r>'. General educators reported greatest
1 iil:.a i!.  with prealgebra (M = 1.07, SD = 0.25) and alge
bra (M = 1.08, SD = 0,32) and least Itr;l.iud'i with inte
48 REMS DIAL AND SPECIAL EDUCATION
Volume 28, Nutmber . JaniaryiFebruar'y 2007
grated/unified high school math (M = 2.37, SD = 1.10). 1':
. i.: .L of topics was .l.IJ . . '. for several variables when
comparing mean ratings of general and special educators,
with general educators 0i .;". i.  ..�  fri.,:; n;t. for each
topic in which statistical significance existed. .iF.. i, ..
.* !:i .,.'. : was noted for familiarity with pre .i, li
t(103.87) = 5.484, p = .000, .. = .46; algebra, t(1 18.53) =
10.827, p = .000, ES = .69; geometry, t(151.67) = 10.395,
p = .000, ES = .64; algebra II, t(141.20) = 13.742, p = .000,
ES = .75; algebra ii ",. ',. ::.i, t(154.67) = 1s r;", p =
.000, ES= .. , and .i.ti... ',i...dilU. . . 1, ':, = 9.647,
p = I. 'II,. ES = .27. S'Yi; f : i',:r was not noted f'..i .; .,1
math or basic "I 111 or for integrated/unified high school math.
Effectiveness of Methods Courses
General and special educators who had taken math education
methods courses (n = 65, n = 56, respectively) and special ed
ucation courses that focused on teaching math (n = 16, n = 56,
respectively) completed followup il. ;' !. ontheeffective
ness of these courses in preparing them to teach math to stu
dents with special needs and general education students. A
4point scale was provided to respondents, with 1 = very pre
pared and 4 = .. unprepared. There was a i r! ii.. 211 sig
nificant. Ilrrl: . in mean ratings for general and special
education teachers on two variables. Special educators felt
more prepared ., I = 2.11, SD = 0.69) than general educators
. 1 = 2.97, SD = 0.88) to teach students with EBD and LD
after completing math education methods courses, t(1 19.99) =
,. p*,,, p = r':,, ES = .47. However, general educators
.1 = 1 . I SD = 00. felt more prepared than special educa
tors. = 2.31, SD = I1  i to teach math to ;:  . r.,l education
students .f, ., taking math education methods courses,
t(112.358) = 3,192, p = .002, ES = .28. There was no sig
nificant !,tI' i.... between special ..' = 1.82, SD = i1 5 '..; and
general educators (M = 2.13, SD = 0.74) :'i teacher prepara
tion to teach math to students with LD and EBD after com
i.liii  special education courses that focused on teaching
math.
Teacher Pieparation and Instructional
Strategy Use
I . research questions were addressed related to teacher
preparation and use of instructional strategies:
1. How prepared do teachers perceive 1,.. are to
use instructional strategies consistent with
NCTM, selfmonitoring, direct instruction,
In. .i , .1 instruction, and student groupings,
and how often do 11 use instructional
strategies consistent with these approaches?
2. What factors contribute to teacher use of
instructional strategies consistent with NCTM,
direct instruction, ri .,, .I'. 1 instruction, and
student groupings?
General and special educators' preparation (i.e., pre
pared or : ., . .. "* was compared for instructional tech
niques in the five ._:. .' categories noted previously (see
Table 2). CO . ...mi. i,. teacher preparation to use instructional
i... 1ri.,:.  consistent with NC' 1 standards, the proportions
of .:.r:,: r.' educators who felt that they were prepared to use
graphing calculators and those prepared to demonstrate a
concept via twodimensional : :.1l;. ., or pictures were sig
lifi ..lil.. greater than those of .... ii. educators. ,: .r...l .1,
three variables were statistically significant in the . ii ;.' of
instructional techniques related to ._. .i.. _ In each case,
;.Il :,1..,,il . more special educators than general educators
reported being prepared. Sr..'i;. 3i more special than gen
eral educators reported preparation to (a) use cooperative
l. rnir. . .. ii. (b) provide 1.. iI1 . .... assistance "' 1L!.:
the rest of the class worked on other assignments; and (c) pro
vide opportunities for peer tutoring sessions.
