THE EFFECTIVENESS OF A GUIDED DISCOVERY METHOD OP TEACHING .../67531/metadc164313/m2/1/high... ·...

THE EFFECTIVENESS OF A GUIDED DISCOVERY METHOD OP

TEACHING IN A COLLEGE IiSATHMiATIC3 COURSE FOR

NOB-MATHEMATICS AHD HOH-SCIENCE MAJORS

APPROVED:

Graduate Committee: — ^ /pf

^ - <7 S*

Hxno.

4 ^ M £

Committee Member

" ' • " • • i J

Dean 01 tii - cnooi of $du<pt.tion

'll r*« rrns*±A? fivh rTTT* ~ V Dean of The Graduate School

THE EFFECTIVENESS OF A GUIDED DISCOVERY METHOD OF

TEACHING IN A COLLEGE MATHEMATICS COURSE FOR

HON-MATHEMATIC;? AMD NON-SCIBNCS MAJORS

DISSERTATION

P r e s e n t e d t o t h e G r a d u a t e Coursc.il of t h e

Nor th Texaa S t a t e U n i v e r s i t y in P a r t i a l

F u l f i l l m e n t of t h e Ho-ûireraenta

For t h e Degree of

D do t or of 7Auc& t i on

By

Den'i i3 D« R3 Isier , M» S •

Den Lor , Texaa

J a n u a r y , 1969

Copyr igh t by

Dennis D. Reimer

1969

TABLE OP CONTENTS

page

L I S T OP T A B L E S V

C h a p t e r

I . STATEMENT OP THE PROBLEM . . 1

S t a t e m e n t of t h e P rob lam P u r p o s e s o f the S t u d y Hyp o t b o s e 3 Background and Sign I f i c a n c e of t h e S t u d y D e f i n i t i o n o f Tarma L i m i t a t i o u a of fell6 S t u d y B a a i e Assumpt ions

I I . SURVEY OP REM TED RESEARCH LITERATURE llj.

I n t r o d u c t i o n Tba S t u d i e s of George Materia and R e l a t e d

S t u d i e s S t u d i e s Using Coding Prob lem3 aa t h e

C r i t e r i o n Task S t u d i e s in W h i c h t h s Or i t o r ion Task I n v o l v e d

D i s c o v e r y of Word R e l a t i o n s h i p s S t u d i e s in Which t h e S u b j e c t s Learned. Sums

of S e r i e s S t u d i e s C endue t e d i n t he B l s a j e n t a r y S c h o o l

M a t b e ro a t i c s Cls a a room S l ' ud i s s Conduct 3d in t h e J u n i o r and S e n i o r

High Schoo l M a t h e m a t i c s C l a s s r o o m S t u d i e s Conduc ted in t h e C o l l e g e M a t h o u i s f c i ca

C lo a s r o om Surras a r y

I I I . BXPKHIMENTAL I)SSI'S!! AND EXPERIMENTAL PROCEDURES . . . . . . . . . . 80

I n t r o d u c t i o n The S e t t i n g of the S t u d y The H?xper irae»ta 1 Dc a ig iT Tho Ts h ch i n g Ke thc-da The T e s t i n g Program Methods f o r T r ? a » l » g febe Data Sumrcary

4 -? f

Chapter P a g e

IV. T H E R E S U L T S O P T H E ' E X P E R I M E N T A L STUDY . . . . 119

I n t r o d u c t i o n

T h e Findings for T e a t X T h e F i n d i n g s f o r T e s t I I

T h e F i n d i n g s f o r T e a t I I I

T h e Findings for T e a t IV S u m m a r y

V. S U M M A R Y , FINDINGS AND CONCLUSIONS, AND R E C G K K E N D A T I O N S . 1 3 2

S m m m r y

F i n d i n g s u n a C o n c l u s i o n s

R Q c o i a r a e s c a t i o n s

A P P E N D I X . . . 1 I } 1

Appendix A A C o u r s e O u t l i n e for C o l l e g e M a t b o i a a t i c a

A p p e n d i x B

Sample L e s s o n s for t h a Guided D i s c o v e r y '

M e t h o d o f T e a c h i n g

Appendix C S a r a p l o • L e s s o n s f o r the E x p o s i t i o n M e t h o d

o f T e a c h i n g

A p p e n d i x D

T e a c h e r Made T e s t a

A p p a n d i x E

TOG Raw Data

B I B L I C G - R A P H Y . . . . . . . . . . . 2 3 8

LIST CP TABLES'

Table Paga

I . C l a s s i f i c a t i o n of S u b j e c t s a t the Beginning of the Exper imen t 88

I T . C l a s s i f i c a t i o n of S u b j e c t s Whose Scores I'ere Used in the A n a l y s i s of Data . . . . 91

I I I . A n a l y s i s of V a r i a n c e Table f o r A ,C .T . # t !**% ... t » % ««». & a

o f t h e Exper iment . . V <1 • * • • • . . . 93

IV. A n a l y s i s of Va r i ance Table f o r Te s t- T X *

. . . 123

V. Analys i s of Var i ance Table f o r Tes t I I . . . . 12I4

VI. A n a l y s i s of Va r i ance Table f o r Tea t I l l . . . 126

VII . A n a l y s i a of Var i ance Table f o r Tes t IV . . . . 128

CHAPTER I

ST A TKKFNT OP THE PROBLEM

A l a r g e p r o p o r t i o n of the e d u c a t i o n a l r e s e a r c h of

today i s cen te red around the s tudy of the t e a c h i n g -

l e a r n i n g p r o c e s s , A g r e a t dea l ia be ing l ea rned about

which t each ing p rocesses a re and. a r e not a p p r o p r i a t e "

f o r e f f e c t i v e l e a r n i n g under given condi fciuna. This

knowledge ia be ing a p p l i e d in the development of new

c u r r i c u l a in many a r ea s of s t u d y . At the p r e s e n t t ime ,

knowledge concerning the ch'trig<*Xoarning p rocess ia

s t i l l l i m i t e d , ka a r e s u l t , cur r icu lum inn ova t i ons a re

o f t e n based on hypotheses which have not been adequa t e ly

t a s t e d . Tills n e c e s s i t a t e s the t e s t i n g of new c u r r i c u l a

and now teach ing methods through c a r e f u l l y c o n t r o l l e d

r e s e a r c h . Such r e s e a r c h can be used to determine

whether now teach ing methods or new c u r r i c u l a a c t u a l l y

accompliah the goa ls which they a r e des igned to

accompl i sh .

One of the t each ing methods which has been developed

as the r e s u l t of r e s e a r c h d e a l i n g wi th teach ing and

l e a r n i n g ia the d i s cove ry method of t e a c h i n g . Thia

method of t each ing haa i i a e a r l y beg inn ings in a t t emp t s

to fo rmula t e c u r r i c u l a and methods of t each ing coapa t i b i a

w i t h the t e a c h i n g s of G s s t a l t p s y c h o l o g y . Al though

r e s e a r c h r e l a t e d t o the d i s c o v e r y method of t e a c h i n g has

been conduc ted d u r i n g the p a s t t h i r t y y e a r s , s e r i o u s

a t t e m p t s to deve lop c u r r i c u l a emphas iz ing the d i s c o v e r y

method of t e a c h i n g were n o t made u n t i l a f t e r 195>0.

Curric-ula which emphasize the d i s c o v e r y method of

t e a c h i n g have been deve loped f o r e l e m e n t a r y and s e c o n d a r y

s c h o o l matberaat ics and s c i e n c e . Programs f o r ma themat i c s

have beon deve loped by the U n i v e r s i t y of I l l i n o i s

Committee on Scbool M a t h e m a t i c s , the G r e a t e r C l e v e l a n d

Mathemat ics P r o j e c t , and the Madison P r o j e c t of Sy racuse

U n i v e r s i t y . Programs f o r s c i e n c e have been developed"

by the Sc i ence Curr icu lum Iraprovcment S tudy and the

American A s s o c i a t i o n f o r the Advancement of S c i e n c e .

A t t empts a r e p r e s e n t l y b e i n g made t o deve lop s i m i l a r

programs in the language a r t s and in the s o c i a l s t u d i e s .

Since prog-:.'are3 emphas iz ing d i s c o v e r y methods of t e a c h i n g

hava been s u c c e s s f u l l y deve loped f o r e l e m e n t a r y and

s e c o n d a r y s c h o o l matbewat ica and s c i e n c e , i t i s of

i n t e r e s t to c o n s i d e r t he use of d i s c o v e r y methods of

t e a c h i n g a t the c o l l e g e l e v e l .

S t a t emen t of the Problem

The problem under c o n s i d e r a t i o n was the e f f e c t i v e n e s s

of a gu ided d i s c o v e r y method of t e a e h l e g ma themat i e s aa

compared to the e f f e c t i v e n e s s of an e x p o s i t i o n method of

teaching matherriaties in a c o l l e g e freshman mathematies

course for non-ma thema t i e s and non - sc ience majors.

Purposes of the Study

During the p a s t few year a s e v e r a l new programs for

ma thema t i c a in the e lementary and. secondary school have

been deve loped . Some of these programs no t only emphasize

n evr con tan t and nsxf o r g a n i z a t i o n of c o n t e n t , b u t new

t each ing methods as w e l l . One of the moa t f r e q u e n t l y

advocated methods i s the d i scovery method of teaching.

In order to p rov ide a r e f e r e n c e f o r r e s e a r c h r e l a t e d to

the d i s cove ry method of t each ing ma thema t i c s , a summery

of research l i t e r a t u r e r e l a t e d to th i s method of t each ing

ma thema t i c s i s p r e s e n t e d . Since many mathematics

educa tors advocate the use of the d iscovery method of

t e ach ing mathematics a t the e lementary and secondary

l e v e l s , i t waa considered to have p o t e n t i a l as a method

of t each ing a t the c o l l e g e l e v e l as v e i l . The purpose of

t h i s s tudy was to a s c e r t a i n the v a l u e , a a determined by

s t u d e n t achievement, of using a d i s cove ry method of

teaching mo thema t i c s in a c o l l e g e freshman mathematics

course f o r non-mathematics and non-science majors.

Hypotheses

The s tudy was des igned to t e a t the fo l lowing

hypo theses :

1 . S tudents t augh t "by a guided d i s cove ry method w i l l

score s i g n i f i c a n t l y h igher on the t e s t , Coopera t i v e Ma t h e -

re a t i c a Tea t s , S t r u c t u r e of the Number System, than s t u d e n t s

t augh t by an e x p o s i t i o n method.

2 . Students t augh t by a guided d i s cove ry method w i l l

score s i g n i f i c a n t l y h igher on the t e s t , Coopera t ive Mathe-

rnstic3 T e g t s , Algebra I , than s t u d e n t s t augh t by an

e x p o s i t i o n method.

3 . S tudents t augh t by a guided d i s c o v e r y method w i l l

score s i g n i f i c a n t l y h igher on the VJataon-Glaaer C r i t i c a l

Thinking Appra i s a l than s t u d e n t s t augh t by an expos i t i o n

method.

1|. S tudents t augh t by a guided d i s c o v e r y method w i l l

raake s i g n i f i c a n t l y h ighe r grades in the course than s t u d e n t s

t augh t by an expos i t i on method.

5 . The r e l a t i v e e f f e c t s of a guided d i s c o v e r y method

of t each ing and an e x p o s i t i o n method of t each ing on

achievement as raeasured by the t e s t , Cooperat ive Math <3-

mat ica T e a t s , S t r u c t u r e of the Nurahag Sâfcorn, w i l l no t be

dependent upon s t u d e n t a b i l i t y .

6 . The r e l a t i v e e f f e c t s of a guided d i s c o v e r y ae thod

of t each ing and an expos i t i on method of t each ing on

achievement as measured by the t e a t , Coopera t ive Ma t h e -

rea t i c s Te s t s , A Igebra I , w i l l not be dependent upon

s t u d e n t a b i l i t y .

7 . The r e l a t i v e e f f e c t s of a guided d iscovery method


achievement aa measured by the taon-Glaser Crit ica 1

Thinking Appraisal w i l l not ba dependent upon student

a b i l i t y .

8 . The r e l a t i v e e f f e c t s of a guided d iscovery method

of t each ing and an expos i t i on method of teaching on

achievement as measured by the s t u d e n t ' s grade in the

course w i l l not be dependent upon s t u d e n t a b i l i t y .

Background and S i g n i f i c a n c e of the Study

The discovery method of t each ing has i t s f ounda t ions

in tha w r i t i n g s of the GeataIt p s y c h o l o g i s t s , According

to Gogne':

This i d e a , having i t s o r i g i n p a r t l y in the w r i t i n g s of member3 of the G e s t a I t school of psychology, ha3 had a profound i n f l u e n c e on mathematics and mathe-mat ics t e a c h e r s . B a s i c a l l y the no t ion i s t h a t p r o -d u c t i v e t h i n k i n g , or i nven t ive problem s o l v i n g , i s achieved by " i n s i g h t / 1 which r e p r e s e n t s an e s s e n t i a l r e o r g a n i s a t i o n of. the e n t i r e f i e l d in which the i n d i v i d u a l and the problem a r e l o c a t e d . What b r i n g s about t h i s o r g a n i z a t i o n ia a ne t of f a c t o r s which tend to s t r u c t u r e the i n s i g h t p r o c e s s — l i k e c l o s u r e , pr agnanz, symmetry, and others.-®-

Iv'ertheimer f e l t t h a t when t each ing i t is preferable

to proceed in a manner which f avors d iscovery of the

"R. 'M. C-agne', "Impliest ic-na of Some Doc t r ines of Ma thaws t i c s Teaching fo r Research In Human L e a r n i n g , " Research Problems in Mathematics Educa t ion , (Wash ins? ton _ - •

6

e s s e n t i a l na tu r e of the p rob lema t i c s i t u a t i o n , ^ of j u a t

what ia needed to so lve the problem, 3 p $ t h a t , even a t

the c o a t of e legance or b r e v i t y , the s o l u t i o n of the

problem ia s e n s i b l e r a t h e r than b l i n d and mechanica l A

Wr i t i n g in a s imi la r v e i n , A u sub e l s t a t e s t h a t in

d i s c o v e r y l e a r n i n g "The l e a r n e r must r ea r r ange a given

a r r a y of informs t ion , i n t e g r a t e i t wi th e x i s t i n g c o g n i t i v e

s t r u c t u r e , and r eo rgan ize or t ransform the i n t e g r a t e d

comoina t i on in such a way as to c r e a t e a d e s i r e d end

p roduc t or d i s cove r a mis s ing means-end r e l a t i o n s h i p , ®

Many e n t h u s i a s t i c claims have been made concerning

d i s cove ry t e a c h i n g . For example, Kersh s t a t e s t h a t

. . » meny would agree t h a t •when the s t u d e n t l e a r n s by d i scove ry ho (1) unders tands what he l e a r n s , and so ia b e t t e r able fco remember and t r a n s f e r i t ; (2) he l e a rn s something the p s y c h o l o g i s t c a l l s a " l e a r n i n g s e t / 1 or a s t r a t e g y fo r d i s c o v e r i n g new p r i n c i p l e s , , snd (3) he develops an i n t e r e s t in what he l e a r n e d .

Although the proponents of d i s cove ry t each ing have made

many claims favor 5ng t each ing by d i s cove ry me thuds , those

claims have no t been adequa t e ly t e s t e d through the use of

2Max Wertheirner, P roduc t ive T h i n k i n s , en la rged e d i t i o n , (Now York, lV59)TpT~ITS — — *

3 r o i d . , p . 9 1 . ^ I b i d . , p . 1 1 7 .

-'David p . i iusuoel , ".In Defence of Verbal Lea i 'ning Res dings io the Psychology of C o g n i t i o n , e d i t e d by Richard C. Anderson cnT 1)1 vTd P^'Fua'ulSal {-^w York, 196^) . p . 90. - . . , ? , ,

A B e r t y . Kersh- i— •

c a r e f u l l y c o n t r o l l e d r e s e a r c h . Accord ing to Cronbacb

In s p i t e of the c o n f i d a n t endorsements of t e a c h i n g th rough d i s c o v e r y t h a t we r e a d in semi -p o p u l a r d i s c o u r s e s on improving e d u c a t i o n , t h e r e i s p r e c i o u s l i t t l e s u b s t a n t i a t e d knowledge a b o u t v h a t advan tages a c c r u e . Ye b a d l y need r e s e a r c h In which the r i g h t q u e s t i o n s a r e a sked and t r u s t w o r t h y scawers o b t a i n e d . ' '

Al though many r e s e a r c h s t u d i e s compar ing d i s c o v e r y

methods of t e a c h i n g w i th o t h e r methods of t e a c h i n g have

been c o n d u c t e d , many q u e s t i o n s reroain unanswered . Many

of t h e a t u d i e a t h a t have been conduc ted have y i e l d e d l i t t l e

u s e f u l i n f o r m a t i o n b e c a u s e the e x p e r i m e n t a l d e s i g n s and

Sxperiiaenfca 1 s t r a t e g i e s employed have been i n a d e q u a t e .

A f t e r r e v i e w i n g r e s e a r c h l i t e r a t u r e r e l a t e d to d i s c o v e r y

methods of t e a c h i n g , Worthsn conc luded t h a t

Koat "d i scovery 1 1 s t u d i e s have been conduc ted in l a b o r a t o r y s e t t i n g s and c o n s e q u e n t l y have d e a l t K i th assai l t ime s a m p l e s , sras 11 numbers of s u b j e c t s , and v e r y d i s c r e t e end o f t e n m a n i p u l a t i v e l e a r n i n g t a s k s . One might a rgue t h a t auch sampl ing of t i m e , s u b j e c t s , and t a s k s i s so r ea t r i c t i v e ' and l i m i t e d in scope t h a t any a t t e m p t to g e n e r a l i z e the r e s u l t s to c l a s s room l e a r n i n g or i n s t r u c t i o n would be s u b j e c t to .a e r i ou3 q u e s t i o n , • *

Sicca ev idence r e l a t e d to d i s c o v e r y Methods of t e a c h i n g

ia l i m i t e d , the e d u c a t o r i n t e r e s t e d in d i s c o v e r y methods

^Lee J . Cronbacb, "The Logic of Exper iments on D i s c o v e r y , " Lea rn ing b j D i s c o v e r y : A C r i t i c a l A p p r a i s a l , e d i t e d b y L a e ' I T I T f C i l H a r / W d ^ v s r ; R , ™ K e i F I i T T c F i t i g o 7 T 9 6 6 ) , p . 7? .

