A COMPARATIVE STUDY OF DEDUCTIVE PROOFS IN GEOMETRY EDUCATION BETWEEN THE U.S . AND THE PEOPLE'S REPUBLIC OF CHINA
By Zhanghai Ruan
Athesissubmitted to the School of Gr a d uateSt udies in partial fulfilment of the requirements forthedegree of Master of Educ ation
Faculty of Educ at:' vn
MemorialUniversity of Newfoundl and May 1996
St. John ' s Newfoundland
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Abstract
During the last twodecades , there ha s bee n a rea s ses sme nt of deduct ivepro.,)feduca tiondue to thest ude nts ' dif ficul tieswith deductive pr oo fwei ti ng . Questi on ssuch as "what are wedo ing? ..
and "why ace we doi ng it 1" ha ve be en thought exte nsively by ge o metrycducatoe s and tea ch ers .
The puepose of this study is to he lp ge o met ry ed uca t o rs and eeecn ers retn Ink ded uc t iv eproof educ at i on. By compar ingst udents' van Hi e lele ve l s,curricula,exami na tions,supplementa ry materi al s, and method of instruct i on in the U.S. andthe Peopl e'sRepublic of China , this study will ena ble educ ato rs and teache rs in both co untri e sto re cogn iz e past failu res inci.e ducti ve pr oo f ed uc ation inthei r own country.
The resul ts of the study showtha t the curr icula of deducti ve proo f s are notwell-accepte d in either theU. S. orthe People's RepUblic of Chi na . The ina de qu acies of te xtbooks, and the teac hers' he avyrelia nce on thelllha ve caus ed lIlanypr o blems inthe U. S. Entr ance exa mi na t i ons and supp l ementary m.at er i a ls impos e he avydemand s onst ude nts 'geome t ric think i ng and cau s e a failu re in the deductiveproof ed ucat ion in China.
More res e a r c h need s to be done onthe factors influencing the st udents' achieve mentin dedu ctive pr oo fedu cat i on. The resultant change s made by res earch will improv e the qual ity of deduct ive proof ed uc a tio n.
~cJl:novle 4geme nt 9
The autho r wou ld like to expresssi nce regra t i tudeto Dr. Lione l Perei r a-Me nd oz aforhi s helpand guidan ce onhis qr-ad u ate stUdies, and withth isresearch.
Aswel l , th eres ea r c he r would lik e toexp r es s app r eciation to Miss Mi che leCook forhered i t ingthe languag e , and to othe rs wh o hel p e dhimconduc t the re se arc h.
Table of Contents
Page Abstract.••••••• . •• • •.••.. •• •• •• • .•••.••••• • •• •• •••• ••• •••• •• ••••ii Acknowledgements.. ... . . .. •. •. ... •...• .••••. . .••••. ••. •..•.. .. . iii CHAPTER
OVER-VIEWOF TilE STUDy •• ••••••• • • •• ••• • ••• ••• ••••••• •••••1 Introductio n•••••••••.•••••.••.•.••••••• • • • ••.• • • •• • •.• • 1 Sta te ment of thePr ob l e m••• •.••••• .•• •••. .•.•..••••• • • ••3 purpcses of th e Study ..•• ••• • • • •• •••• •• •. .•.. .. .•• •5 Significanceofthe Study •.•• ••• • • •••• ••.. ..••••..•6 Limitationsof the Study••••.• •• •.• ••.•• • • •.•••..• • 7 II REVIEW OFTHELITERATURE••••••• • •••••••••••• ••• ••••••••• 8 Deductive Proo f Education•••• • ••••• ••• • .•• • ••. •••••• •.•• 8 Ded uct i v eProo f s •• ..•. ••. .•• •••• •••.•••••.•• •••.• •.8 Problemsin De duc t ive ProofEduc a tio n.• ••• •.•.•.• .11 DebateandDiscussionson
DeductiveProof Education... . . ..•.•• ••••••.•13 TheVa n Hi e l cMod el••.•• • •.•.• • ••••••. •••••••• ••••••••• 15 Th eVan Hiel eMod e l •.••••• •••.••••••••. . .•• . ••••••15 Researchon the Va n Hi el eMod e l•••••• • •• ••.• .• •.•• 17 Rel a t i o n sh i p s Between Hig h Sch o o l Students' Ach ievement inLe arni n g Deductive Proofsand Van Mi e l e Levelsof Geome tric Thi nk i ng •• ••• •• • •••• ••• .. ••••••••••••••.••20
iv
ZZI THEVANHZELELEVELS OFST UOENTS IN THE U.S.
ANDTHEPEOPLE' SREPUBLI C OFCH II~A•••• • • •• •••. . .•. •• 23 The araerLcanStudents' VanHi e l e Levels ••.••. . . . ...•.23 The Chinese Students'VanHiele Leve ls••• • . . .•• • ••. . . . •26 Purposes•.• ••..•.••••...••.••.••• .• .• .•.• •• .• • ....27 Method••... ... ...•. .. .•. .. . • ••• . • ...29 The Results•. ...•.. .. .••...•.. . ••. . . . • ...JO Discussion ••••.•••.•.• •••• • ••• ...•. ..•... .... ..JJ
rv i:lEDUC'l'IVEPROOF EDUCA TION IN THEU.8. AND THE PEOP LEIS REPUBLICOF CHINA•••••• •• •••••••••.••• . .••••• 36 DeductiveProof Educationinthe U.S•••.•. . . ..• . ••)6
Deductive Proof Education in the People's
Re publ i c of China.•• • •• •. • ..• • • • • • .•• • •.••• •.•..• • ...43 Curriculum•.•.• .• ••..., ••. .• . .... •... • •. ... .43 Examination•••.••• • .••...••. ••• . . .. ..• ....•...••..4"
Supplemen tarymaterials .•••• •• • • •••..• • •••• •• •••••49 Inst ru ct i on ••••••••••• ••••••••••••.•. .•.•.•. • .• •..51 V CONCLUSIONSANDRECOMMENDATI ONS •••• ••••••••• . .•••.••••••54
Conclusions•••.••••.• • •.• • ••••• •• • . ••..••• ..•• •••.• ..•.54 Recommendations forFurtherResearch .•...••... ... .... ..55 Bibliography•••••••••• ••••••••• • • •• • ••••.•••.•.•. . ..•..••...57 Ap pe nd i x - The VanHlele Geometry Test•••• • •.•• •..• ••...• ••. ...•. 63
CHAPTER I OverviewOf theStu~y
Introducti on
Before thedevelopmentof Greek culturetherewere no theorems or demon s t r a t i ons ,and deductive pro o f s (Szabo, 1972 : Siu, 19 9 3). The methodsof deductiveproofswere discoveredby theGreeks through the developme nt of ge omet r y from a pract ical science in t o a dedu c t i v e system. Thales (c 625- 547 B. C.), born in Mil e t u s , Greece, be c a me thefirst person to prove deductivelysome ge ome tr ic re l a t i o nshi p s, and was conside re d "the Fa t h er of Deductive Reasoni ng ". He was, however, still un a b l e to organize th e pro po s it i on s intoanydeduc t ive sys t em. Hi s student,Pythagoras of se noe (c 585- 501B.C.l, hon o r edbyth e Romansas"th eWi s e st and Br avest of the Greeks ", proved deductive ly the "f i r s t great theor e m" -- the Pythago rea n th eorem. The methods of deductive proofs wa s develo pedby Pla ta (429 - 348 B. C.), who hadstudied geometry with Pyth a goreans . His greatestcontributionto geometry wa s the discovery of the me t ho d of anal y s i s in co nstruct ing ded uc tive pr oofs. He Lns Lat. ed on ac curatedefi nitio n s , clearly stated as s u mpt i on s, and logi c al deduct ive proo f s . Eucl id (365 - 275 B. C.), a youn gGree kma thema tic ia nan da student at Plato' s Acade my, or g an i z e d all the various is ol ated geomet r i c discover ies an d deduc t i ons of earlier generati ons into one sin g lededuc ti v e sy st em (Lig ht ne r, 1991).
Since ancient Greece, th e methods of deduc tive pro o fs have been considered to be an essential characteristic of mathematics by Western Tho ugh t (Han na " Ja hn k e, 1993). Meanwhile, learning to wri tedeductive proofs ha s been an important object ive for students learning geometry. During the sixt ies and early seventies the Americ an "n e w \l'.ath" rc rcrmsre s u l t ed in a much greateremphasis on dedu ct ive pr oo f s inthe schoolmathematics curricu lum. Educators and mathema ticians (Smith&Hend e rson, 19 5 9;swain, 1963; Lester, 197 5 ) agr eed th a t the study ofde d u c tiv e proofsshouldente r the schoo lcurricul umat the earl iest possibletime that is consistent with ch ild renIs intel lectua l de v e lopment. Lester (197 5) sugge sted tha t "everyeffort be made tode t e rm i ne the most eppr-op r-Lat;e places at wh ich to introduce st ude nts to the various aspects of proof" (p.