Teachers were also asked to report on the frequency that
they used components of five instructional techniques. Teach
ers ranked their use of each technique on a  p. '1. scale
ranging from 0 = never to 4 = . .' In the category of in
structional techniques consistent with \. T'. i standards,
eral education ,.i. = ,' , SD = 1.32) teachers reported
,.iTi�,,r, , trin the use of ,il.i'1,l! . calculators i;i.. :ri '
more ii.. r: than special educators (M = 0.70, SD = 1.13;
t(142.46) = 6.842, p = .000, i .. = .46). In each case where sta
tistical significance was noted concerning the use of tech
niques consistent with direct instruction, special educators
reported greater use of the techniques. For example, there
was a .; 1'vli..1 r.r I ill.....r,... t(146.23) = 3.196, p = .002,
t '. = .28, in special educators' reported use of feedback and
reinforcement to students, compared to general educators
.I = 3.47, SD = 0.60). Compared to general educators '' =
2.16, SD = 1.17), special educators i1' = 2.94, SD = 1.04)
also reported more frequently incorporating mastery learn
rin '.,.. ,; . i. " .., having students advance to the next topic
or skill, t(146.70) = 4.345, p = .000, ES = .34. Special edu
cators (M = :. , SD = 1.27) also noted i. .1, r use of graph
ing student :'r  . .' r.. to make instructional decisions or to
show student progress than  ri,: .I1 education teachers . 1~i =
0.81, SD = 1.04), t(155.71) = 4.202, p = .000, ES= .32.
There was a .r i 'i..ill1 :. �i; ..:i r ,litr.i I in the
mean rating of how often teachers used techniques related to
student selfmonitoring. Special educators had a mean score
of 2.80 (SD = 1.01), and general education teachers had a
mean of 2.24 (SD = 1.13), 1 147.25) = 3.208, p = .002, ES=
,.25. One variable in the category of instructional techniques
consistent with _r. li' r ,.. instruction was also significantly
different for general and special educators. C'.:i.: i . educators
(M = 2.80, SD = 0.92) more frequently demonstrated a con
cept via twodimensional ;.,.1,;. or pictures than special
educators i; = 2.25, SD = 1.24), t(151.78) = 3.203, p = .002,
ES = .248. .'.'::.h i the category of instructional techniques re
lated to grouping, special educators reported the use of same
ability groups I/ = 2.49, SD = 1.10) and .,:11..'i1:1
assistance while the rest of the class worked on other assign
R E M E A L AND S P CIAL E UCATION 49
Volume 28, Number 1 Jaonar'February 12007
TABLE 2. Teacher Preparation to Use Instructional Techniques by Cotegory
Special education
teachers
General education
teachers
Prepared Unprepared Prepared Unprepared
n % n % n % n %
Instructional techniques consistent with \ I 1 I
Encourage students to develop 53 34.9 25 16.4 62 40.8 12 7.9
Demonstrate use of graphing 19 12.7 58 38.7 54 36.0 19 12.7
calculators
Embed math in realworld tasks 70 45.5 10 6.5 63 40.9 11 7.1
Encourage student., i1 i. of 69 45.1 10 6.5 64 41.8 10 6.5
approaches to problem solving
Illustrate a concept via multiple models i. 43.8 13 8.5 58 37.9 15 9.8
Instructional techniques consistent with direct instruction
Provide teacher ; .i.. i,1rir of a concept, 80 51.3
skill, or strategy'
Provide feedback and reinforcement to 80 51.6
studentsa
Incorporate mastery learning/criterion 67 44.7
before having students advance to
the next r. ; 1.1
Provide a review of previously learned 78 51.3
skills/conceptsa
Provide independent practice 78 51.7
Provide cumulative reviews 73 48.0
Graph student ,,..*', to make 52 34.9
instructional decisions or to show
Give regular orientation or advance 56 37.6
organizer for a new lesson
Encourage the practice of basic math 69 45.1
SI ill i. i ,l 'n rl' , 
2 1.3 70 44.9
1 0.6 70 45.2
4 2.6
4 2.6
11 7.3 53 35.3 19 12.7
1 0.7 70 46.1
3 2.0
1 0.7 . 45.0 4 2.6
6 3.9 68 44.7 5 3.3
26 17.4 37 24.8 34 22.8
21 14.1 46 30.9 26 17.4
10 6.5 68 44.4 6 3.9