O B. R . Worthen, UA Comparison of D i s c o v e r y and Ex-

p o s i t o r y Sequencing in E l emen ta ry Mstbc-raatics I n s t r u c t i o n , " 2JI Ha t h e m a t i c a E d u c a t i o n , The R a t i o n a l C o u n c i l of

Teachers ol Mathe tsa t ics T V a s H I F ^ c n . 1967) , p . ) £ .

8

of t each ing must make h i s oun dec i s ions on the b a s i s of

the evidence a v a i l a b l e . In order to l e s sen the e f f o r t

t h a t such a t ask r e q u i r e s , t;h e second chap t a r of t h i s

s tudy con ta ins a co>npr ehans ive review of r e s e a r c h s t u d i e s

d e a l i n g wi th t op i c s r e l a t e d to d i s c o v e r y methods of

t e a c h i n g reathereatioa.

Seve ra l programs emphasizing d i s c o v e r y siethoda of

t each ing have been developed f o r e lementary and secondary

l e v e l courses in me thsmat i c s . Examples of such programs

a re those developed by the U n i v e r s i t y of I l l i n o i s Coram i t t ee

nn School Mathematics end the Madison P r o j e c t of Syracuse

"Univers i ty . Soma who have t augh t courses us ing t each ing

methods recommended by these groups and m a t e r i a l s d e v e l -

oped by thea a groups have been e n t h u s i a s t i c about the

r e s u l t s . 9 T h e r e f o r e , i t i s of i n t e r e s t to cons ide r the

use of d i s cove ry methods of ' t each ing mathmwa t i c s a t the

c o l l a g e l e v e l . This s tudy wag designed to p rov ide

in fo rma t ion concerning the value of us ing a guided

Q 'Henry vfcantiiab, , JSeventh G-i-sdera Volunteer f o r

A f t e r - S c h o o l Glasses in A l g e b r a , " The Mathematics Teacher , L11I (December, I 9 6 0 ) , 6ijO~6ij3. RWs f a y l o r T ^ T O r T T " - " Course in A Ig eb r a - -U, T. <3. S ,M. and .S ,a.—A Comparison."

Ma th em a t i c s Teach er , LV {October, 1962) , )478 -I181. S. E. Sigurdson and i l a l i a Boychuk, "A F i f t h - G r a d e S tuden t S ^ C o o o r 2 ? 6 r o » f ' 3&e A r i t h m e t i c Teacher , XIV ( A p r i l , 1967) , £ * - '^'>'rr:an G s D a i» "An fe'xperlraental Twelf th-Grade Mattoeraaticsngourse,» The Mathematics Teacher , LX ( A p r i l ,

discovery method of t each ing sia thsita t i ca in a co l l ege

freshman matherna t i c s course fo r non-rna themat ic a and

non-science m a j o r s . The s tudy u t i l i z e d four s ectiona of

a r e g u l a r c o l l e g e mathematics couv2a and vas conducted

dru ing an e n t i r e series tar .

De f in i t i on of Terms

1 . A guided d i scovery 'method of t e a c h i n g . A guided

d i scove ry Method of teaching is a method of t each ing in

which each concept or p r i n c i p l e i s t augh t through an

i n s t r u c t i o n a l sequence c h a r a c t e r i z e d by a sequence of

s t eps as descr ibed bclcw,

In the f i r s t s t ep of tb3 sequence the s tuden t s are

asked to work exe rc i se s dea l ing with the concept or

p r i n c i p l e to be learned . These e x e r c i s e s a r e designed to

guide the s t u d e n t to d i s cove ry of the concoct or p r i n c i p l e

During the second s t e p of the sequence the i n s t r u c t o r

teata f o r d i scovery . This occurs a f t e r the s t uden t s have

completed the i n t r o d u c t o r y e x e r c i s e s . In order to

determine whether d iscovery has occurred, the i n s t r u c t o r

asks c a r e f u l l y s e l ec t ed q u e s t i o n s . These ques t ions are

such that i f d i s cove ry baa occurred thoy w i l l be answered

e a s i l y and quickly , out i f d i s cove ry has not occurred they

w i l l be answered with d i f f i c u l t y or not a t a l l .

10

I f t e s t i n g r e v e a l s that d i s c o v e r y baa not o c c u r r e d ,

the th ird s t ep of the sequence i s in the form of mora

e x e r c i s e s . These exere i s aa are d e s i g n e d to g i v e the

s t u d e n t s fur ther guidenee toward d i s c o v e r y of t he concept

or p r i n c i p l e t o be l e a r n e d . A f t e r t h e s e e x e r c i s e s have

bean comple ted t h e r e should be a n o t h e r t e s t f o r d i s c o v e r y .

This sequence of e x e r c i s e s f o l l owed by t e a t s f o r d i s c o v e r y

shou ld be r e p e a t e d u n t i l a m a j o r i t y of the s t u d e n t s bsve

d i s c o v e r e d the c o n c e p t or p r i n c i p l e t o be l e a r n e d .

A f t e r d i s c o v e r y ha a occu r r ed the a tuden t ia g iven

6.xereisea which p r o v i d e him w i t h an o p p o r t u n i t y t o use

what be has d i s c o v e r e d . Whenever p o s s i b l e t h e s e cxare iaea

shou ld be d e s i g n e d t o gu ide the s t u d e n t t o the n e x t

d i s c o v e r y .

The f i n a l s t e p in the sequence c on a i s t s of naming the

concep t or p r i n c i p l e d i s c o v e r e d and in some c a s e s t h i s

s t e p a l s o i n c l u d e s the development of a f o r w a l s t a t e m e n t

of the concep t or p r i n c i p l e d i s c o v e r e d . I f the s t u d e n t s

p o s s e s s the v e r b a l c a p a c i t y t o do ao , the s t a t e m e n t of the

c o n c e p t or p r i n c i p l e shou ld be f o r m u l a t e d by the a t uden t a .

In order to p r o v i d e a d e q u a t e time f o r each s tudent to

d i s cover for h i m s e l f the c o n c e p t to be l e a r n e d , the f o r m a l

s t a t e m e n t of the c o n c e p t shou ld be d e l a y e d a t l e a s t one

c l a s s p e r i o d . In some c s a e s i t may be d e s i r a b l e t o d e l a y

the s t a t e m e n t of the c o n c e p t f o r a c o n s i d e r a b l e l e n g t h of

11

2 . An exposition r;.&tbod of t s a o b i c g . An e x p o s i t i o n

method o f t 6 3 c n i. D- g i;j a ms thod of teaching i n wh ich e a c h

c o n c e p t or p r i n c i p l e i s t a u g h t through an i n s t r u c t i o n a l

sequence chareterized b y a sequanca o f a t a p s a s d e s c r i b e d

below.

The f i r s t s t e p of the instructional sequence c o n s i s t s

o f a formal statement o f t h e concept o r p r , h i c . i p l a t o b e

I s a r n s d . This a t a t e r a a n t is presented either by t h e i n -

s t r u c t o r or in t h e instructional raa t&ri&ls ». I f tb-3 c o n c e p t

or p r i n c i p l e h a s a f o r t i a l asaie, thi;i n&rna ia pressBte-d

a l o n g wiih t h e s o f the concept or p r i n c i p l e .

Tats s t a t ^ i x c n t of t h e principle i s followed by a

discussion or b y a a b o r t lecture by t h e J n s t r u o t o r . The

pjrpcso> cf this disc us a ion c r I c o t a r a is t o c l a r i f y the

concept a n d to h e l p tbe student understand the concept.

T n i a disucssi&n o r l e c t u r e should include illustrations

o f how t h e principle or concept can be uaed t o ;:?olva

spec i f i c p r obleing .

A f t e r the f i r s t two s t e p s of tbe i n s t r u c t i o n a l

scqusnce have b e e n completed the atudents should be

p r o v i d e d w i t h o p p o r t u n i t i e s to uso the concep t p r e s e n t e d

to s o l v e s p e c i f i c p r o b l e m s . In many c a s e s tbe firat two

18 ox. tut» it«ji3-cr uc ziodo 1 sequwiicc &'.•» jf be x cpes t ed

several t i r oes before tbe final s t e p is reached. T h a t ia,

several eoncepta or principle a may b e presented during o n e

12

pr inc ip les may be in the form of exe rc i s e s to be completed

between c l a a s p e r i o d s .

3. A co l l ege ma thema t i c a course f o r non-raa thema fcica

and non-sc ience m a j o r s . A co l l age mathematics course for

n on-ma tberna tioa and noa-acisnca majors re fers to the

course Collage Mathematics 8 t Southwestern State Col lege ,

V*ea fherf ord, Oklahoma . This la a r equ i red course for a l l

non-mathematics and non-sc ience majors and is designed to

serve as an i n t r o d u c t i o n to ma tberna t i c a for l i b e r a l a r t s

m a j o r s . In the teacher education program i t ia a

r equ i r ed course f o r a l l p r o s p e c t i v e elementary school

t e a c h e r s .

i | . A b i l i t y . An a b i l i t y ia the a c t u a l power and

tendency of an organism to respond in a predetermined way

to a given s e t of s t i m u l i . An a b i l i t y ia the a c t u a l

power to perform an a c t and the tendency to perform that

a c t in the a p p r o p r i a t e s i t u a t i o n a . This a c t may be

phys ica l 'or m e n t a l . Since the power and tendency of an

organism to respond may change with t ims , a b i l i t i e s a re

dependent upon t ime. In t h i s s tudy the term "abi l i ty"

ia uaed to re fer to the d a s a of a b i l i t i e s as soc ia ted

tvit/h mathematics „ inc8 a b i l i t i e s are d e o e n d e nt upon

t ime, the time r e f e r r e d to when the term ' 'abi l i ty" ia

used i s tne beginning of the 1968 sp r ing semes te r .

5 . Achievement, In t h i s s tudy the term "achievement"

i s used to r e f e r to toe c i aas of a b i l i t i e s a a s o c i s t e d with

13

t h a t p a r t of mathematics s t u d i e d in College Mathemat ics ,

the time be ing the end of the 1968 sp r ing seines t a r .

Among those a b i l i t i e s inc luded under tha terra "achievement"

a r e thoaa r e l a t e d to the a p p l i c a t i o n of l e a r n i n g in new and

d i f f e r e n t s i t u a t i o n s .

L i m i t a t i o n s of tha Study

1 . Tha s tudy was l i m i t e d to s t u d e n t s e n r o l l e d in

fou r s e c t i o n s of College Matfceraatica a t Southwestern S t a t e

Co l l ege , 77eatherford , Oklahoma, dur ing the sp r ing term of

the 1967-1968 school y e a r .

2 . Tha s tudy i n v e s t i g a t e d the e f f e c t i v e n e s s of only

one of s e v e r a l d i s c o v e r y methods of t e a c h i n g .

.3. The s tudy was l i m i t e d to tha of f a c t a of two

t each ing methods on s t u d e n t ach ievement .

Bas ic Ass urnp t i o n s

1 . I t was assumed thab the four s e c t i o n s of Col lege

Mathematica used in the s tudy were r e p r e s e n t a t i v e of a l l

s e c t i o n s of College Mathematics t aught a t Southwestern

S t a t e C o l l e g e .

2 . I t was assumed t h a t a c t i v i t i e s ou t s ide the mathe-

ma t i c s c lassroom and r e l a t e d to the l e a r n i n g of mathematics

were ba lanced between the groups and t h e r e f o r e d id no t

s i g n i f i c a n t l y a f f e c t the s t u d y .

CHAPTER U

SURVEY OP RELATED RESEARCH LITERATURE


The purpose of t h i s chap te r i s to p r e s e n t a

comprehensive summary of r e s e a r c h l i t e r a t u r e r e l a t e d

to d i s c o v e r y methods of t e a c h i n g . Tho s t u d i e s reviewed

in t h i s chap te r a r e l i m i t e d to those which r e p r e s e n t

conc lus ions based on r e s e a r c h and n o t raara p e r s o n a l

opinion and which compare forma of i n s t r u c t i o n invo lv ing

a one degree of d i s c e v e r y by the l e a r n e r wi th other

Methods of i n s t r u c t i o n . Since t he re a re va r ious

forms of i n s t r u c t i o n through d i scove ry t h i s chap te r

w i l l a t t e m p t to p r e s e n t soma of the form3 of d i s cove ry

teaching and the r e s u l t s obta ined through the use of

these methods . In a d d i t i o n t h i s chap te r is an a t tempt

to provide s r e f e r e n c e fo r r e s e a r c h r e la bed to d i s c o v e r y

metboda of t each ing mathematics .

The s t u d i o s reviewed in t h i s chap te r a r e organised

accord ing to the type of t a sk or m a t e r i a l l e a r n e d by

the s u b j e c t s in the s tudy rev iewed . The review of the

l i t e r a t u r e i s p r e sen t ed accord ing to the fo l l owing

s u b d i v i s i o n s ; the s t u d i e s of George Kafcona and r e l a t e d

s t u d i e s , s t u d i e s using coding problems as the c r i t e r i o n

15

task , atud3.es in which the c r i t e r i o n task involved

discovery of word r e l a t i o n s h i p s , s t u d i e s in which the

s u b j e c t s learned surns of a e r i e s , s tudies conducted

in the e lementary school mathematics classroom, s t u d i e s

conducted in the j u n i o r and s en io r high school mathe-

mat ics c lass room, and s t u d i e s conducted in the c o l l e g e

Ma fcbema t i c s classroom.

The S tud ies of George Katona

and Re la ted S tud ies

George Katona, one of the mors prominent members

of the g e a t a l t school of psychology, was among the

e a r l i e s t w r i t e r s to r e p o r t s t u d i e s in which c e r t a i n

forms of d i scove ry methods of teaching were compared

wi th o ther methods of t e a c h i n g . Katona- advanced the

hypo thes i s t h a t t he re i s mora than one kind of l e a r n i n g ,

This i s in c o n t r a s t to the hypo thes i s t h a t a l l l e a r n i n g

can be thought of as the a c q u i s i t i o n of a s s o c i a t i o n s

or c o n n e c t i o n s . In order to obta in evidence to suppor t

h i s hypo thes i s Katona s e l e c t e d s e t s of f a c t s which

could be l ea rned in mora than one way. i t was then

hypo thes i zed t h a t the way in which a s e t of f a c t a ia

learned w i l l a f f e c t r e t e n t i o n of these f a c t s and the

way in which these f a c t a w i l l be used . The f i r s t s e t

•• •• G e o r g e Katona, Organizing and .Memorizing (New York, 19i*0), pp ,"T~31". .—2.

3.6

of f a c t s used t o t e a t t h e s e h y p o t h e s e s c o n s i s t e d of a

c l a s s of c a r d t r i c k s . S e v e r a l e x p l o r a t o r y and on a group

expe r imen t was conduc ted in which the l e a r n i n g t a s k con -

s i s t e d of l e a r n i n g how . to p e r f o r m the c a r d t r i c k s .2 j n

each expe r imen t one group of s u b j e c t s was r e q u i r e d t o mem-

o r i s e the s o l u t i o n to one or more c a r d t r i c k s . The

p r i n c i p l e invo lved in the a o l n t i o n t o t he c a r d t r i c k s was

e x p l a i n e d to a n o t h e r g r o u p . The fires t group waa c a l l e d

the ineKoriza t i o n group and the second group was c a l l e d the

u n d e r s t a n d i n g g r o u p . In each of the expe r imen t s i t was

found t h a t the two groups were s i m i l a r in a b i l i t y to p e r -

form the c a r d t r i c k s t a u g h t . A f t e r one or more weeks the

s u b j e c t s were asked t o p e r f o r m the c a r d t r i c k s t a u g h t . On

the a e t e s t a the pe r fo rmance of the u n d e r s t a n d i n g group waa

s u p e r i o r to the pe r fo rmance of the memor 12a i;ion g r o u p .

Immed ia t e ly a f t e r i n s t r u c t i o n the s u b j e c t s were p r e s e n t e d

w i t h c a r d t r i c k s d i f f e r e n t f r o m , b u t s i m i l a r t o , the c a r d

t r i c k s t s u g h t . :/hen the s u b j e c t s wore asked, t o pe r fo rm

cnese t r icka 3 the p s r f o r m s n c o of the unde r s t snd in s j <?r dud

was s u p e r i o r t o the pe r fo rmance of fcbo ir,3/ra'>ri,sa b.icn Rroup.

S e v e r a l a d d i t i o n a l expe r imen t s were conducted in which

the s u b j e c t s were shown ca rd t r i c k s and then asked to d i s -

cover the s o l u t i o n t o tne t r i c k w i t h o u t h e l p . On the b a s i s

of t h e s e expe r imen t s i t waa conc luded t h a t :

2 I b i d . , p p . 32-5)4.

17

. . . I f t b e p r i n o ; ' . p l a b f iS h a an d i s c o v e r e d w i t h -o u t a s s i s t a n c e , r s t e n t i o n fa p u n s t o e n d u r e e v e n l o n g e r t h a n a f t e r I s a r m i n g b y ur .d-sr s t a n d i n g . ' f a s t s m a d e t w o m o n t h a a f t e r t h e l e a r n l e g { o r s o l v i n g ) p e r i o d s w i t h a f e w s u b j e c t s s h o w e d t h a t t b e b e s t r e s u l t s w e r e o b t a i n e d b y t h o s e w h o b a d s o l v e d t h e p r o b i era a I o n s . 3

I n o r d e r t o o b t a i n n o r e i n f o r m a t i o n c o n c e r n i n g t b a

e f f e c t s o f m e t h o d s o f l a s t r u e fclon on l e a r n i n g K a t o n a ^ * c o n -

d u c t e d a n o t h e r s e r i e s o f s x p o r L m o a t a u s i n g a d i f f e r e n t s a t

o f l e a r n i n g t a s k s . I n e a c h l e a r n i n g t a s k t h a s u b j e c t w a s

p r e s e n t e d w i t h a g e o m e t r i c a l f i g u r e c o n s i s t i n g o f a s e t o f

s q u a r e s . I n e a c h f i g u r e s soma o f t b a s q u a r e s b a d s i d e s i n

c o m m o n . T b e s u b j e c t v a s a s k e d t o m o v e s cxno o f t b a U n a a

w h i c h f o r r n e d a i d e a o f s q u a r e s a n d p r o d u c e a f i g u r e c o n -

s i s t i n g o f f e w e r , c r i n s o me e s s a s m o r e , s q u a r e s t h a n

c o n t a i n e d i n t h a o r i g i n a l f i i ^ r e . Toes n a w f i g u r e s h o u l d

i n c l u d e t h e o & mo n if r n b s r o f l i n e s a a t h a o l d f i g u r e a n d t h a

s q u a r e s s h o u l d b a t h a s a m e a i s e a s i n t h e o r i g i n a l f i g u r e .

