Preparing students with disabilities for algebra: Kindergarten through secondary school
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Title: Preparing students with disabilities for algebra: Kindergarten through secondary school
Series Title: Gagnon, J. C., & Maccini, P. (2001). Preparing students with disabilities for algebra: Kindergarten through secondary school. TEACHING Exceptional Children, 33(2), 8-15.
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Creator: Gagnon, Joseph
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Acquisition: Collected for University of Florida's Institutional Repository by the UFIR Self-Submittal tool. Submitted by Joseph Gagnon.
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Preparing Students with




Uiisabilities for Algebra





i^-n^ i~cFlu


, ': 'F.. ... ..... . l. . .. ."I-.
.. . . . .. - '.. . . .. : -

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Many teachers share concerns rega .1"
teaching mathematics to students .;i
:'. ,-,t..,. and have voiced these _ -ii
cerns at educational conferences. Th
,:, i,:ii l true for the study of 'i -
bra. r. .' .-.. . school districts 'e
-,i . ri -l , that "' ' ...- i- i':-- r, i --
bra course-or ti-l-, .-! , -..I, 'v _.
that include .'lI, ; . 1.d 2-Ini.'r.
skills- to recei . . .-: , -. ! l ,I i|'pl , i
Further, studc;i. t i.--J .-. .-i -
m ath and i -ii i, s . , -kil.- ' l. 'I .
ful in i, .I' :- li:. '.1i ' -:di
Li,.. '.-h math skills are vital to thi-'
future, many students with .i i,1 *'.- . I
cities ...-: , learning disabilities, emci.'.,
al/behavioral 1- rd..'.i-. experienc- ,l I
hL.u il, with mathematics due to a nuum
her of characteristics that il.."te their
academic performance (Maccini &
Gagnon, -li These characteristics
include 1:,!i- a:r with processing infor-
mation, --if ,,.;i -. in,,, and basic math
skills (see Figure 1). Many students
with learning disabilities and emotional
disturbance- ,_... .-r. , - ,iiif,,, . ,ii ti.
higher-level i.., , . :ii hi o 1 1 t. . '.i'
and ;,. . ,. --. '. . ;' -: skills ( L'. ., i'r.
McNaughton, & Ruhl, I''

8 * COUNCIL FOR EXCEPTIONAL CHILDREN


ii i D i . I li l i I i i.'lii I- ire i .. i.

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.L. 3 ,.'rn I i. I O.s'I : , '-. l- i..r ',:i Ji r.
'31 kt.irm r-i l' "' aiii nl'l -mr" ner;-' it I_
hiOu.1 -.1 ir . ih i. i in [ i. i ah .:.i i , J,_
nil u ,- rim ii - , [,'1 I| i n-nlm i ,n ; m r .i- i ':i "'i-
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', llm.T.u t! .1i L m, . .'i 2!L_ F . i;..1!I ! In. * '-


F*igur. 1. Churacteri4fics *4 Students with Learning Disabilities or
* tniuional Dhurbastoe Related to Mathenmatis

l... l -l it.. . i : i - it n h i .a ! in iro: - l r. - I n.1

* r il ir ..|p, ., 1? ! :-[.'l . l' l,'" t!' - ic.. ' .11:' .) , P t. rn i llt l IP 'ir ", if -o I .ir
r I,-' ' i' i ' *;!lol.[ -_'9-"-L nl , ,- .e if-,. o n * . ] o .'[i: , ,:i, L s" d[I, , 1 1 _' I'- r el) . ei : s,_,3
1 - i-. m . L ;i.'re
* Problems with higher-level mathematics I.i.-i equL-L r,.'.i;ig .tiii prob-
1. !,- ",,' . i ."\k s,.,!.*.*
* Passive learners-reluctant to try new ,.i.:..... i: - - _r to sustain attention
to task
* rilit :Illr. w ith -'* i i , nil.,r::.' " and -_'i r- .:.'i a,,i diili." .;. . Ner i.-:
* DiU,,iilr: with arithmetic, .." -ptir li, -i deficits

Sousre: From H. -i practices for teaching mathematics to .'e, )i:,, r' students
w ith spec,:l I-, d:. I n . r il ' - .in icL l-'i l..'L ' .. -' ,-in' , r- '. 1 '.'. of the
literature by Maccini, P., & Gagnon, J. C., 2000. Focus on Exceptional C h. i
_-"5 1-22.







learning to reason mii.iIthniemaJL -i, The
',r,.',lF,. and Standards involve a
framework of six ,_I n.- ral principles, five
content standards, and five pr..-"s
standards for achieving these goals.