5.170 .023 .18
36.453 .000 .50
0.183 .669 .04
0.025 .875 .01
0.472 .492 .06
 .424 .08
 193 .12
3.532 .060 ,15
 .351 .09
 193 .12
0.031 .859 .01
3.273 .070 .15
1.346 .246 .10
0.845 .358 .07
Instructional techniques related to .ilfrrmnilrirrI
i... I. selfmonitoring strategies to help 66 43.4
students with problemsolving
activities
14 9.2 47 30.9 25 16.4
5.893 .015 .20
Instructional ithriiuniu  consistent with graduated instruction
Demonstrate a concept via two 61 40.4
dimensional graphics/pictures
Demonstrate a concept via three 58 38.4
dimensional tools
Use graduated instructional sequence 53 34.9
dimensional tools
17 11.3 69 45.7
4 2.6 8.384 .004 .24
20 13.2 55 36.4 18 11.9 0.019 .889 .01
26 17.1 52 34.2 21 13.8 0.305 .581 .05
(Table 2 continues)
50 REM IAL AND SPECIAL EDUCATION
Vomne 28, Ntmber 1, JamwarysFebrary 2007
Technique
X2 p ES
Table 2 continued
Special education
teachers
Prepared
Unprepared
General education
teachers
Prepared Unprepared
Technique n % n % n % n % p2 ES
Instructional techniques related to grouping
Have students work in sameability 77 :. ; 3 2.0 67 43.8 6 3.9  .311 .10
Provide cooperative learning activities 77 50.7 2 1.3 62 40.8 11 7.2 7.625 .006 .22
Provide ,r. !! .....: assistance while 79 51.6 1 0.7 62 40.5 11 7.2 10.084 .001 .26
the rest of the class works on
assignments
Provide *. i..' I ;';. for peer tutoring 74 49.3 3 2.0 58 38.7 15 10.0 9.840 .002 .26
sessions
Note, df= I for all chisquare statistics. NCTM = National Council of Teachers of Mathematics; ES = effect size.
"Insufficient cell size to calculate chisquare; p values derived from Fisher's exact test.
ments (MA= 3.12, SD = 0.86) more fr :u.. i:r!, than .:i..:,l ed
ucation teachers (M = 1.77, SD = 1.10; M = 2.15, SD = 1.18,
respectively), t(153.88) = 4 I ,. p = .000, ES = .31; and
t(134.83) = 5.855, p < .01, ES = .45, respectively.
To identify which factors i.. r' ii[isei to teacher use of
instructional strategies, a twostep process was conducted
,: :i,_. the.combined responses of , ir : i; and special educa
tors, First, four i :k ,i.. were used as criterion variables
(i.e., ., 'IT.1, direct instruction, graduated instruction, and
grouping). Each variable consisted of the mean use score on
a T ii. ; scale for several statements. . i:,. 'i..i!I; , NCTM
consisted of three practices (i.e., encourage students to de
velop 5.(r.iiri..'i'. '.di  tilbrni , encourage student ,i1 1 ,:, ,:! of
approaches to .1. 2l, 2 problems; illustrate a concept via mul
tiple models). Direct instruction consisted of eight issues
(i.e., provide .. 1r. . in ',.. . !'in, of a concept, skill, or strategy;
provide feedback and reinforcement to students; provide a
review of previously learned skills and concepts; provide in
dependent practice; provide cumulative reviews; graph stu
dent progress to make instructional decisions or show student
progress; give regular orientation or advance ,r .ri'ii for a
new lesson; encourage the practice of basic math skills and
I ;_ rihn, :' Graduated instruction consisted of three vari
ables (i.e., demonstrate a concept via threedimensional tools;
demonstrate a concept via twodimensional graphics or pic
tures; use a .r .1]u.' I..! instructional sequence to teach con
cepts), and _..,i;r'".:; consisted of four issues (i.e., have
students work in sameability groups; provide cooperative
learning activities; provide r nii.I'1 iii assistance while the
rest of the class works on assignments; provide opportunities
for peer tutoring sessions). Coefficient alpha was calculated
to determine the internal !li.,1,1it of four categories of in
structional strategy variables as i.'.,.  (a) NCTM, a = .67;
( 1 d i:. i i 1 :i,..:1!i a = .73; (c) graduated instruction, a =
.66; and (d) student grouping, a = .75. Internal validity was
sufficient to maintain the criterion variable i., ,i in..