S u c h f i g u r e s c a n b e c o n s t r u c t e d w i t h rcate'n s t i c k s a n d t h a

t a s k c a n t h a n b e c o m p l e t e d b y m o v i n g a l i m i t e d n u r r b a r o f

a a t c h s t i c k s . T h a f i n a l f i g u r e m u s t i n c l u d e a l l t h e m a t c h

• s t i c k s a n d i t i s n o t p a r m i s s a b l a t o p l a c e o;ie mat*-?,a s t i c k

on t o p o f a n o t h e r . A l t h o u g h t h e f l g a r c a w o r e n o t a l w a y s

c o n s t r u c t e d u s i n g Aotui. 1 riiS'con. s t i c k s f t b a o x p e r i i n e n t s o f

t h i s s e r i e s v e r a c a l l e d rca fcch—s t i c k e x p e r i r r i tants .

3 l b i d . , p . 5 0 .

^ l b i d . , p p . 3 1 - 1 0 7 .

18

I n each e x p e r i m e n t one g roup of s u b j e c t s was r e q u i r e d

t o memorize t h e s o l u t i o n t o or;a mafcnb -a t i ck p r o b l e m .

A n o t h e r g roup was shown t h e - s o l u t i o n t o t h r e e p r o b l e m s .

Tha f i r s t g roup waa c a l l e d t he m e m o r i z a t i o n g r o u p arid t h e

s econd g roup waa c a l l e d t he example g r o u p . D u r i n g the

l e a r n i n g p e r i o d t h e s u b j e c t s of t h e example g r o u p were p r e -

s e n t e d w i t h a m a t c h - s t i c k p rob l em and a s k e d t o a t t e m p t t o

s o l v e i t . A f t e r h a l f a m i n u t e t he s u b j e c t s ware ahown a

s o l u t i o n t o the t a s k . Tola p r o c e d u r e was r e p e a t e d w i t h

t h r e e d i f f e r e n t match t a s k s . By s t u d y i n g t h e p r o c e d u r e s

uaed in t h e s o l u t i o n of t h e s e p r o b l e m s t h e s u b j e c t s were

e x p e c t e d t o f o r m u l a t e f o r t h e m s e l v e s ( d i s c o v e r ) t h a p r i n -

c i p l e u sed t o s o l v e t h e p r o b l e m s . I m m o d i a t e l y a f t e r tha

l e a r n i n g p e r i o d tha s u b j e c t of b o t h g r o u p s were a s k e d t o

e x h i b i t s o l u t i o n s t o t h e p rob lems p r e s e n t e d d u r i n g t h e

l e a r n i n g p e r i o d . Then s e v e r a l new t a s k s were p r e s e n t e d

f o r s o l u t i o n . Pour weeks l a t e r t h e s u b j e c t s were a g a i n

a s k e d t o s o l v e tha t a n k s p r e s e n t e d d u r i n g the l e a r n i n g

s e s s i o n and t h e y were a l s o a s k e d t o s o l v e some new t a s k a .

On t h e t e a t a drain ia t a r e d imrnedia t e l y a f t e r t h e l e a r n -

i n g s e s s i o n b o t h g r o u p s p e r f o r m e d s q u a l l y v e i l on t h e t a s k s

p r e s e n t e d d u r i n g t h a l e a r n i n g s e s s i o n , b u t f o u r weeks l a t e r

t h e exiJEjpô g roup e x h i b i t e d s u p e r i o r p e r f or ma nee on t h e s e

t a s k s . On the t e s t a d m i n i s t e r e d irtniedia t e l y a f t e r t h e

l e a r n i n g s e s s i o n and on the t e s t a dpi in i s t e r e d f o u r weeka

19

l a t e r t h e example g r o u p e x h i b i t e d super lor p e r f o r m a n c e on

t h o s e t a s k s which w e r e d i f f e r e n t f rom t h o s e p r e s e n t e d

d u r i n g t h e l e a r n i n g s e s s i o n . I n g e n e r a l , i t was c o n c l u d e d

t h a t l e a r n i n g by examples i s s u p e r i o r to l e a r n i n g b y

menjor iza t i o n .

D u r i n g the l e a r n i n g s e s s i o n t h e s u b j e c t s i n t h e example

g roup were e x p e c t e d t o f o r m u l a t e f o r tbernaelvea t h e p r i n -

c i p l e n e c e s s a r y f o r t h e s o l u t i o n of t h e problema .

I n v e s t i g a t i o n r e v e a l e d t h a t a b i l i t y t o v e r b a l i s e such a

p r i n c i p l e ia not c o r r e l a t e d w i th a b i l i t y to s o l v e m a t c h -

s t i c k prob lems .

H i l g a r d , I r v i n e , and Whipp le^ conducted an exper iment

s i m i l a r t o K3tona*a c a r d - t r i c k e x p e r i m e n t s . The exper iment

was d e s i g n e d to nsoet soma of t he c r i t i c i s m c o n c e r n i n g

K a t o n a ' a a-xparirnsat and t o add a d d i t i o n a l d i m e n s i o n s t o t h e

i n v e s t i g a t i o n of p r o b l e m - s o l v i n g b e h a v i o r . I n p a r t i c u l a r ,

the- f o l l o w i n g g e n e r a l i s a t i o n s we*3 tes ted .

. . . ( a ) The a d v a n t a g e of l e a r n i n g w i t h unders tand ing does n o t a e c e s s a r i l y w s h o w up in o r i g i n a l l e a r n i n g , f o r l e a r n i n g w i t h undera l a n d i n g way take l o n g e r than l e a r n i n g b y r o t a , (b ) He t e n t i o n a f t e r l e a r n i n g by u n d e r -s t a n d i n g tends t o be g r e a t e r t h a n ' r e t e n t i o n a f t e r l e a r n i n g b y r o t e , (c ) T r a n s f e r t o new r e l a t e d t a s k s i s g r e a t e r a f t e r l e a r n i n g b y , unders tand ing than a f t e r l e a r n i n g b y r o t a .

^ S r n e 3 t R. H i l g a r d , Robert P . I r v i n e , aod James E . Whipple , f?oto M e a i o r x z a t i o n , U n d s r s t a a d i n g s s a d T r 3 u 3 f er : An EJA. bene iun 01 Ka tons ' 3 Csrd—Trick l ixper iuients -Jourr-31 2 £ x l v i (Oc tober , 1953) * '286^292.

^ I b i d . , p . 28 8.

20

The s u b j e c t s f o r the experIrcant were s i x t y h igh

school s t u d e n t s . Each s u b j e c t was a s s i g n e d , by a random

p r o c e s s , to one of two groupa . One of the groups waa

c a l l e d the unders tand ing group and ths other group was

c a l l e d the mem or iaa t i on group. Oti tha f i r s t day of the

experiment a l l s u b j e c t s were t augh t two card t r i c k s . The

pnb j«c t s in the memorization group l ea rned the t r i c k s by

l e a r n i n g tbe order of the cards by r o t e memory.. The

unders tand ing group was t augh t a scheme from which tbe

proper order of the ca rds fo r any of a c e r t a i n c l a sa 'of

t r i c k s could be d e r i v e d . Both groups were given p r a c t i c e

and he lp u n t i l the t r i c k s could be performed w i thou t

e r r o r . On the second clay of tha experiment the s u b j e c t s

were given two kinds of t r a n s f e r t e a t s . The t r i e k a on the

f i r s t t e s t could be performed by simply t r a n s p o s i n g the

order of tbe cards neces sa ry to perform the o r i g i n a l

t r i c k s . Tbe t r i c k s on the second t o s t were baaed on

e n t i r e l y new p r i n c i p l e a .

I t was found t h a t the memorize t i on group l ea rned the

o r i g i n a l t r i c k s much v?ore qu ick ly than tbe s u b j e c t s in tbe

unders tand ing g roup . I t was a l s o found t h a t t he re was no

s i g n i f i c a n t d i f f e r e n c e between tbe two groupa in terms of

r e t e n t i o n a f t e r one day . Tbe unders tanding group waa

s i g n i f i c a n t l y supe r io r to the memorization group on both

t r a n s f e r t e s t a wi th the s u p e r i o r i t y be ing the g r e a t e s t on

21

the second t r a n s f e r t e s t . Irs general the r e s u l t s of the

experiment agreed with the r e s u l t s obtained by Katona.

Gorman' conducted an experiment s imi la r to the match-

st ick studies conducted by Ka tona. I t was predicted t h a t

only information appropr ia te to the c r i t e r i o n employed to

evaluate success w i l l be e f f e c t i v e as guidance. I t waa

f u r t h e r hypothesized t ha t where the information is inappro-

priate to the c r i t e r i o n employed no s i g n i f i c a n t e f f e c t s

w i l l ba discerned as a r e s u l t of the guidance. I t was

hypo thesized t h a t , given an appropr ia te c r i t e r i o n , the

e f f ec t i venes s of guidance w i l l increase d i r e c t l y aa the

amount: of j/oforiTia fclon supplied to tho sub jec t ia increflsed,

J i n a l l y , xt was hypothesized tha t the e f f e c t s of varying

kinds and amounts of information w i l l by s imi la r for sub-

j e c t a of higher and lower i n t e l l e c t u a l a b i l i t y . Two

c r i t e r i a for success vera i d e n t i f i e d . One c r i t e r i o n waa

a b i l i t y to f ind a method for solving the problem t i s k and

the other was a b i l i t y to v a r b - l i z e a p r i n c i p l e involved in

tha so lu t ion of the t a sk .

Two hundred t h i r t y - t h r e e twelfth-year s tudents were

given varying awounts of i n s t r u c t i o n concerning the method

of solution to be employed and the p r i n c i p l e to ba used in

Le JO4.U fcioo of ma tch-2 vicl; p-rcbleiss , Thr«»e l eve ls of

arri 5* O o f " a n * *The E f f e c t of Varying Amounts and Kinc.* oi la l or its a t ioc aa Guidance jn Pr cblojn Solvine " £5Z2k£3£fi&2£l M2H£S£££fe£, LXXI (1957), Whole Number l ^ l l

22

i n f o r m a t i o n c o n c e r n i n g the method of s o l u t i o n were i d e n -

t i f i e d . These were no i n f o r m a t i o n , some i n f o r m a t i o n , and

an e x p l i c i t s t a t e m e n t of one method of s o l u t i o n t o t he

p r o b l e m . Three l e v e l s of i n f o r m a t i o n c o n c e r n i n g the p r i n -

c i p l e used in the s o l u t i o n of the problem were i d e n t i f i e d .

These were no i n f o r m a t i o n , soma i n f o r m a t i o n , and an ex -

p l i c i t s t a t e m e n t of the p r i n c i p l e . Bach s u b j e c t was g iven

one form of i n f o r m a t i o n c o n c e r n i n g the method of s o l u t i o n

and one form of i n f o r m a t i o n c o n c e r n i n g the p r i n c i p l e used

in s o l v i n g the problem,. As a r e s u l t , t h e r e were n i n e ' d i f -

f e r e n t i n s t r u c t i o n a l methods v a r y i n g from no I n f o r m a t i o n

c o n c e r n i n g a method of s o l u t i o n and no i n f o r m a t i o n c o n -

c e r n i n g the p r i n c i p l e used in the s o l u t i o n to a s t a t e m e n t

of a method of s o l u t i o n and a s t a t e m e n t of the p r i n c i p l e

u s e d . Each of the n i n e I n s t r u c t i o n a l methods vaa used w i t h

a random subgroup of the t o t a l group of s u b j e c t s .

A f t e r an e i g h t e e n - m i n u t e i n s t r u c t i o n a l p e r i o d two

t e s t a were a d m i n i s t e r e d . The f i r s t bes t was designed, to

measure ehe a b i l i t y r>i the subjecbti to s o l v e problems only

s l i g h t l y d i f f e r e n t from thoae l e a r n e d dc-rIkf the i n -

s t r u c t i o n a l p e r i o d . This t e s t was c a l l e d the s imple

t r a n s j . e r t e s t . -Phe second, t e s t was d e s i g n e d t o measure

the a b i l i t y of una s u b j e c t s to aolva problems b a s e d on the

p r i n c i p l e t a u g n t d u r i n g the i n s t r u c t i o n a l p e r i o d b u t d i s -

s i m i l a r t o the problems t a u g h t d u r i n g the i n s t r u c t i o n a l

p e r i o d . This t e s t wgs c a l l e d the complex t r a n s f e r t e a t .

23

From t h e d a t a o b t a i n e d rroia t h e fj u *<2 3 (j I; vjos c o n c l u d e d

t h a t

1 . I n f o r m a t i o n g i v e n t h e s t u d e n t a b o u t t h e method of s o l v i n g examples ia wore l i k e l y to be b e n e f i c i a l t h a n i n f o r m a t i o n g i v e n a b o u t t he p r i n c i p l e - - a t l e a s t in t h e i n i t i a l s t a g e s of p rob lem s o l v i n g . 2 . Some a p p r o p r i a t e g u i d a n c e w i l l p r o v e mora h e l p f u l to t h e s t u d e n t t han no g u i d a n c e . Leav ing t h e s t u d e n t t o d i e cove r f o r h i m s e l f t h e s o l u t i o n of a p rob lem w i l l n o t p r e v e n t u n d e r s t a n d i n g , b u t w i l l p r o b a b l y d e l a y i t . 3 . The e f f e c t i v e n e s s of g u i d a n c e does n o t depend s o l e l y on t h e siiiount of i n f o r m a t i o n i m p a r t e d . Mors e x p l i c i t forms of i n s t r u c t i o n w i l l p r o v e raoat he, lpf u 1 wifch s t u d e n t s a b l e t o a p p l y t h a ' i n f o r m a t i o n . For s t u d e n t s of l e s s e r a b i l i t y , l e a a e x p l i c i t c l u e s , d e s i g n e d to h i g h l i g h t s t r u c t u r a l r a l a f c i o n a h i p a , msj p r o v e j u s t aa e f f e c t i v e . ^

9

Scandura c o n d u c t e d t h r o e e x p l o r a t o r y e x p e r i m e n t s

which v e r a aonesubat r e l a t e d to t h e c a r d - t r i c k s t u d i e s

r e p o r t e d by K a t c n o . In each e x p e r i m e n t t h e t a s k .to be

l e a r n e d concerned, c e r t a i n t y p e a of a r r a n g e m e n t s o f a a a t

of s i x t y - f o u r c a r d s . The c a r d s were s p e c i a l l y d e s i g n e d f o r

t h e e x p e r i m e n t . Each c a r d had two g e o m e t r i c a l f i g u r e s

p r i n t e d on the f a c e of t he c a r d . The s r r a a g e r a e n t s t o b e

l e a r n e d depended upon t h e t y p o s of f i g u r e s p r i n t e d on t h e

c a r d s . The p u r p o s e of t h e s t u d i e s was t o d e t e r m i n e sorae of 8 l b i d . , p . 1 8 .

^ J o s e p h M. S c e n d u r a , "An A n a l y s i s of E x p o s i t i o n and D i s c o v e r y Modes of Problem S o l v i n g I n s t r u c t i o n , " S S l E d u c a t i o n , XXXIII ( W i n t e r , 19o! |7, i l ^ - l ^ v . ""

2k

the v a r i a b l e s and i ln t e r r e l s i lonab ips vbieb compl ica te ex-

p e r i m e n t a l comparisons of expos i t o ry and d i s cove ry methods

of i n s t r u c t i o n and to p rov ide a framework f o r f u t u r e more

p r e c i s e e x p e r i m e n t a t i o n . In each experiment the e f -

f e c t i v e n e s s of a d i scovery method of t each ing was compared

wi th the e f f e c t i v e n e s s of an expcai tory-method of t e a c h i n g .

The f i r s t experiment u t i l i z e d tiro c l a s s e s of s i x t h -

grada s t u d e n t s as s u b j e c t s . One c la a a , the e x p o s i t i o n

c l a s s , was p r e sen t ed wi th a l l the in fo rmat ion n e c e s s a r y to

so lve the problems. The in fo rma t ion was p r e s e n t e d as

qu i ck ly aa p o s s i b l e and a l l ques t ions were answered as

d i r e c t l y aa p o s s i b l e . The other c l a s s , the d i s cove ry e l s a a ,

waa given ex tens ive exper ience wi th the p r e r e q u i s i t e

m a t e r i a l b e f o r e new m&ter ia l was proa a n t e d . Meaning was

s t r e s s e d in t h a t ques t i ons were asked and h i n t s were given

t h a t 'were d i r e c t e d a t the under ly ing p r i n c i p l e s involved

in the problems. I n s t r u c t i o n f o r both groupa was given

dur ing two p e r i o d s , b u t the Ins t r uc t iona I pe r iods f o r the

d i s cove ry group were longer than the i n s t r u c t i o n a l pe r iods

fo r the expos i t J on g roup . Two types of t e s t s were then

given to each of the two g roups . One t e s t involved r o u t i n e

problems s i m i l a r to those l ea rned dur log the i n s t r u c t i o n a l

p e r i o d and the othor t e s t 1 n /o ivad problems based on p r i n -

c i p l e s d i f f e r e n t .from those l ea rned dur ing the i n s t r u c t i o n a l

p e r i o d s . There was no s i g n i f i c a n t d i f f e r e n c e between the

two groups on the f i r 3 t t e a t b u t on the second t e s t the

25

performance of the d i s c o v e r y group was s u p e r i o r to fcbe

performance of the e x p o s i t i o n g r o u p .