Six General Principies
Consistent with the goals are six ',._' .
al principles of *"*.t:nm.i.iii (NCTM,
2000).
1. The :-r i Eq'.ji-i is the .'.-'niin that
"mathematics is for ill students,
1 "gardlr .-,| l ,',n -, ! ch r.cJe[ ntiUcs,
h-.t>d gnr 1- ,.;r *. . n. 1,hal l eal -t.is_-."
2. _ f::


FreI 'e,.s , cur-
',It rl m n'tathe-
mra ., ,v-'ted .> ,,ir. inte-
S . l' " '. i fatedd number and ipi.raiii:, :hi algebra, (c)
geometry, (d) measurement, and (e)
. ?. i , l. , I H rs data analysis and tro.'Lbabilir': These
S. content strands extend across four grade
effective te Ichihig. requires that band 3-5, 6 8, 9
bands -;,r irad jergari.;t-2. 3-5, 6-8, 9-
".j.l, 'r-. a ic r , d i a, th-=c a![ribl:, s.: (a) a
I I , 'I , i -LC (a) a 12) and have JiilrI.-':M value or *e.,t
deep understanding of math, (b) an ,it.hii each band.
,. t.dei-it.,dihiig of individual student
SIIIIF of individ For :.amn,'ie. the study of number
nlt-oi m eIniiit and how .: hrillre learn
, and how .: lear and openio,:. is hi,.7h emphasized in
math, and (c) the ability to select
the fin ib e gride bands, ,_,.ii'i- ill
strategies and tasks that promote stu- b in
dent .1pI.- ,:- , 2len 2 and 3-5. Interesting-
ly, the :Iu.ud\ of alctbr_ or ils-braii. rea-
4. The fourth principle is the view that onng emphazed n f
soning" is emphasized in :. four -r: ,..1*
'.1'Jd 1]: .lI -W ,.'L , II , t.iii,.l ;ii bands. .-c..,r,.,i to \.jii De A.\.:1.
of mathematics [lh ,I-lgh classes that . algebraic i involves
,2001i. algebraic e.,onintg involves
promote pi 'bi m . 1 i u. W .ii:ii. ! 1 i;
andmote ,, , . 'pi., students to: (a) recognize,
and reasonini! :*
and The sonp i, ".: :. c,.iiid, or Hi i.Ia.L' patterns, and (b)
5. The finSip Slpt,, p J -'^, *`1j%.- oJ 1,1nicate patterns or rei.tunrhi,
and support: Tor - .i'iid I .'L affi tli
of stu n * -... -..i: iii ljiii. - via ilgi. 1 a'., symbol-
of student p,,iw .k .t.M..,. g!a ... ......... ... . -
." ... ..-.-.....- - .- -. --. . . i I ulal,11 '. variables, and func-
(m g,-r-I.nil, - ,. 1 :, 'ts<).' ::. ,

ment of coi.pi; e iiio F've Proces Standards
w world pit bln,. .1: :. : ..
world i, , t ::.; : :. : lI- k 'nrc rti.- and :;.:..i d.. also list
6. ii.t final Dr,..ip. r
.6. i it final . i'. - process standards or ways students
statement h Iif.liF ,ij Ir,- - ,hI:id learn and apply liatrema.i,
stance of + o-.. . 'l1.Nr iL i, the curricular areas and i.i''
calculators) and the realization that bands I it,, fir- process standard, prob-
use ofle.,,.l.n. nl.r,,:,- learn- lem-solving, emphasizes the use of
ing by ,,*, Id;. ,ppuni,. for problem-solving contexts to help stu-
-.i r ,, and ,' IrT '.r -, I dents .-,mid their n.ii,,iuni.ti,:al knowl-
tion. Many , however, rev - edge eerr [[-eiin,- 4 -;nd "doing" math as stu-
ommelnd that r..'.. gd supplement dents solve pi.,il-.em1 I ,IT- is the vehi-
teacher instruction, . [-' ,l 1.. d1 cle for new knowledge. The second
calculations, and mental calculations, standard, rl ,l, process,
I 1tle1 h.4I,, replace them . .. ,,. 1 . , 111 . ... . . .. ....


Five Content Standards
Five content areas or "strands" are
addressed in the NLCI rA. standards: (a)


1lV ,V IU ., I 1in M i I I[ u l. g prou-
lem-solving and c,.:1sideriiih. if an
answer makes sense. Communication,
the third standard, refers to n...-:n
about, dl'-Lr .,nIe.. i .- ii, i;,i im . and writ-


Effqc^a T'chi.nr SWtraNies
fto Ma MoSthlemrlaon
* TRachiog preriqtisne skiU't.
deftRitions, .a1d sarrAqegi!.
* Providinr direct tn.fuiutil ii
probhen reprel.4"laUtWn fi.
problem somttii,.
* Providmg d(hrt. hih'wfion in
selfm.. nitoring p1otfocers, .
* Using oranizers.
* Incorporating mranipulatives.
* Teachig .conceptual knowl-
edge. .
* P1oviding efletive irnisructlon,


TEACHING E... iL -...r 1 11 CHi.DREN * SEPT/OcT 2001 a 9


ing about iii;:'i ci,iL: in. a ,ce._ nl
._ |._;-,-LI ;. i lne, Thy iourth sUta ,dard
,.':i ,._I''. . " r' t relate hi: n ith -
matical ,-.'ai,, hH m-ilhe-niairil
ideas, , .-r-,- ' eas, :m1) wal- .i
situation- i-le r.Senti'i.n. rhe i-.i
',-r \ i.-s -l.tir.;e . rte r,- ., f . ''- -p ,'
math i .. concepts Il,.iuh charts,
, .-.ip i- - ...,.:n '.i ,.; i...t .,! >.. an d -' .n,i
.ilui, , e:i