Next, a regression analysis was conducted to examine
the contribution of the three predictor variables (i.e., years
teaching students with LD or ED; 1 n'. 1l..1~ :. number of
methods courses) on teacher use of instructional .tra,.;;:_:
Two predictor variables consisted of the mean of more than
one survey question. S1..I:iL..:.d, '.....';.'.. consisted of
teacher 1..i.. ,  1:.. of prealgebra, algebra, geometry, general
math or basic i.n , and the combination of algebra solely
and algebra 1TT' i .'r:.'r ,.:r'. I i,. predictor variable number of
methods courses included both general education and special
education methods courses that focused on ;:. I in r math.
For the use of instructional techniques consistent with
NCTM standards, the predictor variables accounted for 12.4%
of the variance (see Table 3). The t test for beta ; :li
was ';.ilirl[...il;. !i!i',' .ri only for the number of methods
courses. Moreover, the number of methods courses was sig
1iti.,, i When entered in the last position and contributed
8.5 ,. above and beyond the ' rl' * ,rijll For the use of in
structional techniques consistent with direct instruction, the
predictor variables accounted for 19.0% of the variance. I...
number of methods courses was the only statistically signifi
cant variable. For the use of instructional techniques consis
tent with ;:.lu.it:. instruction, the predictor variables
accounted for 15.3% of the variance. No predictor variable
was statistically significant when entered in the first position.
However, when entered in the last position, the t test for beta
weights was .;.~1iuc 11i: .; rnti.lnt for the number of meth
ods courses taken. This variable ..,n  :lr; "c..' .. of the vari
ance above and beyond the other variables. Fin ,ii, for the
use of instructional techniques related to the use of  ri.ULL'iI i.
SEMEDIAL AND SPECIAL EDUCATION 51
Voilu'i 28. Nwuber 1, JanuairyFebrtasry 2007
the predictor variables accounted for 15.4% of the variance.
The number of methods courses . :..in was the only statisti
cally :, i !....11 variable when entered in the first position in
the regression equation. In the last position, the t test for beta
weights was statistically i_ :ii. i:i for the number of meth
ods courses, and this variable contributed 8.2% of the vari
ance when ..n rtr .IIl: for the other variables.
DIscusslorN
i ir current national study examined three critical factors that
affect teacher decisions about what and how to teach math to
secondary students with EBD and LD: (a) teacher t il. i i;
with content I i :: ' !il,: and practices; I.i teacher i,'r. r .i
tion; and (c) teacher '.r I . and orientation. Teacher re
sponses are '..1,ii .:.i relative to the N' T.1 standards and
empirically  'il',l.i1. approaches to teaching math to sec
ondary students with I B [i and LD.
Teacher Familiarity
In the current study, teacher !.n ill'.r11: with content knowl
, J.. c and practices supported the results of related research.
For example, Maccini and Gagnon (2006) determined that
,..1. .:i;l., more ;;.:, ri:li educators reported '.Lirr prepared
to teach all secondary math course topics except the com
bined category of general math or basic 1 I. It is of concern
that in the current l,'! . special education teachers reported
teaching math in mostly i  i.. It. ..:ti:i.. and reported that
they felt less familiarity with the math topics than general ed
ucation teachers. These results highlight the importance of
i. r, . .;., rI ' development opportunities for special education
teachers in domainspecific 1. i .. . 1 1: .. Teachers must have a
deep and broad ., r..k . i i..n.lin. of math for students to achieve
at their highest possible level (Koency & Swanson, 21 , I,
This is particularly critical for special educators who are
working in a segregated  ii  and thus may not 1i. it. i from
collaboration with a. .': il educator.
In addition to contentspecific knowledge, teachers need
to be prepared for and use instructional . ii, j i, consistent
with the I; 1 A, of the NCTM standards and ri. ;r..,1 vali
dated '.._hi, practices (NCLB, 2001) for helping students
with EBD and LD (Maccini & G.'i ir,. .. 2'. 1.11' These recom
mended instructional  i. i I..:... include (a) instructional
'.r.ik . i . .consistent with the iI ; [ I standards; (I i direct in
struction; (c) graduated instruction; (d) grouping practices;
and (e) selfmonitoring.