I n t h e second e x p e r i m e n t t h e i n s t r u c t i o n s f o r t h e d i s -

c o v e r y g r o u p were more d i r e c t and attarr.pta were made t o

reduce i n s t r u c t i o n a l t i m e . In the e x p o s i t i o n c l a s s a t -

t e m p t s wore made t o make the l e a r n i n g in ore M e a n i n g f u l . The

s u b j e c t s i n t h i s exper iment were t w e n t y - t h r e e f i f t h and

s i x t h - g r a d e s t u d e n t s . Each c l a s s mat f i v e t i n e s . The

f i r s t t h r e e p e r i o d s wore used f o r i n s t r u c t i o n and t h e l a s t

tiro p e r i o d s were used f o r t e s t i n g . On a t e a t i n v o l v i n g

r o u t i n e p r o b l e m s t h e p e r f o r m a n c e of b o t h g r o u p s was n e a r l y

t h e ae r ie . On a r e t e n t i o n t e a t and on a t e a t i n v o l v i n g

p r o b l e m s b a a e d on d i f f e r e n t p r i n c i p l e s t h a n t h e p r o b l e m s

l e a r n e d d u r i n g the i n s t r u c t i o n a l p e r i o d s the performance

of t h e e x p o s i t i o n group was s u p e r i o r to the p e r f o r m a n c e of

t he d i s c o v e r y g r o u p .

In t he second exper iment t h e i n s t r u c t i o n a l t ime f o r

t h e d i s c o v e r y g roup was l o n g e r t h a n the i n s t r u c t i o n a l time

f o r t h e e x p o s i t i o n g r o u p . A t h i r d exper i raeut uaa c o n d u c t e d

i n which the i n s t r u c t i o n a l t ime f o r b o t h g r o u p s was t h e

same. In order t o accompl i sh t h i s t h e s u b j e c t s in the d i s -

c o v e r y group war a g i v e n a i d in f i n d i n g s y s t e m a t i c modes of

viuCt. iog Las p± c h i e f s . Tue s u b j e c t s In l*.be e x p o s i t i o n

group were t a u g h t an a l g o r i t h m which c o u l d be u s e d to

s o l v e a c e r t a i n c l a s s of p r o b l e m s , i n o r d e r to equate

i n s t r u c t i o n a l t i m e , p r a c t i c e -;o3 g i v e n in u s i n g t h e

26

a lgo r i thm u n t i l the i n s t r u c t i o n a l time matcbad t h a t used

fo r the d i s cove ry g roup , In t h i s experiment t he re were

only e i g h t s u b j e c t s in the d i s c o v e r y c l a s s and seven sub-

j e c t s in the expos i t i on ola s s . A a a r e s u l t no s t a t i s t i c a l

a n a l y s i s was per formed. Examination of mean a corea showed

t h a t the performance of the expo a i t i o n c l a s s was s u p e r i o r

to the performance of the d i s c o v e r y c l a s s on r o u t i n e p r o -

blems b u t the performance of the two group3 was n e a r l y the

ssma on tbs probiems based on new p r i n c i p l e s .

A f t e r an a n a l y s i s of the r e s u l t s of the t h r e e e x p e r i -

ments i t was concluded t h a t perhaps the r.iost impor tan t

v a r i a b l e in the i n a t r u c t i o n a 1 modes involved the " t iming"

of p r e s e n t a t i o n of f a c t s and ideaa dur ing i n s t r u c t i o n .

According to Scaadura , ^ f1A b a t t e r unders t and ing of the

r o l e of t iming and i t s e f f e c t s on l e a r n i n g , r e t e n t i o n and

t r a n s f e r may ba of p r a c t i c a l import in h e l p i n g make the

t e a c h i n g - l e a r n i n g p rocess mora e f f i c i e n t , " Scsndura s t a t e d

f u r t h e r t h a t

. . ^ g e n e r a l l y i t i s assumed t h a t the b e t t e r the t imings tha more concep tua l the l e a r n i n g and the wore t r e n a f e r w i l l o b t a i n . When problem s o l v i n g a lgo r i thms are p r e s e n t e d d i r e c t l y and Mean ing fu l ly a t a p o i n t in time when S- feedback i n d i c a t e s good p r e r e q u i s i t e comprehension, concep tua l l e a r n i n g may be ezpeeted

1 0 0 3 i d . , p . 155.

n i b i d . , p . 155.

27

The s t u d i e s c o n d u c t e d by Katona have o f t e n been c i t e d

as p r o v i d i n g ev idence f a v o r i n g d i s c o v e r y methods of t e a c h -

i n g . His s t u d i e s have a l s o i n s p i r e d s e v e r a l other s t u d i e s .

Moat of t h e s e s t u d i e s have a l s o g iven some support to the

h y p o t h e s i s t h a t d i s c o v e r y t e a c h i n g i s mora e f f e c t i v e t h a n

e x p o s i t o r y t e a c h i n g . C l o s e r e x a m i n a t i o n of t h e s e s t u d i e s

r e v e a l s t h a t the s u p p o r t i s n o t as c l e a r as o f t e n i n d i c a t e d ,

Tba r e s u l t s of t h e p r e c e d i n g s t u d i e s s h o u l d be v iewed w i t h

c a u t i o n b e c a u s e in each s t u d y t h e r e were a t l e a s t two i n d e -

p e n d e n t v a r i a b l e s p r e s e n t and t h e i n t e r a c t i o n of t h e s e

v a r i a b l e s was n o t a lways c o n s i d e r e d or c o n t r o l l e d . One of

t he v a r i a b l e s p r e s e n t i s t h e v a r i a b l e which l i e s a l o n g t h e

con t inuum of a l l p o s s i b l e d i s c o v e r y methods of t e a c h i n g .

This v a r i a b l e v s r i e s from s i t u a t i o n s wbere a l l f a c t s a r e

l e a r n e d w i t h o u t h e l p t o the s i t u a t i o n s where a l l i n f o r -

ma t ion ia g iven in such a way t h a t t h e r e i s no o p p o r t u n i t y

f o r s e a r c h . The o t h e r v a r i a b l e l i e s a l o n g the con t inuum

of a l l t e a c h i n g wethoda which m i g h t be c a l l e d methods of

t e a c h i n g by u n d e r s t a n d i n g . This v a r i a b l e v a r i e s f rom

s i t u a t i o n s in which a l l in forma t i o n i s p r e s e n t e d in such a

way t h a t new infoi ' ina t ion can be a s s i m i l a t e d b y the l e a r n e r

and c o r r e l a t e d w i t h p a s t e x p e r i e n c e t o s i t u a t i o n s in which

i n f o r m a t i o n la p r e s e n t e d f o r which the l e a r n e r has a m i n i -

mum of p a s t e x p e r i e n c e and in which l e a r n i n g i s b y r o t a

lnemor iza t i o n .

28

S tud ies Using Coding Froblema

a a the Cr i t e r ! do la s k

Severa l i n v e s t i g a t i o n s i n t o the t e a c h i n g - l e a r n i n g

p rocess have been conducted in which the l e a r n i n g t a sk

c o n s i s t e d of l e a r n i n g methods of dec iphe r ing encoded

s e n t e n c e s . Such l e a r n i n g tasks have been cons ide red sim-

i l a r to the l e a r n i n g tasks encountered in a t y p i c a l

c l a s s r o o n s i t u a t i o n because coding problems a re of an

a b s t r a c t n a t u r e and a re baaed on c l e a r l y d e f i n e d p r i n -

c i p l e s .

In one such s t u d y , conducted by I ias lerud and Mayors,-*-^

i t was hy op the a ized t h a t p r i n c i p l e s de r ived by the l e a r n e r

s o l e l y from conc re t e i n s t a n c e s w i l l be more r e a d i l y used

in a new s i t u a t i o n than those given to him in the form of

a s t a t emen t of p r i n c i p l e and i n s t a n c e . Seven ty - s ix co l lege

s t u d e n t s , rang ing from fre-ahmen to s e n i o r s , were uaad aa

s u b j e c t s in an expe r imen ta l group in a s tudy des igned to

t e s t t h i s h y p o t h e s i s . Twenty-four s i m i l a r s u b j e c t s were

uaed aa a c o n t r o l g roup . The exper imenta l group was given

Wo t e s t a in which the t a sk was to dec ipher twenty c o d e s .

The second t e s t was given one weak a f t e r the f i r s t t a s t

was g i v e n . The c o n t r o l group was given only the a.scond

tes t . - The f i r s t waa des igned to give the exper imanta l

G-. M. I ias lerud and S h i r l e y Meyers, "The Trans fe r value of Given and I n d i v i d u a l l y Derived P r i n c i p l e s , " The £EEE2®i 2± Educat iona ' l Psychology. LIX (December, 19OT7 293-295. •• • 5 ^ >»

29

group two types of experience in problem solving. For soma

of the problems specific directions for deciphering the

code were printed above the problern while no directions

were given for the remaining problems. Ther e ware equal

numbers of the two types of problems and the type a of prob-

lems were alternated so that each subject would solve

approximately an equal number of each type of problem. A

different code vaa unad in each problem. On the second

teat the subjects vera required to identify the correct

solution to each of the same twenty codes. For each'Swb-

ject in the experimental group there were four possible

scores. The first score teas the number of codes solved,

correctly on test one for which the rule waa given. The

Second score wsa die number of codea solved correctly on

test one for which no directions were given. The third

score was the number of correct codas on test two for which

rules had been given on teat one and tfca foartb score was

the number of correct codes on test two for which no

direction bad bsaa given on tea t one.

It was found tint on teat two the performance of the

experimental group tcag significantly superior to the per-

formance of the control group. On teat one significantly

more codes w&re solved for which 'the rule waa given .than

codes for which no direction waa given. On teat two the

identification of correct solutions for codes for which no

direction had been given on tost one Increased forty-six

30

per c e n t . The i i ic rease f o r codes i'ojf which the rule bad

been given waa only ten par cent® This d i f f e r e n c e i s

h igh ly s i g n i f i c a n t . I t was concluded t h a t t h i s experiment

added strong suppor t to the contention t h a t independently

der ived p r i n c i p l e s a re mora transferable than those where

the p r i n c i p l e la given to the s t u d e n t .

IIrebs^-3 conducted a s t udy very s i m i l a r to tha s tudy

reported by Has lerud and Meyer a * An i n i t i a l t e a t , v ery

2imilar to the i n i t i a l t e s t used by Haslerud and Ke.ye.rs,

v a 3 adraini-s t e red to t h i r t y - t w o n in th -grade s t u d e n t s . A f t e r

t h i s t e s t waa completed a t e a t c o n t a i n i n g coding ifcema

boned on the same p r i n c i p l e a used in the i n i t i a l t e s t waa

adrainis tered. This t e a t waa admin i s t e r ed again a f t e r s ix

days and then aga in a f t e r f o r t y - t h r e e d s y s , Another group

of f i f t y - e i g h t subjects took the Is at th ree taat.3 b u t not

the i n i t i a l l e a r n i n g t e s t . This group '.raa c a l l e d the con-

t r o l g roup .

On the i n i t i a l l e a r n i n g t e s t more codes f o r which the

coding p r i n c i p l e waa given (given p r i n c i p l e s ) were d e c i -

phered than codaa for which the coding p r i n c i p l e waa not

given (de r ived pr inc ip lea ) b u t i l l u s t r a t e d by an example.

On the f i r at 'admins t ra t ion of the second t e s t , the exper i -

mental s u b j e c t s scored s i g n i f i c a n t l y higher than the c o n t r o l

^Stephen Or d e Kr eh a , n A n In v e s t i g a 11 o r> o f Tr a n s f a r E f f e c t s of Given and Derived Coding Pr inc ip le s a t librae Levels of Mental A b i l i t y , * unpubl ished d o c t o r a l etiaseyfcsfcioo, Michigan S t a t e U n i v e r s i t y , Eas t Lans ing , Michigan, 1962*

31

s u b j e c t s . A f t o r s i x days the d i f f e r e n c e waa s t i l l s i g n i f i -

c a n t . A f t e r f o r t ; - - t h r e e days tbe d i f f e r e r . o e bad narrowed

to b o r d e r l i n e a i g n i f i c a n c e . The t r a n s f e r i n d i c a t e d by

t he se r e s u l t s was a t t r i b u t e d to the t e a c h i n g methods and

not p r a c t i c e . An a n a l y s i s of t e a t da ta l ed to the r e -

j e c t i o n of tha h y p o t h e s i s t h a t s t u d e n t s w i l l t r a n s f e r

de r ived p r i n c i p l e s b e t t e r than given p r i n c i p l e s • Teat

r e s u l t s showed no d i f f e r o n c e . The h y p o t h e s i s t h a t the

t r a a a f e r e f f e c t s of d e r i v e d p r i n c i p l e s w i l l be more perma-

n e n t than those of giv-sn p r i n c i p l e s was a l s o r e j e c t e d .

Wit'crock*^- ccinpsrad the e f f e c t s of th ree v a r i a b l e s

upon the a b i l i t y of s u b j e c t s to dec ipher encoded s e n t e n c e s .

Tha "var iables were r u l e , example, and o r d e r . There were

two 1 eve la of the r u l e v a r i a b l e and the example v a r i a b l e .

Tliese l e v e l s were r u l e given or r u l e no t g*ven and example

g iven or example n o t g i v e n . The l e v e l s of the order v a r i -

a b l e ware r u l e g iven f i r s t or example g iven f i r s t . A l l

p o s s i b l e . combinat ions of these f a c t o r s v i e Ida e i g h t d i f -

f e r e n t Ins t r u e t i o r i a l methods . Two hundred nine ty - two upper

l e v e l c o l l e g e s t u d e n t s were a s s i g n e d s i ysndem to the e i g h t

t r e a t m e n t methods . I n s t r u c t i o n waa p rov ided through p r o -

gramed b o o k i e t a . S u b j e c t s were a l lowed to precede through

the b o o k l e t s a t t h e i r own p a c e . The f i r s t p a r t of each

. u . >'7ittrock, °Verbs 1 S t i m u l i in Concept Format ion : Learn ing by D i scove ry ,* J o u r n a l of E d u c a t i o n a l Psychology , LIV ( J u l y , 1963) , 133-193"""—

32

i n s t r u c t i o n b o o k l e t p r e . f a r t e d , Id va r ious sequences d e -

pending upon the expa r imsn ta i t r e a t m e n t . a p p r o p r i a t e

d i r e c t i o n s , s r u l e , a worked example of the r u l a , and an

oxanple of tli8 r u l e to 1)6 worked. j. n3 rqttiuiiiucr of to3

b o o k l e t >ra3 d iv ided i n t o ten a e r i e s . In each a e r i e s a

r u l e waa given or was to be de r ived by the s u b j e c t and an

enc iphered sentence was p r e s e n t e d fo l lowed e i t h e r by the

sesie sen tence dec iphe red or by a space where the sen tence

vaa to be dec iphered by the s u b j e c t . The order of the

stimuli waa v a r i e d to f i t the t r e a t m e n t .

A tbree-waek r e t e n t i o n t e a t c o n s i s t e d of twen ty - fou r

enc iphered s e n t e n c e s , 'Hie sentenced were chosen fco sample

r e t e n t i o n and t r a n s f e r . E igh t of the sentencea ware i d e a -

t i e s ! to thoss p r e s e n t e d dur ing the IsarniDg p e r i o d .

Another e i g h t of the sen tences were new examples of e i g h t

of the ten r u l e s p r e s e n t e d e a r l i e r . The r e g a i n i n g sen -

tences were codaa based on new r u l e s . The sen tences ware

a r r a n g e d ' i n random o r d e r .

On the o r i g i n a l t e s t the groups t h a t were given the

r u l e decoded More aentsncoa than those who were no t given

the r u l e , No s i g n i f i c a n t d i f f e r e n c e e x i s t e d between the

r u l e given group and the answer and r u l e given group

a l though bo th groups wsre supe r io r to the no r u l e and

answer given group, There waa a s i g n i f i c a n t i n t e r a c t i o n

between the r u l a and answer f a c t o r s , Mien the r u l e waa

n o t g i v e n , g iv ing the answer improved l e a r n l a g , b u t when

33

the r u l e was g i v e n , g i v i n g the answer d i d n o t enhance

l e a r n i n g . When time t o l e a r n was c o n s i d e r e d , i t waa found

t h a t the r u l e n o t g iven and answer n o t g iven groups r e -

q u i r e d a i g n i f i c a n t l y more t ime t o l e a r n than any of the

o the r g r o u p s .

On the r e t e n t i o n and t r a n s f e r t e a t the r u l e g iven

and answer n o t g iven group was s i g n i f i c a n t l y b e t t e r than

the r u l e g iven and answer g iven group and the r u l e n o t

g iven and answer n o t g iven g r o u p . The r u l e n o t g iven

and answer n o t g iven group bad a h i g h e r r e t e n t i o n s c o r e

than i n i t i a l l e a r n i n g s c o r e . The o the r t h r e e g r o u p s '

r e t e n t i o n s c o r e s were lower than t h e i r l e a r n i n g s c o r n s .

On the second t e s t a s i g n i f i c a n t i n t e r a c t i o n was found

between the r u l e and answer f a c t o r s , j u s t 0.3, on the

i n i t i a l l e a r n i n g t e a t . However, the answer f a c t o r reade

no 3 ignx i i canfc con crzbu11 on t o the r e t e n t i o n and t r a n s f e r

s c o r e s . Giving s p e c i f i c answers improved i n i t i a l s c o r e s

"Whether or n o t r u l e s ware g i v e n , b u t improved r e t e n t i o n

and t r a n s f e r s c o r e s only when r u l e s were n o t g i v e n . I t

was conc luded t h a t when the c r i t e r i o n i s i n i t i a l l e a r n i n g

of a few r e s p o n s e s e x p l i c i t and d e t a i l e d d i r e c t i o n seems

t o be raost e f f e c t i v e and e f f i c i e n t . Whan the c r i t e r i a

a r e r e t e n t i o n and t r a n s f e r * some I n t e r m e d i n t s amount of

d i r e c t i o n seems to produce the b e a t r e s u l t s .