14). Howe v e r, "ex a gg e r a t e d formalism , unsuccessful teaching exper iencesand a cri ticalpu b Lfc eventually le d toth e demi s eof th isreform, an dinturnto a criticalreassessment of mathc-mat ics educa tion " (Hanna&Jahnke, 19 9 3 , p. 42 2 ). Thisreassessmenthas in f l ue nced the at t i tu de andpra ct i c e of deductiveproo f education at theseconda ry le v e l. Today , the roleor meaning of deductive pr o o f s in school mat hematics is being great ly debated by the mat hema ticseducators and teachers in North Amer ica.
The met h ods of dedu ct i ve proofs we r e tran sl ate d into China by Mat teoRicci (1552-16 10 ), th e mostpromi nentJe s u i t missionary in Chi n aat thetime of theMing Dyn<:lsty, and Xu Guang-Qi (1562-16J3), a famous mathema tic i a n in ancien t Ch ina. Ricci said in his
journal, "n ot h i ng pleasedthe Chinese as much as the volume on the Elements of Euclid. Thisperhaps was due to the tact that peop l e esteem mathematics aehi g h l y as the Chinese" (Siu, 19 9 3 , p , 345)•
The style and contextof deductive proof education inthe People's Republicof China hav e significa ntdifferencesfromthose in North America. In China, Geometr y courses require stude nts to have abilities to see relationshipsamong se v e ra lgeometric figures, to make simple infe r ences , and to pr ov e geometric statements deductively. They require stude nts to possess a high degree of technica l knowledge in formal de d u c t i on , log i c a l thi nking, and logical expression. Mo s t geometrycontentinhi gh schoolsin the U.S. is studied in grades 8 and 9 in middle sch oo ls in China . Thereis a uni f i e d mat hematicscu rricu lum , cal l e dthe NewUnified series , to be usedthrough out the country. The te x t s are more theoretica l th an thos e inNort hAme r i c a . Not onlydo the amount and de p th of mathema tics inChi nese te x t s surpass those inNorth Ame r i c a n te xts, but studen ts in China do far more comp l ica t ed probl e ms than th o s e of the st ude n ts inNo r th America.
1l1At.§mAnt of thA Proh lem
Forthe la sttwo decad es the r e ha s been manydiscussions andgreat debati ea ondeducti veproofs. Some Ame r i c an educators (Shaug h nessy
& Burg e r , 19 85 ) sugges ted that th e st udents' int rod uc t ion to
ge ome t ry sho u l d be informa l withoutdeductivepr o ofs. Cons i de r i nq
that few st ud en ts achi ev e any compe ten c e in ded uctive proof wr i t i ng, and many ha ve ne g at i ve attitudes abou t it (Sank 1985;
Sh au gh nessy I".Burger1985 ) , it maybe hel pful toin trod uc e students in forma l l y to ge ome t r y.
Introduc i ng st ude n ts inf o rma l l y to geoml:!try does no t me,).n an abandonmentofdeducti ve proofs. Prevo st (1985) said that tod a y, ge ometri c co n t e n t is of te n emph a s i z ed on computationnl skills . This results inst u de nts' verywea kgr a s p ofgeome tr1cconc ept s at thegrade ten. Co x (1985) I".McDonald (19 8 9 ) found that stud e n ts enrolled in th e in f o rma l geomet r y co ursewere no t as cap ab le of prod ucing ac c u r ate st r uctures of geome t r ic conte nt as those enro ll e d in the form a lgeometry cours e. Althou g hthe students in the in f o r ma l geomet ry co urs e had above average 1Q's and al l successfu l l y compl etedthe course in plan ege omet r y , the level of understandingexhibited by thesest ude n ts was at be st supe rfic i a l .
Wemus t cont i nu e to tea ch our stud e nts howto wri te ded u c tive proofs and how to read them cri tica l l yinor dertohe lpthe mhave a completeunderstandingof the geo metry tha t they are stUdying (Rei sel, 198 2). Considering that there are man y pr o b l e ms in lear n i n g and teaching dedu ctive proofs , de d uc tiv e proo f s as a ma t h e ma t i c s contentarea needtobe reth ou gh t. We need to think extensivelyab ou t ou r teaching- --whatarewedoing ? andwh yare we doing i t? We als oneedto thinkaboutsuuden t;s ' learni ng sty les , their st a g e s of intellectualdev elopment, and learning environment.
'rc rethink deductive proof educat ion , we need t.osha r e mater ials and methods found to be effective Ll teaching and learning deductive proofs. An internationalcomparative study on deductive proof educa tioncan create a cha llengeto educators to rethink the educational practice within thei r own country. It encourages educators to re-e valuate curri cula, class e taee , teaching materials, and learning styles which are all crucialfactors in any educational process (Lamon, 1971;Chang, 1984) . Lee (1982) &Chang (1984) compared the hi g h e r education system of the pepubj Lc of China to th a t of the United state s, and saw th e va l u e s of the internat iona l exchange programs. This study compares deductive proof education between the U.S. and th e PeopleIs Republ ic of China. It will be an informative ch a ll enge to investigate, compare, and summarize the deductivepr o o f education of the two ccuntr-Len, It wil l he l p us re c og n i z e past failuresin deductive pr oo f education inbothcountriesand present a long-termgoal for deductive proofeducation .
Thisstudywi ll enable usto utilize thebe s t availabl eresou r c es fr om both cou ntries. It will provide recomme ndations and imp lic a t i on s forde d uct i v epr oo f education in both countries. It is designedto achieve the follow ingobjectives:
(1) to investigate, compare, and summarize deductive proo f educ a t i on at the seconda rylevel: and
(2) to enc ou ra g e furt her study. discussion, and po s s i b l e ref i ne ment of deductiveproofed uc at i on in bot h countries.
This st ud ywill provideans we r s toth e followingquestions : (1). Cana st ud e ntinan ur ba nsc h ool in Chinabe assigned " van Hi el elevel?
(2). What perc enta g e of studentsatthe beginni ng of the eigh t h gra de in urban schoolsinCh i na are prepared for learningdeduct ivo proo f s ?
(3). Which le velofinstruct ional lang uagecan be us ed toteach geometry at the eight hgrade in urbanschoolsin China?
(4). Are the r e any diffe re ncesbetween the van Hi e l e le v e l s of mental deve lop me nt ingeometry of qr ade eight stu dents in urban schoo ls in China and tho s e of grade te n stude nts in the United states ?
signit icaDceofthestudy
This stUdy wi l l makethe following significa nt contribut ions to the field of ded u ctivepr o o fedu c at i o n inbothcountries:
(1) tohe lp the understanding of the critica l issues of deductive proofedu cat i o nat th e secondary lev el in both countries;
(2) torecognize pas t fai lures in de d uct i v e pr o o f educat ion in bothcountri e s ; and
(4 ) topres e nta lon g- t e r m goalfo r deductive proof education.
Iim 1totiOD::lof t~
This study is limited to secondary geometry education in both countries. A.study related to Chinese students' van Hiele level wa s conductedinth~urban middle schools in China at the beginr.ing of the fall, 1994. It had the following limitations:
(1). Stude ntssampled were enrolled in gradeei g h t for the first time in the fall of 1994. Students repeating grade eightwe r e eliminated from the sample;
(2). the popUlation from which the sample was drawn consisted of all grade eight studen tsin the chinese urbanschools. An urban schoo l was selected giving a sample size of 95 students whose students' IDnumberswere even; and
(3). studentswere tested during the school day.
CHAPTE RII Revieworthe Lite ratur e
Thisch apterwill br ie fl y describ e how towri t e and r-eed deductive proofsin geome t r y. discussstud e nts ' difficult i e swithdeduct i v e proo fs , and examine the debate and dis c uss ion s onde d uct ive pr o of educa tio n. Anin t rod uct ion tothe vantuere model will be made.
The mod e l willbe used to exp l ainwhy manystu de nt s find lea t-ninq ded uct iveproo f s in ge ometry suc h a difficu l ttask. ce o toth e sho r tag e ofthe lite r a ture in Chi naat thepr e s onttime , thore vi ew will mainly rocueon the litera tureintheu.s.
