Teacher use of empirically-validated and standards-based instructional approaches in secondary mathematics
CITATION
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/IR00000476/00001
 Material Information
Title: Teacher use of empirically-validated and standards-based instructional approaches in secondary mathematics
Series Title: Gagnon, J. C., & Maccini, P. (2007). Teacher use of empirically-validated and standards-based instructional approaches in secondary mathematics. Remedial & Special Education, 28, 43-56.
Physical Description: Journal Article
Creator: Gagnon, Joseph
 Notes
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 (LD) responded to a mail survey. The survey examined teacher perceptions 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) factors contributing to the use of effective instructional strategies. The number of methods courses taken by teachers uniquely and significantly contributed to the number of effective instructional approaches that both general and special educators reported using. Significantly more special educators than general educators reported the use of specific, empirically validated instructional practices. Additional results, implications, and suggestions for future research are provided.
Acquisition: Collected for University of Florida's Institutional Repository by the UFIR Self-Submittal tool. Submitted by Joseph Gagnon.
Publication Status: Published
 Record Information
Source Institution: University of Florida Institutional Repository
Holding Location: University of Florida
Rights Management: All rights reserved by the submitter.
System ID: IR00000476:00001

Full Text




dawS. . . . . . . . . . . . . . . . .


Teacher-Reported Use of Empirically

Validated and Standards-Based

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 high-stakes 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 43-56









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 open-ended problem-solving
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 problem-solving pl-rf .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 -ininrtt-I .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 ...- ;. rrT-i. 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 real-world problem-solving 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 student-centered 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
high-incidence .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 three-phase 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 h--lpI :-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 15-year-old 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 small-group in-
struction and provide assistance while other students work in
cooperative - :..11
The final validated instructional , ii: - of interest is
self-monitoring. Use of this strategy helps students to moni-
tor their problem-solving behaviors via self-questions (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
pre-algebra 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. im-i,, - . i i- view (i.e., a dy-
namic and problem-based 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
real-world context.


FACTORS AFFECTING INSTRUCTIONAL
STRATEGY USE

Given the importance of research-based 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 research-based
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 1i-i,. -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 primary-grade 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 research-based 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., pre-algebra, 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, .1f-llr.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, 2000-2001). 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 nm-c.. 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. 1l-nii.-- , 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 in-1i 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 closed-ended 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 open-ended 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 ,r-I 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 open-ended responses
to assess the F..J;l' of .,1;n ;l .-,r, .. - into categories
(Fink, 1995). The agreement :-.r open-ended 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









Follow-up 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 ; chi-square tests to compare
proportions and i .r,-- i, ..n analysis. For each chi-square
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 open-ended 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 open-ended 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.- .-i-rri., 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 nnir-n 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: .r-i.. .-.
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 follow-up questions concerning
students with EBD and LD in those inclusionary classes.
Within the ',...,l-i 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 in-class 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 t-test
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

Pre-algebra
. .,.-I r :
Geometry
General ni..[h i l: -. I-1.-
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 chi-square statistics. LD = learning disabilities; EBD = emotional and behavioral disorders; ES = effect size.
*Insufficient cell size to calculate chi-square; 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 open-ended question concerning
their views of math were used to address a two-part 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., pre-algebra, 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 4-point
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 pre-algebra (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 follow-up il.- ;' !. ontheeffective-
ness of these courses in preparing them to teach math to stu-
dents with special needs and general education students. A
4-point scale was provided to respondents, with 1 = very pre-
pared and 4 = .. unprepared. There was a --i r! ii.. 211 sig-
nificant. Ilr-rl-: . 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, self-monitoring, 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 two-dimensional : :.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,: .I-1 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 ,lit-r.-i I in the
mean rating of how often teachers used techniques related to
student self-monitoring. 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 two-dimensional -;.,-.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 real-world 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 .ilf-rrmnilrirrI


i... I. self-monitoring strategies to help 66 43.4
students with problem-solving
activities


14 9.2 47 30.9 25 16.4


5.893 .015 -.20


Instructional it-hriiuniu - 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 same-ability 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 chi-square statistics. NCTM = National Council of Teachers of Mathematics; ES = effect size.
"Insufficient cell size to calculate chi-square; 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[i-sei to teacher use of
instructional strategies, a two-step 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 three-dimensional tools;
demonstrate a concept via two-dimensional 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 same-ability 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 pre-algebra, 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 - :l-r; "c..' .. of the vari-
ance above and beyond the other variables. Fi-n ,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 domain-specific 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 1-i. it. i from
collaboration with a. .-': -il educator.
In addition to content-specific 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) self-monitoring.