Three s t u d i e s have been rev iewed in which s u b j e c t s

d e c i p h e r e d c o d e s . Two s t u d i e s i n d i c a t e d t h a t a c e r t a i n

3k

amount of d i s cove ry en tbe p a r t of fee s t u d e n t enhances

r e t e n t i o n and t r a n s f e r wlisrea® aaoth&r s tudy i n d i c a t e d

t h a t s t u d e n t s w i l l not t r a n s f e r and r e t a i n dayivod p r i n -

c i p l e s bet tor than given p r i n c i p l e s .

S tudies in Which tbe C r i t e r i o n Task Involved

Discovery of Word Rela t ionabips

In the p reced ing s e c t i o n i t was found t h a t tbe t ask

of l e a r n i n g methods of dec iphe r ing codes has been con-

s i d e r e d s i m i l a r to the l e a r n i n g tasks encountered in

t y p i c a l classroom s i t u a t i o n s . Another l e a r n i n g taak that

baa been cons idered s i m i l a r to t y p i c a l classroom l e a r n i n g

i s the t ask of l e a r n i n g a pr inc ip l e or r e l a t i o n s h i p s a t -

i s f i e d by four of a l i s t of f i v e words b u t not s a t i s f i e d

by the f i f t h word,

Stacey" conducted a s tudy in which the e f f e c t s of

vary ing amounts and kinds of i n fo rma t ion given dur ing

i n s t r u c t i o n on the l e a r n i n g of word r e l a t i o n s h i p s ve ra

xnves t iga t a d , 'j.'oe e f f e c t s of reward end punisbrosnt upon

learning were also i n v e s t i g a t e d . In the l e a r n i n g s i t u a t i o n

used in the s tudy the s u b j e c t s were p r e s e n t e d with a l i s t

of f i v e English words . The taak of the subject was to

l e a r n to respond to the s t imulus c o n s i s t i n g of the l i a t of

Es the r J . Swer. son , G. Les t e r Anderson, and Chalmers L. S t acey , Learning Theory in School S i t u a t i o n s (Minneapol i s , 19ii9) r p p T T f p l t f J T * — ~ — —

35

five words by saying one of the words. In order to ir.aka

the learning situation meaningful four of the words were

related by some principle. The correct response consisted

of the word which did not "belong" according to the prin-

ciple. A further taak of the subject was to identify the

principle satisfied by four words of the list. For the

total learning task a set of fifty liata was prepared.

These lists involved ten different principles. During the

experiment a list was presented to the subject and the

subject was esksd to make a response. If the response was

correct the subject was rewarded by hearing the response

"right5' from the person who was presenting the list. If

the response was incorrect the subject vas punished by the

• response wi ru;< ." Ibe list of words wss kept before the

subject until a correct response MQ3 made. Then the next

i.ist was presented. The entire set of fifty lists was

presented to each subject five times. .Daylog the first

or 6^ en t.a Ci. i.)x} di t-he set 01 fifty lists each subject wss

given one of five different kinds of instruction. In

hethod A the subject was given no intimation aa to wby one

of the five words did not belong with the other four, nor

was he told that there might be any particular reason under-

lying a correct choice. In Method B the subject was • •

informed that there wag a reason why one of ths five words

of each liat did not belong with the other four words, bat

he was not told the reason. In Method C the subjeet was

36

g i v e n tbe c o r r e c t r e s p o n s e to each l i s t d u r i n g the f i r s t

t r i a l , but be was g iven no r e a s o n vhy the r e s p o n s e was

c o r r e c t . In Method D, during t be f i r s t t r i a l , the s u b j e c t

was g iven the c o r r e c t r e sponse to each l i s t and t o l d tha t

t h e r e was a r e a s o n why tbe r e s p o n s e was c o r r e c t , b u t he

was not t o l d t be r e a s o n . In Method E t he s u b j e c t was t o l d

the c o r r e c t response to each l i s t and be was t o l d tbe p r i n -

c i p l e or r e a s o n u n d e r l y i n g the c o r r e c t r e s p o n s e .

Tha s u b j e c t s used f o r the experiment were one hundred

s i x t h - g r a d e s t u d e n t s . The s u b j e c t s were a s s i g n e d to the

i n s t r u c t i o n a l methods by a random p r o c e s s in such a way

t h a t t h e r e were twen ty s u b j e c t s in each i n s t r u c t i o n a l

g r o u p . C r i t e r i o n s c o r e s were in tha form of the number of

i n c o r r e c t r e s p o n s e s mnde to each l i s t d u r i n g each of tbe

f i v e t r i a l s . Trnmodiately a f t e r a s u b j e c t bad comple ted h i s

f i f t h t r i a l , the Jarr.e a e t of f i f t y l i s t s to which ha had

responded was once more p l a c e d b e f o r e bins. On be ing t o l d

tha r e s p o n s e he had m3de f o r each l i a t , the s u b j e c t was

asked whether or n o t t h e r e had been any p a r t i c u l a r r e a s o n

why t h a t r e s p o n s e bad been made, and i f s o , what t h e r e a s o n

was. A r e c o r d was made of the r e a s o n g iven by each s u b j e c t ,

In a d d i t i o n a s e t of f i f t y l i s t s of -words ba sed on p r i n -

c i p l e s s i m i l a r to t hose used during the i n s t r u c t i o n a l

p e r i o d was used as a p r e t e s t and a p o s t t e s t .

On the b a s i s of tbe injftwna t ion o b t a i n e d from t he

exper imen t t he f o l l o w i n g . c o n c l u s i o n s were drswn:

37

1 . I t makea f o r h o t t e r l earn ing i f the l e a r n e r proceeds by a method of a c t i v e p a r t i c i p a t i o n i n -v o l v i n g s e l f - d i s c o v e r y r a t h e r then by a method of p a s s i v e p a r t i c i p a t i o n i n v o l v i n g o n l y r e c o g n i t i o n or i d e n t i f i c a t i o n of I n f o r m a t i o n p r e v i o u s l y pro-vided him.

2 . The p r o c e s s of s e l f - d i s c o v e r y on the par t of the l e a r n e r weakens e r r o r s or wrong hab i t s and t h e r e b y e l i m i n a t e s them more q u i c k l y than does t he process of a u t h o r i t a t i v e i d e n t i f i c a t i o n .

3 . The imparfeet l e a r n i n g t h a t occurs d u r i n g a p r o c e s s of s ' e l f - d i s c o v e r y ia l o s s d e t r i m e n t a l t o l e a r n i n g than t h a t which occurs d u r i n g s p r o c e s s in which the r e s p o n s e s a r e i d e n t i f i e d f o r the l e a r n e r #

If, The l e a r n e r o b t a i n s es many or wore f a c t a , and d i s c o v e r s mora c o r r e c t r e a s o n s f o r theirs* by a p r o c e s s of & a I f - d i s c o v e r y t h e r e b y a p r o c e s s of a u t h o r i t a t i v e i d e n t i f i c a t i o n . °

K i t t a l l 1 ^ s t u o i e a t he e f f e c t s of minimum, i n t e r -

m e d i a t e , and rcsxlraum amounts of d i r e c t i o n d u r i n g d i s c o v e r y

on t r a n s f e r sad r e t e n t i o n . S u b j e c t s f o r t he s t u d y were 132

s i x t h - g r a d e p u p i l s . The s u b j e c t s were p l a c e d i n t o t h r e e

t r e a t m e n t groups th rough a p r o c e s s of a t r a t i f i e d - r s n d o r a

s e l e c t i o n . The s t r a t i f i c a t i o n was baaed on h i g h , medium,

and low r e a d i n g ach ievement c l a a s i f i c a t l o : n a , The l earn ing

m a t e r i a l c o n s i s t e d of l i s t s of f i v e words In which four- -of

the words s a t i s f i e d soma p r i n c i p l e b u t the f i f t h word d i d

n o t . The s u b j e c t s of the ainircum group were t o l d that , each

group of t h r e e l i s t s was based on a c camion u n d e r l y i n g p r i n -

c i p l e . No i n f o r m a t i o n we a g iven t h i s group aa to the n a t u r e

1 6 lb i d . , -p. 100.

•*-7jack E. K i t t e l l , f,An E x p e r i m e n t a l S tudy of the E f f e c t of E x t e r n a l D i r e c t i o n Dur ing Lea rn ing on T r a n s f e r and R e t e n t i o n of P r i n c i p l e s / ' The J o u r n a l of E d u c a t i o n a l P aye ho l o g y , LXVIII (November, X957T7 3 9 x ^ 0 5 .

38

of the p r i n c i p l e s . The group r e c e i v i n g i n t e r m e d i a t a

d i r e c t i o n was provided wi th a l l the in fo rma t ion s u p p l i e d

the minimum group p lus a v e r b a l s t a t emen t of the p r i n c i p l e .

The p r i n c i p l e s were p r i n t e d ironed i s t e l y p reced ing the group

of l i s t s to which they a p p l i e d . The m a t e r i a l s used by tha

maximum t r e a t m e n t group included a l l the c luea provided the

in termed ia ta group p lus o r a l s t a t emen t s of the t h r e e c o r -

r e c t responses for each group of l i a t s were given to tha

s u b j e c t s .

Tho t r a i n i n g p e r i o d was f i v e weeks in l eng th wi th

nine items based on th ree p r i n c i p l e s be ing p r e s e n t e d and

r e p e a t e d twice each week f o r a t o t a l p r e s e n t a t i o n of

f o r t y - f i v e items based on f i f t e e n p r i n c i p l e s . Ana lys i s

of v a r i a n c e showed t h a t the i n t e r m e d i a t e snd maximum groups

were s i g n i f i c a n t l y s u p e r i o r to the nimiiniun group when the

c r i t e r i o n was number of c o r r e c t responses dur ing the t r a i n -

ing p e r i o d . On a t r a n s f e r t e a t each group was s i g n i f i c a n t l y

d i f f e r e n t - from tba other two. The s u p e r i o r i t y of tha groups '

waa in t h i s o r d e r : i n t e r m e d i a t e , maximum, and minimum. On

the a b i l i t y to d i scover new "pr inciples the gronrf-2 wera In

t h i s o r d e r : i n t e r m e d i a t e , maximum, and minimum. A r e t e n t i o n

t e a t was admin i s t e r ed a f t e r two weeks and again af t e r fou r

weeks.. On both a dm in i s t r a t i o n s :y£ the rafccat lon t e s t the

groups were in t h i s o rde r : in termedia fee,, -pjaxiaiujn, and m i n i -

mum. I t was concluded t h a t s u b j e c t s b e n e f i t from he lp

given them -in t h e i r aearcb fsjr feas&a de te rmin ing correc t

39

responses b u t s p e c i f i c a t i o n of answers in advance encourage

r e l i a n c e on r o t e memory r a t h e r than d i s c o v e r i n g and a p p l y -

ing on the b a s i s of under ly ing r e l a t i o n s . I t was a l a o

concluded t h a t maximum and minimum amounts of d i r e c t i o n

dur ing l e a r n i n g bee one l e sa and l e s s e f f e c t i v e as the

s i t u a t i o n s to which t r a n s f e r i s made become i n c r e a s i n g l y

d i f f e r e n t and as the e lapsed time between t r a i n i n g and

measurement of t r a n s f e r "become g r e a t e r . 'The i n t e r m e d i a t a

amount of d i r e c t i o n i nc r ea sed in e f f e c t i v e n e s s between the

f i r s t and second t r a n s f e r s i t u a t i o n s and ma in ta ined the

new l e v e l of e f f e c t i v e n e s s on the t h i r d t r a n s f e r s i t u a t i o n .

Using word r e l a t i o n s h i p s as the l e a r n i n g t a sk Craig-*-®

t e s t e d the hypo thes i s t h a t i nc r ea sed d i r e c t i o n of d i s cove ry

a c t i v i t y i nc r ea se s l e a r n i n g wi thou t accompanying l o s s e s In

r e t e n t i o n or t r a n s f e r , F i f t y - t h r e e sophomore and jun io r

educat ion s t u d e n t s were d iv ided i n t o too g roups . 'Each was

given a d i f f e r e n t amount of d i r e c t i o n to he lp them d i scover

the p r i n c i p l e s a t i s f i e d by four words of a l i s t of f i v e

words. The s u b j e c t s of one group were t o l d t h a t a p r i n -

c i p l e e x i s t e d b u t were no t t o l d wha t the p r i n c i p l e was.

This group was c a l l e d the independent d i s c o v e r y group . The

s u b j e c t s of the o ther group were provided wi th a abo r t

. s ta tement of each p r i n c i p l e . This group was c a l l e d the

xs hober t O . G:vs ig 9 '"Directed Ve:c s ua Independent

Discovery of & s fc a b I J. 3 h e d Rala t i ons Tha <T OUTDO 1 of Educa t iona l Psychology. XLVII (Anv *1

1|0

d i r e c t e d d i s c o v e r y g r o u p . The s u b j e c t s of t h i s group were

r e q u i r e d to d i s c o v e r now to app ly the s t a t e d p r i n c i p l e t o

the l i s t of words .

In the a n a l y s i s of the r e s u l t s i t was found t h a t when

the c r i t e r i o n waa tha number of o r g a n i s a t i o n a l r e l a t i o n -

s h i p s or p r i n c i p l e s l e a r n e d the d i r e c t e d d i s c o v e r y group

•was s u p e r i o r to the i ndependen t d i s c o v e r y g r o u p , The r e -

t e n t i o n of the two groups waa compared over t h r e e d i f f e r e n t

i n t e r v a l s of t i m e , t h r e e d a y s , s e v e n t e e n d a y s , and t h i r t y -

one d a y s . Cn the f i r s t two t e s t a t h e r e was no d i f f e r e n c e

between tha two groups h u t on the t h i r d t e s t the d i r e c t e d

d i s c o v e r y group waa s u p e r i o r to the i ndependen t d i s c o v e r y

g r o u p . On a t e s t of a b i l i t y t o d i s c o v e r new r e l a t i o n a h i p a

no d i f f e r e n c e waa found between the two g r o u p s . I t waa

conc luded t h a t the h y p o t h e s i s made a t t he b e g i n n i n g of t he

s t u d y via a s u p p o r t e d by the r e s u l t s of the e x p e r i m e n t .

Underwood and R i c h a r d s o n 1 ^ i n v e s t i g a t e d the e f f e c t s

of l e v e l of r e sponse doxninsnco 3nd type of i n s t r u c t i o n s on

the l e a r n i n g of c o n c e p t s . Only tha r e s u l t s r e l a t e d t o

type of in 3 t i u c Li on s r s reviewed h e r e . The l e e r n in gr t s a k

used in t h i a s t u d y was s i m i l a r t o , b u t s l i g h t l y d i f f e r e n t

f r o m , the l e a r n i n g t a s k s used in the p r e c e d i n g s t u d i e s .

In the l e a r n i n g t a s k each s u b j e c t was p r e s e n t e d w i t h the

Benton J . Underwood and. Jack R i c h a r d s o n , "Verba l uoncep t Lea rn ing as a F u n c t i o n of I n s t r u c t i o n and Dominance L e v e l , ' J o u r n a l of Exper i r c e r t a l Ps ^cho lo^y . VI {April , 1956),

1*1

names of four corrmion objecta . The task of the s u b j e c t was

to d i scover what s i n g l e c h a r a c t e r i s t i c can be used to

descr ibe a l l four of the o b j e c t s .

A t o t a l of II4.I1 s u b j e c t s was used in ths experiraeirfe.

"These s u b j e c t s ware e n r o l l e d in an e l e m e n t a r y psychology

course a t the time of t he experiment . The s u b j e c t s were

d i v i d e d i n t o t h r e e equa l g r o u p s . Each of the t h r e e groups

was g i ven a d i f f e r e n t s a t of ins true t iona and then p r e s e n t e d

with the l e a r n i n g t a a k . The s u b j e c t s in one group were

i n s t r u c t e d to r e spond to each l i s t of f o u r words in a f r e e -

a s s o c i a t i o n f a s h i o n d u r i n g t h e - f i r s t p r e s e n t a t i o n of the

l i s t . They were f u r t h e r t o l d that i t would be a good idea

to v a r y the i r r e s p o n s e s on s u b s e q u e n t p r e s e n t a t i o n s of the

l i s t u n t i l t h e y s t a r t e d to g e t some responses c o r r e c t . The

s u b j e c t s of a second group we:?s l e d to d i s c o v e r the type of

r e s p o n s e s r e q u i r e d b e f o r e the l e a r n i n g t a s k b e g a n . A a a

r e s u l t , t h e s e s u b j e c t s had more i n f o r m a t i o n than the first

group c o n c e r n i n g tbe n a t u r e of the l e a r n i n g t a s k . The sub-

j e c t s in the t h i r d group were t o l d the c o r r e c t responses on

the f i r s t p r e s e n t a t i o n of t he l i s t s . Tbey were a l l owed t o

study tbass r e s p o n s e s u n t i l they cou ld r e p e a t them. Fur-

t h e r m o r e , the c o r r e c t r e s p o n s e s were p r i n t e d on a c a r d and

the s u b j e c t s were a l lowed to s t u d y the csrd d u r i n g the

p r o c e s s of the l e a r n i n g t a s k . The l e a r n i n g t a s k cons i s ted

of s i x l i s t s of f o u r w o r d s . Bach l i s t was b a s e d on a d i f -

f e r e n t p r i n c i p l e .

I|2

A f t e r an a n a l y s i s of t e s t r e s u l t s i t was concluded

t h a t the g r e a t e r the £mt?urrc; of infor rae t ion given the sub-

j e c t concerning the n a t u r e of the concepts to be learned,

the more r a p i d the a c q u i s i t i o n of the c o n c e p t .