Dedu etiy ePro of tduc;U1.Qn De d.u cti v eProo fs
Itseem s clearthat a ma t hema tica l proo f is "a careful sequ enc e of steps with each step foll owing logically rrce an assumed or pre v i o u slyprov ed sta tement and from pre vio us ste ps" (NeTM. 198 9 ; Pereir a-Me nd o z a ' Qui g ley, 1990. p , 9). Tha t is. a proof is an exe rciseinlogic (Ot te , 1994). This de fin i tion isocpecLalLytrue tor adeductive proofingeo metry.
Geometry, th e best and simpl e stofall logi c s (Hope &Jack, 19901. isthe stU dyof the propert iesandcharacteristics of certainsets , su ch as line s , angles, triangles, circles, and plan e s. four thousandye a rsago , th e Egyptians we reconcernedwithland meas ures andbecamevery proficient indealingwiththe s epractical aspe cts
of ge ometry due to the annual floods of the Nile River. Two thousand ye a r s la ter, the Greeks moved away from this practical concept of geometry endde velopedi t intoa logicalsystem (wei s s, 1972). Th issystem emphasizesaccuratede fi nitions,clea rlystated assumptions, and logical deductive pro o f s . It consists of und ef i ned te r ms,de f i ni t i o ns, po s tu l ates, andth e or e ms.
In the formal, rigorous development of geometry, the only pro pe r t i e s of points, lines, andplanes that can be ,"<lt h e ma t i c a ll y accepted are those comi ng fr omthese postulates ,defini tio ns, and theore ms(Weiss, 197 2 ). A postulateis an as sump tion consideredto be true wi th out ha ving been proven. A theorem is usually a sta teme nt tha t consistsof the"if" par t,cal ledthe hypothe s i s , or the giveninformation , andthe"tnen" part, cal ledthe conclus ion, or theaspect tobe proved. Topr ov e a theorem ingeometry, th ere needstobe five pa r t s :
1. thesta t e mentof the theorem;
2. a di a gra m thatillu str a t e sthe hypothesis;
3. alist , in term of the figure, ofwha t is given :
4. a list, in te r m ofthe figure, ofwha t you aretoprov e ; and 5. a series of statementsand reasons , suchas given inf ormation, defini tions, post ula tes , and theo r ems that ha v e alread y been pr oved,whi ch lead from thehyp ot h e s i s tothe conc lu sion.
Ba s i c a ll y , th ere are onl y th r e ekin ds of ma thema t ical pr o o fs in geometry:
(a). DirectProof: froma hypothesisor a "g i ven" toproveanothe r statement. The "pr o o f " wi l l take the for m of a se ri e s of statements, the first onedirectl yderivablefro m the hypo t. b euis and eachsucceedingone derivabl e from it s anteced ent.
(b). IndirectProof:sometimesit is difficult or even imposs i bl e tofin d adire ctpr oof , but it maybe easytore a s on indirectly.
Ifthe theorem to beprovenis:a-e-o-b, and we can not think of a way of arguingfrom hypothes i s or given"a" to proveth e conclusi on
"b", i tisus e f ul to know that thecont raposit ion and thecond i tion areequ iva l ent. ThUS ,if a--> b,then: b' -->a '. Thi skindof pr oof always sug gest the possibilityof taking the negation of the statement to bepr o v ed and arguing fromth at to seei f fromi t the neg a t i v eof the hypothesiscanbeobtained. If so, thenwe have proved the the orem . In such a pro ofwepro ve that something can not befal s e by showingthat if it werefa l s e we would arrive at a cont ra diction . A indirectProof is usuallywritten as follows:
1. as su ming temporarilythat theconcl us i onisnot true;
2. reasoninglogically untilyou reacha contr adict ionofa known fa c t; and
3. pointingoutyourtemporary assumptionmust befa ls e, and that the conclusionmust then be true.
(c ). Two-Way Proo fs:sometimesath e or e m isst a t ed inthe form:
a<--- > b rec ogn i zab l e by such phrases as "if and only it",
"nece s s a ry and sufficie nt conditionthat". Thestudentsshouldbe
10
taughtunambigu ouslyto recognizesuch phraseologyand tore a li ze that it always me a n s thatth e y mus t contr i v etwo proofs , not jus t They must prove that: a-->b. Whenthat isdon e,the y must then pr ove that b-e-oe.
All of the three kinds ofproof s are deductive pr oofs . Th e y are deduced by using joqic a Lrea s oni ng. Direct Proof and Indir ect Proofare studied during gradeten inseco n d a ry schools in theU.S . ....hil e Dire ct ....roo f is studied by Chinese students during grade eigh t and IndirectProofdu r i ng gradeni n e. Two-Way Proofs are st ud i e d during gr a de ten in Chi n a. They are not studied by studentsin the U.s ,
ProblemsinDedu ctiyeproo fEdyc;ot ion
Many geometryteacher s fi ndth a t most high schoolstudents have a lot ofdiff i cu l ties withdeduct i vepr oo f s. The y do no t unde rstand the roleor meaningof anax i omat i c sys t e m. Sha ug hne ssy " Burg er (1985) said intheir art icle, "despite our best effortsto teach them, eve n themostcapable algebra stu dent s may struggleand get throughgeometry by sheerwillpowe r and memorization but ....ith littleunderstanding of th elog ica l sys tem weha v e beendev e l o p i ng all year" (p.419). Thereare t....o kinds of problems in lea rni ng deductive proofs: the technical rati onal problem and the psychological-emotional problem.
11
In writ i n g ded u ct i ve proo fs , stu d ents ne ed to pos s e s s somedeg ree of technicalknowl edgein formal deduction, logical thinking , and logi c al expres. sIcn . unf ortunatel y , many teache r s find that in geomet ry courses, a numbe r of stude nts do not know much ab o u t writi nga deductiveproof. Theyusu allywill notbe ableto fol low strictlydeductivereasoningfrom thest a r t. Theyofte nbegin with a more orless diso r d e r period of log i c al thinki ng in Which th e y try to fi nd a se ri e s ofst a tem e n tsand reaso n s inorderto ded uce thecon clus i on fromthe hypothesis (stone, 1971). The techni ca l problemha sbecomea greatbarrierin dedu c t i v epro ofeduca t ion .
The ps y ch olog ic al -emo t i onal prob l em
Another kind of problem in deduc tive proo f edu ca tion is the psychological-e motionalproblem. Muchre s ea r ch indicates that many students has a dualisticview of the natur eofdeductiveproof s. Balacheff (198 8) &Chazan (1993) found that some st udents vie weda deductive pr oof ingeometry as a proof on l yfor asi n g l e cas ewhic h was pictured with the assoc iateddiagram . On an i temwhich assessed students'understand ing of thegene rali zationprinci pl e of deductiveproofs, Will iams(1979) &chaaen(1993)foundth at 20\ of the studentsdid not rea lizethat agiv e n deduct i ve proo f pr oved a relationship fo r all triangles, with onl y 31l of the stu de n ts apprecia t ingthe generalizationprinciple . In Schoe nfe l d's(19 89 )
& Chazan (1993) re s e a r ch, students were askedto prov e results
deductivelyand then to make a constructi on . They fi r s t proved correctlythedeductive resu ltsand thenconj ect ur e d aso l u tion to
12
the construction problem that flatly violatedthe resultsthey had just proven. Fischbein (1982) &Chazan (1993) interpreted thi s resul t as an indication that the students were not aware of the disti nctionsbetweenen inductive and a deductive proof. That is, st ude n t s who held this beliefdid not understandthe generalization principlefor deductive pr o ofs (Williams,19 7 9 ; Chazan, 199])r they didnot: understand thatthe validity ofthe conclusion was meant to be generalizableto all figures which satisfiedthe givens(Chazan, 1993)•
While many students do not appreciate the generalization principle ofadeductive proof ,some students measurea seriesof examplesto prove geometry statements withoutwritinga deductivepr o o f. They believe that an evidence is a deductive proof. Martin (1989)&
Chaza n (1993) surveyed 101American preservice elementaryscheol teachers and foun d that 68% of the sample believed that an induct i ve proofwas a SUfficient"proof", only 6.4%fel t the need for a deductive proof. Thesedualistic views of the nature of deduc t i v e proofs are more likely to oause failures in teaching students to read deductive proofs l:ritically.
DebateandPiscuss !onson Deduc:<t i y e Pro gfEducati on
Al t hou g h geometry has beenthought by many to be a powerful tool to train studen ts to write formal pr oof s , student'sdifficulties with de duct i v e proofsmake geometry a largely debated topic in school curricu la. Fey (1984)& Masing il a (1 9 93 , p, 38) calledgeometry
13
"th e most trou b led and controversia l to pi c in schoolmathematics today". Jean Di eudonne (Masing i la,19 9 3, p , 38) claimed "Eucli d must go !". Stone (1911) thoug h t that "t h e 'Gyn th eti c' method impo s es heavy demands onthe learn e r' s 'geometrica l intu i t ion' an d 'mathematical ingenu ity'" (p. 91). And, "there is a dive rs ity or vie wson what high schoolgeometry is" (Pereira-M~ndoza& Robbi ns , 1977, p , 189). Pereira- Mendoz a & Robbins (197 7) found that teachers tended to zeei that Lnroraar skills in qeometry ...ere imp o r ta n t whil e educa tors fa vour e d deductive abili ty.
