Page 1 ENGLISH OF MATHEMATICAL FUNCTIONS AND

ENGLISH OF MATⅡ EMATICAL FUNCT10NS AND FORMULAE: TⅡ ROUGH

A COWUNICATIVE METHODOLOGY NIlichael C.Faudree

Abstract jκ g網 洸ι″ 関 ″cs Jお お焼ι ιttοdοJθ ′rasι κ″グλ ′9″ J`α jκ g `グ `″ “ικ告 “ jッ “ Jcs'κ ヵ らιεο″協 んjε α′ ιεttssハ θο`ッ.Cο κgκ ι″ ″jヵ わKrasλ α力 らι α′ `Eん “ “ “ “ “ jttυ jOκ ″″どわ ″ 滋の α″ S`グ ル ′ κO′ グ J+f,bα Sた ιJι ικな Jれ 滅お 滋ιttο JJθ ,α ″ “漁 r “ jο ツιttι ttθ Jε α ι9ρ ι ′ 4s.AJttθ gん sι げ εο滋 `ヵ れjε α′ リ 協0″ εθ″ψJ`χ αttι ttα ′ ル “ “ “ “ ““ Jた ′ 滋‐ θ ttz“ ん Jε α ′ Jθ れ s ttzソ ι b`ι sι ご Sθ ι ′ J“ ι ι α s“ b」 iι ε なs“ 6カ α s bα s'ε ε ,謬 わ /arク 。

Pα ″ JJ`′

ら ヵ

ゎ J`α

gJお あげ

“ε “s`ん ″“ ο 協θ.劉り “″α θ“′″‐ λα s bι ルss ε 乃 ″ ごjκ ″溜り sり・ε `κ `9″ “ “ jん 'S′ れ,ε ルrttα ガθ κ,α んどg演9η り ″οrた Oκ た感cr j″ ′ ち ε ο ″脇 κJε α′ι た7`ε ″θ `″ `ι “ `4sjbJι “ ιグル 協 ag″ ″′ げ 6f 協 ι′ λθ“ グθJθ gッ 147α S i“ノ ιttι れた4グα″ ″αs εOJJι ε′ 滋ιεJα ss“ θ

jcS ttα ′ ι λ

Eκ gJJsλ

jッ

“θriん gα 濯 J9′ ακι sι ι gJκ ι

jθ αttι ″協′ s stttcκ な αtt sttθ ″″ ご滋α′洸り ″ιた αbJι ゎ ″εαJJ “ “ jィ Jθ κ .F“ れ ′ Jα J α ttθ ″ br“ α ′ rrL`′ ″ s`α κλ ιιJι グ ′ θごι′ ιrajκ 乃 Js as′ bs″ げ `√ “ j“ jι jε jソ 'S“ ε ヵαJ″ ε Sι α んα ′ θ ごε α んα 力 ち′κ′ε ん ε Cθ 滋 ソ `協 `疵 `α “ “ ““ “

Introduction

Although thc communicativc appЮ ach has bccn uscd for quitc somc tilnc with othcr SuttCCtS Such as basic communications(Terrcll,1983;Krashen and Terrcll,1983;Ch五 and]Bassano,1981;Michcncr and Muschlitz,1979;and Winn―

stison

BcH C)lscn,1977;and many latcr

works including IIcgclscn,Brown,and Mandcvillc,1999),uSe pcrtaining to scicncc rclatcd SutteCtS Such as mathematics has bccn icss common.Thc goal ofthis papcris to dcsc五

bc how

a communicative approach augmcnts leaming English of rnathematical functions and follllulac

in the ESP(English for Specific Purposcs)clasSr00m. Thcse functions include,but are not rcst五 ctcd to,intcgration,dc五 vatives,differcntial cquati6ns,Inatriccs,loganthms,linlits,facto―

rials,and basic addition,subtraction,division,and inultiplication. FЮ

In thc standpoint oflcarn―

ing thc English of FnathCmatics,thc infomation prcscntcd hcrcin attcmpts to b五

ng to light how

communicativc clcmcnts such as comprchcnsible input(Krashcn,1980,1981,1982a;Ochs, 1982,Harkncss,1971),communicative interac■ on(Long,1981;Allwright,1980),clariication