Pour s t u d i e s have been reviewed in which s u b j e c t s

l e a r n e d r e l a t i o n s h i p s s a t i s f i e d by l i s t s of words . In each

s tudy the cumber of r e l a t i o n s h i p s l e a r n e d , whan compared

with the number of r e l a t i o n s h i p s l ea rned ia a t y p i c a l

classroom dur ing an e n t i r e s e m e s t e r , was r e l a t i v e l y sma l l .

In th ree of the s t u d i e s a l l of the ins true t ion was given in

a a ing le s e s s i o n . Stacey found t h a t l e a r n i n g by a process

of s e l f - d i s c o v e r y i s aa e f f e c t i v e or perhaps even wore

e f f e c t i v e than l e a r n i n g by a p rocess of a u t h o r i t a t i v e iden-

i>&±.lc<A tJ.on. & i t t e l l found tha t ijhon loorning by d i s c o v e r y

an intermediate amount of d i rec t i on i s most e f f e c t i v e ,

wnereaa C r a i g , as we l l aa Underwood and RIchsrdson , found

t h a t a maximum amount of d i r e c t i o n ia most e f f e c t i v e .

S tud ies in Which the Sub jec t s

Learned Suins of S e r i e s

Severa l s t u d i e s have been conducted in uh ich s u b j e c t a

ve r a r e q u i r e d to l e a r n formulas f o r sums of ser - iea , 'fhe

formula given by the equa t ion

1 + 3 + 5 + . . . + {2n + 1) - n 2

was used in s e v e r a l of these s t u d i g a . Thia type of l e a r n i n g

m a t e r i a l waa uaed because i t i s a type of l e a r n i n g mater ia l

k3

a c t u a l l y used in the mathematics c l a s s room. T h e r e f o r e , tbe

r e s u l t s of these s t u d i e s , i t has been a rgued , should be

a p p l i c a b l e to t y p i c a l classroom l e a r n i n g .

20

Hendrix conducted s e v e r a l e x p l o r a t o r y expurJmenta in

which the s u b j e c t s were r e q u i r e d to l e a r n the formula given

above . In Method I the s u b j e c t s were given the s t a t e m e n t ,

"Tbe sum of the f i r s t n odd numbers i s n - squa re This

r u l e was then v e r i f i e d fo r one p a r t i c u l a r i n s t a n c e . In

Method I I the s u b j e c t s were asked to f i n d the aum of the

f i r s t two odd numbers, the sum of the f i r s t t h ree odd num-

b e r s , the sum of the f i r s t four odd numbers, and so on

u n t i l the speed of response of the s u b j e c t i n d i c a t e d t h a t

the formula h&d been d i s c o v e r e d . Any s u b j e c t was excused

from the room as soon a3 he had d i scovered the f o r m u l a . In

Method I I I the s u b j e c t s were given the same d i r e c t i o n s as

the s u b j e c t s in Method IX. In t h i s group, however, when a

s u b j e c t g&ve evidence t h a t he had d i scovered the formula he

was immediately eakod to s t a t e the r u l e he had d i s c o v e r e d .

The experiment was r epea t ed three; t imes , one a wi th

e l even th -g rade boya , once wich t w e l f t h - g r a d e boys , and once

wi th c o l l e g e g i r l s , in a l l t h ree s t u d i e s only f o r t y sub-

j e c t s were involved a l t o g e t h e r 11. . . and s i n e s soma

c o n t r o l s on a l l -chrea runs of the experiment were a d m i t t e d l y ? 0

Ger t rude Hendr i x , "A New Clue to T rans fe r of 19ifT) i n?97 giemontary School J o u r n a l , XLVIII (December,

poos* $ a l l r e s u l t s r&ua t be Bza/tiiried w i th & view to f u r t b e r

e x p e r i m e n t a t i o n . " 2 1 On the baeis of t h e s e s t u d i e s the

following hypo theses were o f f a r e d :

1. For g e n e r a t i o n of t r a n s f e r power , the unvorbalized cviareness method of learning a generalization i s b a t t e r than a method in which an a u t h o r i t a t i v e a tatement of tha g e n e r a l i s a t i o n c01?!6c f i r s t .

2. Verbalizing a generalisation iruraedlately after discovery does not increase transfer power.

3 . Verbalizing a g e n e r a l i z e t i o n immediately a f t e r d i s c o v e r y may a c t u a l l y dec rea se t r a n s f e r power .

Using forty-eight e d u c a t i o n a l psychology students as p-i

subjects, Kcrah compared tha e f f e c t i v e n e s s of t h r e e

t e a c h i n g nethods in t e a c h i n g the same r u l e used in the .

a tudy r e p o r t e d by Henelrix. One group of s u b j e c t s , c a l l e d

the no h e l p group, waa r e q u i r e d to d i s c o v e r the r u l e w i th

no h e l p . Another gsoup , c a l l e d the d i r e c t r e f e r e n c e g roup ,

was given some d i r e c t i o n in d i s c o v e r i n g the r u l e . A t h i r d

g r o u p , c a l l e d the y j l e given g roup , was g iven a v e r b a l

statement of the r u l e . An i n i t i a l l e a r n i n g test was

a d m i n i s t e r e d irianedia t a l y a f t e r i n s t r u c t i o n . This t e a t

was roadiainisterad fou r weeks l a t e r . A questionnaire was

adminia t e r e d d u r i n g tha same s e s s i o n in which the re tea t

waa administered. The r e sponses on t h i s questionnaire

2 1 r b M . , p . 198. - 2 2 I b i d . , p . 190 .

^ B e r t Y, Kerah, !1The Adequacy of 'Moaning.' as an Explanation f o r the S u p e r i o r i t y of Learn ing by Independent Discovery," J o u r n a l of Educational Psychoid??. XLIX* ( O c t o b e r , 19OTT~2$2'^92-;

l|5

r e f l e c t e d a d e f i n i t e d i f fex ' - snoe In r s o t i v a t l o n among t h e

t h r e e t r e a t m e n t g roups * M o t i v a t i o n and i n t e r e s t was

h i g h e s t i n t h e no h e l p g r o u p . A a a r e s u l t i t was c o n -

c l u d e d t h a t t he s u p e r i o r i t y of t h e d i s c o v e r y g roup may

b a s t be e x p l a i n 3 d i n t e rms of m o t i v a t i o n .

I n a s e c o n d s t u d y , i n wh ich t h e saraa l e a r n i n g m a t e r i a l

was u s e d , K e r s b h y p o t h e s i s e d t h a t

. . . t o t ha e x t a n t thai ; t he e x t e r n a l d i r e c t i o n p r o v i d e d t o t h e l e a r n e r ia l e s s e n e d d u r i n g t h e a t t e m p t s t o d i s c o v e r t h a r e l a t i o n s h i p s wh ich a r e c o n s i d e r e d e s s e n t i a l t o tho u n d e r s t a n d i n g of a c o g n i t i v e t a s k : ( a ) the l e a r n e r w i l l t e n d t o u3e t h e l e a r n e d m a t e r i a l more f r e q u e n t l y a f t e r t h a l e a r n i n g p e r i o d ( i . e . t o e x t e n d the p r a c t i c e p e r i o d v o l u n t a r i l y ) and as a r e s u l t , (b ) ha w i l l remember i t pnger snd t r a n s f e r h i s l e a r n i n g n o r a -e f f e c f c i v e l y . "*•

I n o r d e r t o t a e t t h i a h y p o t h e s i s n i n e t y h i g h s c h o o l geom-

e t r y s t u d e n t s v:sre d v i d e d i n t o t h r o e e q u a l g r o u p s . Each

g roup was t a u g h t a s e t of r u l e s f o r aumrning a e r i e s b y one

of t h r e e d i f f e r e n t t e c h n i q u e s . The D i r e c t e d L e a r n i n g group

was t a u g h t the r u l e s and t h e i r e x p l a n a t i o n 6 r s t . t r s l y b y a

p rogramed l e a r n i n g t e c h n i q u e • Tha Guided D i s c o v e r y g r o u p

t;aa r e q u i r e d t o d i s c o v e r t b a e x p l a n a t i o n of t h a r u l a a w i t h

g u i d a n c e f rom the exper i m a n t e r . They were t o u g h t t u t o r i a l l y

u s i n g a fo rm of S o c r s t i c q u c s t i o n i n g v b i c b r e q u i r e d each

s u b j e c t to p e r f o r i n s p e c i f i c a l g e b r a i c m a n i p u l a t i o n s snd to

make i n f e r e n c e s w i t h o u t h e l p . The s u b j e c t s of t h a Rote

^'+Bsr t Y. K e r c h , "The M o t i v a t i n g E f f e c t of L e a r n i n g b y D i r e c t e d D i s c o v e r y , * J o u r n a l of E d u c a t i o n a l P s y c h o l o g y , LI I ( A p r i l , 1 9 6 2 ) , 6 $ - i r . — .

Ii.6

Learning group were a imply t o l d the r u les and given no

e x p l a n a t i o n . Testa of r e c a l l and t r a n s f e r were given a f t e r

t h ree days , two weeks and s ix weeks. Each tea t was given

to tan s u b j e c t s of each group . The t e a t s were admin i s t e r ed

in such a way t h a t each s u b j e c t took only one t e s t .

The number of s u b j e c t s who uaed the a p p r o p r i a t e r u l e

to so lve a problem on a t e s t was used as a measure of

t r a n s f e r . - The number of s u b j e c t s who wrote an a c c e p t a b l e

s t a t emen t of the r u l e was used as a rasa sure of pure r e -

t e n t i o n . I t was found t h a t tha r a t e of f o r g e t t i n g d id no t

d i f f e r s i g n i f i c a n t l y ac ross the t each ing t r e a t m e n t g roups .

The Rote Learning group was found to be c o n s i s t e n t l y

supe r io r to each of the other t r ea tmen t groups on each of

tha c r i t e r i o n t e s t a . Upon a n a l y s i s of the ques t i o n a i r e i t

was found t h a t the s u b j e c t s in the Guided Discovery group

used the r u l e s s i g n i f i c a n t l y more o f t e n a f t e r the l e a r n i n g

p e r i o d than did the s u b j e c t s in the other g roups . Prom

t h i s i t was concluded tha t l e a r n i n g by d i s cove ry may not

produce supe r io r l e a r n i n g b u t i t does produce a g r e a t e r

i n t e r e s t in what is l e a r n e d .

Another s tudy using m a t e r i a l s p e r t a i n i n g to the f i n d i n g

of auras of number s e r i e s was conducted by G3gne; and

B r o w n . j n t h i s s tudy the s u b j e c t s were n i n t h and

PC - 'Robert M. G-sgne/ and Larry T. Brown,, ''Soma F a c t o r s

in the Programing of Conceptual Lea rn ing , " J o u r n a l of Exper imenta l Psychology, LX1T (October , 1961 )'j

kl

tenth-grade b o y s . Inat;*uofelon was p r e s e n t e d th rough the

use of programed i n s t r u e t i 0 2 a 1 m a t s r i a l s » The e f f e c t i v e -

ness of the v a r i o u s i n s t r u c t i o n a l methods used was measured

by a t e s t wb5ch r e q u i r e d the s u b j e c t s t o use concep t s

l e a r n e d d u r i n g i n s t r u c t i o n in nove l s i t u a t i o n s .

Three methods of i n s t r u c t i o n with corresponding p r o -

gramed mater ia l s were compared. One Method, c a l l e d the

r u l e and example method, p r e s e n t e d each concept to be

l e a r n e d th rough a v e r b a l statoxaent of the c o n c e p t f o l l owed

by an example of the concept . This metbod was compared

w i t h two methods which encouraged d i s c o v e r y of v e r b a l p r i n -

c i p l e s . One method,, c a l l e d the d i s c o v e r y method , u t i l i s e d

r a t h e r l a r g e s t e p s in the i n s t r u c t i o n a l p rog rams . The

o t h e r method , c a l l e d the g u i d e d d i s c o v e r y method, u t i l i s e d

approximately the same size s tops as used with the r u l e and

example method . The program f o r t he g u i d e d d i s c o v e r y and

the r u l e and example matboda ware c o n s t r u c t e d in a cco rdance

w i t h the p r i n c i p l e s developed by B. P . S k i n n e r . Both d i s -

covery metboda r e q u i r e s u b j e c t s to use p r e v i o u s l y l earned

concept a d u r i n g the process of d i s c o v e r i n g new c o n c e p t s .

In the r u l e and example method each c o n c e p t was p r e s e n t e d

s e p a r a t e l y and no s p e c i a l a t t e m p t was made to r e l a t e a

concep t w i th p r e v i o u s l y l e a r n e d c o n c e p t s .

On a t r a n s f e r t e s t the pe r fo rmance of the t h roe groups

was in the f o l l o w i n g o r d e r , from h i g h to low: the g u i d e d

d i s c o v e r y g r o u p , the d i s c o v e r y g r o u p , and the r u l e and

1+8

example g r o u p . I t was c o n c l u d e d t h a t

D i s c o v e r y sa a method s p p s a r s t o go in i t s e f f e c t i v e -n e s s f r o n t h e f a c t t h a t i t r e q u i r e s t h e i n d i v i d u a l l e a r n e r t o r e i n s t a t e (and in t h i s s e n s e * t o p r a c t i c a ) t h e c o n c e p t s he w i l l l a t e r use i n s o l v i n g saw p r o b -lems . Tha e x t e n t t h a t t he G. D. ( g u i d e d d i s c o v e r y ) p rog ram wsa s b l e t o i d e n t i f y t h e s e c o n c e p t s , i t c o u l d t h e n p r o v i d e s y s t e m a t i c p r a c t i c e in t h o i r u s e , and t h u s l e a d to a p e r f o r m a n c e s u p e r i o r t o t h a t a t t a i n e d o t h e r w i s e . "

I n o r d e r t o s u b s t a n t i a t e t h e f i n d i n g s of Gagne' and

Brown, E l d r e d g a ^ ? c o n d u c t e d a r e p l i c a t i o n of t h e i r s t u d y .

The sample f o r t h e r e p l i c a t i o n was compsed of two n i n t h -

g r a d e a l g e b r a c l a s s e s . One c l a s s r e c e i v e d i n s t r u c t i o n

t h r o u g h a g u i d e d d i s c o v e r y method and t h e o t h e r c l a s s r e -

c e i v e d i n s t r u c t i o n t h r o u g h t h e r u l e sod example m e t h o d . In

t h e Gagne ' and Brown s t u d y a l l of t h e s u b j e c t s were b o y s .

In t h e r e p l i c a t i o n each c l a s s c o n t a i n e d t h i r t e e n boys and

t b r i t e e n g i r l s . On a c r i t e r i o n t e a t no s i g n i f i c a n t d i f -

f e r e n c e was f o u n d b e t w e e n t h e two e x p e r i m e n t a l g r o u p s .

Th i s r e s u l t does n o t c o r r e s p o n d w i t h t h e r e s u l t s o b t a i n e d

b y Gagne ' and B r o r a .

An e x a m i n a t i o n " of t h e p rograms used ,1n the GrSgne' and

Brown s t u d y l e d t o t h e c o n c l u s i o n t h a t s e v e r a l v a r i a b l e s

n o t v i t a l t o t h e d i s c o v e r y method of l e a r n i n g had n o t b e e n

2 6 I b l d . , p . 320 .

^ G a b r i e l M, D e l i a - P l a n a , G a r t h M. E l d r e d g e , and B l a i n e R. Wor then» Sequence Charac fcer i s t i c s of T e x t M a t e r i s l a and T r a n s Y e r ' o f " Leax'B'TjTg j"F2Ft"'*lT " Exger ' i raents IITdIsCOve'ry~LoarnTrig {SA l T T a E e l fI t y 7 19^5) T{~.'l\2T ' ~

1*9

c o n t r o l l e d . As s r e s u l t , an Improved s e t of p r o g r a m s was

p r e p a r e d and a n o t h e r s t u d y waa c o n d u c t e d b y E l d r e d g e . I n

t h i s s t u d y t h e s u b j e c t s w e r e a g a i n n i n t h - g r a d e s t u d e n t s .

T h e r e were n i n e t y - s i x s u b j e c t s i n a l l . The p r o c e d u r e s

u s e d w e r e t h e same a s t h o s e used i n t h e o r i g i n a l a t u d y

w i t h t h e e x c e p t i o n t h a t t h e i n s t r u c t i o n a l p rog rama h a d

b e e n r e v i s e d . On a c r i t e r i o n t e s t admins t e r e d i r o m e d i a t a l y

a f t e r c o m p l e t i o n of i n s t r u c t i o n i t was f o u n d t h a t t h e p e r -

f o r m s n e e of t h a g u i d e d d i s c o v e r y g r o u p waa s i g n i f i c a n t l y

s u p e r i o r t o t h e p e r f o r m a n c e of t h e r u l e and example g r o u p .

T h i s i s i n Agreemen t w i t h t h e r e s u l t s of t h e Gagne ' a n d

Brown s t u d y . A f t e r t h e f i r s t t e s t was a d m i n i s t e r e d t w e n t y -

e i g h t of t h e s u b j e c t s w e r e g i v e n an o p p o r t u n i t y bo e . xp l a in

o r a l l y t h e p r o c e s s b y w h i c h t h e y s o l v e d t h e c r i t e r i o n ' p rob -

l e m s . Then a s e c o n d c r i t e r i o n t e a t was a d m i n i s t e r e d . On

t h i s t e a t t h e p e r f o r m a n c e of t h o s e who h a d an o p p o r t u n i t y

t o v e r b a l i z a tha p r o c e s s e s u s e d in p r o b l e m $ Diving waa

s u p e r i o r t o tho. ia who were n o t g i v e n t h i s o p p o r t u n i t y .

T h i s r e s u l t i s in c o n t r a d i c t i o n t o t h e h y p o t h e s i s o f f e r e d

b y H e n d r i x . ^ ® A f t e r f o u r weeks a r e t e n t i o n t e s t was a d m i n -

i s t e r e d . No t r e a t m e n t d i f f e r e n c e s were f o u n d on t h i s t e a t .

