TOTAl 10

CENTRE FOR NEWFOUNDLAND STUDIES

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AN ·INVESTIGATION-INTO.TH&VAN HIELE LEVELS

'• AND PROOF-WRITING ACHIEVEMENT OF' LEVEL I _{. ;/ J .. . ..}

•

^•

GEOMETRY STUDENTS IN NEWFOUNDLAND .\

•

Greg ^~aaffe, B.A.r , B.Ed.

;

i

A thesis submitted to the School of Graduate Studies in par,tia~- fulfillment of the

requirements _{. '} for _. the degree of

Maste~ of Education

Department of Curriculum and· Instruction

~~rial

universi ty of Newfoundland

. ... . ^{:·~. .} . . .

... August 1983 ·

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.'·."· Abstract

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The main purpose of ~his study was to determitle the

t

vari .Hiele levels ~nd ^pioof ~riting a~hiev.ment o~ ~r~d~_.

-

^{; ... .} ·l~

stude~ts

^{· fn}

~founciland.

^Another.

asp~~t

of the study'

wa~ -

to determine the relationship between s'tudents ¹ van Hiele

. .

^"

levels and their proof-writing achiev~~eht. Other copcerns

of. this study includ~ed trying to dete~ine any sex-relabed

; ... .. ~ • I • I

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differences between van Hi~le levels and p~oof-wri ting ·.

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.,

achievement.. r '· ~

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The sampl~ consisted of 201 students on the Av~~ron , ·-.

Penipsula ,in the province of New.foundl¹and.

...

Each student was

t'

·administered two tests. One. tes't attempted· to determine the s{udents¹ van Hiele level; the other test was given to

.

\--~·-·~·~~~-t~rmine

the :tudents ¹

proof-w~it~ng

^a.chievement. The tests were administered 'Over a two-week period 'in latter part· of May, 1983 • .. . - --- - - -

The,

~iverage

•an Hiele leve·₍ l was"'found to

b~"'

_{. \} ^{2.-'2g and}

.

.too, ' , .

1.45 using the 3 of 5 and 4 of·

. .

5 criteria respec~tVel¥ for the classical van Hille theory. ^{The av~rage} van Hiele levels

' '

for the modified van iele theory were found to be 2_. 22 and 1.45 using the 3 of 5 and 4 of 5 criteria respectively.

A significant relationship was found be~ween the students' assigned van Hiele levels and their proof-writing·

.achievement. No sex-related differences w~re found between

··students ¹ van Hiele 'levels and their proof-writing achievement •.

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Acknowled,ements

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The·, wd"'ter ·

~ish~s·

^{to .·e j r.ess ..}

his~ _a~preciation ~o

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^{. . .}

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all those who pr_ovided assi-stance i.n 'the prepa~tion and

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CGn~uction of. this ~t:udy. · · . _··'·

Sp~ci~l

appreciatio_n fs,

e~:en"ded _{. ..}

^{·to· Dr.}

~F.Jank

^{Rig_gs ,''}

my advisor, for his·constant interest, cooperation, and ... ^. .

..

^·' . . assist:ance _S~nc~re ~ gratitude is· a.lso extended to Dr

.

^{~ .} _{,; ,,.} ^..

\.

^{'·. ...} .. · ^-A. Brace. ^.

··, arid Dr, •. D. Drost for their will,i.ngness :•to serve on my.

· examining committee .. . ^.

..

App~eciation .is ·expressed ·to the many principal's • and teachers for~eir cooperati~n

. ^.

.during t~e ^{. ~}

of the -tests ~or the stpdy.

•

administel;'ing

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_·

_{. . .}

A special word of. thanks goes to the writer•~.wife,

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encouragement w ~c rna~~ ~t possib~~ .. to: devo~.the _t_i_m_e--'---~-"---- necessary to complete .the st-qdy. · ·,;

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Ta~e of

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+ •• t;age ·.

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-r-~ .. Abstract

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Ac~nowledgem~nts • . • • •. ~·.

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CHAPTER I · INTRODUCTION .• • .•

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The Problem ^{•· • 0 • • ' •}

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c;HAPTER I t

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Purp<?s~ titf. ·study

• 't ' ' ,f ' I ' ' ' o ~

MaJor _Hypotheses • • .•

-~i'"ficanc~

^{of Study·}

• 41 • • • ••

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Definitions

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^{. •. ·}

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Sc0pe and Lim:Ltations·

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Description of the Study Out~ine of the ~eport

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• t ••

. RE:VIEW

OF .