'Ii .i..- students with mathematical
tasks and processes, as recommended
by II, M!. the itcr..-., ,. r .f these stan-
dards and documented '.- pi..-'
of how to teach math to students with
i..di *iii- is recommended. ? i'.-
*.il researchers have determined that
certain :n *:'- ii [lrt[' -, l. .l -rt: h-
*nin I ,l'-l '. il ill" .. t i.: l . Ili. sli_,.t2,H ra l ? --
formance of students wit' ir . iJ i and
behavioral disabilities m'-Le M.i.:c.ii eP
al., 1 -,'", . For example, Maccini et al.
analyzed the literature focusing on
teaching algebra to *..,indi., students
with -.'".:ig disabilities and deter-
mined ii.LI successful interventions
included variations of seven critical
components: (a) i.. ,'.- -: i , l~ i. ,i i
skills, ,..ini.'--, and strategies; (b)
*.-ii.ti:-.: direct instruction in problem
representation and problem solution;
(c) providing direct instruction in self-
mt,.,;,-,in, . procedures; (d) using organ-
izers; (e) incorporating manipulatives;
(f) teaching conceptual knowledge; and
(g) Li.- .di , effective instruction.

Teach Prerequisite Skills,
Oefinitions. and Sharegiv.
Before nil .'.. .1 i ; i ro .:; .0 ,.epi u _-
quizzes o r :. i' i. i . : r.,
den ts l,,,. , i..,, ,' . ...... .

skills. Slud'. iIt- i .'i; l.. : ii !e
m ay lact I .,. :c, - .. ii :.ii
terms and operations (Huntington,



Al students need higher'

level math and ieasoning

skills to be successful In

today's techn�ological

SOflety,







'* "'. i . ! i , , ll. ' -u .ients w ith











� .. ..l .:... . Skills , :r: :. '..... & ; ll !;,l
. .. .' ' .', ,, ~I,-, ri passive









, One such math strategy, DRAW
:ji : m . : & M,il, 1 ' p. .;- c, ues stu-
dents to solve math !r .1 i' 1 . : ..
S.'.' i'.. *i.. i tasks:
i the
S the problem.
-.. .. or DRAW a conceptual repre-
sentation of the problem using lines
and tallies, and check.
iy . : answer and check.
Another first-letter mnemonic strate-
gy, STAR, is effective with older stu-
dents with m, id disabilities l . 1. .-.n;; .
Hughes, 2000; Maccini & I-'!l, 2000).
This . s-. cues students to complete

ed substeps and is based <:i 1n l 1i.
iors of . ' .. i . .. (see Fi:
2).
,:1.. ..i~: -i. article shows how this
strategy is used D ..i ' -'-. numbers,
use of STAR may also be generalized to
other tasks r'h ; 1i-. pi problem-solving
skills. I -.. four main steps of the strate-
gy include
-.. w1. the word a11.1. e !I '- . read the
problem ..::. . 11, write down
S . .l . i -.
A. ,,-..I :. ,, ri: article sh w I h w orin
s 't g u II .'. hoi, -I. I11, I.],
other t:ask * ', .. , ..ipti: D i._ p ro . -fli-:ng




-. .~ i . ...r (i.e., reread the
problem, check the reasonableness of
the answer).

Direct Instruction in Problem

Solution


experience . :, : - t. i' ' 'i .. iiI
o 'r I - ; Il l ' . ' . l , . . ; ,, .

i' .,' the solution, 1 .- . i . O'Shea,
Cr, .. & ...4 ." - i . 1 :1,..- . . Thus, you


10 * CONm. FOR EXCEPTIONAL CHILDREN


i - -! -,li ,- - ;1 , $ .r - -1 -r - --l-i.l "-


I . '..i .-t! i h 'i '\.' . facts do I -r. *'.
,.,


', h.: 1 .1 - I ! 1 . i,.


2, Ii.r !' .M O, ir;-, an p-, *.rn. in picture form



i...'. .i! i' Algebra Lab Gear (CONCRFT"T '.2PLI-

rir C. , [.i 'ir .-. l r- r . :.,. I. .i, '.!i-, NCRETE APPLI- ...1i I.- I
\ '. i . I -I.rni. . i i[ ... i ,:. ' 'I *'T '.r r L i: -- I Ir T .,


Addition Subtraction




Same signs Add #s ,i the
, p" l " ,.;. . of
Liii-re - , - , second
.1,i, ,, ,,, , - -. and i S
.'l -IOf term
from zero


4. s ,.'view the solution
(a) J. .,.I - the problem
'1,) '-.. " 'l-r ,:. 1', -- the answer
(c) Check answer



.1, .:h - . t o .-rh [ l, ': F ,. . r: ,
tion (i.e, ',ie.r.,ir', the information
from a .. . iJ r : i i, i .: I' rep-
resentation) and �,.,.Ii... -*.:.,::,*, (i.e.,
.iij'i' . .; . ,i'.'' ' ' i;' procedures to
derive the solution). For example, the
Iir:i it - steps of ll._ DRAW --' ,
and the first two : . - I :e -', : . '. strat-
egy _,* .h _ .-.;- ; ,.,t al�!, ; - s:, at .,,,
To "', .. ,t- both problem represen-
tation *,..l .]..,ii.. 'i ,l.tt -r, provide
-.i-L' . 11. . . . -. I-,l 0 .1. .. : ,i O da
...1 ,' 'r ri r., .i l , r ! '. I- .-exam -
pie, tFh~ prompt i '.i: ,- of ,' C
S. . '. -,- . I -:' ; 1, :ri[ ': and
represent ih. :..-. l. " i.. r!. the
.,i .ii' hl "i . , . . answer make sense?


m ake -':- '_.'t.


answer.