NC'i A Standards. The \.IN [I standards support
teacher i . ' . 1. of a wide range of math instructional tech
niques and r,.i t1i. as well as current research concerning
TABLE 3, Regression Analysis for Teachers' Use of instructional Approaches
Construct entered initially Construct entered last
Teacher Variables R R2 p A R2 p
Use of instructional techniques consistent with NCTM'
Years teaching students with LD or ? P'L .017 .000 .841 .003 .490
.... 1. .194 .038 .546 .032
Number of methods courses .299 .089 .002 .085 .003
Use of instructional I bi hniiin , consistent with direct instruction
Years teaching students with LD or EBD .086 .007 .349
Knowledge .365 .133 .011
': irl, r of methods courses .338 .114 .001
Use of instructional techniques consistent with graduated instruction
'i. r, teaching students with LD or EBD .012 .000 .887
Knowledge .2 ' .070 .138
Number of methods courses .236 .056 .021
Use of instructional techniques related to use of groupingd
Years teaching students with LD or F i I1 .055 .003 .526 .000 .892
Knowledge .266 .071 .138 .020 .807
'.,r.,l..r of methods courses .A. .133 .000 .082 .003
Note, NCTM = National Council of Teachers of Mathematics; LD = leading disabilities; EBD = emotional and behavioral disorders.
'coefficient a = .67. coefficient a .73. Ccoefficient a = .66. "coefficient a = .75.
52 REMEDIAL AMN SPECIAL EDUCATION
Volume 28. Nmnber 1. Janiua~ylFelbrary 2007
math instruction (Parmer & Cawley, ', *). In the present
study, two ,, i! l. it. 1 I. :.. ir.,. were r. I. i across teacher
types, with general education teachers i i :';. greater
preparation than special education teachers for encouraging
students to develop strategies and J1 'l.rilir,,. and demon
strating the use of 'r.irl.ir. calculators, )1.. ,...:. . the use of
:1 ir.lIi I  calculators is a recommended practice for teaching
math to secondary students with disabilities i ,:. ii &
Gagnon, "111'r i. special education teachers reported that ',. .'
seldom to never used the technique. I !.:.: i:rlir, I.. are of
concern, given that students in the United States are com
monly proficient in onestep problem and routine procedures,
but less competent in nonroutine and more complex problem
solving situations (Wilson & l:i .i ,! i .'l', Thus, there is a
need for teachers to provide students with more opportunities
to apply I .. to realworld contexts, communicate mathe
11 i i, and participate in activities that require them to use
mathematical r. I .nir.:. and 'ini... 1.11. and to il. ;,. answers
' ii..,r & Blank, 1999).
I)irtcl Ii[nsiiructirn. Close to 70% of general and spe
cial education teachers reported being prepared to use the
majority of techniques consistent with direct instruction (i.e.,
teacher i, iL..1, 1,_: feedback, mastery learning, review, in
dependent practice, graphing student progress, advance or
; !II; r. practice of basic I ! and I.. I. ..i 1�::. them
fi :' pi I:'r (i.e., 24 times per week to 1, 1I . 111 is promis
ing given that the use of techniques consistent with teacher
directed instruction has been empirically validated for
t: .Li 1t I, rI i1 1i to secondary students with LD (Kelly, Gersten,
& Carine, 1990; Moore & Carnine, 1' " *
Graduated Tiinruction. In terms of preparation and
use of instructional strategies consistent with ,._i..',..l
instruction, general educators reported being prepared .. ii
icantly more than special educators to demonstrate a concept
via twodimensional graphics or pictures. Also, more than
half of the :. :.1 and special education teachers reported
'.,.IL: prepared to demonstrate a concept via three
dimensional tools and to use a .:i .dl,. ..I instructional se
quence. However, teachers reported i, :.i the techniques
once per month to 2 to 4 times per month. The use of a gradu
ated instructional sequence is an effective approach for teach
in; higher level math to students with F. '. ) and LD i l i. 'r
& Hughes, '"11" Maccini & r.; il1. I, 'l', Witzel, Mercer, &
.Il:.I., 2003). However, the technique must be used fre
quently to ensure that students with disabilities understand
the ni i ,i :*:. 1_ concepts prior to ',! I i,.: ir.. to a more abstract
1 1;. (Maccini & Hughes, 2000).