RF!LATED LITEAATURE

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The van~Hiele Theory

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Brief Historica~ P~,~spective \.. • • • Implications of the v \ Hiel~: Theory'"""' Summary

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ii

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iii

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1 2 · 3

9

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10

15 19 25

CHAPTER III DESIGN OF THE STUDY • • •t •

21

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Sample -~ •

Instrumentation

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ol ·l. • •

'• • •

·Grading Procedure. ; •

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^{. . .}

REP:ERENCES .APPENDIX A

APPENDIX B APPENDIX

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C APPENDIX. D

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Statement· of _Analyses Used _•.

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SUJIU'!Iary "

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T~E _Question

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_.. !iESULTS ^. ₁ OF THE STUDY

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Questibn ·2

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Question 3

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^{. :. 49}

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^{. • • I •}

The. Study·

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Discussion of'the Results

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summary of the Diacussion of Result~ •

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Table I Sex an? Category o~ Sample • • • • •

~abl~

^II

~Proof. ~sts

^Item

Analysi~

i .

7 · .

'(Usiskirt,

19~?> • . . •

\

List of :rables

..

. •

~~ableiii Breakd~ \o~

weighte: sums

~ith

^Corres-.

· · ·• · ·1-. · pending Ass:Lgned yan Hiele Levels · .·. • •

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Table IV Numb~ .and. ~rcentage .of Stude~ts Who 'Fit' the va Hiele Theory • . • • ·• . • .

I ~ ' ',

' , . ...:_,, I ' '

. ^' . .. . \ . . , .. '( . ·". .

-

Tabie ·v •· · · Nwi!ber and Pe~9e~tages o.f Students· at

·. Eac;:h v~n .Hiele\Level _ _'·Using_ the 3 of. 5 ·

· Cr~ ter~on , • • \• • . • •. • • • • • . . •

I. :

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^'

VI. · ·

N~er

^·and' Perce\tages

o~

Students. at ·

Each van Hiele Level usin1; the 4·-of 5

·.1 Tabie

Criterion • • • · \. •. · • • .' • • ·, • • • ·• -~ Table

·vn ' .

Means and

s~andai~oeiriati,;ns ^Of

~ssign~d van Hiel~·t-ev.el~ • ... • •

• !· .• ' . ... . \

'Tabl~~V.III ! \t~~ns

^and

Stand~rd ri~vi~~ions

^of

t~e

' 1.'

v_a_~ Hlele Levels in C~.te~ories 1 and 4

I . . \.. . ..,.

Table IX

I

Proiif Test

~

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~tem Analil~is

• .• • • • Tab:f.e X ¹ Number·, I Meims artd stand~rd .Deviation

of Scorbs on Proof Test for Each _{I .} ~ .

· :. Ca teg6ry .· • .• -~ . • • .. • \ -• . ·• .• •

\ ^'• \ .

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P.earson .Correlation Coefficients 'netween

I . 1 the Weighted ~urns, Classical and

·Modified 'Levels wit:h Each' It'em ·and

Average Score on. the Proof T~st • • • •

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• Table XII .

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Tabl~

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Table XIV

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· Table ^.JN.

. Table . XVI·

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... - ^{~--·:· -~·-··· ·-.}

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-~~

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Pearsob Correlations Between the

· Weighteq swn, Classical and Modified

Levels¹ with the ~verage Scor~ on the Proof /Test for Categories 1 and 4 . •

. . I . . \ . . .

The Pearson Correlations Between the Ayerage Score on the Proof Test

fo~ Males.and Females arid the

Cla~~ical· and ~dified va~ Hiele · _ Levels . . . ·· . • . .

~~ . .

Brea~down

^of

Mea~

^{:an H.iele}

·Leve_i~

for oth Males and Females With Resu ts of T-.test Using the .J. Qt'

5 c4iterion ~· • • •• · • • • • .

I .,i

·, '

Breakdown of Meim van Hiele ·•Levels .. for BOth Males and Females·With Results of T-te Using· the 4 'of

s

criterion . ~ : • ·• • •

' •.

(.\

Mean Sco es on the Between · ales an· and.in Ca ries Results of ^T~ t

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^'

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vii

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Pr f Test, emales Overall

1 and 4 With

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54

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CHAPTER I ·

,

.~

THE PROBLEM ..

..

Introduction and Statement Qf the Problem

.Deductive· proofs' are·~ .'at pr~sent,' formalii i'n.tro.duced

'in. Newfoundland.

