PEARSO N -St·ott H1n•sm;m

RES El\RCH SAY S

Supporting ESL Students in Learning

the Language of Mathematics 8 r J I \l C L" \1 \1 I N S. P R 0 F E S S 0 R 0 F C U R R I C l; I. L' M

O~TARlO I NHITUTE FOR STuDIES IX EOUCATIO'\

t.'\1\'ERSITY OF TORO'l ro TOROXTO . 0:--ITARIO C.H • .\0.\

,,,~ :hemaUes and Lang·mg.,

M.lthen-.ohcs can •eg1limater; be ccns cered as a

lono.,og!> 11 <tsalf 1n tha: 11 enploys s•1mbols to r.;oresert

concepts ard to lac•htate ocr t~ inki~g about aspects of

ron ity. Ho.,.,ev<:r. mathematics is also intm<ltely re•otdd to

thO •10iuroii.Jnguage tnat we beGir to acquire o; Infants

ilnd use :o eornmun•cale in a var·ety c f evel)o.~y ~nt1

ooncr.moc: ccnlexts. ~.•amematiCS and lal'gL~e .:re lrt <·

COnr".CCtt!d at :;eo.eral ~·eels:

T.,ac~,rs t..se r.atvallanguage tc exc:lar1 mamematc3

conceots ard c<Jtrf cv: <rattema:IC<ll opNotoens

Students who have mted c:roicercy 11 Eng! <t> rt>QIIre

~.dolt•onat support 1n order to understand rratrerr~u,..ol

concept~ .111(1 oparotior s taug~t trrcugh EnQI sr Among

lhO ~upports that toachers can cse to make 111Sin..ction 1

co~Tprchocsiblo fer Engl sh language Wllnrrs M l

drmo~>strohcns. use of concrete hancls·on

monpul'lli,oo. graphic organ1zcrs. Simplification and

p~t •Pr" 1$ing of 1nstrcctiona11ansuo.ge, and d••oct

tc.lct"•ng Of -.JJy ~'Y·

AS if' other ac.'ld'>rroc SIJ0!9C~. matl'err:uics u•es a

' MC•O.O:Cd techrocaf 'IQ" ... ab ... ary to <"P'!'Sef1t cor~ "~'·

md des<:rlte open:uoens As early as grace t. st "ICI-,r,ts

~ro rcq. uroo :o leatn the ~ean ng of wcrcts S..ICh ~~

ad Mirm. su/.;l:tJc:ron, svm, addend. and so on. th.lt .:vo

hkCiy to t:(l found only in matrerratics C1scourse Otrnr

trrfl'IS h:.lvo ::poe-fie m~n1ngs in mathom. :tiC'~ dir.~.:''\ .P e

Article used with permission for:

Preparing Middle SchoollEP Students

for

Success in the Math Content Classroom

TESOL, 2010

:hat diNer 'rom the r meo~ings In e,, ryd.1y csage and in

oth"r SL.btect orea.s. Exampl~s nclude ·.,crds such as

:able. prodi.:'-'· e • ...,, odd. Md so er1 Hormprores

sun as s= and tCfT'e r::ny a·<;O bO ccrt .. sng for ESL

~Jdents. Grad<> t sl\.a'JentS are recu rOl<lto 'eam

co-.cEP:s such as "<.'ft!bcf tx:s IV1t18dQ.II()I1 ser.:ences at .:1 t me ,•.hen rr,ny ol :•.um (JIYJ Nr"•c<.klrt; ESL

studentsj may rot kr.ow tne oroadcr mea~•ngs ot

v.orcls such as 1acls ard Sflr:tences

, ------------------------------------~

PE-'RSOI\C -S<'ott Foresman

r .cJd+t or ~o the tecl".niet'll \ .. cc;!bulal'; oi mo.therrauc.a,

~r. ... ~· Jl.age 1ntersects wt"'' mat~rr 1~iC!I ~ rr.e br....adt~

feo,l)l ct gMe<al /OCOOI..Iat'/. synt.lX. scmamlcs. or.d

~1scc:urse. Mcst mathematical problems reqwe S1Uderts

10 underS:aro ptopcs tens aro ~ ..;.atoons tnat a-e

excrcsscd Ncugi' '.ar guage. Cors.der ti'1s prctlem .at

we. <JrJ.de .:. level:

~\eno'y Jav9 a to:al oliO t~ts ro-~ ~ho ga~11 her larr:;e dog 2 mere treats men she

0-''19 .-cr •mibl dog. how muny :reJts d:C S!l'J

g<•e :0 e <e~ oar;?

Hore students reed to ~nderst.J.rd tcr be at:•a to figure

out) the moonings of ..... ords sucl'l as :c:iii and !reats

They need :o ~nde<sland the logical relation exp•essed

by the conotr.1Cl10n moro , , , t/'lvn and they nc>ed to ln:ar

t~.at WeN} I orly l'<lS th'O dOl;>;. dl:f>Oo!1ltrliS s 'lOt

expressed d11oct1y •n the problem. Clearly, ti'e lange~oge

::Jemonds ot the ma~o curricvlum ncrease a. otll<:!cru

PfiJ!)'eSS throug' the gaoes ard ~'lis c.•n cause

particular d1fllcult.r$ for ESL students

Tho ESL Cnn!Jonge

Numorcos resea-c!l st.Jd,as 'lil>O OOmo<·stn:<KI :lv'>• ESL

stud.lnts M">oro ~~ re<;u ro at lp.·,~t f.ve years to CD tel up

to nat1ve ~peakers in aco<.Jem1c I<Jrlguaoe pror.crency

(e.g., reacing at'Cl ..,..,;:,..g .ldls) (se-e Cu~M><IlS, 2C01 '"'

a <OV1ewl In mathematic~. ESL students ohen make

gcOd pr09ress ,,.. ocqu r"'ll b.:l~oc comp..natiCt'l slo IS "'

:he OO<Jy sra<Jes; however. they typiCJllly exp(·l ionce

greotur d.rticulty n conying out wora problems, and t~1s

dlfo;u~y ineroosl"' In t"< later~ f elcm 'l\:3/y

schools as u·e wcrd croblemn boccme mere

I ngLI•$t ca ly and conCCPtll<:ll:y comol<'x.

T'le'...e OO.otopmer·t<J p.:lttor-.s C<¥1 te t..rOO";tood il'l

retallco to lhr,e 'cry <lliiNeot a <poets or t::.ng11agc

prohc1ency.