(Long,1981;C.Brown,1985),and gЮ upwork(Long and Portcr9 1985;Klippel,1992)bcncfit L21eamers. The purpose of the rationale is four¨ fold,emphasizing the interactivc spcaking, listcning,rcading,and writing domains. In thc past,most FnethOds that included English of mathematics were geared for students to work alone and thc focus was p五 manly on preparing thc studcntto bc ablc to rcad and mcmo五

ze thc cxprcsslons through d五 11-typc excrcises. In linc

I gratefully acknowicdge thc studcnts from the mathematics and enginceHng departments ofToktt University for participating in this study

57

with the more basic communicative suttectS,this paper describes a natural approach(Terren, 1983;Krashen and Terrell,1983)to leaming the English of this essential sutteCt used in the scicnces.

The Role of"Comprehensible lnput"for Mathematics English NIIost bcginning to intermediatc lcvel Japanesc university studcnts who study the sci―

cnces arc not familiar with thc English vocabulary of inathematical functions and folll.ulac. and physics.Therc―

MttOriCldS ofstudy includecnginecHng,medicinc,mathcmatics,chelms"ら

forc,like any othcr suttect,prO宙 ding comprehensiblc inputis neccssary p五 orto implemening intcractive approachcs,in this casc,to use the language of rnathematics.」

For teaching methods,

studies have been done by Kiashcn(1982b)whiCh COmpare rnethods that present comprehen‐ sible input with those that insist on perfect carly production and focus on follll. ThuS,conllnu―

nicative methods such as the Natural Approach,Suggestopedia,and the Total Physical Response, were found to be cffective in enhancing communicative skills. Through interaction,other lan‐ guagc skills were acquired. Other inethods such as audiolingual,Silcnt Way,audiovisual,or

Community Language Lcaming have becn viewed to emphasize carly production and focused on grammatical follll before the language is actually used for communication. For comprehcnsible input,slow and clcar speech is used by the teachcr resembling a stylc wcll― known

as"caretaker specch",and also known as"forcigner talk",or"motherese"(Glcitman,

Newport,and Glcitman,1984:Snow and FcrgusOn,1977).ThiS type of speech stylc is typically used for Ll childrcn and]し 2 1eamcrs. Krashen'sヽ lonitor Theory of L2 acquisition takes this

notion of"caretakcr speech"into account stating thatitis very important for the communicative classЮ om,Ll,and L2 acquisition as a whole.This stylc is particularly important for sutteCtS such as inathematics,、 vhich involve new and specialized vocabulary. ■ erefore,in the ESP and other language leaming classrooms,the goal of the teacher is for the new infollllation presented

to be comprehensible so it can become what Krashen refers to asintake in the L2 1eamers.This intakc helps assistthem in using the vocabulary and expressions in subsequent comlnunicative activitics and also outsidc the classroom.

Thc ncw vocabulary and exprcssions wcrc introduced in thc classroom to the studcnts by practicing thc corrcct pronunciations with the teacher and thcn with thcir fellow students. Ac‐

cording to Krashcn(1982b),and Others(Long,1981;C.Brown,1985)speaking with othcrs hclps reinforce the comprehensible input efficicntly.Inf0111lation is presented and reinforced by the studcnts in a step― wisc mannet from the more basic,to thc more complex. In other words,

once the leamer has lnastercd the English of expressions at one level,hc/she is ready to advance

to the ncxt level,or as(ashen calls it,the"j+f" level.

