Mathematks Instruction in Secondary Interim
Short- and Long-Term Alternative School
Joseph Calvin Gagnon and Brian A. Bottge
ABSTRACT: Some of our most d,..lk-a i T. youth are educated in
interim, short-, and 1..i .. i.- :T alternative school ..rrin,:- Many of
those youth have emotional disturbances and learning disabilities
and have experienced academic and social failure in other school
.-..;. Federal mandates require teachers to provide mathemat-
ics instruction to those students with d. - .i- , :-i.. that is consistent
with r. -,.1 r public schools. _., :ii characteristics and institu-
tional barriers however, make that a .t' J ..:. task, In this article,
the authors ihi.i 1, ii i - :r.i,C.:,.,i issues that .it: ., instruction for
secondary school-aged youth in exclusionary school - - i, r. The
authors also provide specific examples of ' i'i:_ * .: instructional
.Lri i, ,.!., in mathematics, such as enhanced anchored instruc-
tion, hands-on activities, and other integrative approaches.
KEY WORDS: .. ..... secondary education, special education
7 H.I-:.'l'S IN 11i.'.1! I IM, short-, and long-term alterna-
tive school settings are faced with the difn.. ili task of meet-
ing the needs of high-risk students while also mi .i:'.d:-.in;
high academic standards (Aron & Zweig, 2003). Ti,: N.
Child Left Behind Act (2002) places increased accountabil-
ity demands on teachers; while at the same irn,. they are
charged by the Individuals With D -.ai'ti.; Education Act
([TDF.\1 2'." ili with r: ,r,-il.ilir, for pr.-. ili- , students
with special academic needs and problem behaviors access
to the general education curriculum. In i. , i:r . organiza-
tions such as the National Council of Teachers of Mathe-
i.,ij,. r ,!n-a.., (["'.!TM]; 2i1 1',) urge teachers to accom-
plish those tasks in problem-solving contexts that reflect
Amid calls for reform, .i.J1',.-.rw' issues confront teach-
ers who must plan and provide mathematics r.s :i ,ii for
youth in interim, short-, and 1- cll fer-m alternative school
,.,irn*. In this irri:l-, we identify several of those issues
and provide examples of how research-based instructional
approaches in mathematics can be effectively implemented
in exclusionary school .. ttiiq.i Specifically, we hope to
ac!inr- the range of alternative school .'ti[n_, id.inif; the
institutional ..ilk .,: that impede instruction for students
with severe academic and behavior .irt i.' Iir-., d-.. rib, the
characteristics of youth who attend those schools, and pro-
vide examples of !,. ri. c instructional approaches in
ri lll-itil. that can be implemented despite the difficul-
ties ripov I-, by those challenges.
Defining Interim, Short- and Long-Term Alternative
School /- ,'.. . ,..,
An alternative school is described as, "a public elemen-
S. '.- i.i n.1.!: - school that addresses needs of students that
typically cannot be met in regular school, provides nontra-
ditional education, serves as an adjunct to a .' - il r school,
or falls outside the categories of regular, special .J_.: J.ii. .n
or .::..nal education" (U.S. Department of Education
[DOF1, ,'irC'., p. 55). Youth are placed in alternative schools
for a number of reasons, in;- i ;lr, (a) behavioral problems,
(b) emotional concerns, (c) academic failure, (d) law viola-
tions, and (e) as a proactive alternative to increase the
chances of academic or behavioral success or both.
The definition of alternative schools may lead to some
confusion because considerable overlap can exist between
alternative schools and other types of schools, such as (a)
therapeutic day treatment schools, (b) therapeutic residential
schools, (c) juvenile correctional schools for detained; ...,, i,,
and (d) juvenile correctional schools for committed youth.