Deb ateanddiscussionsondeductiveproof education continuetoday.
But there seems to be a general agreement thatdoingde ducti v e proo fs in geometry is the most difticult topic of school lIathernat i c s (Ca rpe nter et aI., 19 80: Us i sk in, 1982; San k, 1985 , 1989). About 85 pe rcent of high school graduates intheUnit e d Sta t es have not mas tered de ductive pr oof s that unde rl Les the st ructure of a standard geometrycourse (s enx , 1985). Man y Chinese stUdents have rejected the topic of deduc t i v e proofs, and few st udents can prove de d uctively a geometric problem on a high e r sch ool mathematicsentr anc e exami nation .
We ne edtothi n kexte nsivel ywhy many stude n t sin both countries fin d learn i ng deduct iveproofs such a difficult task. The van
!!i el e mode lcanbe used to expla i n the students' difficul t i esw1t h de d uct i ve proo fs an d to help us recog nize past failures in ded u ctive proof educ a t ion.
14
TheVan MieleHoOel
The van Hiel e model was pr opo s ed by Dutch educators, P. M. van Hiele and his late wifa, Dina van Hie l e -C a ldo f in 19 5 7 . As experie ncedmat hemat i cs te a chers in Mont e s s or i schools, theymet with th e sa me di ff i cul t i e s th a t we all encounte r in teaching deducti v eproo fsto our stud ent s. Basedan class r oomobservations, the van Hiele s believed tha t "thestudentsmo v edse quent i a ll y from onelev el of thin k ingto the nextas the ir capabi lity increased"
(Gutier rezet a1., 1991 , p, 237 ), andthat inst r u c tio n playe d an importa n t ro le inraising the geomet r ic think i n g lev el of th e ir st uden ts (Gutierre z et al. , 199 1).
Theya n Hisle M~
Thevan Hi elemodel isa me ntal imagemodel. Itis composedof two main aspect s . The fi r s t cons ists of five le vel s of geometr ic thinking , and the second is concern ed with the devel opmen t of intuiti o nin stude n ts.
~ ! !at' geometricthinking
Thefirs t aspectof the va n Hielemodel isrIvelevels of ge ometric thin kin g. The vanHieles thoughtthat all eeueent;eente re d at the ground level with the ability tonameand id entify the common geometric fig ures , butUsiskin (1982) &pe g g (1985) reported that so mest u de nt s di dnotent e r at thegrou ndlevel. The maj or i t yof
lS
American studiesuseda 1to5 sc ale. They alloc ated a basele v e l, level 0, for the students who di d not have the vis ua lizat ion abi l it y (Pegg, 1985).
The five levelsof geometricthinkinghav ebee nde s c ri b edby Ho ff e r (1981) and Shaughnessy 'Burger (19 8 5 ) asfollows:
Levell: Vi sualiz ation. St u de nt sat thi s lev elhavethe ability to name and identi fy the common ge o me t r i c fig u res, but they only see a whole geometric figure, noattention isgiven to its components.
Level 2: Analysi s. At this level, st ud e n ts have the abi li tyto think of apar ticu l a rgeomet r i c fi gu r e and no te its pr opertie s that i t mu s t have (neces sa ry co nd i t i ons ), bu t the y do not have the ab i l i t y to see howone figure relatesto other figures.
Level 3: Informal deduction . At thi s level, stu de n t s have the abil i ty to, see relationship s amo ng several geo me tr i c figur e s, appreciate the ro l e of general defin itions, and make simp l e inferences.
Level 4: Formal deduction. At this level, students have the ability to reason formally within the contextof amathemati cal system, co mple t e withundefined te rm s, postul a te s, an under ly ing deductive system, definitions, and theorems.
16
Level 5: Rig o r. At this level, students have the abili ty to compare deduc t ivesystems based on differentpostulates, an d to study vari ous ge omet r i e s intheabsenceofconc re temo d e ls .
Tho 4e yel o p mentofintui t i o n inStlll'ent5
Th eseco nd aspectof the van Hlele model concentrateson the phases oflea r ni ng by Wh i chmeansa teacher can assi st th e growthofth e st ude n t sth rough the variou s le vels ofthi nki ng . Pr ogres s fr o mone level of thinkingto thenext is more dependenton instruction than on students' age or maturation(Fuys &Ge d d e s, 1984). VanHiel e emph a s i z e d that instruction should be directed at the st u d e nt's le v el. Henoted thatma n yfai lur e s in teac hi nggeometryresulted froma langua ge barrier:theteache rused the lang u ag eof a higher level thanthat which wasunderstoodbythest ude nts (Gutierrez , et al.; 1991).
Re search OD~~
since the 1960's , the van Hl ele model has ca us e d en t hu sias tic responses. Rus sian educatorsconductedextensive re s earchon this model, and made resultant ch a n g e s in geometr y curriculumin the early1960's. In197 6, Wirszupintroducedthismodel to American matihematifcseducatorsbygiving a detailed account of themodel in the lightof the experiencesin Russia. In th e19 8 0 ' s , Australia n mathematics educators did SOme informal investigatory work about thismodel (Pegg, 1985). Furthermore, there has been a growing interestin re s earchon the model in the last10 years (Hoffer ,
17
19831 se n x, 19 8 5 1 Fuys et al., 1988; Gutierrez & Jaime, 19691 Gu tierrezet al., 19 91 ) . Much of there s e a r chwa s re lated to the van Hi ele levels, the studen ts ' le vel of thinking, and cu rric u l um an dinstruction.
ThevanHisle lev el s
Pr e v i ou s studieshave drawnsome conc l usio ns rlli e v an t to the van Hiele lev e l s of developmentin geometry. Mayberry (1983) , Fuys et;
a1. (19 8 5) ,and Pegg (1985) found thatthe van Hi e Le levels appeaar'
to be hie r arc h i c al innatu r e. That is , "what is viewed by tho pup ilas of paramount importanceat one level is subsumed by a new perce pt ion at the ne xtlevel" (Pegg ,1985,p, 6-7). The van Hi e l e levelsalsohave a logical struct u re: th e re c o g nit i o n of a figure in Le vel 1is essentia l tothe de ve l o p ment of its properties in Leve l 2. The se pro pe r t i e s are re qu i r ed in Level J to form th e essentia l prerequis i tes for the unde r s t a nd i n g of a mathematica l syst em at Level 4. The pr operties in Level 4 are essential to compare systems based on diffe r ent ax ioms and to study various geome t r i e s in the absenc e of conc re te models in Le v e l 5 (Pegg, 1985).
Tbe 8t.udentsl LeyelaofT ~
One of the impo r tant foc uses of re s e arch onthe van Hiel e mode l is todete rm ine the stu den tsI lev els of ge o met ric thinking (Gutie rrez et a L,, 199 1 ). Manyre s e arch ers us ed a writtente s t todete r mine a stud ent 's vanHie l e level (Usis kin , 1982; Gutierrez '" Jaime,
18
1987; Guti err ezet al. ,1991). In 19 82 , Usiskinconstruc teda 25- itemmul tiple-choi c e Van Hie le GeometryTest and adminis te red it to allth e stude ntsenrol led inthe lOt h -gradegeometry course in13 schoolsin the uni tedStates. In hi s stud y , Usi::okin (1 9 8 2) found that overtwo-t hirds of students taking the test could be assi g ned a van Hiel e level . He noted that whe n the fi f th le vel was excluded, th e 3 outof 5 criterionassigned 85\ of thes tude ntsto a vanHie l e le ve land the 4 out of 5 criteri on assig ned92% of them (Wils on, 1990). Ba sed on thes e findin gs, Usi s kin (1982) concluded that ind i vidua l studentcanbe assi gneda vanHie l e level. This co nclusi on ha s been conf i rmed by many oth e r studies (Usisk i n&
aenx, 19 90).
Ins t ru c t ion al l!l::Jpec ta of theyap Hiel e mode l
On eimporta n t inst r uction a l asp ect of thevanHie l e model is that stude ntscannot be expecte dto unders t a nd ins truc t i o n at ale vel higherthantheirleve l. Shaug hnessy&Burger(1985)foundthat it wa s ver y likelyth a t the teacher and the st ude nts were re asoni ng e.Scut;th e same conceptbut at different levels. This indicates that stude ntsmayha v evastl y di f ferentgeomet ricco nce ptsinmind than te a c he r s thi nkthey do vhen teachersareteachi nga course in ge ometry . Davey&Holliday(1992 ) foundthati finstructi onwa s at a hdqh az- level ofmentaldeve lo pmen t tha n the students ' lev e l , some students mig h t re jectthe SUbjec t ,andot h e rsmaywish toplease the tea cher and jus t accept wha t the tea chersay s without any unde rstanding.