58

j+f for]燿 athematics English:Step‐ Wise Comprehensible lnput Parallel to that of lcarning mathematics,itis wcll documented that clemcnts oflanguages are also acquired in a ccrtain ordcr9 thc lnore basic arc acquircd bcforc others: This holds truc

for many asp∝ ts oflanguage,for example phonology(Eckman,1977,1991;Mttorand Faudrec, 1996;Faudree,1999,in press). COngruent is

Кttashenis(1980,1981,and 1982a)Input Hypoth‐

esis,which states that L2 1carnersi progression from thc present stage,called'11"to thc next stage,or"j+f",is inastcrcd by the leamcr being able to comprehend the input at that level. ■■口 ■■■

hereforc itis necessary for the lL2 1eamer to first have an understanding ofthe basic vocabulary and functions before buil(Ⅱ ng up to the ncxtlcvel. In this way,learning languagcs,which en‐

compasscs leaming the English of inathcmatics,is actually sinlilar_lo leaming mathcmatics itseli

lt follows that the more basic lunctions and fo.11lulac are presented flrst,beginning with basic addition,subtraction,division,Inultiplication,and also powers,roots,and scientiflc nota‐

tion. Examplcs such as the fomowing are presented to the studcnts:

15+50=65

"flfteen plus flfty equals sixty― five.",

y=χ 4

''y cquals x to thc fourth powen", and

y=γ χ

:'y cqualsthc nth

Юot of x."

Vocabulary used for addition,subtraction,division,Inultiplication,powcr9 and root functions

such asshown ttК ,田3 the building blocks used for more complcx opcrations.FЮ m these we can de五 vc the English for fomulac contttning intcgrals,dc五 algё braic

vatives,limits,fκ toHJs,and largcr

expressions.3pical CXamples given to thc students include:

y=∫

(χ

y=ltt

3+22_3)魏 _1 χ

"y is cqual to the integral of x‐

cubed

plustwo x‐ squarCd ininus thrcc,dx.",

and

''y is equal to thc lilnit as x goes to inflnity of x―

squared plus onc all divided by x nllnus one."

Again,as with the basic functiOns and formulac,students wcrc instructed to articulate morc complex follllulac such as the above widl thё

tcachcr and then with their partncrs as thcy were

introduced. As inentioned carliet the purpose is forincreased comprehensible input.《

〕f course

comprehensible input is needcd before it is used for communicative interactions,which rcin― force the knowledge base ofthe L2.

59

Conllnunicative lnteractions Oncc a sufficient amount ofinput is reccivcd and undcrstood,communicativc intcrac― tion,or using thc languagc to conllnunicatc can bc achicvcd. This,of coursc,is the purposc of languagc.For thc classroom,itis wcll― known(Richard― Amato,1988)that Va五 〇us communica― tivc activities elicit the nccdcd communicative interactions. Popular incthods long used for other thcmcs that can be adapted to thc tcaching of rnathematics English include the following:

pairwork(Hegelscn,Brown,and Mandeville,1999;Faudree,1997;among many others), groupwork(Long and Portcr,1985;Klippel,199の

,inding missing infollllation(HCgelsen,

BЮ wn,and Mandcvillc,1999),bOard‐ advancing gamcs(Gershan and Marcs,1995),jazZ Chants (Grahanl,1978),and CVen songs(Pallncr,1971). In addition,othcr FnCthOds can be adaptcd. For cxamplc,cach student writcs an cquation on a card. Thc teachcr collccts thc cards and redist五 butes. Studcnts are then instructcd to rcad thc cquation on the card they are carrying to

their partncr whilc thc partner w五

tcs it down withoutlooking. Using thc above rnethods,thc L2

1camcrs usc the language to intcract with others and gettheir rncssagc across.

Communicativc tcaching mcthods focus on just that,communication,or"getting thc mcssage across".(]Ctting thc rnessage acЮ ss is actuaHy thc focus of attention,or lnotivatlon to progrcss(】 Gashen,1982b).HcnCe,in this stuoy L2 1earnerゞ

focus on the contcnt,or mathemat―

ics,and through interactions acquirc thc L2. Undcrstanding spccch clcmcnts■

ot yct lcarncd

comes fronl cominunicating known follllS and attempting unknown or partiany knOwn folllls.

During thcsc instanccs,L21camcrs arc oftcn correctcd during conversation. This,Krashen bclicvcs,is the way languagcs arc acquircd. Thc rcsult ofthe corrections are enhanced commu― nication ability and a wider L2 knowlcdgc basc. Thc corrcction pЮ cess includes the notion of clanfication,where one spcakcr asks qucstions to undcrstand the lneaning of another speakeris

message.