-, L-. h-lI,.:- some key distinctions are evident. For exam-
ple, therapeutic day treatment and residential school pro-
grams t: K:-.:. have a *ic;:iKant therapeutic component
that may not exist in other alternative schools. Day treatment
schools provide mental health support to students, social and
Joseph Calvin Gagnon is assistant professor in the ( ..'. , fi .:. . -
tion and . . . .. . ... .-. ,-,.:* Mason University, Fairfax,
Virginia. Brian A. Bottge is professor in the Department t.' '..'' .
ration: *..,'. :'., and Special Education at th. U.,. i, .v it .
consin-Madison, 1 ,. ,, . .-. Copyright � 2006 Heldref Publications
40 Preventing School I ,,uwr.
clinical support to their families, as well as general and spe-
cial education (A r' i ri,:. ('I.. --.. .; & Palma, 1992). Resi-
dential schools have similar therapeutic and educational
characteristics, but students also have 24-hr ,.r.,lit.,!tn;
(Kauffman & Smucker, 1995). Alternative schools differ
from juvenile correctional schools because the latter are t..; .
ically secure care and include youth who are .1 I.l con-
t.I.. for 24-hr per day, 'i .Irh are arrested and held in a
secure detention facility until a ilri,- is held '. 'r, a
judge. A-youth's stay at a detention :f..'i'it may continue
until he or she has a .ii- [1"', .ii i'.r..i hearing, and even longer if
a placement is unavailable (Snyder & ..:l.nuri..,. 2006).
Juvenile correctional schools for committed ;. .'-ud provide a
secure care fi.:'i'r for '... .iTi who have been adjudicated
(i.e., the juvenile version of being found _.uir I).
Institutional C. r', ::-
High student turnover combined with .. .t 1i and safety
issues contribute to the challenges associated with design-
ing and delivering instruction in interim, short-, and I.----
term alternative schools. F-.. h,:i- in juvenile detention
facilities are most ,n:..:.:--d by student turnover, ;*-!'r' :I l.
when the average ::..n-rh of stay is just a few days, .Ilti.. h
detention may extend to a week or several months in some
cases (Austin, Johnson, & Weitzer, 2005). Students in alter-
native schools commonly stay from 1 month to 1 year,
although a third of students attend for more than 1 year and
even until they graduate (Lehr, Moreau, TI .r-. & Lanners,
I, Ii-l. The ,_-nthb of stay is approximately "' months in
therapeutic day treatment schools and 15 months in resi-
dential schools (Gagnon, 2nOth',. In juvenile correctional
schools for committed youth, the reported average' - nth of
stay is about 9 months (C ::'n ,n 2006a).
Another _i,.r,-'r for teachers in those schools is the
competing interests of other professionals within the facili-
ty. For example, issues of safety may be a significant con-
cern r .r youth in juvenile corrections and in he'r:a '.Jt
school programs. Although safety is a valid concern, it is
also the r. -; i ..i,-i iti of educators to ensure that students
can maintain access to an appropriate education. Due in
large part to safety concerns, there is often lack of adher-
ence and enforcement of the IDEA . ' ' -4i and No I h ilI
Left Behind (2,'I',' in juvenile correctional L. i'a.-.: and
therapeutic day and residential programs (Gagnon &
h 1.. L au,:I., 2004; Leone & Meisel, 1997; Wolford, -'l""',
' . r[.i hb , exclusionary schools may operate outside the
typical school oversight mechanisms and may not be held
accountable for student academic progress.
Characteristics ,, . .',. in Interim, Short-, and Long-Term
Alternative School Placements
Perhaps the most relevant and consistent student charac-
teristic across all interim, short-, and long-term alternative
school settings is the :. I, .1 1. status of students. A rli :I.