Expliiit anstruir- Moniioring Procedures
Many students witlih L,.',iih J.- :._,ite-
experience l!,iF. '. with *..,. F.Fi...
their [[I.i 1, ;i -.1A- ',; behavior (Monta-
gue 1 .011 l-':H ';j students to ask
IL,-IF- - ;.' ..lu,..:E[i'.,... - 1.11 problem -
solvii:; F, i-' i' n-: ow to use
prom p * r . iF,.F ...- ir, ni a - H'i, .FrF l]
w ork s h,.-..' '. , 'i , . _. i il 1..i! :.
r ..li'.._ and answering questions
aloud), as students observe the self-
...... : i- process (see 1'.-.. p. 12,
IF I .tF IF' .F' F __-,FaFse'. ' p- i.


3 A , t I ll
3 . .; S.o i " .r iii p i ..lil.-n i








Teaching students a first.

letter mneoonic strategy

enhances recall of general

problem-solving steps with

computational skWls.


Fiurr 3 l an 1 -armple of a
,ri .red . ', lik _.I.. b.-t :l-d on the first
..:-. R .ir , e:2 ,. ,, shown, the
"i l..t, Ihte -trdlt-i:. Mt-'" and
. ..p-- oc sItor -in. to "check


Organizers
Many students with rrild disabilities
experience jilh'tull, r'.nem'hern. i or
.' ,i lt i ' ,r unil ', ,...r Il n-. tL1 I .-f
'la:. 1996). In idilitiLii. 'hOil-. :[uLe[,
may have li fii rl, il Lihifil, i.4' relevant
information within a problem and
SLA 3 r, i.- 1n P, the information.
Using visual .igati.g C!. such as
structured worksheets, prompt : J 3I or
p3iii orarniz.r:i. helps .' students
analyze and solve problems. These
Sr.-'.,iz .' hellJ students remember gen-
eral -ir'.blhr-.i-,-'. Mrl steps/substeps
and the information within the problem
(see [Fi.r e. 4 and 5 for some examples
of problems and the �sep- '.jd'-nt. use
to solve them),


Manipulatives
Teachers may also it .,-,r:,'.r.m the use
of i.,bir.ct: or other visuals to help stu-
dents with problem representation. For
example, students can use items from
their trM iroiiItr:l' (i.e., iit'iv, i.s within
nature) when in .e:-'.itin: patterns to
, ,i,_ o.j . ,-b ,: le.\r, -l, s ..llc:. [oo
�lnJeil"- .:dii tt._' A flb 1br l ,l-rki, ,.co a
l , . gei r. L: a i G,.I r (4[, .,-_clcoi . ,,ni i .

to ihip tir i ,iu.i.i'r boiuh nuileiic
and v i_-bk .ini'tni- (see I-ipur. ,1i

Conieptual Kno-wledge
A S.,. ..t'~ --.mi,' n r-eM ab.ir.ic[ 1'. - -
A) i.trui..onaj ! iequenci ' :ul r'ii.,rlI stu-
iear>' :lJe'!rata.-,.q il !?. ,! t irid'-r inrg
111.-l ll t., h .* b, l,_ ,_ cpI' _.-le lll., r lie. '
(i.e., visual to abstract repte- 'fntati.I'..
see Fig.:,.- Si. As V\ii ['i \V.ile 120011
4-11101. il1 % t' e nilipll.i it Oill, !h- pi:iro
�:edir.-l ruil , trii i- little reason for
.,tud l'li. ILo 31,i rid tE .i1i *, oln iep .l jus-
ifiti.a: ,n., D[ 1 t., b. .' nt. nr n till li ,h
ir.''.'r., J1. l.iV demand (p. 425).
During the iiiai! phase of instruc-
tion :' e, conccreii. students ,'pr.--[i
IEt plo ,l--'i i11 ' h ,., tehC l Sr.Lud '[i i; h-?i
.id' ai3.w ' Ir the :- n1ic..'ltl'ere pi.:'L' of
inirnf.._u-_.n and draw or use pi._,-otil
t.p~itnlati,"i..n of the qlua;rjtiii The
ab-lr.ILT phli. of instruction involves
nfi '.,ri.. Ier'prese tal ionli. , instead of i,
[<,ri.i ! di.-l u >. I lu -, I [15ti uuni nillh
_',-iiersic: can be slj' .:.ful,- embedded
within a -rilI,_mni .],iitg strategy. For


example, the C-S-A sequ-jenic, . iintgra
ed mt.v, the STAR ,rJl ', iJ idt -0t6 les
cue students to lrllow the igrajhijed
sequence.