F i v e s t u d i e s h a v e b e e n r e v i e w e d i n which s u b j e c t s wsre

r e q u i r e d t o l e a r n f o r m u l a s f o r f i n d i n g suraa of a e r i e s . I n

e a c h of t h e s e s t u d i e s , w i t h t h e e x c e p t i o n of t h e s e c o n d

<Z0 H e n d r i x , HA KTew Clue t o T r a n s f e r of T r a i n i n g , "

p . 1 9 3 .

50

s tudy conducted by Ker sb / -^ waa concluded t b a t aoiae

form of d i s cove ry method of t each ing i s mora e f f e c t i v e

than a d i r s e t method of t e a c h i n g .

S tud ies Conducted in the Elementary

School Mathematics Classroom

The remaining s t u d i e s which a r e reviewed a r e s t u d i e s

t h a t have been conducted in t y p i c a l c lassroom s e t t i n g s

and t h a t have u t i l i z e d the l e a r n i n g m a t e r i a l s t h a t were

r e g u l a r l y taught in these c l e s s rooms . In t h i s a e c t i o n

those s t u d i e s t h a t have bean conducted in e lementa ry

mathematica classrooms a re rev iewed.

One of the ve ry f i r s t s t u d i e s in which a d i s c o v e r y

method of t each ing waa compared wi th an e x p o s i t i o n method

of t each ing waa conducted by T. R. Hc'Connell in 19.3^»

KcConnell-^"5 i n v e a t i g a t e d the e f f e c t s of two teach ing meth-

ods in t each ing the one hundred b a a i c a d d i t i o n and the one

hundred b a s i c s u b t r a c t i o n f a c t a . In Method A the number

combinat ions were l ea rned by sheer r e p e t i t i o n . A s tud ious

e f f o r t was made to keep the c h i l d from d i s c o v e r i n g or

v e r i f y i n g the answers to the number coiribina t ioaa . I f a

c h i l d made an e r r o r dur ing p r a c t i c e the t eache r 5.'0;KGdiatsly

2 % e r s h , "The Mot iva t i ng F f f e c t of Learning by D i r e c t e d Discovery , " p . 69 .

3®T. R. KcConnel l , "Discovery va , A u t h o r i t a t i v e I d e n t i f i c a t i o n in the Learning of C h i l d r e n / 1 S tud ies in Education, , IX (1934) , 11-62. '

51

c o r r e c t e d him by s u p p l y i n g the c o r r e c t an swer . In Method

B a l l the number comb in a t i ona were i n t r o d u c e d through

c o n c r e t e s i t u a t i o n s . Each one of the two hundred number

f a c t s was p r e s e n t e d m e a n i n g f u l l y through the use of p i c -

t u r e d s i t u a t i o n s . The c h i l d was expec ted t o sae th rough

an a c t i v e p r o c e s s of d i s c o v e r y and v e r i f i c a fcion t h a t the

number ccrnbina t i o n s a r e t r u e and r e a s o n a b l e . The c h i l d was

expec t ed to d i s c o v e r the answer to each of the number com-

b i n a t i o n s .

Pour hundred f o r t y - o n e s e c o n d - g r a d e p u p i l s s e rved aa

s u b j e c t s in &u experj.it;ent d e s i g n e d t o cottipora the e f f e c t s

of Teaching Method A and Teaching Method B. These s u b j e c t s

were s e l e c t e d from e x i s t i n g c l a s s e s b u t the s e l e c t i o n of

s u b j e c t s was made in such a way t h a t matched p a i r s e x i s t e d

w i t h ens s u b j e c t from each p a i r in each t r e a f c - e n t g r o u p .

S p e c i a l m a t e r i a l s were p r e p a r e d f o r each group and e x t e n -

w i r - a r j u a l a were p r e p a r e d f o r the t e s c h e r s p a r t i c i p a t i n g

in coe ti>.pei iinen t . Tnese aiQnuaxg . p r e s c r i b e d the c o n d i t i o n s

wnxch the io& uc t i o n s j. ins t e r i s l s v s r e to be used ,

in i ee s e t s ol i n t e .v p o Ir? ted t e a t s were g iven d u r i n g the e.x -

p o r i w e n t 3nd seven, t e s t s were g iven s t t he c l o s e of the

e x p e r i m e n t . These t a s t s were des igned t o measure a c c u r a c y ,

speed j, t r a n s f e r , a b i l i t y t o s o l v e v e r b a l p r o b l e m s , a b i l i t y

t o d e t e c t e r r o r s , a b i l i t y to l e a r n new s k i l l from a s i l e n t

r e a d i n g l e a s on, and m a t u r i t y in manipuLa t i n g number f a c t a .

52

I t was found t h a t the g roup t a u g h t by Method A was

d e f i n i t e l y s u p e r i o r to the group t a u g h t b y Method B on

t e s t a of speed b u t the group t a u g h t by Method B was d e f i -

n i t e l y s u p e r i o r on t e a t s of m a t u r i t y in h a n d l i n g number

f a c t s and t e s t s which p u t a premium on d e l i b e r a t e work and

t h o u g h t f u l m s n i p u l a t i o n . The group t a u g h t by Method B was

s u p e r i o r on a l l t r a n s f e r t e a t s a l t h o u g h o n l y one of t he

d i f f e r e n c e s was s t a t i s t i c a l l y s i g n i f i c a n t . I t was c o n -

c l u d e d t h a t tha r e s u l t s of t he e x p e r i m e n t d i d n o t s u p p o r t

the e x t e n s i v e c l a ims made f o r t he s u p e r i o r i t y of t e a c h i n g

methods s i m i l a r t o Method B.

T h i e l e - 1 conduc ted a s t u d y which was an a t t e m p t t o

improve on t he s t u d y conduc ted by McConne l l . The d r i l l

method used in t h i s s t u d y was s i m i l a r to Method A as d e -

s c r i b e d by KcCom-fSll. i n t he g e n e r a l i z a t i o n method t h e

s t u d e n t s were not. on ly r e q u i r e d to d i s c o v e r tha number

f a c t s b u t t h e y were r e q u i r e d t o d i s c o v e r r e I s t ionahIDS

among the number f a c t a as well® In t h i s s t a d y , when tha

c r i t e r i o n was knowledge of t he b a a i c oca hundred a d d i t i o n

f a c t s , i t was found t h a t the group t&ogbt b y t he g e n e r a l -

ization method was s i g n i f i c a n t l y s u p e r i b r t o the group

t a u g h t by t he d r i l l method , Thia s u p e r i o r i t y h e l d f o r

s u b j e c t s of a l l l e v e l s of a b i l i t y , 1/han t h e c r i t e r i o n

*0 fh*3??; k; Thi!1!/ 3.® Of Generalisation

-il2 1BJL LedjfliPjg of toe Add it It on P a c t a {We\fYoFk^'"T9JBT7~

53

t e s t waa a t o s t of t r a n s f e r of t r a i l i n g i t was-again found

t h a t the g e n e r a l i s a t i o n group was super ior to the group

taught by the d r i l l method. I t was concluded t h a t the

achievements of the pup i l s taught by the g e n e r a l i z a t i o n

method were g r e a t l y super ior to those a t t a i n e d by the pu-

p i l s taught by the d r i l l method.

Another study in which second-grade pupi l s were taught

the one hundred b a s i c addit ion- f ac t a waa conducted by

Svsnscn.^ 2 The purpose of t h i s s tudy waa to compare the

e f f e c t s of three d i f f e r e n t methods of i n s t r u c t i o n on

i n i t i a l l e a r n i n g , t r a n s f e r of t r a i n i n g , and r e t r o a c t i v e

i n h i b i t i o n . The ch ie f v a r i a b l e among the three methods of

l e a rn ing waa degree of ewphssia upon organiza t ion and gen-

e r a l i z a t i o n in the l ea rn ing p roce s s . In one method of

i n s t r u c t i o n , the g e n e r a l i z a t i o n method, fcbs add i t ion f ac t a

were presented to the ch i ld ren in groups which were d e t e r -

mined by a b a s i c un i fy ing idea or gene ra l i za t ion* The

f ac t a vhich cc-nt^r sd around a g e n e r a l i z a t i o n were presen ted

together in such a wcy t h a t the teacher could , by s k i l l f u l

i n s t r u c t i o n , lead the pup i l s to t h e i r own formulat ion of

the g e n e r a l i z a t i o n . Children taught by t h i s method were

allowed to r e f e r to concrete objec ts as often, as they

3Ê3ther J . Swenson, G. Laster Anderson, and Chalmers L. Stacey, Lejarninrr Theory in School S i t u a t i o n s (Minneapolis , 191+9) »"*ppT SP39.

Si*

needed as a i d s ID s o l v i n g the s b s t r e a t number combinations®

In ano the r method of i n s t r u c t i o n , the d r i l l method, each

a d d i t i o n f a c t was p r e s e n t e d as a s e p a r a t e f a c t to be memo-

r i s e d . Wo atterwpt was made to p r e s e n t the f a c t s in an

organized f a s h i o n and the c h i l d r e n war© d iscouraged from

reason ing out answers to number combino fcions . The c h i l d r e n

were t o l d fcbe answers to the number coiubinations and i f a t

GBJ p o i n t a c h i l d h e s i t a t e d a t g iv ing an answer the t eacher

immediately supp l i ed the c o r r e c t enswer . In a t h i r d meth-

od , c a l l e d the d r i l l - p l u s method, aorae a t t emp t was made to

make the a d d i t i o n f a c t s ransningful. At the time of i t s

i n t r o d u c t i o n each number f a c t was p r e s e n t e d by p i c t u r e s of

conc re t e o b j e c t s . The c h i l d r e n were a l s o given ths oppor-

t u n i t y to v e r i f y the a d d i t i o n f a c t a by man ipu l a t i ng

conc re t e o b j e c t s . In t h i s method a l i m i t e d amount of or ~

ganiza t i on was us ad in do termining- the order of p r e s e n t a t i o n

of the a d d i t i o n f a c t s .

A t e s t on the one hundred a d d i t i o n f a c t a vaa admin-

i s t e r e d f i v e time aT once b e f o r e the exp&r iraen fc began ,

aga i a a f t e r the complet ion of en i n i t i a l s a t of a d d i t i o n

f a c t s , aga in a f t e r the complet ion of a second s e t of f a c t a ,

a f t e r the Christmas v a c a t i o n , and aga in a f t e r the com-

p l e t i o n of the f i n a l s e t of f c o t a . IXu:-ing ths t h r ea o loas '

p e r i o d s a f t e r the f i f t h admin i s t r a t i on of the a d d i t i o n t e a t

t h r ee t r a n s f e r t e a t s were a d m i n i s t e r e d . I t was concluded

t h a t the evidence ob ta ined from t h i s s tudy lends suppor t to

5$

the hypo thes i s t h a t o r g a n i s a t i o n sad genera l i za t ion} dur ing

the l e a r n i n g p rocess a re d e c i s i v e f e e tor a in f a c i l i t a t i n g

t r a n s f e r . I t was a l s o concluded t h a t when the l e a r n i n g

r e s u l t s a r e viewad as a whole tha g e n e r a l i z a t i o n method i s

s u p e r i o r to the d r i l l method and the d r i l l - p l u s method.

I t v;as a l s o found t h a t the t each ing method t h a t was accom-

panied by the most t r a n s f e r was a l s o a t t e n d e d by the l e a s t

r e t r o a c t i v e i n h i b i t i o n .

In a s tudy conducted in t h i r d - g r a d e a r i t h m e t i c c l a s s -

roo.fiis, F u l l e r t o n 3 3 compared the a f f e c t s of a conven t iona l

t each ing method and an e rpe r i r a en t s l t each ing method on the

l e a r n i n g of the m u l t i p l i c a t i o n f a c t s f o r whole numbers. In

the exper imenta l method s t u d e n t s war3 l ed to d iccovar tha

need fo r tha p a r t i c u l a r phase of a r i t h m e t i c to ba t a u g h t .

This wag u s u a l l y dona through v e r b a l problems. Once tha

proolam was i d e n e x f l e d severa 1 s o l u t i o n s w3rs s u g g e s t e d .

The b e s t method, based on the a r i t h m e t i c background of tha

s tudents- , Vos tnen s e l e c t e d . Rules or s 1'?, tti; 'crjts of or o —

cedurss were formula ted by the p u p i l s , e i t h e r i n d i v i d u a l l y

or as a c l a s s . 'Phase s t a t emen t s of r u l s a or p rocedures

were made a f t e r the p u p i l s had gained exper ience wi th the

p r o c e d u r e .

_ , „ , •io Ksrr i u l i e r icon ftA C o:^por i s on of tha E~lect- .vene3s of Two P r e s c r i b e d Methods of Teaching I i u l o i p l l c a t i on of Whole Numbers / ' unpubl ished d o c t o r a l

i s a e r t a t l o n , S t a t e U n i v e r s i t y of Iowa, Iowa C i t y , Iowa,

56

An experiment; was conduc ted in order- t o compare the

r e l a t i v e e f f e c t i v e n e s s of the two t e a c h i n g m e t h o d s . The

s u b j e c t s f o r the expe r imen t were a p p r o z i r a a t e l y 770 t b i r d -

grade s t u d e n t s in 28 c l a s s e s . Tba r e g u l a r c l a s s r o o m t e a c h -

er t a u g h t the l e s s o n s and a d m i n i s t e r e d a l l of the t e s t a .

The m a t e r i a l v;a3 t a u g h t in e i g h t l e s s o r s . A p r e t e s t waa

g iven b e f o r e the f i r s t l e s s o n was t a u g h t and a p o s t t e s t was

g iven a f t e r the e i g h t h l e s s o n . Another t e s t waa g iven t h r e e

and one-ha i f weeks l a t e r . Cn the p o s t t e s t t h e r e was a

s i g n i f i c a n t d i f f e r e n c e in f a v o r of the e x p e r i m e n t a l group

in a b i l i t y t o r e c a l l the f a c t s t h a t were t a u g h t . There was

a l s o a s i g n i f i c a n t d i f f e r e n c e in f a v o r of the experimental

group on a c s a t of ' c rc i j s fer of t r a i n i n g . On the d e l a y e d

t e a t the expe r imen t . i l group m a i n t a i n e d i t s s u p e r i o r i t y in

terms of t r a n s f e r of t r a i n i n g . No a t t e m p t was made to

compare the two groups w i t h r e s p e c t t o a b i l i t y t o r e c a l l

l e a r n e d f a c t s .

Anderson3*f .compared the e f f e c t i v e n e s s of two methods

of t e a c h i n g four t -o -gra d e a r i t h m e t i c . One method of t e a c b -

ing was b a s e d on conaoc t ion i a f c t h e o r i e s of l e a r n i n g and the

o the r t e a c h i n g method was b a s e d on f i e l d t h e o r i e s of l e a r n -

ing and emphasized u n d e r s t a n d i n g end g e n e r a l i z a t i o n . The

expe r imen t was conducted In e i g h t e e n f o u r t h - g r a d e arithmetic

c l a s s e s . Each c l a s s was t a u g h t by a d i f f e r e n t t e a c h e r .

. o , swenson, Anderson , and S t a c e y , Learning' Theory in School ji_tjj£ti^pj2£. p-p, lj.O-73. ^

5 7

Ten c l a s s e s were t a u g h t by the method based on c o n n e c t l o n i a t

t h e o r i e s . This method was c a l l e d the d r i l l method and the

teacher spen t appr oxlinotely e leven minutea each day on i n -

a t rue t i on and twenty- four mini: tea each day on d r i l l . E igh t

c l a s s e s were t augh t by the method "based on f i e l d t h e o r i e s .

This method was c a l l e d the meaning method. In these c l a saea

the t e a c h e r s spent approx ims to ly twenty-seven minutes a day

on i n s t r u c t i o n and e igh teen minutes on d r i l l .

P r e t e s t s f o r the s tudy were made in November and the

expe r imen t s ! t each ing methods were i n i t i a t e d a f t e r the p r e -

t e s t s had been a d m i n i s t e r e d . The i n s t r u c t i o n t e rmina ted

the f o l l o w i n g May. At t h i s time the p o s t t e s t s were a drain-

i s woi'tid. Unit jtss fc-j were admin i s t e r ed by the t e a c h e r s

throughout the i n s t r u c t i o n a l p e r i o d . Tes ts of a d d i t i o n ,

s u b t r a c t i o n , rau'itiplics felon» d i v i s i o n , and unde r s t and ing

of s o c i a l concepts in a r i t h m e t i c showed i n s i g n i f i e a n t d i f -

f e r e n c e s between the two t r e a t m e n t g roups . The performance

of the group t augh t by the meaning method wsa s u p e r i o r to

the performance of the group t augh t by the d r i l l method on

a. t e a t of compu Rat iona l s k i l l , a l though the s i g n i f i c a n c e of

t h i s d i f f e r e n c e waa b o r d e r l i n e . On a t e s t of mathernat ical

t h i n k i n g a b i l i t y the performance of the group t a u g h t by

the massing rr.athod was - s i g n i f i c a n t l y s u p e r i o r to the

performance of the group t augh t by the d i r l l method, A

q u e a t i o n n a i r e was given a t the beg inn ing and aga in a t

the end of the exper imenta l p e r i o d , Thia

58

q u e s t i o n n a i r e p u r p o r t s d to isaasura the degree to which

p u p i l s l i k e d a r i t h m e t i c » An ana lys is of tba responses to

the q u e s t i o n n a i r e showed t h a t the p u p i l s t augh t by the two

methods did no t d i f f e r s i g n i f i c a n t l y In t h e i r r a t i n g of

a r i t h m e t i c as compared to cthor school s u b j e c t s .