most students with disabilities in those exclusionary set-
tings are classified as emotionally disturbed (ED),
researchers have reported that over half of students identi-
fied as ED may also be eligible for a .. .y...' I learning dis-
.i, i; (LD; (.i , -".- . Hooper, & .1 ,''i..'. 1999; Got-
.fr,:,... Gottfredson, & Skroban, 1996; I' .], Gonzalez,
Nordness, Trout, & L.pI. .. i .2 i, . Estimates concerning
the number of :''ith with iL .it-.l.. . in alternative schools
range from !1. -1 '., with over 50% of those students
'... tr I:. as ED (Gorney & Ysseldyke, 1993; Lehr et al.,
2004). il! I r therapeutic day treatment and residential
schools, one half to three fourths of the students in day
treatment or residential schools are labeled ED (Duncan,
Forness, & Hartsough, 1995; \ 1Clr:', F -r'.1 ',r, Boodoos-
ingh, l i,,.;.,'- & r *. r.i.. !': .' ' and 6 times more youth
are .1 ' ti.-J with L1 i than are youth in any other disabili-
ty c.,ir - .r. (G.,'rn,. '21 .ii ':'r .hi . In juvenile corrections, more
than -.0 . of youth are classified with a disability, and over
-.: of those children, are classified as ED (C.L;:r-r:
2', n,,., Quinn, i: itIl int i. & Leone, 'I I,:, Quinn, Ruther-
ford, Leone, Osher, & Pi..r,.:r 2005; U.S. 1 itii,:rt of
Education, 1999) and another .2 - are classified with LD
(Gagnon, 1_- it-..
School failure. Another common characteristic of most
S,,I'' ih i n trim, short-, and long-term alternative school set-
tings is a long history of school failure. A primary focus of
alternative school educators is to provide support to some of
the most at-risk students, to prevent them fir .r -, ..rF'r1in.- out
(Raywid, 1994). Given the high i .... ,:.- of youth with
I.I and LD in alternative school placements, it is important
to consider that 65% of youth with ED and 38% of students
with LD drop out of school without a .ill rI I' (U.S. Depart-
ment .' ..ii ii .n. 2'" 1'. Youth within those exclusionary
.-,.:i-i. , ril .'E'. function academically below peers in reg-
ular public schools '7.'.-'1 & N1 lr,, 2001). If one considers
the research on youth with ED or LD, it is p ..- -I.: to get an
accurate picture of ....il. academic fr:ii..Tri'inii_ in interim,
short-, and long-term alternative school ain,; .: For exam-
ple, youth with ED and LD L. pi.: 11. function below .-r r.l,-
level in mathematics and have ni.. iultic with basic *ki;l~,
hii zh -,.c.,-I concepts, and cL'rll*k..n-: L1 i', (. :h'.:,:' .:,
O'Shea, Crews, & Stoddard, 1987; Greenbaum et al., 1996;
Huntington, I '-', Hutchinson, 1993; Maccini & Hughes,
2000; Maccini & Ruhl, ?'i , 1, Reid et al., 2004; Trout, Nord-
ness, Pierce, & Epstein, 2003).
Behavioral .' "..' ,': .Mj youth in interim, short-, and
long-term alternative placements have -r nifi.: n ir behav-
ioral issues that have had a considerable impact on their
academic and social success. In general, youth with ED are
4 times more ii.kl. to be suspended or expelled than are
youth with other disabilities (W7 r n, r. Kutash, Duchnowski,
Vol. 51, No. I
;:'.I'..: & Bottge 41
Epstein, & Sumi, 2005)., '. .'.'. affected by 1 I.i' are about 13
times more likely to be arrested than are youth without dis-
abilities; .pi] . ;:'.t.:1 . .11 are arrested, in detention, or
on probation while enrolled in school (Doren, ii iL1. &
Benz, 1996; Silver et al., 1992; U.S. DT ,-, , it..'. r , .1 t ....,.
tion, 1'- .-,i. V -.il, with LD i, .. have similar difficulties.
For example, youth with LD are 3 times more likely to join
a gang than are youth .,:f -ti,.. I by LD 'S ,- i. & Sick-
mund, '-.ir-.i .