ClariFlcation

IInportant in all communication is clarification(Long,1981;C.Brown,1985;and andevillc,1999).ThCrc arc many classroom methods thatinvolvc Hcgclscn,BЮ wn,and Ⅳ【 clarification including"gucssing gamcs"(Richard― Amato,1988).ThcSC act市itics usually in― volve pairs working togethet wherc onc pcrson has thc answcr and does not divulge it,whilc thc

partncr has to guess whatitis. An cxample is,"I'm thinking of an anilnal,can you gucss whatit is?:: Ъ cn thc partncr asks yes/■ o questions such as,"Does it swiln?'1,or"Does it have stnpes ?“

. From these,the first pcrson answcrs''Yes."or"No."until thc first pcrson has enough infor―

mation to figure out the corrcct answcn Some vcrsions involving nlinling arc less vcrbal. In all cases,if the point of a rncssage is not sufficiently understood,the"listcncr"Inust

ask qucstions so shc/hc can understand. Thc qucstions thcmselves have to be asked in a way

60

that is understandable. The two pcople then communicate back and forth until both reach an understanding. This is especially true when using a foreign language such as the L2,L3,ctc.,

andeven morc me when spcaking about specialized sutteCtS Such as mathcmatics.Evcn people speaking in their Ll oien nced to clarify their messagcs by discussing back and forth so they can understand eachothe■ 1llis is universal for all pcoples and all languages of the wond. Clariicatio,for understanding encompasses all su切

∝tS

from work and day‐

to― day

liv―

ing to rclationships and hobbies.Like in any other suttect,languagc can somedmes be ambigu― ous,and hencc cla五 flcation of thc rnessage is needed. Given this,a clarlflcation cxcrcise seemed

to bc a benefiting activity for the students. 1lls activity did not emphasize absolutc corect

pЮ nunciation or grammatical follll,but rathcr9 successful cxchange of infollllatiOn and gctting Jhe message across.

一 一 市 い 嗜 嗜

The activity was as follows.Students wcrc given different fo..1.ulac that could be de― sc五 bed

with the samc words. One setincluded:

y= y=

,=辱 1.ulac could be descnbed wiJh the same English sentence,1.e.: Ъ e above three foュ ニ

'ly is cqualto the squarc

Юot of X times cosine x(五 vided by log x", whilc

a fourth equation which was also silnilar included:

y=端 S√ )gχ

::y is equalto x tilnes the cosinc ofthe squarc root of x all divided by log x."

Students were explalned the above phenomena and then told to flnd a parenen For each pair of students,6ne student was instructed to look atthe board while the other studcnt faced the back ofthe classЮ om. h teacherthen wrote,for example,one ofthe above ambiguous follllulas on

the board. The student facing thc board explained the fo...lula to thc othcr student only by speaking= As the other student wrote he/she asked clarification questions to thc first student about the follllula. A list of clariflcation questions was givcn to the students,which included:

61

"Where is the squarc Юot?",

'lAll square root?",

1:Bottonl all square root?",

"Top all square root?“

''Only the x?::,

■ e students interacted and clanfled until the student not looking understood the correctinfor‐ mation. In addition to the above,other silnilar ambiguous equations wcre given involving othcr

functions such as roots,powcrs,logarithms,ctc. One example including the power function was:

y=

(bCOSχ )3 χ_1

For these other cases,thc clariflcation questions were not wHttcn down for the students. They extrapolated initial given questions to fit subsequcnt equations given. For example,for the abovc fomula the clariication questions asked by the smdents include止 "All cubed 7:;"Top江 1 to the third powcr?11;'lAll lninus l?'1;and"x■

linus l all on bottoln?'1. In my class,there was

one student who was hcanng impaircd. For the above scenario,this studcnt and his partner interacted purely by w五 ting. Aftcr the acti宙 ty was repeated a number of times the partners switched

Юles.