Effecive Teaching
ii ,.oi 'r.ptin e lii. ient inc. ffC:rtvt:
I.-. [l' . :,.'inrnpo.n iin irke I... te:-ching
lrlhtit- pr n'- 'i student learning and
101 li.,,l i fPll( el-tIhin .'- Stevens, lc.,
...iet im i I. I ,It ri l l '- [ -',:'1 [ . '),
Ret.' n ici- hi,.e Il..-d the following
s H'p . t-lc' t e ll mIII I m.-Itlh. Ilrl I IIon of
. ,ii.'.s h t ,i.l irl iii ', Jls.i |ies."- m od-
hig, iliN ii-k. p[,rinlin'. giiJei and
n!] ('id pi' fi'ilril p j,'i,2;- , IItr .]iLe 'tu n re ' ..i - .
5iid , tiri 'ln .mai i p.i-ILuJ' fe -.1',b
I.N t. L I ' el [ .il.. I')'I'" , 1.11. L I : ,
Hugtg..s 2'UO1. Mal, 'ri & Rulil .00t.
.lercer & Mill-r, 199.2 e Figur.- 4i
Teaching Algebrair Reasoning. Th,-
Idrel act ltej.hllin, .ilg.L!al, . leakhl.i. tLo
-tuden's '.ith m ild d,:i. itii l^ l: e.r[,
S kirindergarti. .:-. r:CL mnilrideJ b..
N tIM1. nul- '\ Sr..n lii.1E I.l-t -ll i .
golil BuI it e iC'OiMl-i --t i .1igr . 'L.-
"onrihg .1. th. stiid% :, e re ng.
,nrrlizinmg,. termnialimnig) of pallt'i
,vLniin niathmai-,icL. (\. in De Walle,
2.1 .ll ;w. .:an _.er[aliin ' appr,,ach this
type nf reasoningi at inamant levels.
T-, ',arnpl=- simplee repLutl!g' pat-
terns are :c*mmonl% itmght to .rldilhjer
i earl; a l kirdergarier. "I hes pa[len: i:.
nis irt ilId, v'erb.l patterns (i.e., the
mii*,-al iii.t do, do, mi, mi, do, do),
m,.woerneijr and visual patterns using
! inip,.lj~-i'.'ri St_.d s in the e.i,' ele-
MIi-ni.inr grades regularly ,-n,ieae in p.,i
terned c:.' , motor movement t i ii % 't, ,..
Such lessons can be linked to patterns
as the basis for future exploration in
algebraic reasoning.



Representation, an NCTM

process standard, refers to
expressing math

ideas/concepts through

charts, graphs, symbols,

diagrams, and

manipulatives.


TEACHING Exci *'-.* -.L ::; i!:,.i . SEPT/OcT 2001 * 11


.. .......,... . . .t .�. . ............. ....


Figiur 3. Straclrud Warksheot

Strategy Ouestions W ile a theck (/) after
completing each lask or question

Search the word problem

(a) 1;,.id tit.: problem carefully

(b) A. ' v.*_ ur-eif q,'-.iin "What /
facts do I know?" "What do I ____---_---
' '-ri In ri i il "

(c) Write down facts I know I have two ','. (+ 7)
and...


6..'.: ,' ..;ii & Hue0 -. 2ULi00. Maccini & Riil .'i : i.1








Irennffm S .rotgy Steps . .
Step 1: Provide an Advance Oroalan w
Provide .ij advance organ met k'lal connect A ew ifsaraikon -to -pr.t..im- y
learned skils, (b) ww iate te iw skill . -bk-learned. and ti) lprcvi A fadst.
aik intrudwurg the flew topic.
Step 2: Provide wtneher Modeft .
POvide Leacter 'iIntling" thtriugol Iwu atib.jdP. First, ".thIlc altedi 10 -t.
dents while inildvcing a strategy . Then. "foAde< leX edurt- ta-edI r pmnjrt$
whl.e htivorimi. ro.J,?uts iift appliantor of the tratlgy. FiM t example. fo{knVing
the teacher MNii del.. 8lLttfntts A (Te-rqoespiis and wite down thetr response-s
using idw garap ir, rf.ga'r2.ti l sr sitrurctqrd wo-;:Mtets,.
Step A:Pwrvikfe Gtuidedl Prl-ct-
h,'idjq opportunities' or ti.dents ti, praet-e ihe now strategy with teacher
aaSistCl Faride Pt;'bet .iwmtanmt uf otil otudetis can ;k'rfofnr the task inde-
riteLelnuy. - ' '
Sflp 4: PtfO le Student hadpeudtt Practice
.s',, sY.n.Arit 'mr.istWr- of t r.Atfl; bv parvding pmbnleui, witfow teacher .
f'{ T .tpr.las/ m ant ee. . . . .
Step 5: ProvIde Feedbackk
Pr'td& pjsr-itive and corrmnvtv fbedback thrughiou the lexscw Wi (iN-e steps:
* nrt-nfmo;t stheidt pdfratincc fe.g., cal.xtlate the lper'eaen e St.l.t).
* Ttrgl w n panettrt?/incorrett answers. ." .
* Rok'ach, if -ner'sary.
* Prrvide sWidrnt pra'tke with simhktr ptibllems and UAtitor :itudwtid t tr-
If.ln _a, .. " �
* Close with poth.i. feediat-k- . ..
Step 6: Provide for CeneralizaMtiou
Provide polmpts or questtotis to pItftfsn #ierAWAtIou t Mtnt
q PriAem-stWlv ig stations
* Content areas
* Real woarlid situations,

&ur.';e. adaptedd f"oin Maccird & VHtIls, 30O0i Ma"itm Ruli, AiflO; Mercer
& MiWe-r, 199 ..