Co nve rsational rtueney is 111e a:>lly :o ~:my on a

conv0rsat.c·n in ra!T'.ili3' foce-to face :.ituaticns. fh1s +s

the l<u ·d c l profb 11cy tn. '' tne v ISl r-" 11on:y of nat .

~~~er.kt:r'3 :t Erg'ish ,..o·:"J .e,:e.op\.'d .. ., .. .:~ rhi .. y ent ·r

x.'lCOI li J9e i.•.e . il r~"' J;;e or I' <)• 1raq,.. :'ICY

worcs and s rrple gro.mmil11Cal cor sti'\Jcl ons.

ESL •:<.cents gerer •• ly ca-~cp b.;soc f~.,ncy n

ccr\"lfS.:lt.or.al a;c ..c:s o• E1gl sr .vitru· a ,ear cr

two of exposvre to tr e la~o~.age a t1er n sc11cc or

n t'>e erwirorr.art.

Discret e lartguage skills refloct ~pccif1c

phonologico.l. l<mcot ord grammut cal t-r.owt,;oge th1t

.tuc~or·ts can acQlire in I'NO "'"Y''; (a) a,. a r¥~r. of d•ect

inslructicn J.od (b) beth tnrmDl M d inforl'T'a.l orc.ctice

(e.g .. teae1ng). Somo ct tre:;e di;;<:re:e tangu?q., sic liS

&e acquir<ld eati'tln :;chOOI.rrg :lnd some cant nue 10 be

acq"lred throughout scl10c rng. The d•SCI\lte '<lnguage

;kPIS &equi<'ld e'\'ly nck:de krO'N'COJO ot :lla ~ llers Of

'h9 d'pi'acj)l. thO SOL.nds represented by nd'vrdual

etters and ccmt1n.:rtons of lttters, a"ld t~e al:rlty to

decooe v.Tn'cn w«r~s ;nto ;)poropnate cour.e, •. ESL

~tudonts Ciln learn 1t·ese r.pecoflc lang1.~ctge st-1 Lata

rolat1 ,..,., ea"\' st.1Q8'" tM11 acQI.Isr.on ol En<] I:./''

1n 1ac1. these ~k1l's can be lcJ.rned ccncurrantty w'tn

the~r dovelo~r-ent Of tasiC vocabL.IarJ •lf1d

r.cm:\~F..ationaJ :::ro"' ..;;e.ncy.

1"'1 mnt~ernatlcs, 'novo doscrcte longuoqe s"1ll!i 111C1Lid~'l

knOV~oledge of tne symbol> that rep•es. nt basiC

ma!llrma:ocal oper>!Jors (e.g .. +. - . etc). we :r rms u:;ect

to rotur to tnese symbols and cp(lraticns ta<ld. suotract.

plus, mo~...::. ·:c.), Jnd tn. ws'c tcdlrw:a 'F.>CCb.tary of

.,..athtmat .. s Clearly, the -.bil .ty to decouo w11t1or tnxt is

a1so o neces:wri ~b .... t rot s~.-tticier.tJ cc.nJ ton t~ carryng

cut wcrd proOI..~ ''Pf"''sed r ""~~'<~ ...nsuar,e.

Academic language proficiency u·~c'ude')

knowl~ugo cf ti'e less irequ.,nt vocebclar; or Evf:h J.S

we'l a; 1/'e <Ooitt to nle~H ar<J vee~ ·ncr ~~rqty

complex '.\ir,lter. language. As s tudent:l s:.rogre~s tnroug~

the gr3deS. tl1 y er•~Pt,r t:.r tr~re 011 lf1XIl• rcy

·:.c~1• :J:flr".,-dy tnm G·t":!~ and Lat;n •.or,r~es), ccMplex

::,ynta.x (e.g., cass~., .. u vOt\,;0), and ~tst:a ... l exprt.::J'J!Cr •. J

tr.at a.-e .,ru11t1 rl"'.'flf re1r:1 .n ""•e<)Wf cc··,.,sation.

PEARSON.

S<·ott Foresman

'i!LC!'Ill3 l<P. :'e<':~r d 0 A"CV;.lOfld "')ui.iUC<IIy afld

CC<'Cep' ;Jl Ly C:.Jmllllo: 1"9 !, :> r tne ::on ant .reas

g. o<tra' .x~. SCCiru s:..does. scio<'c~. m.Jll"err<J:.cs)

ro rc use tris ',rg· ..,__-e ";,r acwate end ccro;root

Ha"J in ,~ 01.1'\ •'•fiiii"'.C

"~'"9 :~Cole ""'C <ll'n.w'fliS cNJ;,n.;•·g let at

_tudef'ts Fer elClm~te. ~ools spi!OJ at~~ · 2 yew;

tr,ong '0 e;;tend I~ ~Cf1·• r• l!IONI Ltn<]-1&)" Ira: 'lat••e­

~ctak,ng en IO'en brorg to SCIIOOI nto •ne•<J more

com1=tex 3ca, temic tong~-J ~chcres. It 1s h~ruty

surprsing. tre•efore :hal •esearrh no.s repeat"Ciy shown

1101 ESL stu cents ..os1.a ~/ req~"• ot 1~:1sr 5 1e3r~ of

llXPOSure to ocooam•c EnrJhSh 10 ~atch up to nalive·

speaker ncrfl'S tn adell lion to ntorr•(ll ug ncreo,lngty

complex 3cadem•c tangunga. IESL SM:Jcnts mus: catch

!..JO to a moving tar.:;et. Ever, yeor, nat~ve~~H:::eo;o<ers are

making large ga1ns 111 lt".eir re.ltJinq and wr I nv ac•hties

Dfld in their knowlodg~ ef 'ocobu'ory. In ord;,r to catch

cp to gr.:t,je norms w thtr. 6 yeo.rs, ESI. s:Jdoo ItS must

rn<1ke t 5 montns' <Jilin •n e·.er1 tO· month scr ocl year.

By cor.trant. the t)pl~at rot 'e·spe.,kl~g ~tudent Is

cxpoctad ~o moke 10 mon:~s· llJin in o 10 month

school year tCcll er t T>oomas. 1 9991.

"t :nrc1! a• .pee'~ oll3"!)c~~ P<Cflei"<>Cy..re omoortant

f-k:oftove<, l~'l'1 'VO 1r$C;0.¥1'1 ~'L''ld t:y t)Oicy 'T13i<&<S

,.rd tne mach for e•,vnpiiJ. ~ s somet mes c~ med

tMt ct.l<lren accuro laflllL39<o racicly and rt101 orO·fi)JI

cf It" ·t~ ora s.repert •S $0..tfl¢• ,. , 1•1~>~•-.clo:t ESl

s• Jd.:rt• to catc'1 up ~· 11 "'r~<> !J, ~ili'Y ESL

st.lder'ts ... ro ,..,.,~ a<:Q.J•ed tuerv COI"MS.ltoorill sk.ls

<lo"e .,t>l a ong Na/ rrom groJe h.c•,el P<:rformance ir

nc:oct.;lm<: ln"9' ·"'l"' prr ''~ tii"C'i (e () rean•·g

r.omr.r;r.e-os.on •n ccnt"nt Jr~s n• och o.; m.:.:h).

$ m 'Jf",o: :ne 100""1 "l(J •'' rhSCrOtO c.'lt'l•jl :1-:JP S-<.ll:s tlOtJS '10t

q~rer"liza autom1:•CJI!y to C•Ca.:l~rl'lc l~r.gl•·1Su

r:-.rc rit:.ency. ESl (and ~OliVO sc£1.1kln'1) :.;:u-Jerts ..... no cc1n

··renl1 Ll f"'Hllht..t"J'\.11!<'\"1 ,,r:-ml.:tM fhNrt lly m~y h.wo only a

v€~rt'i,.... iled untl(1r .o~ir1n£ l1nq 01 ~no ···.orr I:; :"'"lcl Sl:lmer-ces

lt1c'Y r,o~ deccr.'P.

T:\O..S. ESL stude11s ~ay 1JQtJ'9 ex:e11<!A<! '<'~'Ql JG'l

sJpccr ~ ·~,.. tta dassrOCI'Il .1rl or.:.ef tc C,..'lhl"'~ !C

ma<e ; ·n kr;e ::rogrec..s n cement aret~s SUI.t' as

matt"~cs ~c~a :re 'ac· that -r .. , rave xcured

o,r.o;rs<::c<>a t'Uerc-1 n Erg'ist: :C\)e!Nf wm t~

matl"ema: ca -~a~c~ ar.cl CCr'l!lUt~l cnal ~is. s~Jde~:s -rr, s:l 8'P€1'iE'lCe sacs n ,.,. .. 1-rt:v.'Bclr;~ ()/

more sc=~slicat?d .cc::bl.l<.r,, S'jrt1.<, l-~ ()scC\Jf:e

'ea:ures of tro:1em.Jtoeallangwge

7~aeh l ng lho l .a<lgunga o f ltl.at ha.'l'Uillee

::rom an inst·uct~cnll perspect ve, tr"IO ,.,'aliorshio

~etween language and matrer'1atics is two-way and

reciprccal. Ma:nematicat kncwtedgt: •s deveiOC<lO

It-rough language and targuage otil ties can and should

be developec through onat~ema:ics 1nSlfUCtlon

Because l""athema:icdl conceals ar d op&rotlors are

embedded in language, the spocontizect vocooutar1 of

matrematics aM ~~e d scour"e rantL.rDs of maone'!'ahcat

or·~positions must be tauGht exp icitty 1f 5tudents ora to

make strcng academ1c progress in rrotht~m~u cs

Equally impo11an1 howeo. eco, is the fact t"al 1n t .. ac:-nrq

mat~err.atics ·;.e are <>!so Cfl\oetcplng and r&ln'etCif\G

sr.derts' gerqal academiC langua<je P<Cticienc',. Fer

exat'"pe. ll'r< atX>..tlfe ta.1g1.;aga lcarn.ng II" at ,.ill <~ly

OCC\I as tt-e tea<:her e'<,Pairs tho> lo41o-... rg grace 1