Research concerning youth across the various alternative
i-:-i;.- indicates similar behavioral concerns in each pr,
A.. Oin. In one study, for -,. unpk 22% of students in alterna-
tive schools had irt'. 1 .: ni-. . i ri a law enforcement agency
(Davis, Brutsaert-Durant, & Lee, .'i.ii'1.. ri-iblLr[,r .r in
' ,r n..- ,t .i. students in alternative schools were 3 times more
l!..-., to commit acts of vandalism, assault, or -t.,,tltin;
,J .11 ct.n, Harrison, & i.-.r, I''-. Fulkerson and col-
leagues also reported that compared with students in regular
public school, students in alternative schools were 2 times
more 1l..:1'. to be involved in .:n~- In a 7-:. .i-r study,
Greenbaum (l'-,Jr.., noted that 4'- .- of youth .tfol tni by
ED who attend therapeutic residential schools were arrested
at least once and 34. i' - were ...l i, I i.-I Even elementary-
age youth in -hIr. '...ik day treatment and residential
schools have a higher than .' 'r ._. involvement with the
juvenile justice system (Gagnon & Leone, 2006).
Example of Effective Instructional Approaches in Mathematics
P.r. ...i, r:nt of the newest wave of reform in mathematics
education -n, ,.1u- .. teachers to deliver instruction that pro-
vides more opportunities for students to enhance their com-
'piii and problem-solving skills in concept-rich contexts.
Government and professional ..A"ip. have hii...i the
emphasis on pir ,.Nrl,--.'-ii,; by urging educators to help
all students acquire and jr':. 1111-. especially those .'. II.
necessary for successful employment (e.g., Goals ., '',,
Educate America Act, U.S. Department of Education,
1994a; Preparing Students for the 21st Century, Uchida,
Cetron, & M.,lc.ti: i-, .-'r,". No Child 1, fr Behind Act of
.-*'"!) A..., ,li:,. to the NCTM I ''ii,. math instruction
should enable students to deal with novel problems in a
* .ri,:i, of settings.
-.,, :. ,,. Anchored Instruction
One promising method for tr.: irl.r. the intent of those
initiatives into practice for students with LD and ED in
alternative schools is called enhanced anchored instruction
(EAI). Based on the concept of anchored instruction (Al),
popularized by the C r. !it;>.n and T.'Thn..h..:. Group at
Vanderbilt _'i; .: -it, (1990; Goldman, Hasselbring, & the
C, :nti. i. and Technology Group at Vanderbilt, 1997), EAI
uses a combination of nitii.:;b.i -1,j.,o, problems deliv-
ered on CD-Ri''I (called anchors) and related hands-on
Pi' i:.. (e.g., designing, '.Iiii'n;. and 'i.in on hover-
crafts). As in most authentic problems ( :. , ir.r:'. Secada,
& Wehlage, 1 ", ' '), and i.iii !. - most school-based tradition-
al problems (e.g., word problems), each anchored problem
consists ,f several subproblems embedded in a:,' II .:. and
motivating context; students usually take five to ten 60-min
class periods to solve -1h. ...- problems.
A series of experimental and quasi-..: ;-n'.j studies
with n,i.ici and high school students, i:,..ii;, low-
achieving students, have yielded moderate-to-large , :..
sizes on r':.. bk--., ,ding tests (. i .81) and transfer tasks
(.37-.62; e.g., B ". :,. 1999; B.,il.- & F-i.,....'[1 _. 1993;
' .rr:.:, Heinrichs, Chan, & .',:rlii 2. '1; B..- . Heinrichs,
. lhri & Hung, 2002). An unusual ..l'., of EAI prob-
lems that is ..,:,. i ..- ., important when working with stu-
dents with ED is that they motivate students who have pre-
viously shown a dislike for mathematics to -n.;: in
i1 T-i C :-ft it, in contrast to most traditional text-based
problems (Lesh & Kelly, "I'L'; The i.iii .... i pi.'l'..
those applied p-i' .4-: re ; . skateboard ramps, h,. ' :ir.' it
ti. *rLd helps students understand the importance ande n ,.-.t
of l-.uni iin, math. In EAI, the amount of reading required for
solving the problems is limited, a ..tr-.:.t'. ni 1t rIt for
many students who experience J ii!i t,-1 in math and read-
ing i-uch, & Fuchs, "".. Geary, Hamson, & Hoard, 2� i ',
Study Boosts Problem Solving V./.'