19
Relation sh ips Betw e e n Hi gh Bch o ol Bt ude: n.U..!.
J\gbiayementin LeuDing ped \l gtt yePrQQ.fL andvanHiele Leyelsat Ge o metricThinking
Onprevious parts of this chapter. it waa shown that to Le nrn dedu c tive proofs in geometry is to understanda system of deduc tive reasoning, to organizeresul ts into a deductive sys te mof axioms, majorconceptsandthe o r ems ,andmi no r resultsderi ved fr omthos e (axioms , major concepts and theorems). Th a t is, to learnded uc t. t vc proofs in geometry is to thinkge ometry at van Biele' level 4 (formal deductive revea).
A major que s t i on needs to be answered: whi ch van Hiele level is necessar yfor stu d ent sto learnde d uc t i v e proof s in ge ometry ?
In19 BJ , Senk conducted a research that consistedof751 students whohad st Udied deductive proofs in geometryand fitted th e va n Ili el e model in the CDASSG pr o j e c t (Usiskin, 1982) _ By administering"CDASSG Proof Test"•...0 her sa mpl e, Senk (19 89) found that van Hielelevelaccountedfor25\ of thevariance in deduct ive proof writing achievement for students who took th e "v e n mete Geomet ryTest "in the fallof 1982, and 34%of the variance inthe spring(gc .00 01)_
Ba s ed 'In th e s e findings, senk (1989) indicated thathi gh schoo l
20
students' achievement in learni ng deduc t ivepr o o f s in ge omet ryis positive l y re l a t e d to vanthe Hie l e le vel of geome tricthinking.
Shenote dtha t a stude ntswho started ahigh sch o olge ome t ry course wit h Level 0 had li t t l e chance of learning to write deducti ve proofslater intheyear. Ast ud e nts whosta rtedwi t h Level 1 was likel yto beableto do somesimplededuct iveproofsby the end of the year, butsuch a student ha d le s s thanonechance inthreeof mastering de d uc t i v e proof writing. A student who started with Level 2 had at least a 50-50chance ofma s t eri n g deductiveprccrs by theendofth e year, and a student who entered withLevel 3 had an evengreaterchanceof masteringdeductiveproof writing. She concludedthat "a l t h ough th ere is no individual vanHiele level tha t ensu r e sfutu r e success in proofwt"i t ing , Level 2 appearstobe thecrit ica l entry le v el" (1989,p, 319).
By alsoadministering "Test for Kncw.ledqe of Standardcontent" to thesame sample of studen ts , Senk (1989) found th a t thestudents' enterin ggeometryknowl e dg e accounted for 37%of the variancein proof writing achievement when it was used to examine the relationshipsbetweendeductiveproof writ ingac hievement and thei r van Hi e l e level inthe fall , stude nts ' en t e r i ng geometrykn owledge and van Hi e l e leve ls accounte dfor ab ou t 40 % of the variance in deduct iveproof writi ng scores inthe fall and60\ in the spring.
The r efore, "much of a stUdent's ac h ievement inwr it ing geometry proo f s is due to factorswithinthe dire ct cont ro l ofth e teache r and the curr Icu Iu m'' (Senk , 198 9;p , 319) .
21
Research on the van Hi e l e model suggests somechanges th atseem app ropr i ate for the ex i s ting school qecmet r-y cur ricu lum at th e se conda ry level. It suggests that instructionshouldbe di r ec t ed at the stude n t 's level. It also suggests th a t t.eachers should assist studen tstodevelop thei rgeometric think ing le vel s inor d e r to prep arethem for learningded u c t i ve proofs. Therefo re, it is nec e s s a r y forus to th i nk ofthege ome t r i cthi nk ingle v el of our students in order to improve the quality of deductive prcor ed uca tion. Thene xt chapte r will attempt to find and compare the van Hiel e leve ls of the students in the U.S, and the People's Repub licof China. The students' van Hie l e level found will be use d to analyze and compare deductive proof education inboth cou ntries.
22
CHAPTERIII
Tbe Van HieleLevelsof8tuelents intb e U.8.
aneltb 8Peop 1 8'BRepublicof Cb i na
TheAmeriCAn Students·VAnHi~
In 1982, Usis kin cond ucted the Cog ni t iv e Dev e l o pment an d Achiev e me nt in Se c o ndary School Geometry (CDASSG) project to acdreesa va rie tyof que stions re1llotingto the van Hiel e model. One of the re s earch questions inhisstudy was"h owar e ent e ri n g geometrystudentsdist ribut e dwithres p e cttothelev els inthe van Miele schemes " (Usi skin, 198 2,pi). "TheVa nHi e le Geometry Test"
wasconductedtodeterminethe distributi onof theva nMiel e Level of the studen t in the U.S.. (As the only van Miele instrument availablethatcanbegroup-administeredto as s i gn studen ts' van Hiele level sin the United sta t e s orCa nada (Usisk in&Se nk,1990),
"TheVan Hiele Geo metryTe st- ha s made avaluable cont r i b u tio r to res ea r c hon vanMi e l e lev e l s (Cr o wl ey , 19 9 0)]. The test has five su b t e s t s, each of whichconta t n s five ite ms. The ite ms with ina sub t e st were written to corres pond direct ly to characteristic eenavtcurs th at stude nt s exhibit at each le vel, whic h described bythe vanHieles (Usiskin, 1982; Crowley, 19 90 ) .
InUsiski n 's (1982 ) study, ast udentwas assigned avan Hiel e le v el base donthe se quenceof su b t es tsmastered. Masteryof a subte st was dete rmin ed by the followi ng rule s: it a stu de n t met the
23
criteri on (th e re weretwo crit eriaforpa s s ing alev el: 3 out of 5 items correctand 4ou t of 5 it emscor rect )for passi nglev elsfrom o ton, and failed tome etthe criterio n for le v e l s n+ I and above, thenthe stude ntwasassigned to thele v e l n; i fth estudent couldnot be ass igned to anylevel, thenth at student wa s saidto not be fit into a level.
By admi nistra ting "Th e VanHie l eGeomet ry Test" to 2699 students enro l l ed in a one-yea r geometrycourse (grad eten) in13 schoo ls sel e c te d fr om throughout the Un i t e d States to pr ovide a br oad representation of communi t yocctc - e conc n t e s, Usi s k in (1902) fo u nd that astudent in the U.S. coul d be ass i gned a van Ili ele leve l, and over 70%were at lev el0 (30.7%:) orle v el I (4 2 .7 \ )us ing 4 outof 5 criteri on. The numbe rsand pe r centage sof students at eachof the van Hiel e lev els foundby CDASSG are presen ted in Ta ble 1.
Also, the nu mbe r of st u de nts who were bel ow the vi s ual i z a ti on leve l . end tho se who di d not fit the theory are incl uded. The cri t erion used in Table 1 is 4out of 5 (80\) items correct.
CDASSGstudyshowed that it wa s possib le to classify91.9% perc e nt of the st ud e n t s intoa va nHi el e le v el or bel owthe visualiz ation level. Th e majorityofstude nts were at orbel owthe visuali za tion level. This res ult sug gested thata forma l lan g ua g e cannot be usedtoteachgeometry at th e beg inning of gradeten in theU.S.
2'
'/',.111_,I
Ulllnl...·r:; ..nd l'p.cr; ,-mt<l'-.u::; of Stud e n t s at Eachva nHi el e Level Using A .,(j'IL'-~f ~Ccit.P.rion
I,.:v e ]
No'i t
'rot.at s
Numbe r Percentage
19 1 8.1%
726 30.7't
1008 42. 7t
338 H 3%
93 3.9%
0.2%
2 361 100.0%
~: The dat a in tabl e 1are from "v a n HieleLevels And Achievement InSeconda ry Schoolnecmetry- CDASSGProj e ctby uniskIn ,Z. ,19 R2. (ERICDocumentRep r od u c t ion Serv iceNo.ED 220 2R8 )
fllllwug h the COASSGpr o j e ct WdS con d uc ted in 19 82 , its resul t s :'1iII rcpr cacnt t.11l~ enteri ng geometry studen ts di s t ri b u t e d wi th 1I'[:p.!I :\. to the leve l s in th e van Hiele schemes. usiskin& Senk 111;')0) u.r l d illt1Wi l'art icl e :
'l·h., rr-sL l'l'heVC':l1lIio l eGe o me tr yTe st) has been morewi d e ly
2>
.,
us ed than we wouldever have imag ined. Ove r 10 0 ind i v iduals have formal ly requested and received perm ission from us to dup licateit. Th e te s t has almostuni ve rsa ll y been used to de teI'llli nevanHi _I e lev e lsforaset ofind ivid uals (Us iskin ,Senk , 1990 ; p. 242).