prcblem to a Qt'OI.P cf ESL stuckirts:

Is 3 - e g.-oore£ tl'.on 10, flql..:P to 10,

St..certs wol ea-. oct orty ~'1e specolic tnrorn}S ot ..,.,

~arms gteat~ t.l't8"t, ~ to &"''d ~s3 t~ ... 1t'. but .1 so Sf"llO:,ms tQf th.ase ~efms te d . a synorym for gtev: tS

b•g arc t~e rrP.ani~g of grcat.:r 1.""" IS s m lar to t~e

m"3'an ng of t;:gger thar:,: The :rccle• n •·; ra.so ':'In

op"ortunity fer :~e teacrer co leach studen::. tro :J nP•OI

corcPCt cf ccmparar:ves and the gNlfJr:'l r1.,.lv fer fo• rrlf'ILJ

cc.~m~.:ratr..:cs ·~e.g. , grrxa, qt~1tt•.r; ;Jff\J!•'~:,. t..'g Olli$1N'.

biggeSi}. i t'e fact t1at ,ct ~~ comoarallvO~ ~C:hO XUCt~

PEARSON -Sl'llll Foresm<m

;h.~ fcrrr c~n atsv b'l ta•ogllr n ·eoator tc :ess, 1es-. .wJ:. F.n3'~t. the -r.e:<:r.g cf tre wcr::l bcl3in GM te

ta•J<)ht ,e g., cescr.te. teil &cet.t, etc.) and •elaJA.d to ts

"'"-' n o·ne< S..;CJCCI areas je .~ .. science\.

rnc -ecoprcc.,,f ntero~erce ct ~:~r.go..ase and

mntrematics is ''$1Y e¢'1>0US 'rom peru5ill c l lll"'f

mamemauc> textb.."'Ck. Muc~ of wl'>at s:udents olre

e'~octed ~o learn "matrematocs .s presented 1n writer.

texl St.JCerts are raouirOCl to read the text 1n ctr.Jer to

devc.;cp their u.nderstonding of rnath concep:s ard corr1

cut •nath orooterns. Froque~Uy, tney are otso reqclfed to

oxpiJon orally cr In vvr.t1ng hew they so.ved the problem.

00111ou~ty. toacr.ors and students '"'" di;;cuss mese

concepts, but w thout stTOf'S rea.:t.ng sk.Js. studer>ts will

~nd it vur1 d•ff•cu.t to ocq~..re :~e con:en1. Wltr.c.~t strong

"'"""9 S< !s. they"'' h<Jo,e or rut>, de"''<'stra'"Q ~

kr>O'Nfedge of t"e concej:ts. Thus, ~ffe<:uve readu1g anc

1m1rg 91<lols .:.re nece!>Sa'Y for s:udents to make

progess r matrematlcs. part'.cutarly as trey progress

tt.rougi' lhe eJemE:Iltary SCheel grades By the surr.e

tclleo. the ta,ch t'g of rnatr em<Jlics crowJe~ 1mportant

oppcmur lti(!S fer teachers to mcdel academic language

in the-ir in•eroctlons with srudents ar,d also tO te~cn

features of academiC language U>roc:ly (e.g. reaJ1r>g

comprehen~:~on s:ra199ies, compa<otove w,ac:.·.-'15.

''ocab~IJr;: and so or.).

Effective academic language •nwuc:hon for ESL

studer.ts across the curncu~ '"' 1S buo.t en tf'ree

f\Jr,aamenta• piii<Us:

• AclrJatc Pr.or KncwlodgO/R•.fd Oad<(lrc,ro

Knowledge

• Access Cc'>ntent

• Extenc:t Language

In d9\oeloping matnema(cal know'!Xlge througn

language, ano language ac•lities througn motr-.emaN:~.

we can apply the~e tr.ree instruction a' pmc pies n

powerful ways.

A :'th~ ~~ .Prlof i<.no•,~tla.Q~e/!a*.JII~

Bac kgro<Jn<l Know! ~ dg.,

A. Pr;or knowledge as the lounc!atlon of

le a rn ing. Th.;re Is gonenl aGr·~~nt amcr~ coqnr..~

psyCMIO~rs:; that we tearr c-1 ntegr.:n.-g new i~ or to

cur B!<lSt ng C<JSn l1•-e Slru,l•re$ Cr SCI'.err>a:3 0V r;rict

ex;:f!rance ;:ro-.-'Ces <he fo.nda' .on 'or nteop•~ti-.g r.ew

1nforrrat~Qr1. No oearrer IS a ~:iSM ;;<ate. lr fact. 1earrjrg

cnn t:e dtfrl!d 3S tr.e crocotss 01 re:all'g new

infounatoon to the irfoonatlco w'J a.r•ady possess. Wl1en

·.·.e read a matre<"'.a:.C:.t problem, for axOIT'ple. we

ccnstruct meanirg by brir.girg cur prior knowledge of

l:lt'guage. ma;hematics. and of the world •n generJ.I to

the text. Our prior '<nowledge enables us to -akc

.nferonces about tne moan ng ot wofi:;IS ana express.ons

that we may net have ccmo ecrcss be:cte. As our pr.or

knowledge expands l~rc.~o~h n<:w lea<rlf'll. v.e a-e ~

to understand l greater ra~~ge of matf'ematc<ll COOCEJ:ts

and otso ltle ODng~e ma: »prt!SS85 tr.ese concetts.

r~us. a ma;c• ratiOnale fCif o.crvarng studc'l'ts' fY'C'

know100Qe. or If ttrete IS mirlorn.'lf prror knowlec~e on a

part c.tor toooc or ~SS~.Je, buokt'ng r. y,it/l ti'e Sluden:s, ''

to m.Jlo.e :ne ....... ng proce .s "'<)re erociMt It is

imccrtant :o xava:e studerls' pr.cr knowledge bocnuse

sn.chtts ma, net oxolle:•tly real'za wha: lr~y know obout

a carticut& tooic or isaue; conseQuelitly. their pnor

kf'oviedgo may not foc1Hate lcarmny un1ess '' 1s orought

to consciousness

B. Prior knowledge and ESL students. In a

cfass•oom w1th se.cond language learners from c1versa

backgro1111ds, prior knowledge o.bout a ponocuiJ.r t~p«:

rnay va•v v:ide·.y. Thus. s1mp1e tro.nsm.ss.on of the

•nfOf'T18110n or sl<!of will fa.l 10 connect •~th tre prior

~now'.edge anu previOliS ex;:eroer.ce cf r\Jrf students.

As o result. :he Input ''-~' be m.oc~ IE>ss cor'lp-.!Jje

for tnese s:ucrents. Some stvde<'ts rrai have "Jeo,am

nforrna'oo in thc1r firSt 'a.1Slk!3~ ~,)bot ror r ·al'zg ~~

lt-~e:-tt .s any (..orlf"'.et.tk:n .·.ith 'lwflal tr~·v are .earrJng :n

Engisll (12). :.. otrv ~. Ire l)lgc•it~rr~ aJ'd

PEARSON

S<·olt Foi'\'Snl<m

~lrll l&g1es ~hat Sl!. 1C'ents huv~ Jcqu1r:ld fer :a~r-;ing :u~

rl'utll opcra:ior s ir tre r co~rtfles of origin may diHer

~ :cr'~idfornbi)· from the ;:>(l")t,.:eCUo:.:l tl",ey are now be1ng

tnut)nt Ck<lr~J. t~is drscr~s;Lncy c:;~r ::aJSe confus!Vr

fOt stucents.

h ~o.'\Chlng ma:h to ESl :i!udel't~. n s mpcrtan: tr<~t -.e

;'lrtor'"'pt to ~onnec~ tr e rr>strvetiOn both wrth studen:s'

IJOOr experience otlearn•ng rrolh ond with lrerr

l<nowlodge or the world in general. In btuldir g up our

krowtedge of students' educutlonaJ ar d c~lwral

C. rckgror..nds. we can cc labOrote with ESL :eacrers

·M>O may r <~·;e greater accesi to this nlormauor and

nt;o with amrTUltt 'IOI...,til<l~S whO c;:~n ot.en prowle

<1'.alt.able insigr.:s about otudents' poor learrlr.g ard

Cultural knowledge.

Lois Meyer (2CCO) nas exprossod clearly rne lmportarce

of prior Knowledge (farrrl ~rlty wltn the topic) in reducing

too cogniti••e load c f the •n• truct on for ESl srudents.

She notes that the not on of cogtM>e :cac refers to tne

I'UI"ber and comptex.ty of ~ cx:ocepts emtedded

on J lesson 01' text. llis depends not on't on the text

•t~c II but also on the ttudents· pnor ..now:edge of

ttle content

If lro Er>glish learner I lOS 1•1110 entry knc·wledge

obout the SJbject mmt.;r, tho cognlti'e lead of the

lesson wei be heavy, for mJny ccncepts w1.1 be new