.r., , " ; .; , Bottge, l'iiu ..la, and S nki i t._.,r, 1. '11ni r] :--:1'.
used EAI in a study of students at a public charter transition
school (l "I'S) c. -. ii.: at-risk high school students from 16
ni._i;-t,, iir.; school districts. All of the students (10 boys, 7
girls) attending the C'I , at the time of the -�r.i! were includ-
ed: 6 students were in 12th grade, 10 students were in 11th
grade, and 1 student was in ,', grade. The participants were
C- t,.. i in, except for one Hispanic student. 1.- i students
were identified as ED and 2 of those students had additional
diagnoses, 1 with attention . ..i.-ii-,. ic:.,,.: it. disorder
(ADHD) and I with LD.
'", . t of the students had been suspended from their home
school at least one time for a, .r.-%' of serious behaviors such
as extreme aggression and in'i. .mplih..nc Almost all ,'-' )
of the students had been involved in court actions; serious
alcohol and drug issues ," 1 . i; some students were homeless
and lived on the streets (24%); and a ;':'.- students were par-
ents (- ,. '.. .-i A of i- students had been unsuccessful in at
least two high schools I-. .. their referral to CTS.
EAI methods. The students were taught two complete EAI
problems (i.e., two multimedia-based anchors and two
.jrplid problems) in 75-min class blocks over 28 days
B,ri .;. Rueda, & S,:i.-ii,..,n . Iir., The EAI problems
jI:'nc'. closely with several expectations ;. Tin,,'1..! in the
42 Prevenling School I 5h.or,.
NCTM t.in. Lia i for grades 9-12 related to problem solv-
ing, measurement, data .r. i1 . and ;i. l...tll,. - .'- r,,. .rr ,
and algebra. D;ir;- the multimedia-based instruction, the
teacher .i1 i r -1. ., .! the EAI problems on a 1 'L screen at the
i:.. ri of the classroom ,.' using a video pi.,. .I. i connected
to a laptop computer. Each day, the teacher asked students
to summarize the main problem in the video and to review
the i-f;:.. n.i , .'i they had entered in their problem-solving
packet. The packets contained a set of forms that asked stu-
dents to .list the main questions posed in the video and to
identify relevant information for i ii:.- ' :'-.: the questions.
Space was also provided for students to compute their solu-
tions and describe their procedures. B'ri , rni- were welcome
to search the CDs to access important information at any
time in the class period. For example, students could navi-
gate the video for important ii, I. : nii . 'r compute solutions
to subproblems, and suggest tentative solutions. The teacher
went to each group, checking students' rn ,,.,-- *. redirecting
them back to scenes in the video when ":1... had questions,
and encouraging them to keep trying to solve the problems
even when the solutions were J: 1'.. u. Ii to find.
One EAI problem c ill,:- Fraction of the Cost, p.q.ti .
three teenagers who want to build a skateboard ramp but
must find a way of I1... ,. it with the amount of money and
lumber available to them. To answer the subproblems in the
video, students needed to calculate the -r- ..,r i,- of money
in a savings account and the sales tax on a purchase. Ir\ -
also had to read a tape measure, convert feet to inches, deci-
pher building plans, construct a table of materials, compute
mixed fractions, estimate and ;..ip.iti-- combinations, and
calculate total cost of -ii niLr the ramp. Several learning
tools on the CD- i.?,,'. I helped students understand concepts
in the overall problem. For example, one module showed a
Slii .._ -.J i , ,.. . i r i.i i ramp that students could rotate to see all
sides. I h.. 2 x 4s (i.e., .*lI.'.n. ni r - of the lumber) used in
'-i.ili;:i. the ramp were color-coded to enable students to
see more clearly which lengths corresponded to which parts
of the schematic drawing. In another module, students
could build the ramp by dragging Ikn- 1.d.- of 2 x 4s out <.f a
stack of lumber and !i.]inI _- them in the correct way.
dniiik', accomplished two applied projects after solving
the problems in the Fraction of the Cost video. For the 1:: .
rp .ju�t. students had to build the skateboard ramp by .-n.