This process of ,p, ii: aa .p;* *: to
a d .'1 * i I: I :11i I!* W ill ' :!P'! 11 .-. lI -
im portant .i * r-i _i-,ir ,l: - 1 i i- , -.i
can be maintained - ' n -. i1. mnate-
rials :--: ,,: t .... [ h- .v !l .!._ : .i!.


Trnslate the words of a
problem into an equation
in picture form (i.e., choose
a variable, identify the
operation, and represent
the problem through
manipulatives or picture
form)A


12 n COUNCIL. FOR EXCEPTIONAL CHILDREN


Algebraic reasoning can be a focus of
I11 .- nf|. 1i1'i ; h.i ,' i i ilIi[ , .- .I iL', ,- i .1 _ ','."
ties that focus on repeated pr:r-,.
f ...- . 4 and 5 provide extended exam-
ples of (aj repeated t' r: I . f , '.-
11,2 patterns; see -I gur. 4: .,ii- I.. -I' i%-
ger operations that '-. ':- ir- students for
more formal .,.l c.. symbolisms and
,,;ni',i:.i F'. (i.e., solving word prob-
lems with ':! .-: numbers; see Figur-e
5).We pr.:' i.1- these examples of teach-
:..'-bi.'": reasoning to ILt, iui-. the
i'.:- ,-,_ of n:!..'_, .. 0. u Ai.n inalge-
braic reasoning. Though not exhaustive,
these * i- ' i-.t .i- i , .'! . ',.' s're i' 'i f p
tion of NCTM Process Standards and the
.. ki..;,.- :- of ..cUe.: math instruc-
tion, (e.g., the components of effective
teaching, general .r.-.i.i -, ' i c,. m, , '.i .
gies, and .-r.:.I,.;- :;


Provide students with

questions or prompts on a
card or structured
worksheet.


BEiilting on Prior irwlegdy.. In pre-
rr- -ii r. problems like these, your initial
:-wt is to assess the "',r, *. .; .- skills
.'-e"ted by students to nir:'l.: r- ti-
I.- , hiem. A brief e.amjiinanuun t i..--
pI : L,' i . i,',*.'.al the n-ed i:,:r -T. .illen
',, rIave experience with nui;.l! I! I!,-'.I
.ini division, the concepts i-! r, ....
:nJ recursive patterns, and writing sim-
I',: equations. In .iL ii. . iJI' ,ir:.
"i.:ild have a working knowledge of
' .'i'T ,', ' and some . p- -.- *'. to
' -, a : ..' organizer to solve word
'pid-b.ems. These concepts can be infor-
in. jly i--.t --'.1 oiiii',l Ji , .1 . i, ii and
r.:'- w or more '-. i.' - ,.u-.: a
-Ih,, t quiz.


it... ..d-.- presented here may prove
wi,!f i when designing a..,l ..'. t.:r -,;
:lIgebra i'ltii.i.tiin to your students
,ith 1,!i i; .i- :: ..- Students at all
- ale levels can learn to reason alge-
.; . via ..i:; n -ei',i; in problem-
:.o.-ing activities that :r. L idk , r --.ic.'i
Iy-' ilidated practices, - -i-, .1 the- A
continuum, g-PIi *S organizers, explicit
ir .ir..-]-i.'p . r ]. i,,l'l l.,iin :, a .; i t .'t ,'
instruction. ..- L.r. . -.-i,-n have the
,-i-ll ,-,-, ,. ,, '. t : -.L: '.r . - rH , L. : '-
practices and thi g..1- '! lh, NCTM
standard s within ie r 1 . -.r.. .. ' .



A concrete-semiconcrete-
abstract (C-S-A)
Instructional sequence
supports students'
understanding of
underlying math concepts
before learning "rules,"








Figure 4. Growing Pattern: Sample Problem
Sample Problem: Today, i'.'.I''er borrowed $2.00 from his dad with the nil,,�r.*-TLLdiLhL' that he would have to ta. his dad
inr.-!-t ' each day. On the second day, Walter owed his dad a total of $4.00, on the third day, a totil -f S6 00 and on
the fourth day, a total of :"- i If Walter didn't get his paycheck mnil the .5in day, how much ".I,....1I h- .' his dad?


Phase of Instruction

1. Concrete .Applicdaoln:
a. ' ;.l,,,i.s use paper money to represent the problem. Use the
graphic organizer and count the money owed each .v -r up to
the .: day.
b. Look for the il-l,;i. shi' . - .,I' ; 11 .
* between each day and the number of dollars owed
* across days (i.e., from one day to the next)


. STudents to:A
,F F. u , students to:


i-,liiah problem (read i. r-i0fui.. ask questii.-.jr
write down facts); Tranr-;.ie ill. problem
using paper money; Answer th- [ i Lbi L-
using Pi,,n,, iini R . ,.. the ,.ii. i,_h. i.'Ie nii
the problem, check .. al. i , e- and .-
culations).