and urJamirar T~e studMt "~I ~1M! ~w bass from

v.lltch to generate hypoti'Os-.IS reg.Jrc<~g me m<>anongs l'1e teacl'ef •s cor.vGylng thrc-ug'> E.1Sfrsh.

If tho st~cem's enlrf knc.w1Hc1qo of t1e topic s

considerable. this will llor.ten tho cogritive load

Learners can draw on t~e·r K•10wledge to nterpret

hn9c•stic snd non -lingu•~l•c cltJCS n tre lessen

'" order to make ecucatod 11 lj._:c-$ obc:M...t tt,e

rre:lnings of tile teac~er·s t<~.k and text. ,~,~~-e·,

20CO, p. 223)

Cleany. tre cogn '"' IOctd of mary matl'lemat col texts

Is cr.ns.cerable. partlcLtliJ/Iy lS students crcgre~s

thfOL£h the grades. F ncllt'Q OLrt wnat students '<now

,Ocut J partrc.:lar topic a~'¢v.• the teac~er to supply

"""''ant conc<;<:ts or -.ccabo..larJ that scrre or all

ot\A..,.,ts may oo taG1<r19 bo..t .,,l'och ....n be ~art lOt

cncel">tarding the uoccm•ng •ext or lessor. Bl.rldong tt'is

context per.:r:s studo•ts tc Lrderst?nd mere cornp ex

language ard to pursue mere cogrltively derrordng

act.w'es. It leS$ens tho C•'IJn•ti'e load of the text and

frees up bran power

C. Strategies for activating prior knowledge

and build ing background knowledge. T>lrae

·y~es of prier kno'.•AOO!)e aro rcte>ant to COI'S.dcr II"

taocr irg ,...,atre!Nltics:

Knowlodge of math concepts and operations that

students have previously been taught. For examplo.

rn grade t we migrt activate students' i<ncwledge of

CCtJn:•rg as a preoooe to teOlehirog them to ~~ cou~t"")

on as a 'OCI f<:A' adObOn. Or Clt tNt grade 4 te-.el, "''8

~ogr.t activate s:ooeo:s 1\nO'., ooge ol basic

~r.o t plcat.cn fa<;ts ir or~.,, to rernf.:rce tt'e four.daton

'or teacring more compr<>x fT'ultiplicatron operatocns

Knowledge of the world that students have acquirod

through their pnor experiences. For example, at a 'ery eJrty age most c~•'c!ren develop nn irtUttve sense of

'fa rress • ard an ao.lty to judge "'he:her goods cf

v:li10US lands :e g • C<ltldres 01 treats) ha'.~ ooan

Cl>tnbu:ed equa'l'f or f~r·"y Wr ·an US& e<a•ns:C<JT.-.g,

tole play, and s n-u .a~IQn w1"h ccncrete rr arupul.ati·it•s to

cnrry out a vo:icty c l rnath ectivrtres I hat tap into

students' re;tJ.. fe expurrenco of ec;t ral (or fair) distrrbution.

Stm IJJ1Y. v:e c3n ~nd cut from ctudents •.·,hat act~".·· ties

tney engaGe r. ooA~:Je )I tha "l<.t'<lot ccnte:<~ aod ~'11<

r'ati 'EI"'atics rlf::uctiOil t~ tt>e::e acti.o:ocs fe.g. stucle< ·s

v.no engage ir. 'latiOUO flO'I• can c:wy cut a varl€1y ct

C<>icvlatrons relcv.n: to tho~t' ·PQ!!~, ~ucr as batting

aver<~ges).

PE'ARSON

Sl'lltt ForPsman

\Ne CJi ~lso Co proacU"'e in ~rat1:•tng ~xpentinC~lb ·or

'.tud< nts 'hat w:ll prc<"ota mathemat.c".l know'GCJ~e

Jrd SK:II. Fer a~emple. we m:g'll ~tng-.(ie carar.t as

.;ollc.Oorators Oy h.l"rg the<" ¥;ork w th :oor c"'loret' 10

CJ'cw.a~no tl'\e prcpcrtJ<ln ol ... -oo•ly food ''penatu ... s

~~ .... : tl'\e t~rril't spEnds or !I'J> •'31l01..s food grcuos.

:- {)y ra.-torc.rg both '1CCi.lJ sruoies 3nd math

ooroecpts.

Knowled{;e ol the wor1d that students have acqulnd

through secondary sources such as books,

television, movies, and so on. For exampl9, r.olat.'oe~,

few r:eople :n North Amerca have ever oeer '" a 1ung'e,

~ut most adults ana cn.ldren coutd descrloe the man

leatures of Jungles as a ri!$.Jit o: s<:!COf'ldary e'l)l.<ierces

c l 'arous sons. 1n the classrccm .... e can use •teratJre,

l'tig1rtaras: e"')CS•tory rex:s. and ether terms cf mecia

(o g., vodeotapesl bolt to acwat() studerts' prior

kr.cwJedSo of math and also t:uHd bacl<;rour.d

kr.cwlodge. tn some cases, this w:ll .owc:ve usa or :,wios ml\1 ha•1e been spec.llca.ty se:ected becausa

tney como:n re.evam math content, •n ot~er ca54!S, ,,e

,,,,n connect mat'1 ccnc~p:s and ep;raticns to oll-6

• •bteet matter actOSS tne ctsncul\lm. For P'<3ITIP'e. we

m.gnt l.ri< math to a soc-.al stud· a~ •.JrJt en gove••1rror.t

3nd whe·~ 't ge:s ihe 'urCs to operate by rav.og

ct~derts calccl3te tre so•es tax that their lnm lies pay

lor vanous ourchases.

We c.~c also '"'k mo.th to the deo..elcpment of crit•cal

ll'\inkh1g by havirg stuaoms c"rry cut projects that go

beyond 1~0 currcUium :n various ""ays. "Ot example. •n

u ctass ..... :h ma'1y ESL •tudents we mognt ra'e s:ud..-Js

tarl') ot.Jl a sur ;ey cl tl'.e ;r.erago "'-mber 01 tan(lua9Cs

:hat >tuder.ts r1 th! c.ass (Of tre N'tire SChOol) ~<r>ow

rod how \he'J :~· know tl' ese languar.es. In aratyzmg

:lata ti'at •r fleet their cwn experiences and :dentities,

,tudPnts' mo';Va~ ,on to explor~ eftoctrv·e annl·7t~c

; traiA<;)<eS nnd preser.t<;t:on IOOoS (e.g., gral)hs.

romputerlzed ~hde snow:it, und ... ~en) IS ,tJ.·~;: y robe

cor:-.~1tfer;~ y;latt."' •!!al"'' ,\t'!e:1 ·re ar.:~'l' &s ~' 1r.cr\!

.,s:an tror tr<Jt e>"~ ar-d 111reres<s.

"'he esSe!'ltoal rem !'l€'8 S tr<Jl :l'e mer~ ~nrec:IC<1S W~

C3l1 mal<e !::<:ill :o studei'ts' excerlerCL'S and interests

...-.d to cl~ •1$$ cl tre curticcJum, the more r'(!levorce

m;;.:n is ''ke.y to as~ume n st•Jderts' minds ard r,,,es

Th's, in :ur,, will result:n mone powerful 'eJ.rnir.g o• matll.

An add't1Ct1al ~cnsldera110r. o act••,at:ng ESi.. sh.dents'

crior knowledge :s tr.ar th:s ~:roce~s ccmm<.n·co:es a

•ense of r~spect fer wrvt st~den:s a~•.>cy 1<11ow 3nd ar

lntere.;t 1n their c<..:vra• cackground. Th s artm>atio<' of

studo)nts' l(!,rtrty ;ncreases ~ludet'ts' PllfSO"al :ll1d

acaeh!"'oc con"ioor<:e and ":lDt•vates !I' em to ln\/Cst the:r

idar\t.r.esmo<e st:o<'IJ'fln cursuing academic success.

.AcceS$ Co~ta n !

How Cl)!1 teach()rs make the comO' ex lang<Jage 01

rr.atrert".a::cs comp~t"hensillle n s:uderts ·.·.he are sN in

tre process 0' teatr"'g cingo'h? Hew can ~tuderts be

enablc>d to :like o.· .. ".er<.hc c l thelf .earr:ng of math

con<:41)ts arc "petal oos ra\~er than jusr ;earning rote

r rocecu'&s? One :mponam strategy has already been

noted n the pr~"ous soct:on. Ach•,a:.ng and bu •C<ng

~!<.dents' background ~n.;wi<'Cfge s J l essent•dl P".>lt ol

the process ol hefpong sr:J<!ents •o cart.cipa:e

ucaoomocally <.nd ga.r ar.cess tc tl'e mear.ng. W'>er we