Teamwork Another classroom acdvity that can be beneicial and yct fun forthe studcntsis groupwork,

orteamwork(RiChard― Amato,1988).In thiS Case each group,or team was instructed to come up with an English nalnc foritseli Each tcam had flve members,with each inember designated

as"Person A,B,C,D,and E". The group organization is as follows. A represcntative fЮ

nl each teanl,beginning with

all pcople dcsignated"Person A"wcre instructed to comc up to the fЮ

nt of the roo■ l and de‐

sc五 be a follllula w五 tten on the board by the teache■ When flnished,all::Person All students sit

back down. Next,only students dcsignated Person B come up to thc front ofthc roo■

l and are

given a different follllula. Next are Persons C,E),and E,before the cycle begins again startlng

with Pcrson A,B,C,and so o■

.

For each group of students atthe at the front of the roo■

l attempting to desc五 be a for‐

mula,the teacher wagers the points An electЮ niC game,called"Hayao Shi Pin Pon Bu"made

by Yonczawa(Party Room 21)was uSed.In this galne,cach student holds a button.After thc teacher says"Go!"the irst studentto push herrhis button gcts to attemptto answer the quesdon. I needed,students are given hints by the tcacher or by their tealnmatcs. If the student's answer is underst∞ d by the teacher(aNS)to be COrrect,points are awarded.For ambiguous fo二

62

.1.ulac,

thc tcacher rnay ask qucsdons such as"All square root?",or"Only the'xt has a square

Юot?".

Thc teacher thcn givcs that studentis team thc approp五 ate amount of points according to thc difficulty of thc formula presentcd;partially corcct answers are awardcd with partial points. Every student gets a chance to participate and thc team wiJh thc highcst amount of points wins. Students appeared to e可 Oy this method of recalling whatthey have learned through teamwork.

STUDY ofINFORDIATION INTAKE One wcck aftcr the inathcmatics English classroom lcsson,an anonymous survey was taken to detellllinc how much of the mathematics infoll.lation was"takcn in"by thc students

from thc communicattve approach desc五 bed in this papc■

Partidpants and Tasks Fifty‐ seven

Japanesc NS studcnts fЮ m Tokai University,Shonan Campus in Hiratsuka,

Japan participatcd in this study.Most were freshmano Studcnぱ

ields of study were:1)math―

ematics infollllatiOn(22 participants),and 2)enginCe五 ng of acrospace and astronautics(35 participants). MOSt WOuld be classified as beginning levcl in conversational English,although thcrc were somc cxceptions at highcr levels. In addition,a control group of four engincc五 ng students frolln another class who did not participate in the mathematics English lesson were included. Prior to this study all ofthe groups had virtually no exposurc to thc English of rnath‐ cmatical functions and follllulac. Although all students studied English in high school,chanccs

to communicate with English NSs in thcir homeland were rarc. For the proccdure,thc two classcs compriSed of the 57 studcnts first participated in a mathematics English lcsson,which lasted two class pcriods,or thrcc hours. The mathematics lesson was in the communicativc format as desc五 bcd in this paper and focused on the English of rnathematical functions and follllulac.《 Dne weck latet students were instructed to complete a follll to detclllline thc amount ofinfollllation they could rccall.

Students wcrc instructcd not

to look at supplcmcntary matc五 als or books while completing thc follll and WCrc givcn appЮ

対―

mately 20-25 minutcs to finish. The fornl had four icvels of recall cxcrcises,which ranged

from basic mathcmatics English usc to thc more complcx(sce Appendix for more dctails). Level l contained thc four basic functions:addition,subtraction,division,and multiplication. Lcvc1 2 contained powcrs and square roots. Lcvc1 3 contained intcgrals,while lcvc1 4 contained

a limit,which rcquired more English.There werc ninc questions in total,i.e.:[lcVCl l]X4+

[lcvc1 2]x2+[lcve1 3]x2+[lcvC1 4]xl=9.

63

Dependent VaHable ■ Ю dependent variable included in this study was the percent of students thtt produced a correct English desc五 ption ofeach fo.1..ula.Hcrc"correcr rcfers to an English NS(in thiS Casc,

the author)being ablc tO understand whatthe student wЮ te.It is noted here that pcrfect spell― ing,gralnmatical,or syntactical folllli was not needed for the answer to be tagged as corect. Ъ e importance here is that a NS can understand the follllula with the words given by the NNS, i.e.that the message gets across. For example,consider:

y=χ 4+1. Of course,"y equals x to the fourth power plus one."would bc tagged as correct,but also counted as corrcct were l:y equalls x four power pruss one.'1,or"y iqal x to fours power all added one.:: If spoken,a native speaker can understand these statements.