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 one-step 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 real-world 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., 2-4 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,- r-I i1 1-i 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 two-dimensional 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 I--irn
more prepared to use cooperative learning activities, peer tu-
toring, and small-group assistance ".. li Il. the rest of the class


worked on other assignments. Peer-assisted 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 .. i-r:l. ..iii[i..n,. L iF-m.r i ri .r . is an .t..:..
approach to help secondary students with il I- ...ti..:. in math
to monitor their pr F-P. 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-
lem-solving 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
- r--ii.. . 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 jn-r. 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 f-l .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 standards-based 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 .~i-1t .1 academic content standards (S.
Thompson et al., 2003). Furthermore, the reauthorization of
the Individuals with Li. :i-j,1-1.-.. 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 in-depth discussion. Thus, future research
should further analyze the :.. 1j;; .". t-i. between the focus of
-:It .1 and special education math methods courses (e.g.,
constructivist vs. traditional, rule-driven .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'"':n-m.i' and field-based 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 ..lf-i,-;.'..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 domain-specific 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 problem-solving skills 1 -l.int: & Dal-
kilic, 1992). This may be a daunting .ail n -: for students
with a history of math failure-particularly 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 ..iq-F,'.' ..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 high-incidence 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;
e-mail: 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.

REFERENCES
Algozzine, B., O'Shea, D. J., Crews, W. B., & Stoddard, K. (1987). Analy-
sis of mathematics competence of learning disabled adolescents. The
Journal of Special Education, 21, 97-107.
Allsopp, D. H. (i ' - Usihg classwide peer tutoring to teach beginning al-
gebra problem-solving skills in heterogeneous classrooms. Remedial
and Special Education, 18, 367-379.
Bernstein, B. (1997). Message and meaning: The Third International Math
and Science Study. Educational Horizons, ' i .23-27.
Blank, R., & Dalkilic, M. (1992). State policies on science and mathematics
education. Washington, DC: State Education Assessment Center, Coun-
cil of Chief State School Officers.
Bottge, B. A. (1999). Effects of contextualized math instruction on problem
solving of average and below-average achieving students. The Journal of
Special Education, 33, 81-92.
'. I' .: I , & i , , .!- -.,. T. S. (1993). A comparison of two approaches for
teaching complex, authentic mathematics problems to adolescents in re-
medial math classes. Exceptional Children, 59, 556-566.
Bottge, B. A., Heinrichs, M., Chan, S., & Serlin, R. C. (2001). Anchoring
adolescents' .i1 i ; -..':i , , i n i... . ;:.i in rich problem-solving en-
vironments. Remedial and Special Education, 22, 299-314.
i. - , B. A., Heinrichs, M., Mehta, Z. D., & Hung, Y. ., :- ', . ., I ..- .;..
benefits of anchored math instruction for students with disabilities in
general education classes. The Journal of Special Education, 35, 186-200.
Braswell, J. S., Lutkus, A. D., Grigg, W. S., Santapau, S. L., If i.,,., i S. H.,
& Johnson, M. S, . I. The nation's report card: Mathematics 2000.
. :.i:,i; . DC: U.S. Department of Education, Office of Educational
Research and Improvement, National Center for Education Statistics.
Brown, A. L., & Palincsar, A. S. (1982). Inducing strategic learning from
texts by means of informed, self-control training. . -, . . in Learning and
Learning Disabilities, 2(1), 1-17.
Calhoon, M. B., Fuchs, L. S., & Hamlett, C L. L. ' 1 I. Effects of computer-
based test accommodations on mathematics performance assessments
for secondary students with learning disabilities. Learning Disability
Quarterly, 23, 271-282.
Carlson, E., Lee, H., Schroll, K., Klein, S., & '-i'; S. ' 'I . ,ofper-
sonnel needs in special education (SPeNSE) final report., ...'.'
MD: Westat. Retrieved October 26, 2005,: ..- !.:,;. :- . :. - � .. :, -..i'
spense/FinalReportMethods_Pub.doc
Carr, S. C., & Punzo, R. P. (1993). The effects of . ..., i,.t n.i:, of aca-
demic accuracy and productivity on the performance of students with
behavioral disorders. Behavioral Disorders, 18, 241-250.
Culatta, R. A., - .;. ! i. I 1. :. Werts, M. G. (2003). Fundamentals ofspe-
cial education: What every teacher needs to know (2nd ed.). Upper Sad-
dle River, NJ: Merrill/Prentice Hall.
r -, l-..' i-!.i:,. .� L. (2000). Teacher :II' and student achievement: A
review of state - II.. evidence. Educational n ., Analysis Archives,
8().