octtvare s:o.lden:s' priot k'lOw:edgo .ve atter"ct (0 mocs::y

the SOil so tr.: tro seeds ol mea:1ing wi lt1ke root.

Howe·'er. •,o,e c:M~ al:.o SL.PPO<t or scaflo10 Gt•Jdcr.ts ·

'~>.arr ng by :rco:tying the :rput itself. We provide :nls

scar.o•c"'9 cy ombeddi~g the content 1r a r.c~ly •edunoJm con: ext wncrd t~e•e are mut:iple routas !O the

onean:ng ,n &.1J•!.on to tne ra~ ~ .elf.

The IOI'c""•rg hst poes~nts a var.;ty ol ways ~I modl1ing

the ptescr·wi~n cf mathema'~Cal ccr.:.:rt tc ESl

; tucfo:nts so :tlm tl' , ::an more eflecl".et; !)Ct access

to :roe ,.~ani"!) .

PEARSON -Scoll rhn·sman

Oemon.stnt lon/mcdellng. For ~xnmpte. te<~CMr:;

can •·ke .t,cents t~rcugh 1 .vcrd p·cbtem r 'YIIlt~

oor~n t·atlflg s:'lp·O', ·o:=o P<OCOOJ'" ard s:ntvo; liS r u c ();lr Lnd exp1ie11 marner.

Use of ltands-cn manlpul~llves, tools, and

technology. In t~o early ~raC:es martp~latwes r-oy

rdude coon:ers and btccks '~": eratle stt..dcnt~ to

co.r-y ~~.t a 1'1athemal'cal operatoo. l'eral~, "'t, rrt r

hardG, ar·d actual~; see :lle concrete results or II• Is

cpaa· on. At tro<9 adl.ar.ced ·eo.'els. rreasunrg toos

SJC" as ruler3 and p•otroctors and technolog•cal .at.Js

sucnas caculators and cC""Cuters w II t:e ~sac. Tre

enectr,eress ct these toots wol te en~anced ff they .1re

used w thin a orcject that students ore ntnnsical\y

roc:Lvat<l(J :o perO<m.

Whole c lass and small-group p~Jact work.

Wc<<I1S "'the< as a v.nc a Class cr in ht. ter~enacus

grcvps 0< 1=a•s. ~~"darts can enga;;e w.th real·'.'e ·r

s•rru•ated prc,ects t~ot requiro a~pltcatton ct a variety or

ma:hMI<.· ca sl<l;s. Oiaz·A.co .oo WP."rd 120021 ~as an example a prcroct in which students are told the

Classroom reeds tote re·carPQted. They ~rst e~·~1'nate

t~ ar•a, 1t1en ch<lc:k theft <*bm..1tes woth meastrtng

tees. Worhirg it1 groups, st~,;dents could .llso calcutole

the cos: or floor C<Mlnng usi'.g po:::es lor va"IOUs t;pes

or 1\oor cc,enng ootatne<:t from local ca!alogues.

Use o f visua ls. We cor-IT()(IIy r.e(lt :he exp<essoen

"A picture is wort~ 3 tncusund words." Thera is a IcY. or

t '\J:h to this in teac~lrg acarJ" rric contvl\1. Visuals

enacle •tvclents to ·see· tt-eJ»;ic ccoce;::t we at~~ '') og

!o teacn m.Jch more effecti,:ey than 11 wu rely on yen

woros Cnce Slvder'lls have the concept. trey ar~ much

'"'ore tkNr to be ot: e to ftg.ro cut tne m<w>.r>J of tr.e

wcr'ds \VA u<i~ to ta'k abo1.1t h Among tho visuat!.:o we co.n

li..<e irl r:rlSIJ<'~ <nalh center' I a-e:

f),c: :.~res/photcgtUf)/'1~. rt.¥.'' obit.'CiS. grspt):c O!f)<-1t:i.-'(Jrs

o1r.,ith~ en c.ett'tt.J'KJ ao;vr:''t:to-s cr :Jri-;J.~.,..e .-r.a~""'· r-s.

-.r c so IJn. Graphic .. rgarizerG ~ra pantcutarly USil\11

tecat:se '""''can be csed by te<•chers not on~J to

1=r%a1t concep:s cut 2.1.,o tv s:uc.r:s trem~e \..S to

t£.k& 10teS, or>)tn<Z9 r•,e r ideas in IO<;J cal coteyones. and

St..mrrorte ·r» 'l!Sl.i".S cr 9'C<-P br:!rs'Otrllng on

oai"\<Clll~r ISSlleS. Some c;rophtC ors~nizers that are

u<;ef\.j 'C< IS~"'J rr:t~ .... ~ 1/e."·' Dngr3>r.S Pie anc 8dt Char:'2, K·W·L Cl'd"j t,_.Vt'a: •Ml ,..no .. ·,., 'hnat .ve

want to '<now. and • ...-ot we have lf>Otnecl, T-Cnor•s

•:a.g_ 'or contr&~s!. ~jSolu!icr. Ct-.ar.s l,!a:n 1'Cea

2.nd Oeta .. ~·s cnar.s; Cat.'$(1/Ctfect ChdrtSI SOO!i&f'lce

C.'lart:;. 7:/TlO /...f'es, ard so Of'l

Lang uage Clarification. Ths categc~; lrclude.l a

, .. ar'ety or strat~les and lar.guago·c~erted acttvi:tes th:lt

clarrly lne rllfl<l""'9 of rew ,,"Ofl:!s ard ccrcepts.

-eac~ers can rrcotfy their to.nguage to students by

N"!!t:J/Vas:r.g ceas afl(j ~g ~ cor.cea•s ar.a

wt;tcJs. Thev can o..cplnin n&N v:ord~ by ::ro'vldtrtg

synor.yms. ar.tJnyms. arl) de'n•t·ons ei\n~r In '"ngltsh or

"' ·~ her-e arg.;..ge of s· J<ients. rl they l<r"W ot

tm~o~o.nt vocabu'ary c.~n be repeated and racyc'ed as

~an of the pari)j;lvas."lQ of k!eas Tea~ers s~ c sceak

ira na:U<al nytt-'!1 but envrc ate cleor~/ ard actr~st

speech to a rat~ that ESL stude~ts wi t find ea<:ler to

<-r'<l'!t•:m Tilt rneay,g can also ce ~ted

ano/or reinicrcl<f througn gestures, bOdy ta.~guage, and

demc~strat ons

Beca •se of :h., r ccn-mon roots on Lat•~ ano G<eek.

rnuch ot tl' e lechr tCal moth vocabul .. ry in Eng ,. ,, ~as

CO(}'I.ItiiS In Romance l(l•<pages s.oc~ as S;:Ms/1

(e.g. oddittcn • adicionJ. :3tuoents w~o k'>Ow these

tang.~<:<..., be er>'Xl.Jiatl&:l to l"ake tl'ese cross·

ling~<. :to tirl~g9s •s a me<ln• of r~trlorctng !he ccncept.

9.1i'10cal at·d Cnghsh·on y uicttornrto~ can also ba useful

IOOis 10< largu.J9e Cla'f.Cat en. 1)<1'1 C~liwly 'O< old..,

et~mcntaoJ gr,1dos students.

Dramat ization/ Act ing Out, Ft:r bi>Jnrirg ESL

stuoorts, Tote• 0~ys:ca1 A, •spon:e whore st.tden!s act

out comMand~. ca,, oo ~lql·ty errect ''"· Math

ca ki' ors C.ll' "" 61Tlt-vcec tr rr..: ~O""tr ,1"':1$ that

stuuents act 01 :. For ex,1mpte. sturJer •Is -::an prowess

~·":'.· • -:·. ' ..... . Yf'

PFAR$0~ -Srott Fon•sman

from ru ~~ act1ng cut :ne command "'T~e .. \le steps

fon•.vd a."'CJ 'V.'O r lecs t:aGJ<wero" •c caJQ.IJt'ng in We1r

her.d~ ,nat they nCtld cr.ly 1ke tnree st8ps for.>~<Jrd :c

reo<:h t~e Cb;toatioo. Aacillonajy. t"e mc.r.r.ss .;I

ind''"CI..C\1 words can oe domons:rated through Qvsrures

E..t~end l.nrg,J;:J'.) t

A s1 •tcmatic focus on ond exploration of langl..tlge is

CS$Cntial rt Students ore to c:evelop knowledge o! the

sp;ollc voc.'lt;Uary ord dls<:otr se par.err.s wil~1r the

genre or mJthematlcal language. As r c tOd abo·Je,

im.-e< Ugat.on cr tt.e ·~ of ~tl"em.•:oes can also

develop In studento a curiosity about 1<Jnguage and

dceoen the>r OJnelcrstandng ol hew ..crds wcri<. Three

st.,:egies for extend.rg studems' knowler:lge of tne

lar.gu3ge of mat, emotics aro OLif1rod below.

A .• CteaUng mathematica l language banks

Students can sys~ematica'ly collect the moanir gs of

wcrds and phtases t~ey er.courter n r-athematrC.ll telCis

"' a perscnru cr groo.IP language t:or <. ldeolly, t, .e

larguage b3r'\k WOLild be createo,n a ser·os of Illes

,.;,~..- the classroom compUter but t can also be CCM in