An incorrccttoken would be“ y equals x plus one to thc fourth power". Although it is granllnatically correct,it does not desc五 be the Hght foll..ula,1.e.,it descЁ bes"ッ =(χ +1)4".

Likewise,:'y iqual x pruss one four power'l would be tagged as incorect because it desc五 bes the fo..1.ula b=o+1)411 and also desc五 bcs the follllulaり =χ +14“

.

A third case would be l:y fours x powerto plus one". lhis would be tagged as incorrect since it is incomprchensible.

In summary9 an answer that was grammatically incorrect but could be understood by the NS as thc correct explanation was deemed better than an answer that was gralnmatically correct, but dcsc五 bed the wrong fo111lula.

Independent VaHables ¶he independent variables in this study wcre the ninc ques■ ons themselves and the math‐ ematics English level as desc五 bed in the Pa」 饉dpants and Tasks section. ■he nine questions

compHsing four levels were analyzed separately.

64

Results and Discussion Figure l shows the rcsults for the 57 frcshman students for all levels of difficulty,i.c.,

Lcvcls l,2,3,and 4. In addition,questions l through 9 wcre groupcd separatcly,whcre lA' rcprcsents addition;'Si rcprcscnts subtraction;'I)',division;'NI11,inultipHcation;'P',powcr;'R',

Юot;T,integral;and:L'reprcsents a mathcmatical limit.

Percent of students that produced a corrcctEnglish description of each

Figure l

fomula according to level and function:57 students

ヽ １ 日 ● 雪己 ■ 劇盤 劇Ｌ

100 00

"

96 “

m 40 20 0

S

D

P

M

￨

R

ヨ

Leve1 3

Leve1 2

Leveロ ロ

=57

QueSt10n D■ 1lcuヨ tv Leveヨ “

The results in Figure l show that close to 100%of thc students for the basic"Levcl l" produccd corrcct answcrs for the first three qucstions. Addition was at 98%correct,subtraction

was at 96%,and division was at 96%。

Multiplication had thc lowcst percent corcct of Lcvel l

at 84%. It followcd thatthe power and root functions at Levc1 2 had 88%and 79%of thc studcnts produce correct answers,respectivcly,whilc integrals,at Leve1 3 wcrc at 72%and 75%. For the inost dilficult qucstion at Lcve1 4,which was thc dcsc五 ption of a lnathcmatical linlit,539♭ of the studcnts had corrcct answers. This was considered quite remarkable by the

Japanesc NSs despitc the fact thatthey wcrc at the beginning levclin English and this was thc firsttimc thcy wcrc cxposcd to lnathematics English. Thc factthatthe data collection waS takcn onc weck aftcr thc lcsson showcd thatthc studcnts wcre ablc to rctain the prcscntcd English of

mathcmatics aftcr somc timё had elapscd.

Control Group lBiSeline) Figure 2 shows thc data for thc control group of studcnts who did not attcnd thc lesson. As for thc questions reprcsentcd in Figure l,'A'rcprescnts addition;'S'represents subtraction; 'Dl,divisiOn;'NII',multiplication;:P',powcr;'R',■ oot;

65

Figure 2

Percent of students that produced a correct English descHption of each

fomlula accordng to level and fu"lon:4 students(COntr01 Group)

４ロ

ｏＬ

∞ ” ∞“ ” 。

颯 〓 Ｗ Ｅ ● ●   ︹墟 尋 僣Ｆ ● 堂

100

100

S

D

LeVell

0 P

M

0 R

Love1 2

Qu●●t10n DEffacuttv Level lI',intcgral;and'Iゴ

0 0 ￨ ￨ L● ve1

L Love1 4

3

‖ =4

reprcsents lilnit. Notablc is that rnost of the questions,i.c.,4 thЮ

ugh 9,had

O`ろ acCuracy. Ъ is indicates inost ofthc Japanesc enginee五 ng students who participatcd in this

study werc not prcviously cxposed to English ofthe mathematical functions and foII..lulac.Thcre― forc,the control gЮ up was considcrcd as a baseline. This baseline lllustratcs thc fact that the 57 students who participated in thc lesson in Figure l werc able to rccall a substantial amount of infollllation.