REMEDIAL ANO SPiCIAL EDUCATION 55
Votlme 28. Number 1, J'nuart ycbruw 2007










Ernst, P. .! . . I I. The impact of beliefs on the teaching of mathemat-
ics. Paper presented at the Sixth International Congress of Mathematical
Education, Budapest. Retrieved May 16, 2004, from ' -.
.uki/PEmest/impact.htm
Fideler, E. F. (2000). State-initiated induction programs: Supporting, assist-
ing, training, assessing, and keeping teachers. The State Education Stan-
dards, 1(1), 12-16.
Fink, A. I. ' - The survey '. .. .' ... -I 1). Thousand Oaks, CA: Sage.
Gonzales, P., Guzman, J. C., Partelow, L., Paihke, E., Jocelyn, L., i. :i -..,
D., et al., i .' I . . . ':, from the Trends in International Mathemat-
ics and Science Study (TIMSS) 2003. (NCES 2005-005) Washington,
DC: U.S. Department of Education, National Center of Education Sta-
tistics.
Guntner, P. L., & Denny, R. K. (1998). Trends, issues, and research needs re-
garding academic instruction of students with emotional and behavioral
disorders. Behavioral Disorders, 24, 44-50.
Graham, S., Harris, K. R., Fink, B., & MacArthur, C. .*..* : r'i.. grade
teachers' institutional adaptations for weaker writers: A national survey.
Journal of Educational Psychology, 95, 279-293.
Hogan, S., & Prater, M. A. (1993). The effects of peer tutoring and self-
management training on on-task, academic, and disruptive behaviors.
Behavioral Disorders, 18, 118-128.
l i.irl;i-i .i rF. J. (1994). Instruction in concrete, semi-concrete, and abstract
representation as an aid to the solution of relational problems by adoles-
cents with learning disabilities (Doctoral dissertation, University of
Georgia, 1994). Dissertation Abstracts International, 56/02, 512.
Hutchinson, N. L. . * -"; Effects of ....:; : . strategy instruction on alge-
bra problem solving of adolescents with learning disabilities. Learning
Disability Quarterly, 16, 34-63.
Individuals with Disabilities Education Act of 2004, 20 U.S.C. � 1400
et seq.
Kelly, B., Gersten, R., & ', .,. i.* (1990). Student error patterns as a func-
tion of curriculum design: Teaching fractions to remedial high school
students and high school students with learning disabilities. Journal of
Learning. . ' , .23,23-29.
Koency, G), & Swanson, J. i .' I ," r 1! i The special case of mathematics:
S: content knowledge a major obstacle to reform. Paper pre-
...N .1 . ,a ii .,,i ' I.. ,;.., -of the American Educational Research As-
sociation, New Orleans, LA.
Maccini, P., & I.. :- r. J. C. _'-1 1 - Best practices for teaching mathemat-
ics to secondary students with special needs: Implications from teacher
perceptions and a review of the literature. Focus on Exceptional Chil-
dren, 32(5), 1-22.
MacciniP., *. ..i, I r I C. (2002). P . .. , - :.,.' i. i.. . P. .i. r'
standards by special and general education teachers: Implications for
practice for secondary students with emotional and learning disabilities.
Exceptional Children, 68, 325-344.
Maccini, P., & Gagnon, J. C. (2006). Nature and type of mathematics in-
structional approaches by special and general education teachers. Ex-
ceptional Children, 72, 217-234.
Maccini, P., & Hughes, C. A. (2000). Effects of a problem-solving strategy
on the introductory performance of secondary students with learning
disabilities. Learning Disabilities Research & Practice, 15, 10-21.
Maccini, P., McNaughton, D. B., & Ruhl, K.' --.. I~.. . ....... for
students with learning disabilities: Implications from a research review.
Learning Disability Quarterly, 22, 113-126.
Maccini, P., & Ruhl, K. L. (2000). Effects of a graduated instructional se-
quenceon 1 .- i-, . i' i,,i .i:i. r. of integers by secondary students with
learning disabilities. Education and Treatment of Children, 23, 465-489.
Manouchehri, A. (1998). Mathematics curriculum reform and teachers: What
are the dilemmas? Journal of Teacher Education, 49, 276-286.
Monk, D. H. (1994). Subject area preparation of secondary mathematics and
science teachers and student achievement. Economics of Education Re-
view, 13, 125-145.
Montague, M., & Bos, C. S. (1990). Cognitive and metacognitive character-
istics of eight grade students' mathematical problem solving. Learning
and Individual Differences, 2, 371-388.