= One dollar


trame Day 1 Day 2 Day 3 Day 4 - 25
Sof Do"ll.ir J L D I I-DI






2. Semiiiconcrele Application: Prompts students to:
a. Draw pk,:.;.t ec- of the money owed within each day of the Search problem ' :,ir .. iul,'. ask ..-iu-.ri.:.,.
i," hi,: organizer. write down facts); 1 ri -I, i-. (represent) the
b. C ,,.I L Lli-.i' in each frame and write the total owed per problem via di.\;n,-.s. numbers, and an equa-
frame (circled). ,..n. Answer the -'r,_'-.1.ii- .1inh- drawings and
c. Look for chr irin .,p p a'erl-.- numbers; and Review the solution irre.id the
* between each day ai:..1 -i. miiii. i of dollars ,.ved ',.., . problem, check i- i,,- i-,l,.nr- and calcula-
day 1: ire ri. 111 t owed is $1 + $1 = $2, or D + D) tions).
* recursive [jliir!:- 1 It . fr.i- ,-,n'- day to Trb next, such as
one day's cost plus $2 or C + 2). Write the numbers
under the organizer.


Frame d0 1 Day 2 Daiv 3 Day 4 -- 25







__ _ _ 0 0


2


2


2


TEACHING EXCEPTIONAL CTILmREN w ri 0 C'. 2001 . 13


$.. ' .










FiPgre 4. *rowing Ps." rn Msasmpe Pro:Mem Jafffti.d.


3. Abstract Application: r n;;.- students to:
a. ' ;i"- c. tail :i ..'-d P-7r frame in t1 .. ',pIh-c 4.-'. problem (read :. .>.It. ,., ask .-:- '. - write
organizer down 7,,' i Translate the problem into numbers within
b. Look for r ii.',i;': "r., r:i: and write ronmnrri.i the graphic ":ni,.r and an ,i.I.ar.i'i Answer the prob-
representation: lem; and. Review the solution (reread the problem, check
* between each day and rh. iir.L,, 1'if di'.:.. reasonableness of the answer, and calculations).


owed *'e .c . i . u:i i n'.-': - ",...: " - ' . + $3
= $6, or D t Dj
* across i.,', (i.e., from one day to -i,' i- .: -,:h
as one day's cost plus $2 or C + 2- .'.- 'hn
numbers under the ,r., ,,-. ,
c. Apply the rule for the growing 'i ., rni to obtain the
answer. Reread and check the answer for reason-
ableness.


1HOPIP- S.I. Olvision- Sample roMem with iP90gers

Sa .'- Pri-.i_.M - ,-',e th._' te':npel.;ii.. , .._ r, 1 . 1- f -20F per hour. The total temperature b.' e .:s -
' F. How many hours did it take for the temperature to change?


1. Concrete Application:
-.0*. , . use blocks to represent the problem.
General c',r', i.e .? ij.. *-' operation o 1. i:ip:1.- ii ;11


Algebra Tiles:


= 1 unit;


= 5 units;


PI'. r -.r- r students to:
Search problem (read ....-liii . ask questions,
write down facts); 'ftanslate 71. ri.'i:r, using
blocks; Answer the problem O. . ,- :i, -ii,.. and
Review the solution (reread the problem, check
reasonableness, and .' i..


- 25 units


2)
1) Student; I .i
with no tiles on the
workmat



a. Count the number of sets
of -2 needed to obtain -16.


2. Semiconcrete Application:
Students draw p1 .-.e . " ii- , c-p i ,.,i "...-






3. Abstract Application:
Students first write numerical ,:--'. ni .i i
-16 " -2 = x, apply the rule for 0i. ,1. .i. , ii.;.. 'o
obtain x = +8, and reread and .1 i ;i.- i-..,,


~(~I.'


b. Students add 8 sets


3)






c. Students count the number of sets
needed .


p.: .1 students to:
.s..i. ' ' -ie., i -. .1 .i- .i. i , ask ,.,e iri iii: write down facts);
Fr 1,I, ' -1 represent) the problem vi . .*i , and write down
the *-. si.i.. Answer "th problem in,; .ir.,' irin and write the
answer; and Review the solution (reread the problem, check rea-
sonableness, calculations).

P r .,.- students to:
Search problem (read:. - .11,, ask q_,:-,.n- write down facts);
Translate ..- !,'i. -. into an equation; Answer the problem
ri rrl. the rule for ii, i -;., integers); and Review the solution
(rere.J il. 1e p . bem, check reasonableness of the answer and cal-
,_i , '.i,,d -


Source: Adapted from The algebra, lab by Picciotto, H.. 1990, Mountain View, CA: Creative Publications.