Desp itethe low relia bil i tycoefficie nts .the resul ts of the CDASSG study ar e rnt herrobust . In retu rnfo r permiss i on to duplicatethete st (the Va nHi e l e GeometryTes t), we ask to receivea copy of theresults. Non e ofthe studie s tha t ha ve come to ou r atten tion ha s fo und per-rcraanc e in high school geometrysign i tic antl y dif fe r e nt fromthat of the studentsin our study (Us is k in, Senk,199 0;p. 244).
TheChinese StudentsI VAPRiolo ICD.1..a In ord erto tho r o ughl y ccepa re de duc tive proof educ r.cron between theU.S . andthe People 'sRep ub lic ofChi na,wecOj ,a ~ct eda stud y focu s Ingon the vanhi elelevels of thest ud en ts atthe begi nni ng of grade eight in urban sc h oo l s in China . The reas on s for conducting the study only in urban schools are that the urban schoolsinChina aremoresi mila r to those of the schools inthe U.S. thanru r a l ones inChina, and thatthe greatdifference s in the sectionsof educatio naland socioe conomi c co ndi t i o ns betw e en
26
the urban and rural areas in China. For example, th e educational level of rur allab orersin China is thelowe st of any group. The Thi r d Pop ulation Survey of China (1982) found that for every thousand peopleemployedin the labor force, only0.4 personshad a colleg e education , 53.4had a seniorhigh SChool education , 214.7 had a junior high school education, 371.6 had elementary school education, and 359 .9 were illiterateor semiliterate (Wu, 1989 - 90). Moreover, most rural sch ools in Chinaaresma lle r than the urba nones. Th e y cannot offer the breadth and depth of curricula tha t urban schoolsdo. Intheextens ivemoun t a i no u s and pastoral regions, many rural elementary schools are taugh t by literate peasants and do not extendbeyondfour of fiveyears.
This stUdy was concerned wi t h fou r research questionswithre s p e c t to stude ntsI le v el of men tal development in geometry at the beginningof th e eighthgrade in urban schools in China. Th e s e questio ns, along wi th the correspondingstatist icalanalysisused to test thehyp o t h e s e s, or descr i be the data collected, are given below:
Cana student in urban schools in China be assigned a va n Hiele le v el ?
Qnestion 2
Wha t percentageof studentsat the beginni ng of theeighthgrade in urban schoolsinChinaare pr e p a red forle a r n i n g deductive proofs?
27
Qllestio n J
Whichlevelof instructi o nallangu a ge can beuse d tote a chgeome try in g:radeeig:htin urb an schoolsinChina?
Ques tjon 4
Are there any differ ence s between theva n Hiel e le v el s of men t al development in geometryofgr a de eightstUdents in urbanschools in Chinaand tho seof gradeten stude n ts in the Un i t e dsta tes?
Null Hypothes is : There is no si g ni fic a n t differenc e in the Hiele lev el s of me ntal development in ge ome try of grade eig h t stud e n t s in urban schoolsin China and tho s eofgr a de te n students intheUni t e d state s.
Questions I, 2, and 3 we reanswered by administer ing a modifie d version of the Va n Hiele Geo me t ry Test to 95 gradeeigh t stude nts at FuZhou 'iian 'run middleschool in Chi n a onOctober 12 , 1994.
Table 2 was constructed to sho w the nu mbe r s and pe r c enta g e s of st ud e n t s at va ri ous va n Hi ele levels us ing4 outof 5 crite rion .
Toanswerque stion4, the null hypothe s i swas te stedby usingthe chi-squaretest for homogen eityof the vanHiele le vels ofstud e n ts in urban schoolsin China and tho seof the stude n tsin the Uni ted states. A cont i ng e nc y table was con s tructed for 4 out of 5 criter i aby usi ng th evan Hielelevel s ofstu d e ntsin urban schools in China and thos e of students inthe Un ited sta tes . The fall result of the CDASSG project (Us is k i n , 1982) was used for the
2.
lat te r groupof students. The level of significance selectedwas .05 inbothinstanc es.
Designoftbe~Y...o..
Thisstudygat he r ed data on 95 stude ntsingrade eight at Fuzhou yian Yun middle school. The school is anordinary urban middle school and generally rep r e s e nt s urbanschools in Chinawi t h similar educa tiona landsocioeconomiccon ditions.
~.
The instr umentused in thestudyis a modified versionof the Van Hiele Geometry Test (Usiskin, 1982). The criteria used in the te s t is 4 out of 5 items cor rec t because it reduces the effec t of guessi ng. The modified Van Hie l e Ge ome t r y Test consistsof the first 20 itemson the original te st ; that is, the items dealing wi th thp.first fourle ve l s. The la s t five itemson th e original testwere excludedsince"th e existence and/or tes tabili t yof le v e l 5 (Rigor) hadbe e n questioned" (Usiskin, 19 8 2 ,p, 79). Aco py of the modified Van HieleGeometry Test is containedin Appendixwit h app ro priate instructionsand answer sh eet.
Tenadministration
TheVanHi el eGe ome tryTe st and answersheets....ere sentto FuZhou vde nYu nmiddleschoo l inCh i na on Oct ober12, 199 4. The following instruct ions were given tothete a chers:
1. they we re asked to administer the tes t before the end of Octobe r, 19 94 :
29
2. thetest was admin isteredto aU grade eig ht studentswh o s e stu d e nt s ' IOnumb e r swer e even in the schoo l :
3. the time al lowedfor the test was exactly30 minutes; and 4. th e answe rsheets wereretur ne d immediately afterthe testwas comp let ed.
TheRQ!mU s
The nu mbers andpercentagesof stude n tsatea ch of thevanuteto levelsare pr e sent e d inTa b l e 2. Also , the number of studentswho werebelo wthe visua lizationleve l , and thosewho did no t fit.the mode l are include d . The criterionused in Table 2 is 4 out of 5 ite ms correc t at each level. This study showe d that it was po s s ibl eto classifyBO percentof the studentsin t o a van
me
re level or bel owthe visualizationle v el. 2 atucent;sor 2.1 percent were be lowth e vis ualizatio n level; 28 st ude nts or 29.5 percent we re at the visua li zation lev e l; 26 studentsor 27.4 percent were at the ana l ys is level: 16 studentsor 16.8 percent were at the informal de du c t i v e le v el ; and only4stude ntsor 4.2 perce ntwere at the fo rmalded u ctiv e le v e l. Thisstudy showed that , exclu di ng not fit stude n ts,there were 48 . 4perc e n t studentswho were ready for learningde d uct i v e proofs atthe beginni ngof grade ei gh t . The ma jo r i t y of students we r e at the visualization, analysis , or informal ded u c t io n le v el. This study suggestedthat an informa l lang ua ge canbe us e d to teac h geometry in grade eight in urban sc h ool sinChina.Table 3is a contingencytab l e for th e 4out of 5 criteriontote s t
30
h' .rOf....ren ei t.v . 'rtie chi-squ a r e value wasfo un dtobe126.96I>11.07 , l! ".O·j) wh ich r'~~u ]t(.'<l in re j e ct io n of the nul l hypot.he~is of (/I1.·:;l.ion4.
Numl,.,rsund Pl ' n;:(.'n l tH! C Sof St ud e n t s at Eac hVan Hie leLeve l usi n g
A-1(Jill:1')[ S cd l'!rion
Nofit
Totals
Number Percentage
19 20 0%
2. 1%
28 29.5%
2. 27.4\
rs 16.8%
4.2 %
95 10 0 . 0\
31
Table 3
Cont i ng enc y Ta b le fo r 4Out of5Criterion toTt' s t IloolOlJ(' IIt, it y
Level
Nofit
Tot a ls
SAmple
China th e U. S . 'r ot,ll n
19 (2 0.0\ ) 191
,
8.l'f.) 2102 (2.U ) 726 (30.7\) 120
28 (2 9.5%) 1OOB 142.n) 1O]G
26 (27. 4 %) 318 (14. 3't, ) 364
16 (16.81.) 93 (3.n ) \09
4 I4.2%) 5 I0.2 '#;)
95 2 361 7.015 6
The fou r research qu e s t i on s can bean s we r edfl!lthe ba:;h;ul 1.1,••
aboveresul ts :
1. a student in an ur ben scho ol inChina callbl'!<l:'~iqncd iI V.1I 1 Hi eleleve l:
2. exclu di ngnot fi t studen ts. t.h er earc 48 .1\pe r c ent ofnt.uch-utr:
who are re a d y for learningdeduc tive proot n .11.th e beqi n n lll<j"I gr a d eeightin urbanschools in Ch ina ;
3. an informa l languag e ca n be us ed Lo l.(~il(:b qeomctr y <II IIi"
ei g h t h gr a d e in ur-banschools inCh i na :an d
4. there arerelat i onsh i p s betw e e nthest.udont:;'V'HIHi',l,·1,··,.·1 :;
32
of mental deve lopment in geometryof grade ei ght studentsin urban schoo ls jnChina and thos e of grade ten students in theUn i t e d States.