Grimupini P'r.irici.. I r,'. variables related to student
grouping were statistically significant, and in each case, spe
cial educators reported more preparation to use the instruc
tional strategies. 'S:. il i;", special educators noted Iirn
more prepared to use cooperative learning activities, peer tu
toring, and smallgroup assistance ".. li Il. the rest of the class
worked on other assignments. Peerassisted practices help
students with EBD and LD with computation skills (Calhoon,
et al., .1.n 1',) and algebraic .., I within heterogeneous math
classes (Allsopp, 1997). As such, varied grouping practices
hold promise for ,1.I. :,. the varied and diverse needs of
students in a general education math classroom.
4r1f,Miniiirinii.. :. .*f,. 'n, more special education
teachers than general education teachers reported f. : i;,.' pre
pared to use .. ir:l. ..iii[i..n,. L iFm.r i ri .r . is an .t..:..
approach to help secondary students with il I ...ti..:. in math
to monitor their pr FP. ii : .1 ii.; behavior and is a necessary
component of successful i1 ' I r., performance (Maccini, Mc
",._ _i,,ii & .:.'.!I, 1999). Thus, teacher training and pro
fessional development opportunities should address how to
integrate the technique with students in secondary math
classes. For : . ,:ir , students can use a structured worksheet
or prompt card with cues to help students with Itr.,_i  prob
lemsolving behaviors i,' 1., 1 f, ". FP , 'i "., !, cues are
. i!,.,. ,i...., for setting up and solving for the solution to a
problem (e.g., '.1 i..a do I know? V', i i do I need to ;.l ').J
Predictor Variables and
Instructional Strategies
In addition to the preparation and use of instructional strate
gies, we also examined fia.. i. .: that .i *r itr'l Il.:1 to teacher use
of (a) instructional t, .  ;.;. , consistent with the NC T I stan
dards, (b) direct instruction, (c) ;. , . .1.t. . instruction, (d) stu
dent groupings, and (e) .1f r,, .r:,r:,;i. The number of
methods courses taken made a significant contribution to the
use of instructional techniques consistent with NCTM stan
dards, direct instruction, student grouping .,.1 r:..:j .;: i, .. and
;..,J, ir.J. instruction, i : ... F,,.!:, .. are similar to those of
related research " Tl.i.ir, & Gagnon, .1'i' in which the
number of methods courses taken by general education teach
ers was the strongest predictor of the number of instructional
approaches (i.e., recommended and empirically validated
 rii.. . implemented by teachers). The number of methods
courses for teacher preparation is an important variable for
preservice teacher education. Teacher education programs
should consider this issue when .1: i'1,h, r :. . i jnr. that ad
equately prepare future teachers to use the empirically vali
dated instructional strategies necessary to help students with
special needs.
Teacher Preparation
In terms of teacher perceptions of the effectiveness of meth
ods courses taken, there was a significant iti' I ii... across
teacher classifications. Special educators fl .i Iifi..ri,
more prepared to teach students with EBD and LD, whereas
general educators felt iln ti :ni.. more prepared to teach
*._ :n! .1 education students. [li, is particularly interesting
given the previously noted lack of familiarity that special ed
ucators have with the typical course topics other than general
math and basic skills. Furthermore, the mean response re
EMEDIAL AND SPECIAL EDUCATION 53
Volume 28. NMnher 1. January FPebrwary 2007
ported by general education teachers was close to '  .1I;.
somewhat unprepared" to teach math to students with EBD
and LD after ' ,'1:1 'tli: math methods courses. These results
are similar to those of a related study (Maccini & Gagnon,
2006) in which . i1. ill significant !.!!...:. were
found between general and special educators . L 1. 1 i i their
i i.' r. i.i . to teach math following math education courses.
Maccini and Gagnon also noted that special educators felt
more prepared to teach math to students with EBD and LD,
and general educators felt more prepared to teach ,i:r il
education students.
Concerns about teacher feelings of being unprepared
should be considered in I , i of recent legislation :.CLB,
' I 1, that calls for teachers to be ;:i.'1i1 qualified in the area
of instruction (i.e., state certified, high 1. _.. .:I f domain
specific '.!r. .l... .:' According to S. f1i..,.r ..1. Lazarus,
and 1I ,il. ,... .'i ":i, "a iu.,li'. education for students with
!,..~li i... in a standardsbased environment requires high
j.i. .ir:. teacher preparation and i; ir.;r., (p. 9). There is a
need for teacher preparation and 1li iiaii  to help educators in
struct students with II.. I'1iii,... in ways that support the
achievement of :r .~i1t .1 academic content standards (S.