a poper-and penc1111otebook.

Parado<ICafty, the corrptex ty oi tn;~tllematrca! 'at'lliuage

pro•ides some omportant oppor:unitics for ranguBgo

eJq:loration. A$ menr.o~ed obcve, a large percentage of

the te:;s frequent acade"'IC and lecl>nieal vocabt.olal)' of

English derives from La:1n anJ Groek roo:s. One

rmp~Qroo or thiS is !hal ~.\ltd formation '<lllcwS oome

vert P•edicrot1e patterns. Tliese patterns are sm rl:1r in

both Eng~sn and ~

IA">ec1 ¥!~dents mow some ol tre ru'cs Ot cor\Crtocns

of hew academic words am formed. t gives tr em un

edge in extendog lht:r \'OC3biAart. 11 helps :l'em r.ot O('jy

figure out I he meanings of lr dividuol words t:ut olso hew

:o fO<m d'ffe<ent part:; of qJeec~ 'rom tt>&.;o •.<'01'0:>

A c.:ntro. .!SpOCI ol X'"C:er.loC lar<]wge IS nor:rit11''i<J;.Xt>

f'h'S -efers •o the crc.cess wr.erc-cy 'lbstr-.cl r o•Jr a ore

rc....,.~d 'rom •.erbs and ac!JKI•.es Take. lor e:<:~mple

let;r ccrr.mcr verb• hat occur in r~e rr.o l~ currlcu.um:

iexc udrg ·.'('tb 'orms a-a purals) for each cf these

v,crds are oresenrod be>O'>v:

Verb

1Tl(!asur9

equal

eqwlize

Noun

mu~plr.a:ron

multiple

mr..l; plieny

dv~

dividend

moos-.remenl

aquaJ,zer

Adjective

dMded

equal

eqLitat:IO

We nee in tr.eze to<.r word families. several commcn

ways n w'11Ch 100 Eng.sh tanguaoe !orm~ rCU1S lrom

verbs One pattern is to add the sulfix lion or ·ion to tno

ver'o rem·. as 1n m~,''•p.:•cnt.On, dv:s~en, and many o~-,er

mathomatJC81 ;eons SUCh os estima::'Ot", ro:a:ion,

oporot;on, a•d so on. Another ponorn 1S to add t"le su~ x

-ment as 111 mees<J'emem. ·.·.-~ a trord pat:ern .s to aoo

the "uttix ·1ty or · ly as in OQvaJ;/y, capaclly. croperty, and

prooabl!ity ~~1"en "'0 Cerny.,loly he,., thiS acOOemic

13r.gu3>9 wo1ks, studenls are mord likely to recogn,ze

parts of speech in . ~o;.r re~d'"'9 of c:omptex text actoos

tt·.e curriculum and to become more adept at nlerr ng

meanings frorn ccntexl. For example, v-,hen a slr..cent

recogn,zes •1'~1 accoleriWon s a noun {r:nhlr tNtn a ve<b

or aUjoctivc) he or 4he " as Ioken a step closer 10 the

me~n ng o' .'1e :e'1111n :ne t cntext cr a CJrto..tar

sentence cr !ext.

.)·- )..,---... , ., ~ ~~

··~ . , B · .~~:! .

Slcdc-rts wn b<! E-rccura•Jed to 1..59 C'icl cna: os tir oclh

Ergis:- 31¥.1 :lleor Ll hl'tlfl ; ·,afcbe •o e•x~ tte mete

zuct e mea"•"<:;s 01 t'\ese mat~ro'\al~ •MJtd' cor

~xompte, Wey co~Jid ce asked to v.~r'< n pars or sm<.ll

groups to work o~t :he di~er~rces n rreonlng cerween

equal and equal:ze cas •,e(!)S), e<;:Loat:~l· eqt~al, <nd

<->qUO.o.;:er .roeunG), vd be~Mler equal 'lf'd er.o.::~ble

(acjectivesl

Thos ncm nalizatoen process also permo IS us to tr ir k on

terms of abstract rooloties or stares and rouse higr er

level cogrotive hJnCIIOm that reou·re uses of LDr guage

·,ery c ifferent 'rom the ccnversat.onal or ·•;; a~;rourc"

anguage that we ac~utre n a-.-er J(l~ s ~.loltoons >l-is

pont is made cte:vly by Pa~lire Gobbcrs·

The playsround sltua·ion does not normally cffer

children tho oppcrturi~;to uso such languogG as:

If we incroo.r;e tl"'o 0r1g/e by S c:egrecs. \1\-13 ccu.'d cut

the cvCOJ"f,..,tJilCe <::o eqo,at parts Nor dOes n

nor-n<:lf', requ re 111<ll.'lnguage a!se<:iat.:!O "";n me

h.ghet orJ"' l" tr' ktng sk~s. such ;~,; I'P, pe!hesozng.

evaluating, tr fcmniJ, genera izing, prcdocling or

ctassi!y'ng. Yet ;hoso are the .anguogo functcns

which are tel3tf.ld to tearotng and the <.levt.:IO~Mer.t of

rogn~oOf\ the-1 'X:c.Jr on aot areas of tl'le c Jric\Acm,

and wittoul r~ a ct*l's potentra " academoc

areas canoot be rcahzed (G bbons, 1 '111, p 3;

Sha goes on to poonl out that e)<J:)Iicil modo lrg of

acadeor ic tanuuaQo ts porticutarly mponant .n nchools

v.oth tar;e numbors of ESL studerta:

In :;uc.., a sr..nool r •s "'"'! easy to falr~to t~e r<bt

or coostanlly < l'tlch!y-rg our tangu.J()O bocaJSe we

expect not to oa ur derstocd. But of we only ever

usc basic IU11gu:.qo SLICh as pu: m or :a:..e OlJf or go tasrer, scme ct".ddren v.'lll r et n.we nny crpc1tun,ly

to leJm Ot~« wa;-s o' express ng tt.e.;e <leas, SOCh

as .nsert OK ffr.XY,-e cr accelera;e. A'ld tho$u a·e rre

WOldS Wl'iCh ,re needed tO re~er 10 th.J ucr~r(li

ccncepts rein led t., the 1Ceas. Sdch J~ r,-.moval,

inrert.ion o.nd arr., ~lumtior: . (1991 , p , 18)

In S.hcrt. 'lvner stucerts k~w s:me c f tt"'b l..'oa or

ccr>encns cf ·on xarll.'lle ··.ads ar~ 'om~<oe rt \F•'9S

tr--m M ~e ;, e.,t :f'<:rq treO' ·=~"'"'· r: '* ps

tt't rn ret on~; 1go_ n o..rtt•o~ meano,gs of indov•cual

'WCrds bet ~.lso r ow to rorm cWerer t ;::orts cf s;::eech

I roM tl"ese \Nerds. Or"t woy cf crganizing stud' niS'

~ase de:ecti·.e t.c.r< '" mat'1€11'18.t cs is :c foet;s

SG!:IIl >taly on r:~ar.;NJ. lctm. and~. V.'Ori<I'IQ tn pars

OK SMall J'CIJPS. ;h ... errs c:1n oe encolll'ased ro co.~o~et

and e>clore or e mothematocs word per cay foc•Jsor•g on

lh9Ce categories

Focus on meanln; . Categories that car tle ~xptcred

:.ttl' n a Focus on ~·e:llllng rolJde; ~:a·oem.~rocal

~ttrlg. £vel)'day rrn> rr.x:g. ~&'llr-9 ;r. o:hi3r s:.oC,ect

Clt~s. L ~ eq:JiviJ.'Onts fe,'ated .vords ;n L ~' (cosnatesJ,

synonyms. amor;yms, homonyms, med.n/ng cr Ct'Cflx.

meaoir.g of :'OCt, and mwnlrg cf suffix. Not oil of t~ese

categories ·,,.i'l ~e rPII",,nt tor every wc<d b": t~ey

ll'O'i<l<l a mc'll) "' .•octoons that a~ e.<plorabc<l ol

mEan ng mght p,.rsue. Taloe a possoble e>t:'OfJ:ron o1

the word s~btract:

Mnthematical mooning: toke one nurr bor or

qJantity from oi"'tllGr

L1 equivalent (Spanish): •tstar. Sl.SI'aer

Synonym: deruct

Antonym: add

Meaning of prefix: under C·r aw:l·l

Meaning ol root: 1rom :he L.ariro for 'r>u~'

Focus on form. ~IOSI of troe root "ords .n

m;..:nema:tcs t~at come from Laton and GreeK foon cot

Jll$1 or·e part of speech; w& COl1 make nours. verbs, and

odjoet ves rr;:fn many r:f fl"orn. If w e kno·.-... It' A ' \fnicol

f.-:.tte<ns 'O< 'Otm.ng r< or< and adj:o:t •.-as from tr.o~.e

.crbs. ;,e can fOC().1 ze rn.;se pa't.l 0: spoech ''·~"""

ttl!:'; appear in text Th<;l mp ocat crs fer expand ng