the problem-solving skills -ih,' had just learned in Fraction
of the Cost. Sriuk '.i helped carry the builiir, materials
(e.g., lumber and screws) and the tools (e.g., . ill saw, elec-
tric screwdrivers, and drills) from the .. ;rin. lot to the
classroom, an activity that was highly r,.i.i'..1i1n because
the students all at once realized that they were - .i.- ,, have
the opportunity to apple ' what il't. had just learned. The 2 x
4 pieces of lumber were precut in the exact k1-i;:h., as those
shown in the video. The lumber served as full-sized manip-
ulatives, which the students used to measure, cut, and
assemble into a skateboard ramp. After the students had
built two of the large , i-i:. '1 .*' tried them out with their
For the second applied problem, students had to plan, draw
to scale, and construct a , ..i, . cage out of PVC pipe for a
ih... i.-r'i :-, (see - I-. 1). As in the viId:.. h.., '- problem, stu-
dents interpreted schematic plans, drew them to scale, and
figured out the most economical way to use the pipe. Stu-
dents attached the pipe to 45% or :i', .. connectors, which
-,': had to incorporate into their plans. >\ ,. :i the cages were
completed, the students lifted them onto the bases of the hov-
ercrafts that were powered by a , ! F blower inserted into a
hole in the platform and that inflated plastic attached to the
underside. The big event occurred when students drove the
I.:r.i.. up and down the hill. i.. of the school.
The second EAI problem involved one of the episodes in
the Adventures of Jasper V..... I.l-. (Learning T,. - lI.-.-.
Center at Vanderbilt Un.i . -rr,. 1997) called Kim's Komet.
In contrast to Fraction of the Cost and the hovercraft appli-
cation, in which the students focused on linear measure-
ment and fractions, the purpose of !, - - 's iA-. ', " is to help
students understand the relationship between rate, time, and
.i, .i,.:-. Students had to interpret figures and data from
tables and - A,-,i i'. construct their own tables and graphs,
recognize relationships .Li.: r those data, and make predic-
tions on the basis of their solutions. For example, one set of
items asked students to calculate the speed of their cars on
the straightaway for each release point on the ramp, plot
those points on the -1 .i. 'ih. and draw a line of best fit.
Ai'.-r students solved the problems in Kim's L. ; ..:.' i--.
competed in their own pentathlon, using a full-sized ramp
and events like the ones shown in the video (see Figure 2).
Students made their own cars out of blocks of wood and
timed them from several release points on the ramp. I n..,
used an infrared detector to measure the time of the cars
from the ':.e .innin.- to end of the straightaway. After stu-
dents made their graphs and plotted their times, the teacher
revealed to the students the range :f speeds that their cars
should be traveling to negotiate each event. . -'. . r had to
construct their own graphs and lines of best fit to predict the.
speed of their cars at the end of the straightaway for every
release point on the ramp.
'.,'., '!-:- O'.r.Tll test results showed :tfti:Lt sizes (Tr2) of .75
and .77 on the Fr...'. ".. .*" '. Cost and Kim's . ... ,-rt problem-
.,-. !i,:- .tests, respectively. Concept analyses indicated stu-
dents : .h1r,--.i Ii 4ii levels of understanding for each of the
concepts on the problem-solving that posttests. On the test
measuring concepts in the Fraction of the Cost problem,
almost i!i of the students could read a schematic plan, two
thirds learned to read a tape measure, and 8 out of 10 students
received at least partial credit ' ,r showing how to cut the
wood in economical combinations and how to add and sub-
.j. 51, No. I
Hovercraft F i.i'. 1'. ,
7 ig \ - - - L
12" 25"" -
FIGURE 1. I'lanniimi. hbuilrini, and riding. hovercrafts.
tract mixed numbers. M .,. students .1- 'p.' 1 a materials list
based on their calculations and calculated the total cost of the
project. On the Ki . '7 Komet test, more than 80% of students
could construct a line --i -rh from a data table, label correctly
the x- and y-axes, plot the rates computed from .11 .. .: e and
time, and draw a line of best fit. Scores on .taJindLr..-;:.d
posttests (including computation and problem i,,I inr'. how-
ever, showed only a flihit p,.....ir trend, and mean scores on
a fractions-computation test . I u i a.,. decreased.
r,.-l, the most r, '- -..:l findings :-rier'.] from the
student interviews at the conclusion of the study. A domi-
nant theme centered on the relevance and importance stu-
dents considered the curriculum to be in their daily lives.