Thompson et al., 2003). Furthermore, the reauthorization of
the Individuals with Li. :ij,11... Education Act (IDEA) of
2004 mandates that states provide more avenues for :;i:f..
sional development opportunities to keep educators current
on prr..mi in: teaching strategies, skills, i tI." r'. and tech
nology. ,M'r j . , adequate teacher training is critical, given
that content 1.r. 1,LI by teachers is linked to increased stu
dent achievement in math. i ' .r.l . '1
Teacher Beliefs and Orientation
;,ir:inl.I to previous research (Pachnowski, 1997), we found
that the most popular teacher definition of math in the present
study was math as a necessary tool. A distant second was the
ikt.nlt:,,r of math as a 1.i''i., : F.trm r,:,. , liii., .1 it
was anticipated that teacher orientation to math would predict
use of instructional strategies, low correlations existed :,r
teacher orientation and use of instructional r i.. :i .. consis
tent with NCTM, direct instruction, ,.,~lI;:.l instruction,
and student groupings. The lack of apparent relationship be
tween teachers' 1:' .  and definitions of math and their in
structional practice makes it necessary in future research to
also examine the instructional setting, student characteristics,
and teacher 1 an  1.1 ; . Discrepancies between reported be
liefs and actual instruction ri ., also be a result of one or more
,r,,'iin, issues, such as conflicting teacher beliefs and
outside influences (e.g., .: i._. school administrators) that
make it necessary for teachers to subordinate their beliefs
(A. C. i�i, rnl.p r.. 1984).
Limitations and Future Research
Although important implications emerged from the current
study, it is important to note two limitations. First, generaliz
ability of the results is limited due to the .11l .nl.'l . size
and the low survey return rate. The return rate remained low
despite two mailings and multiple reminders. ,. ., ".i to fur
ther enhance the validity of findings, future research should
include comparisons between respondents and nonrespon
dents to determine possible differences. Unfortunately, such
comparisons were not available due to <. .Iit i. n1. i.i I . restric
tions from the survey company that '. i..3 the :,, iilii'
and returns.
The current study provides salient information on which
to base future research. For example, the evidence has sug
gested (Manouchehri, ; *'.'i that teachers with more tradi
tional r: ,irir._ and those that have taught longer with this
approach were often more resistant to constructivist activities
and r ir:1 used indepth discussion. Thus, future research
should further analyze the :.. 1j;; .". ti. between the focus of
:It .1 and special education math methods courses (e.g.,
constructivist vs. traditional, ruledriven .ipi'. h i, ''.  1n
over, the number of courses that teachers take ... i i:,. will
affect i i.:r i i L . i.. , i . with current trends in instruction. Also,
fi:r1n : studies should include case study research to il,,, cap
ture teacher use of instructional practices consistent with
" i. T . standards and .rpiii. '1. validated strategies for stu
dents with 1:1.: and LD. Furthermore, the use of observa
tional data within a case study design will 1.,1p to validate
teacher perceptions. l r.. i:, as the number of methods
courses completed by teachers contributed to the use .:.r in
structional techniques consistent with the oi TM standards
and ,_ [Irl,..011 validated i;,,1 ;.' . (i.e., direct instruction,
.i .1.., i .' 1 i in ir.. r,.nri and student :,,l.in.:''. it would be
:I,.i. .ir;._ to examine the focus (i.e., constructivist, behav
S.I l nature of course : i'"':nm.i' and fieldbased experi
ences, delivery of instruction (i.e., special education faculty,
general education fi. . _.., cotaught), and extent to which in
structional practices in methods courses 7.ll. . best practice
for '. .:!,ir;, students with special needs. As Parmer and Caw
1'. (1997) noted, it is important for university instructors to
model  i :..i : teaching practices.