;tLtdt nls' vocabulory , re cleor; •a:her than IO,l'll no ,US!

or·o wryrd 1n isOIC\t on, r:o t~.;denis are en(lblod t~ lciirr.

., ... ;.P • ·:.. / . , ••

PEARSON

&·oft 1-hrf:'sman

lrtre .VC/d ~orr .... , :S. a :rocess th~l r;an drJrrarically

t. .. (ce.ro th:1ir v ... c"'-<•rg vocotlial"l.

. ..,J~cr"S that can te explcred w tn•~ a =ocvs C<'l ""'"

1nclude. wora fJtr:•tv and gr;JfT':f"l,?::r::IJ' ,oattr...,:,s, worels

'Adh 5811"\G pref,y woros Wltf'l same t"Xt and wor:=s w.·cn vo.~me :;uifix. Ccn.;.~•der a~:.llf'l the word subtrJct.

Wcrd family/

grammatical patterns: subtract. subtracts.

&;b:racted. ,. .o::acti"'J

(verb forms)

suot•actic'l, s tl:ract cr.s

(no~n forms)

Words w•th snme prefix: sutJo;•r.ute. suctotal.

sub•~rban, sub"'ay etc.

Words with same root: t•acttlr, lfaCIIC)(I ................. .............. .............................. ......... ..... , ............ ......... . Focus on use. SIUd€1'1 can e'I>IC<e the range cr uces or porticulor words through brains:crming as a

class '::t small group. IOOl<.ng •.-,.orcts 1.4> in do<:t cnar es.

oncyclopod•a.s. or t~.esaun, or ask•ng parert, or ether

Jcuts C1t<lgCI(.,•Ihat can be e•ciO<ed .,, l~in a Focus

on Use Include ooneral "scs, IC•'oms. metaot>o,;ca/ uses,

~·srbs. art~·en~l¥'t"ents. ~ns. anc 1'Ckes. For t"".e '•'"crd

&vbtract, mosl stucJents w 11 not finO ,-uch u·ot """to 1

wilt'in ll"t'OO ca:<!GC<'IeS other than tru ca:egor; ot

Genera,' t.'Se. Hcvoleve<, '•'•'ilh some 01 the more frequent

v.or~s n ,..,athemat.cal discourse tnat deri· ... o 1•om thfl

APQiO·Saxcn lexicon ol Engrosn rat~~' t".an tnt­

Gree-«Lahn •exicon. many or t"'esA ether ca:&gor;es wilt

y c d a Mvliitwde ot examples Ccn•d•' ttl< mu~·~"'

mean ngs and fgu•at ve c·>Os c f words s~ch as greo• (as

'" ·greater tha-1 '). b>g aot.'ble. and so oo tt\.11 studen::;

n1•ghl exc.crg,

11'1 st~crt , when !;h.c,Jt;i"'il$ c:>.plore rhe f<Ygt.. ... ,•JfJ of

m<l:hemot<:s by corect ng •pecime~s of m~:ncmatl~ll.l

.;lng~ag~ tn a sy"emanc o~ cufi1\A ,:,.,-e ·:.a.:. trey

xrar•d ltl'lt OPiy t~c r •Jnderctand'n'-l of f"1:n~matic~

terms ar(l CC'1C~pts ll\Jt o!so tl"atr <now~: ~?G .t 'Oil the

Fr~!ish ar{).;as-e •.vcrKs ~e !J .. 1he 'cct tnat iiC3troct

rouns of"J .lion •ell"' ..:1 tn ErQ.i::n bv acc.rg . .;n :c ·toe

verb). The .Jo•.e c~""er.t of lal"'qutgt: J 1voreniol'Ss in this

wny ,.,d t.,. It ~~.<: ~!s' '1! lOng COI'npnnP.ns<ot> ar<J

·Mt ••9 ab IIIIas ac,-oss t~e c•"rrlc~lum.

B. Taking ownerahfp of m;rttlemati<:al language

by means of "reporting bac:k"

It stuoents J·o :c .a~e o·t.1e"'~'P :f mJt'1Erra·;car

larj;uage. we mc~t prcvide aMple c~por''Jrtties and

er.t:owrageMent fer t~em to use '"'s lang;ag• fer

aut~er.tc purposes .n the cla~srocm. In :ne acserce

cf JCti·•e use of :~ .~guage, st.Jdents' grasp o~ ;he

tnathtmatical register Is hkely to remain shallow

arJd passi,-e.

Resea~cners (a .•. Swain. 1095) havQ noted th•ce way~

•n '"'h ch L2 .lcc;u·s ten rs st mJared by act '•& .JSe o!

tnG la~guage:

• St"dents 11ust try to ~£Ufe out scph s: cared Jspects

of tre INJEIIaogua~ i~ crder to express what they

w<ltl: to ccr-mun!Gate;

• I' tmgs home to .tuder.ts 'V'd to lheir teachers whal

o•pec:s of I(J('gUa!)'.l they need <ISS• tance wtth;

• 'l r-·cr~.oos ·eacl"Prs ..... itr tho ccccrti.J'ty ;."' pro•.· oe

correcti,e lee<:ltack to tuld laPguage a"ar ness and

l'tolp s:ud<!nts r g• Jre cut how the ·ar,!;~age works.

Ore examplo ol how tr s pro~css op( rates '" ti'A

te~r_...g cf ('..cf' 'ef't lr~as s ILh as ,....iltl ·emat.cs eS

provided b·i G•tccr,s ( 199 1 ). She emoha.s zas It-o

irri)Ortance cf '6;lCI'1lf'g back as a s·ra1egy fO< promot ng

acader-•c 1-'lngc<.ge oeve'cpmenl. Fur exa-rpre, aier a

con<;i~;te '13"<:5-c•r grOUP ~xr• r·ence or prc,ect. st~,;en;s are d:<e<J to report ll<'1Ck to the crass ora~'

aboul ·,nat 'hey oid ~nd C·b~ed and t~er lo "'ri:e

eoout n A~ ..,ludcnts pfogr-e ~ C(.;tlt""'P.tA ~· os·on

exper.enco to rrcro 3bstraC1 ot; l anrJ wr•tten lnrguage

u•.e. tre-t n•JSt ncll.lc~ suNrr ~r· nf:r.,..~tiOI' v. 1~ . .- tt>e

•ar,r,u,,Ge it-;cll for '1e r"ea11ing to be urderslMd by

PFA RSOI'Il -S<:otl Fon•sman

t~oce .v~o ·=•Ci net sn.:-.r:> r the ori>J>.oal axper ence.

::;no notes that:

.,..,...,1le t'lanCJs-on excel'ief'Ces ur.:t a iery v~h ... ac'e

st.,o; :x:int fer iarguage "ll\o;q:rcrt. t"ey do

not. on "'leo< CV<'I. cner cr. tel~ acec.Jate

~potturities to ::ew.c.p the rrcre 'ccn:ext-froo'

L.onguage associat9d ·~·tl1 'lltloJ ng :and 'M t ng.

. . . o reporting t::ac>< s>tJat>On is a or•d,;e into the

more torrr<JI demonds of llter~cy. t1 a 'ows chiicren

to 1r1 01.11 in sceech - h a rea istic ard authentic

s ruation- :he scrt c.f language tr.ey rreet in bcoeks

...rd .... Hell trey need :o dtv .. ~o n ti'Y>r v.r trg

1•\l ·cre chld:'en$ r:Hir ~uage tac:kgrC<I'ld Ms r>Ot

led to thiS exters.or cr ora. lan;r .... lge. ~ cec=es

even more 1mpo.cant tor thO c J;sroom to pro·,·de

such oppor.Jnilics. (t 99 1 . p. 3 1)

In s11crt. students ceccme moro aware of the cosnc"e

processes ar d st·ategies they ust to selva matn

~<otlems. 8110 they are Mabtec to ta<e O'.'r,erShlo of :110

~lrguage tt-at re1'3Cts dld IJCio:..tes these cocyi:i'J9

processes. when the etltrieo..lum ~<O'rde; ext ens 'e

Ot:~ortuni:ies fJr them to explain ora ly .1'10 .n wnt:en

form what tney d:d and how trtrt cl d it

c. Mast ering the langu ago

o f mathe matical ass e s amo nt

1-'>gh-~t..kes test ng ~as beCome a fact of l fe r

cta ·Stocms ac-oss -ne Uroted St.nes a'ld ccn..eeQUel't'y a

~Jue majonty o' CIJtloet;klm Mat•:)rldls •'lC~ not only

for"'T'I.oltlve asseSSMer'1t tntegr~!~ Y.lth t'le cu'"'O .... .il .. m t .. mt

bul a so practice oriented to p.:>r'ormJnca on state·v.>ee

otanderdized tests. Cons stQnt W>lh ~he errphasis on

provlelir·g oppcrtun•: es for StJ<id·1tS tc take ownersnip of

the lar guage of ma:h~>rnar.cs t,no•Jgh act ve ~se of :nat

I.JI" ~u...._..)e .,.,e can dlso ercour~g.a '"t j(jerts toga,,

ons.gN ana contto <M lr the lY .;.;age or ma·hemat.ca