They .-- ..,':i1.: Ihi...c the air:-ii .n i i problems (i.e., build-
ing the skateboard ramps, pl-.rn n. and constructing the
hovercrafts, and .;r .ph,.ii the performance of the cars they
made) because, as one student stated, "It's like we're learn-
ing math and ... I didn't even understand until we built that
Gugnon & '. *i'
44 P.,. n iri. 'ii hr.il I j.I.r,;
6 I i -: 1 -'I I i I I ', '1n ', - .. 'I" - ,
, , , , -. - . L .-K
1 11 11 11 11 f IT l If
FIGURE 2. Ramps, tricks, and competition with AMin', Ainm .
skateboard ramp ... To be able to get out and actually work
with your hands and when it's done, like wow I built that
using the skills that I learned with math." He was a kines-
thetic learner and rarely had the p- ..prli:nii . to learn with
both his brain and his hands in his regular school.
One girl :c,:,r'i-,n:dJ herself and her pi ,i'l:i- -, 1., part-
ner as the girliest girls in the world and thought that the pro-
;,-' . were more guy-oriented at first. Then, as she began to
work on the ramps and the hovercrafts, she believed that the
pk,.i.:. posed ;i.'-ii-,; Ie..'ing opportunities for them.
She also !-.., :;'h that i: .r.,andirn:_' the mathematics
embedded in the problems (e.g., r..:.-iir.,; schematic plans
and drawing plans to - .Ke' was -.p. i.ll; important as %'..
constructed the hovercraft. She reported that her friends did
not believe her at first when she told them she was using
power tools to build skateboard i., i [,. and hovercrafts.
Explanation of results. Elements in the Key Model jB .ti,
2' i l) surfaced as important factors in improving the prob-
lem-solving performance of both average- and 1 ., .-.i.:hi- in;
. . - - - " ~ * e
',-: 51, '1. 1
11] i : I :] 1 n U ] 1 il 1
Gagnon & 'io;,t.: 45
students in .-iir.-. with EAL According to the Model, foun-
dation skills (e.g., adding and -*.,r i, ,_. fractions) should be
taught to students in direct ways (e.g., explicit i: ',.tr- ii. .",;' as
J .l: work on motivating tasks (e.g., video-based problems)
and meaningful -'i.i, ,..-: (e.g., t-,,.lill , I ,-: c, i.,ii , Earlier
research has clearly shown important links within each pair
of elements in the Key Model 'c - . an interesting problem
leads to motivated students; strategies for .I' ._- video-
based problems transfer to applied problems).
A ll:.u1.. h elements of the Key Model helped to explain
improvements in learning with EAI, they also enabled
instructors to identify reasons for students not making
more ,itiit. irt gains in skills that are more procedural.