Implications for Practice
Key implications are evident from this national study of sec
ondary special and general education teachers of students
with EBD and LD. The implications noted relate to the
importance of teacher training programs and professional de
velopment .'i' ; .i. : r.  for iri;,i educators to use instruc
tional practices that are effective for helping students with
special needs in math. For example, in the current study, spe
cial education teachers reported less familiarity with upper
level math topics and reported limited use of certain instruc
tional practices that are supported by \.i I ', 1. such as the use
of graphing calculators. Also, _. i:!. education teachers re
ported less I.,nrll.i. with and use of student .;'i'..iiin
arrangements (e.g., cooperative 1 i.': in , activities, peer tutor
ing) and ..lfi,;.'..rit:: techniques. Both techniques have
been empirically validated for students with special needs
54 RM EDIAL AND SPECIAL IOUCATION
Volumne 28, Natober i, JauarylFebr1ary 2007
(Allsopp, 1997; i . 11. . . et al. 2000) when focused on com
,' ti. ,r. and .il..r ' i skills. Based on these results, methods
courses for preservice teachers and professional development
opportunities for general and special educators in pedagogy
and .1 ...;.. .  i . knowledge are critical.
Moreover, because the number of methods courses taken
by , ,. i 1 education and special education teachers was
found to be the .' r.n, .: t i I:i .:.. i of the use of instructional
techniques consistent with NCTM and. nipir; ;:. validated
practices (e.g., direct instruction, graduated instruction, stu
dent groupings), there is a need for more intensive, .,:,., .;r.i.
professional development opportunities for teachers that ad
dress both content knowledge and [i:.1I .,' , (Quinn, 1997).
F, r instance, '.Vi ;ir i:1 .r. (2002) stated that more than half of
the high school math teachers surveyed reported the need for
more professional development opportunities in (a) 1I. irnir: '
to use i,.i ir"  .,i.. 1 and investigative methods of teaching
and (b) t:...li[!n math to students with special needs. Quiur.r,
reported that a methods course that addressed both math
pedagogy and domain knowledge had a significant effect on
preservice teachers' attitude toward math and. L 1i 111 ..11 i in
creased their domainspecific knowledge of math. The course
involved methods consistent with the goals of 'N: 1 I stan
dards C' i.I . 2000) and focused on conceptual understand
ing, use of .. r. ,1 r , and cooperative learning , ir.,i. ;. If
math education reform is to be successful, teachers must have
adequate knowledge of math content (Quinn, ''" ,).
S. Thompson et al. ..I l I:) also noted the importance of
professional development or "teacher i!!, 1 i.,:i '. opportuni
ties for new teachers and the positive impact on retention of
i  i l teachers .' .. !r. to Fideler . ll' .l, , teacher
induction programs should focus on assistance and support
for new teachers and include the i. !I rl ri*. (a) a multiyear
teacher induction process, (b) a supportive environment for
new teachers, (c) mentor training and support via stipends
and course release time, (d) facilitation between schools and
teacher preparation programs, (e) information regarding ef
fective '.. r Inr i practices, and (f) I '".:I inr resources to en
sure quality induction programs.
CONCLUSION
Many states have math standards and assessments that reflect
the N .' I I standards and require students to be proficient in
higher level math and problemsolving skills 1 l.int: & Dal
kilic, 1992). This may be a daunting .ail n : for students
with a history of math failureparticularly for students with
EBD and LD. To help these learners achieve, educators must
be prepared to teach math via : 1r.pili T.' validated tech
niques and the use of instructional strategies consistent with
the NCTM standards. Il current study brings to li ii the
importance of and improvement needed in i' , i' .ii current,
relevant, and comprehensive preservice and inservice training
to general and special educators on effective ..iqF,'.' ..hi , to
teaching math to secondary students with . 1:1 and LD. z
JOSEPH CALVIN GAGNON, PhD, is an assistant professor of special ed
ucation at George Mason University, His current interests include math in
struction for youth with highincidence disabilities and curriculum,
assessment, ,( ' .... .i..1 i. f .j . i. i and practices in exclusionary set
tings. PAULA N1 \'<'I "I[. PhD, is an associate professor of special edu
cation at the University of Maryland. Her current interests include math
instruction for secondary youth with learning disabilities and emotional dis
turbances. Address: Joseph Gagnon, George Mason University, Graduate
School of Education, 4400 University Ave., MS 4B3, Fairfax, VA 22030;
email: jgagnon@gmu.edu
AUTHORS' NOTE
This research was supported by Grant H324N990029, CFDA 84.324N, U.S.
Department of Education, Office of Special Education and Rehabilitative
Services, Office of Special Education Programs. The views expressed herein
do not ,... .:;:! represent those of the U.S. Department of Education.
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56 REME5 IAL AND SPECIAL EDUCATION
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