Note also thatin thc control group in Figurc 2,100%of the students produced correct answers for the addition and subtraction functions. It appears that rnost Japancse enginecHng

students know thc English words"plus"and"minus",i.c.;"three plus seven equals tcnl. Ъ e accuracy for the division follllulac was at 75%,whilc all othcrs wcrc at O%. In te.1.ls ofthC data in Figurc 2,this low pcrcent corrcct by the non―

attending studcnts is

cxpected,howcvcr9 it Could also bc explained by the literature,1.c.,that non―

acquisition will

occur without comprehensiblc input.A great support for the lnput Hypothcsis of Krashen was

made by Long(1981,1983),who fOund thatlanguagc learners who werc cxpos,d only to TV (SnOW9 Van Ecdcn,and Muysken,1981),radiO,mo宙 cs,or ncwspapet i.e。 ,only NS‐ NS

intcrac―

tions did not acquire the language. They only acquircd a small amount of greetings,sayings, etc.In additton,Sachs,Bard,and Johnson(1981)found that hearlng children of dcafparcnts did not progrcss in spoken language ability by watching lヽ こ Thcy progressed when thcy began talking with othcr children and tcachcrs. Thc quality ofinput,i.e.communicativc interaction in these cascs and in this study was inostimportant facto■

Therefore,the 57 studcnts who wcre

exposcd to the comprchensiblc input and used it to intcract recallcd thc English for the functions and follllulac.

66

Learning Process:Mathematics English and Mathematics ltser From this study,the lcaming proccss Of mathematics English appcared to have differ― ences and also silnllarities with that of leaming inathematics itself.

】 〕iffcrences were cvident

from thc standpoint of the solutions obtaincd.In mathematics,there is typically one ottcctiVC,

logical solution.Mathematics English,howcvct is sutteCiVe in nature,where thcre are many correct ways ofcxpression with the goal of getung the messagc across. In addition,the proccss of lcaming mathematics English appcarcd to havc sil■

ilaritics

with that of rnathcmatics itsclf. 13oth can bc describcd fЮ nl thc viewpoint of【 ashen's notion

of"i+1"stcpwisc comprehcnsible input(198∝

1981,and 1982a)fol10WCd by practical use.

This was shown through the co■ 1lnunicative fhethod prcsented herc,where basic elements such as addition,Subtraction,division,Inultiplication,roots,and powcrs werc lcarned beforc they werc utilizcd for inore complex opcrations such as integrals,dcrivativcs,11lnits,diffcrcntial equations,and larger algebraic expressions.

Conclusion(s) For the classroom,this paper presents a communicative approach to teaching the English of lnathematical functions and follllulac. In addition,the rcsults of this study show that thc 57

Japanese enginec五 ng and mathematics students werc able to recan a substantial amount ofin― follllation fЮ

m thc communicative methods praCtiCed here.Thc fact that the students did not

havc pnor cxposure to mathematics English atthe dmc ofthis study was re■ ccted by the control group which was used as a bascline. As expected,in the group of 57 students,accuracy ratcs followcd order of difficulty for all the qucstions and levcls. It is noted that further research is needed for the results of thisふ tudy to bc compared with those utilizing othcr teaching inethods,

such as audiolingual or Community Language Lcarning. In addition,further rcsearch is necded to dctellllinc if the communicative incthod presented in this paper can actually increasc lnath proficiency.

67

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θttss‐

Appendix:QueS■ Onnaire of Mathelnadcs English Recall Dont write your name on this pape■

Please don'tlook at any matenals.

Love1 1

3+7=10 α+b=′ α/χ

=7

3×

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Lovo:2 4+1

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3+χ 2_1)と

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Love1 4

71