Montague, M., Bos, C. S., & Doucette, M. (1991). Affective, cognitive, and
metacognitive attributes of eighth-grade mathematical problem solvers.
Learning Disabilities Research & Practice, 6, 145-151.
Moore, L. J., & Camine, D. (1989). Evaluating curriculum design in the con-
text of active teaching. Remedial and Special Education, 10(4), 28-37.
Mullis, I. V. S., Martin, M. O., Gonzalez, E., r., .., K. D., Garden, R. A.,
O'Connor, K. M., et al. (2000). Trends in International Mathematics and
Science - . : . . . 1999 international mathematics report. Chestnut
Hill, MA: TIMSS International Study Center.
National Council of Teachers of Mathematics. (1989). Curriculum and eval-
uation standards for school mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics. (2000). Principles and stan-
dards for school mathematics. Reston, VA: Author.
National Institute on Educational Governance, Finance,: .I. . i, . and
Management. (1998). What the Third International Mathematics and
Science Study (TIMSS) means for systemic school improvement, Wash-
ington, DC: U.S. Department of Education, Office of Educational Re-
search and Improvement. (ERIC Document Reproduction Service No.
ED 425 938)
No Child Left Behind Act of 2001, 20 U.S.C. � 6310 et seq. (Reauthoriza-
tion of the Elementary and Secondary Education Act)
Pachnowski, L. M. (1997). Teachers' definition of math: Creating and im-
plementing an instrument. Paper presented at the annual meeting of the
Midwestern Educational Researchers Association, Chicago, IL. (ERIC
Document Reproduction Service No. ED 417 941)
i .i:. . . . ,, & Cawley, J. F. (1997). r. - .: . teachers to teach mathe-
matics to students with learning disabilities. Journal of Learning Dis-
abilities, 30, 188-197.
.., ii: , Education Data (QED) School Personnel Database. 'i' .i.i ,. ,
Denver: Scholastic.
Quinn, R. J. (1997). Effects of mathematics methods courses on the mathe-
matical attitudes and content knowledge of preservice teachers. The
Journal of Educational Research, 91, 108-113.
Rosenshine, B., & Stevens, R. (1986). Teaching functions. In M. C. Witrock
- t : . Handbook of research on teaching (3rd ed., pp. 376-391). New
York: Macmillan.
Rivkin, S. G., Hanushek, E. A., & Kain, J F. (2000). Teachers, schools, and
academic achievement. Retrieved December 3, 2003, from i:',-

Thompson, A. G. (1984). The relationship of teachers' conceptions of math-
ematics and mathematics teaching to instructional practice. Educational
Studies in Mathematics, 15, 105-127.
Thompson, S., Lazarus, S., & Thurlow, M. (2003). Preparing educators to
teach students with disabilities in an era of standards-based reform and
- ..* . (Topical Review 5). ( .'1i. Park, MD: Educational Pol-
icy Reform Research Institute.
Took, D. J. (I ' I Student teachers' mathematical backgrounds and attain-
ment of their secondary students. ' . House (Menasha, WI), 66,
273-277.
'' ... .A., Coleman, A. B., I-. i. RJ., & Phelps, R. P. (2003).
teachers around the world I li.. Information i .. ' ... Princeton, NJ:
Educational 1: i.: Service.
Weisberg, H. F., Krosnick, J. A., & Bowen, B. D. (1989). An introduction to sur-
vey research and data analysis (2nd ed.). Glenview, IL: Scott, Foresman.
'.. .mi. :,1 ;. D. (2002). 2000 National survey of science and mathematics
education: Status . . . . : Chapel Hill, NC:
Horizon Research.
Wilson, L. D,, & Blank, R. K. (1999), Improving mathematics education
using results from NAEP and TIMSS. Washington, DC: Council of Chief
State School i iiT,. State Education Assessment Center. (ERIC Doc-
ument Reproduction Service No. ED 431 001)
Witzel, B. S., Mercer, C. D., & Miller, M. D. (2003). Teaching algebra to stu-
dents with learning difficulties: An investigation of an explicit instruc-
tion model. Learning Disabilities Research & Practice, 18, 121-131.
Woodward, J., & Montague, M. :' '- . Meeting `,i .iil'.. : of mathemat-
ics reform for students with LD. The Journal of Special Education, 36,
89-101.


56 REME5 IAL AND SPECIAL EDUCATION
Volhme 218, Number 1. JantuaryiFebruary 2007




University of Florida Home Page
© 2004 - 2011 University of Florida George A. Smathers Libraries.
All rights reserved.

Acceptable Use, Copyright, and Disclaimer Statement
Last updated May 24, 2011 - Version 3.0.0 - mvs