The fi rst major conclusion ind icate s that a beginni ng gra de eight student in an urba n schoo l inChi na can be assig ned a van Hiele leve l. However, theperc e nt age of studentswho did not fit the van Hiele model was greater than thoseof st u de nt s inthe U.S. One possible re a s o n for the resul t coul dbe the la c k of training that Chi nesestudents ha din ta ki ngthemu l t ipl e-Cho i c e standardtest. Stevenson (1992) indicated that "r esul ts from cross-national studies can be gr eatlydistortedi fthe researchprocedures are not compa r ab l e in each area and if the test materia ls are no t culturallyappropriate " (p. 70) . The la ck of trainingmight ha ve some Chinese students, whowe r e slow th inke r s, spend a little valuable ti me to familia rizethe te st. The n, the y had to ru sh the tes t in or der to comple t e more items wit hout careful th i nking.
Thi s might resultinmastering nonconsecut ivelevels.
Anothe r po s s ibl e reason for the re s ul t cou ld be encouragi ng memo r ization and probab l y ratelea r ni nginurban schools in China. Items 14 and 15 onthe Van Hi e le Geometry Test (Uslskin, 1982 ) can be memori z ed without unde r sta ndi ng. Inth i s stud y , 12 out of 20 te st tak e r s who didnot fi t the van Hi ele model ma stare d le v el 3 whilethe ydidnot masterlevel 2.
33
The second major conclusion indicates that with the aid of instructi on ,abou t halfatthe students (the pe r c entagemight be greater if studen t s whodid not fit theva nHielemod e l had been considered) at the beginning of gr ade eigh t in urba nschoo ls in China may raise their geometric thinking leve l to the forma l deduction levelbeforethe end of grade nine. However , nno re is st i ll 31.6per cent of thest Ude nt s (the percentag e migh t begr e ate r if the students who did not f i t the va n Hiel e mod el had be en co ns i d e r e d ) who ar e no t ready for learn ing deduc t ive pr oo f s. Considering that Chinese te a chers oft e n pr ese n t geometry in an abstractmenner,thesestuden t s will not likelybe able to mas te r deductiveproofs by the end of grade nine .
The thirdmajorcon clu s ion indicates th a t geo me t r y ins truct i on in grade eight in urban scho ol s in China sh o u l d be in fo r ma l. Aft er one yearof study, it may be pos s i b le to usethe fo rmal la ng uageto teach deductive proofs.
The fourthma j o rcon c l u s ion is related tothe van Hi e l elev elsof students in gra de eight in urban scho ol s in Chi na, and thoseof student s in grade ten in the Unitedst a t es. ThevanHiel e le v el s of students at the beginning of gr ade ei g ht in urba nsch oo l s in China are signif icantly higher than those of st ud e n ts at the beginning of grade ten in the unitedSt a tes. Th is res ultmay be due to hard work of Chinesestudents. Chinese st ud e nts believe that achie veme ntdepends on diligence. stevenson (199 2 ) indica ted
34
tha t "the idea that increased ef f o r t will lead to improved perfo rma nce is an important factor in accounting for the willingnessof Chinese and Japanese children ,teachers and parents to spend so much time and effort onthe children'sacademicwork "
(p.74).
Th i s chaptercompares the van Hi ele levels of thestudentsingrade te n in the U. S. to tho s e of the st uden ts in grade eight inthe People's Republic of China. The comparison is made witho u t conside ri ng the students' ag e. HoweverI it may be re a s on a b l e becaus estudents ' progress from one level ofth i n k i ng to th ene x t is more dependent on instruction than on their age or maturatio n (Fuys'Geddes, 1984 ). The next chapterwill attempt to compare de du c t i v e proo f education between the u.s. and the People 's Republicof China byusingth e van Hie l e model and the van Hi el e levelsof students in bo th country.
35
CHAPTER IV
Deductive Proof EOuctltionintheU.S . andthe People's RepublicofChina
pe""QtiyoProof EdUQotionintho JJQ
The precol lege educa tio n system in the United States consists of twolev el s:elementary school and secondary school . Howeve r , the educ at ion systems aredifferentamong states . The most popular syste msare:
(1 ) 6- ) - )systems(sixye a r s of elementaryschool, thr eeyearsof jun i or highschoo l, andthree ye a r s of seniorhighschool);
(2 ) 6-6 systems (six years of elementaryschoolandsix yea es of se c ond a r y school);
(3) 6-2-4systems (sixye a r s of elementary school , two yee ra of ju nior hi gh school, and four yearsof seniorhigh school); and (4) 8- 4systems(e i g ht years of elementaryschool and four yee r a of secondaryschoo l)
Al l stude nts ar e re qu i r e d to stUdy na t he ma tIc s in secondar y schoo ls. Howeve r, the mathematics curr i c u l umof the eeco ndary school va ries incontent. The r e are thr ee levels of mathematics course s for stude nts to choosefrom. The firstlevel consists of alg ebra, tr igo nometr y , pla negeometryI and sometimes, calcu lus, st a ti s tics, and compu te r science. The second le vel consists of ele mentary al geb ra and ge neral ma the ma t i c s. The third level consists of la boratory mathe mat i r:.s, consumer mathem a tics, and
36
business mathematics. Most college-bound students choose the first level, and students with lower mathematics ability choose the second or even third Level (Chang, 1984).
Textbooks define limits to the content of the curriculum (Brown, 19731 McKnight, et a1., 1987; Westbury, 1990; Chandler&Brosnan, 1995) and provide structure for 75% to 95% of classroom instruction (Tyson and woodward, 19891 Chandler & Brosnan, 1995). Most Mathematics and science teachers rely on a single textbook for instruction (Weiss, 1987: Flanders,1994). The U.S.has never used a unified set of mathematics textbooks. However, textbook publishers in the U.S. are careful to investigate statel.'ide curriculum goals and the Curriculum and Evaluation Standards' guidelines in developing unified goals for mathematics instruction
(Jiang&Eggleton, 1995).
Both statewide geometry curriculum and the Curriculum and Evaluation Standards (NCTM, 1991) require college-houndstudents to master deductive proofs in geometry in grades 9-12. For example, in the State of New Mexico, the geometry course requires college- bound eeueenes to have:
(1) knoWledge of two-dimensional and three-dimensional figures and their properties;
(2) the ability to think of two-dimensional and three- dimensional figures in terms of symmetry, congruence, and similarity;
37
(3 ) the abil ity tousethe Pyt hagoreanTheorem and special right-trian gle rel a t i ons hips ;
(4) the ability to dr a.... ge ometri ca l figures end us e geometri cal mo d es of thinki nginsolv ing probl ems: and (5) appreciat i on of the roleof proofs (New Mexico Comm ission
on Hi gher Ed u ca tio n & New Mex ic::o Sta te Dept. of Edu c at i cn , 198 7, p.1 1- 2).
Textbookpublishersal s ost ress the impor t a n ce of deductive proofs.