·scho~ls

in the

gr~!i~ . ~in~ ·.' adva~~ed'

mat he-

• ' o ' ' I

matics

.

progr~· which uses^. . the tex~: Geornt~r¥ (Moise ·, Downs,· ·. 19 (f2) • . . . Most· high.-school students. . . do· ·not. : study proofs in . j ., ^~ geom~try .. until do.ing 'ac~de~ic'. mathematics at ~~vel· I (grade 10') in t~e text Math 'l:s Geometry .- Grade. 10 (Ebos, F.;. Tuc.k,

I

'

B.; Hatcher, G.; ·oros~, ·D.,_ l98f>·· s~nk; (19'82) stated "An

understanding<~ of· the concep.t of

. P.ro;,f

^and. th.e facility to

.,1

write proofs are fundamental to success in;the study of

'

,

higher mathematics" (p. _1). ·she a'lso said t~at i t is

be.l,ieve.d writi~g geometr'y ·w:-oo.fs is an are~ in whic~ students

' ..

.

^{. ... . .}

I

experience lit~le ~success. Hoffer· (198;1.) ·and Freudenthal

,(1973)

in~ic, ~ted < that

^high

sch~~l geome-:7~

^{includes . .}

m~ch m~J.-- .

than prqof and that too much ~mphasis is being placed on · · doing formal'proofs.

.

^{. . . .}

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This study c~ncerns the ques~~9n Qf student readiness to reason. deductively .an'ci t~ write

. ^.

geometric. proofs and· it will also explore the que.s1:ion as t6 . whether or not·.~ormal proofs are introduced too early for·

( :.

'·

students·to understand them.

. .

The

van

Hiele theory, which.WaB developed by two Dutch mathematicians in the late 1950's, has be.en use·d to try to

.·r., • . . . . , .

explain why students

!'a~e .

diff

i.~.u~

^ty

~

with hrh schoo\geometry,

.

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p~:rt.'icularly

^proof. The vim Hiele

~~eory

c'ont:ins tf1:re_e

m~i·rf

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. . . ' . . • i ... • . . . ~ .

components_; ( 1) h ve sequen t~al · t~ought levels~ · (i~) proper~ies

. of the~'\e~els; (~ii)· ~h~se~

^,of

·l~ar~in~ ~

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^(i982}

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stated: . -~~ - _. .

'The

va~

^Hiele

theo~y

^{has been}

~ied.

^{-to. . .}

e':'p~air_i why many studez:tt~ ~ave difficu~ty ·· · •.- " ]· . '

w~th h~gher· order cogn~t~ve processe~, ., ·

part~cularly pr~of, required for success · . ·

in high• school geometry; l t has been . . · ~ . th9'Jrized that 'students who have· trouble .· ·-:·.-· . ·. ,,._

are beihg taught at ~)higher van Hiele •; ·. ·· · ·. ·.

!eve"! than they are at or ready for. · The ·. ~ .~. · '· theory also' offers a remedy: ' ·go through .

·'

·the sequence of levels in a specific way. . · · ·· . ,

·.(P~ 1) . . . . .·· . • ' .. • .

' .· ..

. In· this E{tudy -the van ~iele ·levels' 'iinc1 tht· proof..;· ·

•' ' I •

· .

t· ^.,

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writing .achie,;~~erit o·f It~v~l I (grade. lO). students· in '

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Newf.oul).dlan.d ar.~/lrwestigat,ed • . (Level·! ·st.~dents re~erred·

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to) in. this study are equivalent to graqe_ 10 s_1;:.uden:ts,- . This.'_

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is' _. due to the, _{. .} renaming 'of grades 10, ll and i2 .. _{. .} . to ·Levels . I,

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~I1 and III in the revised high ·s.ch?ol .. system in -~ewfo~dland. · .

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To· avoid .any ·confusion· with t·he van H.iei'e le:veis, -her~l.n

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Level

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students will be· referred to·as gr~de_

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,.. Purpose of study

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.. Thi's ... study. was c~~ried out to· det'ermine·. thellltvarr · Hieie

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level of

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r~latio~ship t~

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th~· s~ ~t~dent's ·.

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. . ··rn· ~rd5 .. ~o ?o· _this, ·.~he. fo~~ow~ng:. -~sfiohs ·wer~~ :C~nsidere~-~ . 1) · wha:t_ 'are ·the. ~an Hiele 1eve~.s o·£ geom~~r~ ·st~d~~ts · · ·

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at tb~ end of. g~~de. lO? ·

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