ll.""essment. We <:<r clo ttls 1:'1 ~·. rg st•JCients crea:,;

tt>,lf own -nuthole ·C,•l.C:C (or o•ntv rc ~vi.lnt) :ests 1n

rntJ\t''.umotic.s rJtt'!Cf lhan a.lw:1ys t~Pi(j on the recerJirg

er d c f tes:s tr3· adci·s ,ave crr1dte<l. -he process '11>gn:

wcr~<. as fc,lows:

n crder t~ 'ard anze 9ttJe 1 ·ts w ~h the pr::cess ~:ard

a so P<l\.e :;crr.e fun r 3 lr·erdly ccrrl)e:~r;e

c~ntext), .,.e W1 ro,e them .,..,rl< n ~eterogereous

<)'OOPS :o cct's·r ..~~;t :her cv.n test.>. >r n.altf ;;ro

topics •.•.it, ..vh =~ tl'e~ ~rG lamuar or on ,.,r ch they

'1a'l43 carrie:.t ot..t reS~:3r~h Fer exa"""'ple, tt'e ~eact-er

rrigh: exp tin oow rnultlpte·oholce items are

cors~ructed (a. g. the role of distroctcrs) and each

Q<C<•P rrogrt corst'\JCt a sat o• ac~<cximati!ly 5

iters on t()!:icS su:/1 as ba~J. CCll'o!ar rr ...so: •

'efe>.lSio, ptO!Tams. pof)<.JIJr •<lf19 and sc <lf'.

T~es.; •:er:s a<e ~~~ pooled and tre entire s-et cf

ttems is adrrlf'istared to tr,e t nh:"9 cl~ss

S~t)sequert•y, each sro~p m>ght resoorcn asoects

of a porticutar content area ar .d construct ,te>TS

based en their researcn. In tne context c f rrat,,

gre<.ps ccufd ccrwJCt r.ems tr.at 'ocus en :r.e urr.

cf study (e.g . 'rac:.ons. I!OCmals) :l'at has jt.st toon

corncteted An 1'1C4'f'lr .... s,~t!ll" coo..ld be

onst,tuled s .. ct- tr ,t t··e grouos .._, , po>,ts

based on their P"rf.::tmnnco " " the pooled test

thot leads uliirnatP,ty to some reward

Tt-o ralicnale for th1s reversal of roles ts that construct or\

o' lOSt items is rrore o;;n>l>V91y cMIIer.g•rg :and

ongagorg) tnan smply Pl'f~ir1<J ·cs: i:e.-.s In O<do< to ::orne ..p ·•·. n i:&ru that v. I be Cl'<~\;ng.ng lor t·oe otr.;J<

fiOUC.S. Sh ... ::ferts m.IS~ k."'W>,'J 'he C:Qnte'i~ C( trte 0'11\ Ill at\

o.-;ltve ta~h~r tron !l oo-sr.lvc way - t-c w:th n-s;r::'-P

d>:,cus~ioo and colt ~bormlc,n n gene rat ng the items and

cit~lractors is a so 1 kely to rt"'intorce botn lt:r guage and

Contont ~nowl€d;;e for ~II .t>,O~nls in tne orocp, but

p&rl>eularly 'or lhe:.e SU<h a~ £Si. • tudC<'ts ·:,nose gra,p

01 t'>e CC<'tent m.,, be ·rag •'

~"'~~''"' t-.s COOCfp:ouO, stanc <dtZed ma'h tesls are

v>·wed as or•e pa1ictllf '] •rr~ ot l.lrg ,,ge. Students

.,hcul::: be t.ami!,or w1th lh(l corvnn:~ons of th•S genre f

ll1t ir acndom~c v/':)(lh s to bP rucounizCC In fJeneJ3Ii,O

PEARSON -Scott f (>rt•sman

mulli~1e·o:hOI~ terns, st~dc.r.t~ are coveiC~Irg l.,nguo.<;e

·.v,J:ene"" .n eJe ccrtilx: ~~ ..1 ""iq'I'J :rzter~ ng tbl.t

J•ngoglno) o:ogmtive octr, ity.

"71'\e sane prinC'~Ie C.ltl oo ocpi€<:1 to the c;ra~llclf'", cl

ct~oer 'orms of ascessmer t that 1ap bo:h mat'1 or.d

l.;l"l)'J(l9e cor.cects. For exa.-.ple. "'e ccuc IIJ•-e

.t~eems c•cate multiple-cr.oice cloze ser tences :nat

,,,•oct bee, e.er,'d.Jy ard rr<:lir-scec.fc rr.?...r.rss cl

m r hemotical •:oCllbul<:ry

Target words: plus double equals negative

' F/16 --- six _ ___ e;c. Yl.

2 On the _ _ _ side. my share _ _ _

3. 011/!e ___ sx!e. /':is shale,. _ _ _ t>V'e.

4 Nurr.J:Jers less Chon zero are cailccJ _ _ _ _

,..,.,berS.

Conclusion

~'<~tretm•<:s . .,'ill a~ relovaroca to studerts and b&

leorned much moro ettect1vety v;hen stvdems can reli1te

Ire ccn:er~ :o :heor prior ex~ienoe and currert

Interests. In uddihcn to acwuting stu<lents' pttcr

knowledge and bu.Jd ng baokgrOLnd knowledge, we

mJy need tO mcd1!y o~r lr.stru<::•on .n speclf.c: wa-1 s to

m<tke the content access•blo> to ESL students who aro

Ct1ll1n the process of catch,rg up to '1dl':e·S!l<'akers •n

lCademic Cnglish IDr.guage proficiency. T'1is catch-up

process''"" ~p.c:ally tal<e a; ·east tr~e years, J:<l~

beelluse students are ClltCI"irg up 10 a mow1g target­

n.Jt~W·Spcakers ol English :VO oo: SIOl'ldng s•l wa~.ng

lor ESL students 10 bridge tho gap. ·r~us, ovm ESL

st.,de<lls v.ho are rPIJtNely B..ent .n Enif.sh may raqt..lre

~pccrfic S~I'PCrt in uccess'11g rrat~.CmJ:•cal concepts

Mel prob'ems expressed tnrough E~>Sh. r,...,_e

cupports £110uld locus not only on mt\1<1ng lhO

m..tnemahcs cortent comr>< !he'!Slbe to sn.-~onts bl.t

also on oxter d,ng l1'8ir awore11ess of how thEI :anguaoo

cl rnat,..;m(lfics 'NO<ks. In ths w~rf, • tuden:s Cif' de~<etop

ir" .£191'"tz at:o1..t ~1cedcm.c lanou: 1ge tr.at ., .. ill bear tru1t in

o:hPr ccn:c'1' ... -ecs :« 9 • o;w.oq ccrrprer.eruor n

lanq~., age art:;, \10CJbu ary '::lLui<.J1r g ir socic:J studies). A

goa; c;ltoo orccsss cf ex:enc "il st.Jd<"nts' COI'I'mard of

acJdtJmic lanouage 's to arable :hem to lake owrershrp

cl t'lO ICI!lQt.aQe cl t'"4 curlC\Ji.tll arc ure r. fe< au:hentic

purposes Thus. tney wilr t:er er· trcm opport~r 1 es to

carry <X.t croj<lC'.3 ana explaJI'\ v.t'at tney d d bet~ c•aUy

and IM wntten tcr-1. A~ the a~c,cnce t:eccrres more

diStant (e.g .. in tl'e case c f a more rcrrral 'Mitten re~rt)

str •dEnts nre r~<;uired to use rrcre at:struct, a:<pt(;,t, and

preo~ Jarguoga :o cc,.,muOI<:llte tne r rreanir9. 'h'1en

we 1ntegrate these ae1111c uses cl Jango.rage wnh the

rrathematics curr;e~l~m sJUdcrts cer eH t:oth w1th

respect to Ma!hernat~<:s ar:d langlrage Elbtlrt~es.

References

ColhOI, v. P. ond ThorMA w. P. t1999) ~'al<lng v S schools

ellae'Nit tOf Eng1Gh 'anguage ''""""", PGrt • TESIJI. •\l.2~Bf<, 9.4 iAugusVS.p·cnte~. pp 1 3rd 5

C..tnMins (200 1 ~. •Vegot.ating :de"t.''.'es: E'oUCD!:or; tot sny;cwermtw o11 e QO<t'I"S<) .l<XI<l'Y. 2nc ec:lircn lOS Mgoles: Cah'ornia Associu~lcn for Srll.,gut)l Edveaton.

Claz·P.ieo. L end V;l!ed.l<. Z. i200ZI T1>e ="''"'•'· /lirglJV!]e, artd ~.c <te•JetJ()If"'Cf;t ,.~,..,- A CGfflcle~e

K- 12 retctenc., ;;u:c:e 2r.c edit:on. Boston. Allyn & Onccn

Gitoons, P. (19!11; L<.lllf""'fl:C "'"".,a SOt:«<!~ r-..ewtcwn, ~l!)trDira . Fr.m,,lt')' Engll"lh Teaching AssoCIJl<>r.

VP'f"' l.. (200~' Bamors 1c mcar.-.gtul "'" IWIICO for Cog"h ~e.:~rr,,-rs. The&> rr.ro Pract<e, 34:7.), op. 228-2:36.

~ - -.r• 't , l , i:·-.· .. ,\ 1: 12' . ,' 'I ' ' • '

Cp8~0V]236 "1/\J'ri =cP')Illjtlt P~·"•:t'l 01(':tiiiCI'"', Inc. • • "' .... ,_ ,... ..... ........ .. e.~ .... " "" . ...... l'r '•' •1 ~._..., ,,. t'lt Amerc n