For .:- rp,- the lack of improvement in .'.n,.'.ini: with
fractions in the study just described was the result of stu-
dents' intense, ,n-. ri. reactions to 1 i i,-_ basic skills
and taking t.:,.liti.,. 1...1.1.: tests. When the teacher
attempted to teach fraction .,-... I'n-i, . the students refused
to listen and threatened to halt their work on the EAI prob-
lems ll . :C ;h,-r
Si-,.:i lack of motivation in response to p. ', instruction
on basic - .i'. has led to current research aimed at discov-
ering more effective - i . of teaching fractions skills more
: ili. within the 'e..'.in,, .' , -i .- o ', - of EAI. The
method used to accomplish that goal relies i ..- . . . on the
concrete-semiconcrete-abstract (C-S-A) sequence. DU: i ,
the first week of instruction with ''.. ::. ; of the Cost, stu-
dents make their own !' i I, i.! strips (i.e., concrete i -..L'i to
help them understand the equivalence and value of common
fractions used in measurement (e.g., halves, 4ths, 8ths, and
16ths). They also learn how to add and subtract simple frac-
tions. Di i ,' the semiconcrete stage, students work on a
:i ,. . ... .h L instructional 11 ..-;r -ii' called Fractions at Work;
a p.r. n, designed to help students bridge their under-
'. diiu. from concrete manipulation of fraction strips to
more abstract .J._-.ri ln . I i.t new software enables stu-
dents to manipulate fractions, thereby l-'lpl them under-
stand concepts related to rational numbers f, *.v, a measure-
ment perspective. The software also provides teachers with
a sequence of instruction during this phase that appropri-
ately addresses the misconceptions inan,. students typically
retain about fractions lhr..,'Ihi high school. Ir ruiii'l, I11
EAI, students revisit and further build on their understand-
ings of fractions as teachers review in a cyclical pattern the
As just ..i :ii'..], the use of 'C.hnl'.'l,/ and problem-
based 'rjnii,-, as .l,:i ,red in the EAI problems can be
-ffe' c .- in boosting the problem-solving skills of youth
with LD, L D1', and BD in alternative settings. The EAI exam-
Il.' provides opportunities for youth to practice working
co. pr-.!i' -p 1. for a common purpose; skills that those stu-
dents often lack. C ',i' i.i.I- :t.iri,"'i . provides students
with .-.It ;.i nr.i., and structured interaction with peers. The
combination of ,c..,..!. -. and collaborative problem-based
learning may improve the academic achievement of .,1,,
with ,j,. i-, 11ii.:. and also set the stage for further positive
interactions among students (PT. m..!i & H1- ! .i , rI'*-,
The examples we provide also integrate worksheets that
require students to I L.11, i f '. and write key questions asked in
the video-based problem as well as to identify key informa-
tion. The forms include an area for -.., . :i,.,i 11.n the solution
and describe the procedures used. Those f' ,r,.- , allow stu-
dents to systematically approach the problem and focus on
both problem representations and solution. M., ' youth
with ED and LD do not use a n', "h ..11 .!1 approach to prob-
lem representation and solution and may even skip problem
representation (I-b- .i.:; Nuzum, & Marzola, 1987;
Hutchinson, 1'". Maccini & ITH-c.-, 2n',. Montague,
Bos, & Doucette, ,',Il i.
1 !.il.. ,, iii,'i the example provided, ., '.ir are -,r'r.ri-
ed by working on skill development and remediation with-
in a .*: Ioi. :.J sequence model (Cf instruction. In that
approach, activities progress from the concrete ,.i -.: (i.e.,
the students use manipulatives), to the semiconcrete stage
(i.e., a focus on drawings), to the abstract stage (i.e., a focus
on numerical representations and rules). Use of a C-S-A
sequence for instruction and L.i- is critical for youth
because n, ..i . of those students may lack conceptual under-
.i i.Ii.._n. of foundational math skills (C., r.-i . * . t.l:;
rllri ,' &" F ... 2005).
The academic and behavioral needs of youth with ED or
LD in interim, short-, and 1 .n;-i.t: r ii alternative school set-
,ir.. - are complex. Current educational reform : it..t . insti-
tutional barriers, as well as student academic and behavioral
characteristics make i.i r, .. t, i those youth in mathematics
difficult. \\: have provided examples however, of' fe.. i .
instructional approaches that are also hii._-1. rn...ti' .ain, for
students. The examples provided illustrate that instruction
can move beyond the common .pr.. ,.il to mathematics
instruction in exclusionary .,li.- . an approach that is
often f, ., :-1 primarily on practice of basic nm -h facts and
the use of worksheets (C .-ft -. & Gemignani, I ",I.. Teach-
ers in each setting will need to adapt their instruction and
the pi i:... - on the basis of the unique needs of the -tt-in,
and the available materials. N-' C..i-th. . it is -l':.r 11h.1. use
of EAI, hands-on activities, and the irT,' .!r :i,1.'n of. t.d. t. il..-
al .ffe.:i - .' instructional approaches holds great promise for
some of our most -h.ti.i n.:ci". ,'.ri
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