Jurgensen&Brown(1990) thoughtthat "te a ch ingstuden t s to wr ite proofs isoneof the toughestjobsof a ge ome try te a che r "(p.T55). They (1990) mad e th e fo l l o....ing sugg estions forte a c hing deductive proofs :
Obj e c t ive 6: unde rstand ingthe Organ izatio nof Proof s . This si xt h objecti veisto ass i s t st u dentstoundersta nd the organizationofproofs . To read a dir e ctproo f ofa theorem, stude nts need to se e the organizat io nascon tai ni ngat least these co mpo ne nt s: a statement of the tbec r-en, a figure, a
"given", a "prove", and a pro ofwith statements and reasons listed in logical order. The key to readingthe proo f is dev ele-ping an interrelationship amo ng these pa rts, whi ch involveslookingupand down andfr om sideto sidema nytimes. Although many the o r e ms arepr-esentad wit houtpr o o f in thi s textbook, it is impo r t a n t for your stude n ts to de ve l op an ap pr e c i a t i on of this orderly method of reaso n ing . The partially proved theore msand exercise s will help stUdents
38
~J<.liflnkijl in writing proofs in twa-columnform . Some of the c xc rcl rsen call for the~I r i t i ng of complete proofs. Ind i r e c t proof j !; presented as an Application. Your better stude n ts may cnjoy exper ime n t i ngwLt.h thi sme th o d (p. T3'1).
bigll sch o o l geometr y textbooks presented meter-Le I at or above Vim uloIo level 3and had problemstha t oftenjumpedfromle vel0 to lovo l3{Goddess etill, 1982; sneucnnessy&Burger, 1985). Th e folIowinq pro blems in a text bock (Jurg ensen & Brown, 1990) uen<> rilily r-cnrcsenc requirements of ded ucti ve proof writing in
U[iJl!l1 Len in the secondary schoo l in the U,S. They requi re
nrudoutsLoforma llydeducethe conclusionsfr o m the hypothesis,or toLhiIlk at v an Hielelevel4.
otvcr nL1~ .L2;AN'It
i:B -,
Proven: L4=: L3 (p,T91.IrJIitod compl oto proofin two-column form.
Gi Vi'11: V'I'
f.'
SRi \iR' ! /TS l'ln,,<,: . / \ IlS' J'-a:... /\TVR (p.1'101.39
AT'" AV (p. T11).
V---- ---
T "'T
/ 'r-
fZ Ii~ //
cive n .
L 3
=L".
Prove:Th e LevcL of the lan g u a g e us e d for instruction is cft.cn a luovol.h,~
ana lysis levelingeometrycourses. The majorLt y of t.e.rchorsw:od the Traditional Lecture/De monstration method to t.oach t.huir- clas s es . Seve ntyeight pe r c e nt of th e tcecheru lJ:Jf~(J<jt:r.,nlf·t ri(·
soli ds le s s than once awee k. Ho s t high schoort(~(lcher:; r'/-,porl',, ! that: they taugh t 5 mat.hematicsc La s ses per clay, IitliJY~~per wed.ill the secondary school. Th us, litt.le cuc ontLon could 1J'"qlvon I.', individ ua l needsafter classes (ETS f. NAF,1J.1~91).
Consideringthat over 70 percentof secondaryschool students have only levels 0 or1of geometric thinking prior to taking geometry (u s i s k in, 1982), and thatlevel 2 is necessary for students to learn de duct i ve proofs (Senk , 1989 ). thege ometr y tex tbooks and ins t ru c t i onwerenot well -designedfor secondaryschool stude nts in the U.S.
Currently, research on the van Hiele model has made resultant changes in geometry curriculum and instruction. Educators (lo: i r s z up ,1976;FUys, Geddes, &Tischler, 19 8 5 1Senk, 1989) have demonstrated vaya to raise the van Hiele le ve l s of students in elementary and secondary schools. Standards (NCTM, 1989) advocated educatorsand teachers to consider th e importance of sequential le arn ing as expressed by van Hielemodel:
"Evidence suggests that the development of geometric id e as progresses through a hierarchy of level. studentsfirst learn to recognize whole shapes and then to analyze the relevant properties of a ~hape. Later they can see relationships between shapes and make simple deductions. Curriculum de v e l opme nt and ins tructionmust consider this hi e r a r c hy"
(NCTM, 1989, p. 48; Teppo, 1991: p.214).
Th e Standards (NCTM,1989) made the fol l owin g suggestions for geometry curr iculum and instruction:
standa rd9, "Ge omet r y an d spatial Sense " forgra des K-4 (NCTM , 19 8 9 1Te ppa, 1991), calls for studen ts to be ableto "de s cribe ,
41
model, draw, and cl as sify shape s: inves tig ate and pr e d ict the results at:combining, sUbd ividi n g, and cha ngi ngsh apes; dev e l o p spatial sense" . It re comme nds that students lea rn to re c ogni ze geometric shape s by usinga variety of "everydayob jects and other ph ys i ca l materials ." Thi slearningrepresents st ude n tslge ometr i c thinking at thevan Hie l e's visua lizat ion le vel(Tep po ,19 9 1) .
St a n d ar d 12,"Ge omet ry" for grade s 5-8(p. 112), call s for stude n ts to "identify, de sc ribe,compar e, and cl a ssi f y geo me tr i c fi gures:
visu a liz e and repre sent geometri c figur e s••., expl ore transformat i ons of geom etric mode ls:unde rsta nd andapplyge ometrLc prope rties and re l a t i o ns hi ps "(NCTM, 1989: Te ppe , 19 91; p.215). These learning activ i t ies co ntinue th e dev e lopment of geomet r i c thinklng begun in grad es K-4. They help "defi n i tions bec o me meaningful,rel ati onshipsamongfigur es beunde rstood , andst ude n ts be prepared to us e th es e ideas tode v e l op inf o rmal ar gume nts "
(NCTM, 1989 ,p. 112 :Teppo, 19 91; p , 215 ) . The ydeve lopstude nts ' an a l. ysis and infornal dedu c t ion ab ility.
within the span of grade K to grade 8, students dee pe n their understanding of concept s of geometry and are provided with essential preparat ion for the stUdy of ded uct ive proo f s in secondary sch o o l geometry . The systematic geometry ins t ruc ti o n before the secondary schoo l isnece ssaryto insu re thest uden ts ·
la t er succ e s sin learni ngdeductiveproofs. "T he spec ific lanqu age
of th e standards,withthe incl usio nof .examp.Le s of activ i t iesfor student s, servesas an excej.Lentblueprintfor the incorp o ra t io nor
42
the vanHi e l e theory into Ame r i c a n mathematicseducation " (Teppo, 1991; p. 215). They help to ensure tha t effective classroom instr uc t i onoccurs.
peductiye Prgo!EdJlCation in thepAopJAI:J RepUblic of China l:JI<ti.<lI1lII.
The educa t ional system in the People's Re pub lic of China is modeled after the Russian system. Precollege education in China is organized into thee e leve ls: pri ma r y school (grades 1- 6 ) , junio r midd leschool(grades7-9),andsenior mi ddle school (gra d e s10- 1 2). StUdent ages7-1 2 at tendprimary school; ages13- 1 5 attendjunior mi d d l e school; and ages16-1 8 at tendseniormiddleschool.
A scho ol's curriculumis determinedby theMinistryof education.
Text booksare commissioned by the ministry of education and written by university facult y and commit tees of expe rience d te a c her s in 'ke y 'schools, whichserve as college-preparatoryschools forhi g h- achievingst uden ts. Studentsattend schoolssixdays a we e k , for 9- 1 0 mont hs each year . The curricul umof the mi ddl e school is quite dem.andfnq, Both or di n a ry and 'key' mi dd l e schools offer fourtee ncourses: Chinese,Ma thema tics , Fore i gn lan gu ag e s (English bein g the most common, but also Russi a n, Japanese, Fr en ch and Cerman ) , Phys i cs, Chemistry, Biology, History, POlitics , Geography, Healt h Education, Phys i ca lEducation ,Music,Ar t,and la bor Skil ls.
Mat hematics has an impor t ant role in pr imary and secondary educat ion be caus e it is conside red tobe thebases for the studyof
43
all ot h e r aubjacts , St ude nts study mathema tics from grade7 to gra de 12 in mi ddle scho o ls. There is a un iried mathe mat i c s curricu l um,called th e NewUnified series, tobeused throughout all of thecoun t ry. The mat hematicscurricu l um inclu de s a studyof algebra, plane andso lid geometry ,tri g o nometry , analyticgeometry, probabil ity and sta t ist ic s, and one semester of calc ul us (differe n tialandin tegral).
The tea ching of geometr y beg i ns ill the primary school. Ingrade thre e an dfo u r, te a c hers teach studentsto cons t r uct, prot r a c t, measure , and class ify bas ic ge omet ry figu res. Most geome try cont e nt in secondaryschoo lsin theU.S.istau g ht in gr ade8 and 9 inmiddl e schoo lsin China. The following isthe geometric syl labusofmath emat i c s cu rricu l umat the secondary level in Chi na.
I tha sgivenmuch more emphas istodedu cti ve pr oo f sthan thos e .in the U. S :
Geo met ry in Gra d e8:
L Fund ame nt a l concep ts (16class hou r s ) 2. Parall elandperpe ndicu la r lines (18) 3. Tri angl es (40)
4. Qua d r ilatera ls (2 0)
5. Area of polyg onsandPythagorastheorem (8)
Ge o metry inGrad e9:
1. Simi l aries (36clas sho urs ) 2. Circl e s (4 8 ) (Leung, 19 87 1p.41).
• 4