14 a CouNci. FOR EXCEPTIONAL CHILDREN


'


- ------------ __









*a00 not be content wHfi

right answers; always

demand expionotions.v


Algozzine, -* .*.1 -1. i ,_r .'. ,v .
Stoddard I . : "i . , .1.jl\- . ,., ],:-,r
m atics i_. i.l r-nc - i. 'r in.i .1, bi el
adolesc ,.i ','-1 . ' "i. '** .-. ,, r 1 ...'i , ,
21, 97-107.
*'J'irl:, ..-, D.. J. ','-1.4 Instruction in con-
crete, semi-concrete, and abstract r, [.-
sentation as an aid to the solution of rela-
tional problems by adolescents with learn-
ing disabilities V.-'- r.- dissertation,
University . *i 1 .i-'. I'.4,i Dissertation
Abstracts ",: '. .:.- -. '_ I 2.
Maccini, P., & Gagnon, J. C. (2000). Best
practices for teaching mathematics to sec-
ondary students '* ,:. '.t -.il needs:
it.i,!' . i 1;, from teacher pi ....i-ii. and
a review of the literature. Focus on
I.' I ; :. C children, - . -, 1-22.
Maccini, P., & ";!ui-.. C. A. (20001. F.:..]i'
of a prr,ul i..,,i -0 'w i -,:. v r. -n *h intro-
ductory i . -1I i r m*.-r i I ' :..1 .ir,
students with l. ,ri ' ., ilr i ,IL.'. -
.- ir,,. . Disabilities Research & Practice,
15, 10-21.
Maccini, P., ::, ..;hi,.. D., & Ruhl, K.
(1999). Algebra instruction for students
with learning disabilities: i-ip;-. ui. r:s
from a research review. Learning
,,. , . ," .. . , ,. .'.. l i ,-126 .
Maccini, P., & Ruhl, K. L. i 2'i ,.'li Effects of a
graduated instructional - .I. -n:.. .. the
algebraic subtraction of integers by sec-
ondary students wit- :. - . - il .1,. -..hi:[,- .
Education and reatme:' -' , .-,'. 23,
465-489.
Mercer, C. D., & 1f.l- : p. i I.,_ Teaching
students with learning problems in math
to acquire, understand, and apply basic
math facts. ;,'- ., .: and Special
EdIcation, 13, 19-35, 61.
Montague, M., ir. . C. S., & Doucette, M.
(1991). .I-. :1r . .�,gl.q ,, . lid i Wi.::A:.. -
nitive l[ri;,or,: Of i l . ir .. r ill. -

'..... ': Research & Practiceo 6, 145-

National Council i T iLh'. '. .l., 1iti.,.i -i
.* .I .I .I i : -' .11. Principles and standards
for school ,' . i,':.. ,. R . [..'i VA:
Author. *
O lson , J. L .,, rl,, . . 1 ['. 1 -" -,' : ,;,',, . .
children and adolescents with special
r-: . ,i l ed.). i.'V I I LA,.dl f' .u , NJ:
Prentice-Hall.*


Parmar, R. S., & .- . .i.. J. F, . I'-,'.
5,Ji cl dl l>-''a_" l r i-,J '-." 1e;1i' C ,,i "-
!'.-[ *l :.,-I~. 2 -1*11.1 -pc..i111 l.Ij'.uT i. [. f' 'l
, ,';s r ,n ,, .' .r,', ,',.ir, i r .1 . i,','i r. .2 ,- - -
50-66.
lI., ..... , i I ..'i ,i, Cr n, hr , Itb
S. n . . kil * C .\ iCi p . bP n l,i ., - -,

T:el iil[, functions. In M. C. Wittrock
iE.-1 " i., ,-f A*4 .*1 *- .arch on teaching
I - ! , 'pp 3 . -. 1 WI .'. York:
10 J,' II U '
V11. F.-* 'V.II.. '. 1 i'" l .,',. . *-' and
'... ; school mathematics: ].',. 'i:,'
.1,'' ',.,' "'' .', h '.h J , New York:










*To order the book .. .,, ", !,. ai. asterisk '
please call 24 hrs/365 days: 1-800-BOOKS-
NOW (266-- .-. or (732) 728-1040; or visit
them on the Web at http:!/wvuw.
clicksmart.com/teachingi. Use VISA, i -
-4 \'' . '. i ** or send check or money
c '. . - ' : ' , - .'L each add't item)
to: Clicksmart, 400 Morris Avenue, Long
'r.>. *i. MN 07740; (732) 728-1040 or FAX
(732) 728-7080.

Joseph Clhlin Gagnon, Doctoral Candidate;
and Paula Maccini Assistant I-.c't.:.
. " , ... i of Special Education, University
S,',i! . ' ", ,...'.'c:. t rk.

Address correspondence to Joseph Calvin
Gagon, :. . , of Maryland, Special
. ....,-, , 1308 F ... ..,. .* P,'.'.,, . College
Park, MD 20742 e-maill: ...".' , ,* ;'r-
ailr,.i . r c:- '


TEACHING I .- p .rid
No. 1, pp. 8-15.

I .'i. .' . .' '*i CEC.


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TEACHING EXCEPTIONAL CHILDREN a SEPT/OCT 2001 * 15













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TITLE: Preparing students with disabilities for algebra
SOURCE: Teaching Exceptional Children 34 nol S/O 2001
WN: 0124400442001

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