In s i x t e e n s i x t h - g r a d e rr.atheniaties c l a s s e s , W o r t h e n ^

compared a guided d i s cove ry metbod of t each ing wi th at) ex -

pos i t i on in a tbod of t e a c h i n g . Bight of the c l a s s e s were

t augh t by the guided d i s c o v e r y method and e i g h t of ih a

c l a s s e s wore t augh t by an e x p o s i t i o n method. S i g h t t e ache r s

p a r t i c i p a t e d in f a s s tudy and each t eache r t augh t one c l a s s

by each of the t each ing methods. I n s t r u c t i o n was through

" q u a s i - t e x t u a l i n s t r u c t i o n a l progrmaa,M Thcsa prograiaa p r e -

sen ted the fo l lowing metbemat ica l concep t s ; (1) n o t a t i o n ,

a d d i t i o n , and m u l t i p l i e s t i o n of i n t e g e r s ( p o s i t i v e , nega-

t i v e , and z e r o ) ; (2) tha d i s t r i b u t i v e p r i n c i p l e of

m u l t i p l i c a t i o n over a d d i t i o n ; and (3) e x p o n e n t i a l n o t a t i o n

and m u l t i p l i c a t i o n and d i v i s i o n of mnnb.irs express ad in

e x p o n e n t i a l n o t a t i o n . These t o p i c s were p r e s e n t e d dur ing

s i x weeks of i n s t r u c t i o n .

In order to determine the r e l a t i v e e f f e c t i v e n e s s of

tha two t each ing ir.ethodi four s u b t e s t s ware given dur ing

the ins t rue t i o n a 1 p e r i o d . Faoh s u b t e s t was dea ipned to

3 . R. Worthen, "A Coraparisou of Discovery and Expa i to ry Sequencing in d lemantary Katbsrnat ics I n s t r u c t i o n , ' Research In MsJfeei>j« fclcfl Educa fcion, e d i t e d by Joseoh M. Scsndura (Washington, ^ iW?lT 'ppT" i i i -59 .

59

c o r r e s p o n d w i t h tbe i n s t r u c t i o n a l m a t e r i a l completed j u s t

p r i o r t o tbe a drain i s t r a t i o n of the t e a t . A t e a t d e a l i n g

w i t h a l l the i n s t r u c t i o n a l m a t e r i a l was a d m i n i s t e r e d t w i c e ,

once f i v e weeks a f t e r i n s t r u c t i o n and a g a i n e leven weeks

a f t e r i n s t r u c t i o n . T r a n s f e r of h e u r i s t i c s was measured by

two t e s t a and p u p i l a t t i t u d e toward ma th ems t i c 3 was a s -

s e s s e d by two a t t i t u d e s c a l e s .

An a n a l y s i s of da ta showed t h a t tbe d i s c o v e r y method

d id n o t produce s u p e r i o r r e s u l t s in terms of i n i t i a l l e a r n -

ing b u t d id produce s u p e r i o r r e s u l t s in r e t e n t i o n f i v e

weeks a f t e r i n s t r u c t i o n . On the t e s t of r e t e n t i o n e l even

weeks a f t e r in a t r uc t:l on the Wo t r e a t m e n t groups were n o t

s i g n i f i c a n t l y d i f f e r e n t , "oa r e s u l t s from a t e s t on con -

c e p t t r a n s f e r showed no s i g n i f i c a n t d i f f e r e n c e between the

two g r o u p s . 'The r e s u l t s from the a t t i t u d e s c a l e s showed

no s i g n i f i c a n t d i f f e r e n c e in a t t i t u d e toward mathemat ics

between the group t augh t by a d i s c o v e r y method and the

group t a u g h t by an e x p o s i t i o n method . I t was found t h a t

the d i s c o v e r y £roup was s u p e r i o r t o the e x p o s i t i o n group

in a b i l i t y to t r a n s f e r h e u r i s t i c s . I t was conc luded t h a t ,

in g e n e r a l , the f i n d i n g s of t h i s s t u d y s u p p o r t e d many of

the c l a ims made by p r o p o n e n t s of d i s c o v e r y methods of

t e a c h i n g .

In t h i s s e c t i o n f i v e s t u d i e s conduc ted in e l e m e n t a r y

mathemat ics c l a s s rooms have been r e v i e w e d . In each of the

s t u d i e s , w i t h the e x c e p t i o n of the s t u d y conduc ted by

60

McConnel l , some e v i d e n c e was f o u n d which supported the

h y p o t h e s i s t h a t d i s c o v e r y 01 g e n e r a l i z a t i o n methods of

t e a c h i n g are s u p e r i o r to e x p o s i t i o n or d r i l l methods of

t e a c h i n g .

S t u d i e s Conducted in t h e Junior and S e n i o r

High S c h o o l M a t h e m a t i c s Classroom

In t h i s s e c t i o n e i g h t s t u d i e s t h a t h a v e been c o n d u c t e d

in j u n i o r and s e n i o r h i g h s c h o o l c la s srooms a r e r e v i e w e d .

S i x of t h e s e s t u d i e s d e a l w i t h t he l e a r n i n g of m a t h e m a t i c s

w h i l e t h e r e m a i n i n g two d e a l w i t h c o n t e n t l e a r n e d in t e c h -

n i c a l e d u c a t i o n *

M i c h a e l - 0 compared two methods of t e a c h i n g p o s i t i v e

and n e g a t i v e numbers , t h e f u n d a m e n t a l o p e r a t i o n s ' w i t h them,

and t h e s o l u t i o n of s t a p l e e q u a t i o n s in n i n t h - g r a d e a l g e b r a

In Method A the s u b j e c t s were l e d t o d i s c o v e r and under-

s tand t h e f u n d a m e n t a l p r i n c i p l e s and r e l a t i o n s h i p s t o be

l e a r n e d th rough e x e r c i s e s b u i l t a r o u n d f a m i l i a r s i t u a t i o n s

such as t h o s e d e a l i n g w i t h t i m e , money, teiv'pera t u r e , and

o t h e r s . No s t a t e m e n t of r u l e s of o p e r a t i o n was made b y t h e

t e a c h e r or p u p i l s in t e a c h i n g or in p u p i l d i s c u s s i o n s .

Method B errcphasized t h e use of a u t h o r i t a t i v e s t a t e m e n t s of

t he r u l e s of o p e r a t i o n combined w i t h e x t e n s i v e p r a c t i c e or

^ R . C. M i c h a e l , "The R e l a t i v e E f f e c t i v e n e s s of Two Methods of Teach ing C e r t a i n T o p i c s in N i n t h Grade A I r - e b r a , " The M a t h e m a t i c s T e a c h e r , XL1T ( F e b r u a r y , 191+9), 8 3 - 8 7 ,

61

d r i l l . No a t t empt was roads,. b e f o r e pract ice began, to ex-

p l a i n why the r u l e s operated to give the c o r r e c t r e s u l t s .

The s tuden t s i n f i f t e e n n in th -g rade a l g e b r a c l a s s e s

w a r e u s e d a s s u b j e c t s . F i f t e e n d i f f e r e n t t e a c h e r s p a r -

t i c i p a t e d i n t h e s t u d y . T e s t a w e r e given t o m e a s u r e

computa t ional a b i l i t y a n d a b i l i t y t o g e n e r a l i z e . A

q u e s t i o n n a i r e designed to measure a t t i t u d e toward a lgebra

was a l s o s d r c i n i s t a r e d . These t e a t s were used as p r e t e s t s

and a s p o s fetes t s . The r e s u l t s ve r s analyzed through the

use of a n a l y s i s of cover i a n c e . The a n a l y s i s i n d i c a t e d

t h a t Method B produced s i g n i f i c a n t l y g r e a t e r gains ia

c o m p u t a t i o n a l s k i l l ' and a b i l i t y to m a k e and u a a g e n e r a l -

i z a t i o n s . Method A produced b e t t e r r e s u l t s in a t t i t u d e

f avo rab l e to a l g e b r a , the s u b j e c t of immediate i n t e r e s t .

In another s tudy conducted Kith n i n t h - g r a d e a lgebra

s tuden t s Sob a 1^7 compared en a b s t r a c t , v a r b a l l s e d , deduc-

t i v e method of teaching in which concepts were de f ined and

p re sen ted by the t e a c h e r , fol lowed by p r a c t i c e exe rc i s e s

with a c o n c r e t e , nonve rba l i zed , induc t ive teaching method

in which s tuden t s were guided through exper iences invo lv ing

a p p l i c a t i o n s , t o d i scover and verbs 11.? 3 c o n c e p t s . Tha

e x p e r i m e n t was c o n d u c t e d d u r i n g t h e f i r s t f o u r w e e k s o f

t h e school y e a r .

A . S o b e l , "Concept Learning in A lgeb ra , " Th< M a t h e m a t i c s Teacher . XL1T (October , 1956), ~

62

Spec i a l c r i t e r i o n t e a t s were developed f o r the ex-

per iment dur ing two p i l o t s t u d i e s , A f t e r r e v i s i o n s were

made two forms of the t e s t a were comple ted . Each t e s t

con ta ined two p a r t s . One p a r t d e a l t wi th concepts and

the o the r p a r t was des igned to eva lua t e fundamenta l s k i l l s .

On. t b i3 t e a t i t was found t e a t the performance of the

s u b j e c t s wi th high a b i l i t y and t augh t by the i n d u c t i v e ex-

p e r i m e n t a l method was s u p e r i o r to the performance of the

s u b j e c t s wi th high a b i l i t y and t augh t by the deduc t ive

c o n t r o l method. For s u b j e c t s wi th average a b i l i t y t h e r e

was no d i f f e r e n c e in performance between the s u b j e c t s

t augh t by the two d i f f e r e n t methods . Three months a f t e r

the o r i g i n a l t e s t was g i v e n , a t e a t was admin i s t e r ed to the

same s u b j e c t s in order to determine the e f f e c t of method

of p r e s e n t a t i o n upon r e t e n t i o n of c o n c e p t s . On t h i s t e s t

i t was again found t h a t the performance of the s u b j e c t s of

h igh a b i l i t y and taught by the i n d u c t i v e exper imenta l

method was s u p e r i o r to the performance of those s u b j e c t s

of high a b i x i t y and t s u g n t by the deduc t ive c o n t r o l method.

For s u b j e c t s wi th average a b i l i t y t he re was no d i f f e r e n c e

due to t each ing method u s e d . Q

Wolfe i n v e s t i g a t e d the problem of whether an a b r u p t

change to e x p o s i t o r y methods :! 11 have any adverse e f f e c t s

38, J"Marein S y l v e s t e r Wolfa, " E f f e c t s of Expos i t o ry i n s t r u c t i o n in Mathematics on S tudents -Accustomed to Discevery Modes," unpubl ished doc tor©1 dlsdtsr fcat ion, d n i v e r a i t y of I l l i n o i s , Urbans , I l l i n o i s , 1963.

63

on the per romance of tha s tudent vrbo baa been accustom

to lea rn ing by discovery methods. The a ample for the study

waa taken from a populat ion of secondary school s tudents

who vera taught ma thema t i c s from the Univers i ty of I l l i n o i s

Coram i t tee on School Mathematics ma te r i a l s during the p r e -

ceding school yea r . Three hundred s tudents were chosen

from c lasses in three junior high schools and two senior

high schools . One hundred s ix ty of tha sub jec t s wars in

tha n in th-grade and one hundred f o r t y of the sub jec t s were

in grade t en . Within a given c l a s s , one-half of the sub-

j e c t s received exposi tory i n s t ruc t i on and one-half of tha

c lass received i n s t ruc t i on u t i l i s i n g a discovery method.

I n s t r u c t i o n was given through s e l f - t e a c h i n g programed

m a t e r i a l s . Each program era to In ad approximately ten

pages of i n s t r u c t i o n a l m a t e r i a l s . On the day a f t e r the

programs had been completed by the s tudents an achievement

t e a t was a dm in is t a red . On tha next day a t r a n s f e r t e s t

waa adminis te red . Ho s i g n i f i c a n t d i f f e r e n c e was found

between the group rece iv ing exposi tory i n s t ruc t i on and

o>is group *-coi i? viig i n s t ruc t i on through a discovery method

with r e spec t to achievement and with r e spec t to a b i l i t y to

t r a n s f e r knowledge to new s i tus ' c i cns . Xk was coneluded

tha t a change to exposi tory ins true j i a - for "indents who '

are accustomed to discovery methods w i l l not eer ioua ly

i n t e r f e r e with the a b i l i t y of s tudents to learn ma the -

re a t i c s .

6I|

Howitz39 conducted a study in which tha r e l a t i v a

e f f e c t i v e n e s s of a guided d i scovery method of teaching and

a conventional method of teaching were compared. The

subjects used in the s tudy were 290 n in th -g rade general

mathematics s t u d e n t s . Most of the subjects could be con-

s ide red "alow l e a r n e r s . " Sevan teachers p a r t i c i p a t e d in

the s t u d y . Each teacher t aught one c lass us ing the guided

d i scovery method of teaching and one c l a s s using the con-

v e n t i o n a l method of t e a c h i n g . A textbook was designed to

be used in the c l a s s e s t aught by the guided d i scovery inetb-

od. The conten t of t h i s textbook was not s i m i l a r to the

con ten t of t r a d i t i o n a l gene ra l mathematics t ex tbooks . The

textbook used i^ t h j clauiios uiughfc by the conven t iona l

method b a s i c a l l y presented-a review of a r i t h m e t i c . The

s tudy was conducted throughout an e n t i r e school y e a r .

P r e t e s t s and pos t tes ts were given and the re su l t s wera

analysed us ing the a n a l y s i s of var iance and a n a l y s i s of

covarianca t e chn iques . For esch t e s t a throe-way analys i s

was used with t e a c h e r , t r e a tmen t , and sex of s u b j e c t as the

th ree dimensions . On a a t andara ized sehievsment t e s t there

was no s i g n i f i c a n t d i f f e r e n c e in achievement as a r e s u l t of

the t each ing methods. On a t e s t s p e c i f i c a l l y designed to

t e s t achievement of topics taught by the guided discovery'

3^Thomas Allen Howitz, "The Discovery Annroacta: A Study of I t s Re l a t i ve Ef fec t iveness in Matheroatica unpublished d o c t o r a l d i s s e r t a t i o n , U n i v e r s i t y of Minnesota, Minneapolis , Minnesota , 1965.

6 5

method the s t u d e n t s t a u g h t by the g u i d e d d i s c o v e r y method

showed s u p e r i o r ach ieverasnt as compared to t h o s e t a u g h t by

a c o n v e n t i o n a l i ce thod. No d i f f e r e n c e wss found b e t w e e n the

two t r e a t m e n t g r o u p s in t e r n s o f a t t i t u d e t o w a r d ma thercat ica

A s e c o n d s t u d y u t i l i z i n g g e n e r a l m a t h e m a t i c s s t u d e n t s

a s s u b j e c t s was conducted b y P r i c e T h e s t u d y was c o n -

d u c t e d in t h r e e t e n t h - g r a d e c l a s s e s . One c l a s s , c a l l e d t h e

c o n t r o l g r o u p , wss t a u g h t b y t h e t r a d i t i o n a l t e x t b o o k -

l e c t u r e - r e c i t a t i o n method u s i n g t h e n o r m a l c o u r s e o f s t u d y

f o r t h i a c l a a a . The two o t h e r c l a s s e s , c a l l e d the e x p e r i -

m e n t a l g r o u p , were t a u g h t b y a t e c h i q u e d e s i g n e d t o f o s t e r

d i s c o v e r y . B e f o r e the s t u d y b e g a n t h e s t u d e n t s were a s -

s i g n e d t o the t h r e e c l a s s e s b y a random p r o c e s s . The

s t u d e n t s in t h e e x p e r i m e n t a l e l a s s e 3 f o l l o w e d t h e same

b a s i c c o u r s e p l a n as t h e c o n t r o l c l a s s e s e x c e p t f o r soma

a d d i t i o n a l t o p i c s t h a t we re t r e a t e d in t h o s e c l a s s e s . One

of t h e e x p e r i m e n t a l c l a s s e s a l s o rcsde use of s p e c i a l l y

p r e p a r e d t r a n s f e r m a t e r i a l . These exercises we re u s e d t o

p r o v i d e e x t r a e x p e r i e n c e w i t h problems i n critical t h i n k i n g

i n non-roa thema t i e s ! area a.

C r i t e r i o n t e a t s were standardized t e s t s? d e s i g n e d t o

m e a s u r e m a t h e m a t i c a l and t h i n k i n g a b i l i t y . These t e s t a

v. e r b x i r s t g i v e n ss p r e cc 3 c 2 d u r i n g the 1 i r s t w e e k of

1 J a c k S t a n l e y F r i c e , " D i s c o v e r y : Its E f f e c t on t h e A c h i e v e m e n t and C r i t i c a l T h i n k i n g A b i l i t i e s of T e n t h Grade G e n e r a l M a t h e m a t i c s S t u d e n t s , " unpublished doctoral d i s s e r -t a t i o n , Wayne S t a t e U n i v e r s i t y , D e t r o i t , M i c h i g a n , 1 9 6 5 .

66

s c h o o l . The t e s t a were a d m i n i s t e r e d a g a i n a f t e r one

semes te r of i n s t r u c t i o n . In a d d i t i o n a q u e s t i o n n a i r e con-

cerning a t t i t u d e toward the c l a s s was g i v e n . Prom, tha

r e s u l t s of t h i s t e s t i n g program i t was concluded t h a t t h e r e

was no d i f f e r e n c e in ach ievement among the three c l a s s e s .

The e x p e r i m e n t a l groups showed a g r e a t e r i n c r e a s e in ma the-

m a t i c a l r e a s o n i n g then the c o n t r o l g r o u p , but the d i f f e r e n c e

was n o t s t a t i s t i c a l l y s i g n i f i c a n t . The e x p e r i m e n t a l group

r e c e i v i n g s p e c i a l t r a n s f e r m a t e r i a l showed a s i g n i f i c a n t

i n c r e a s e in c r i t i c a l t h i n k i n g a b i l i t y . F i n a l l y the groups

t a u g h t by t e c h n i q u e s d e s i g n e d to promote d i s c o v e r y bad a

p o s i t i v e change in a t t i t u d e toward mathematics b u t t h e group

t a u g h t b y lbs t r a d i t i o n a l method abowed a n e g a t i v e change .

N i c h o l s ^ a t t e m p t e d to a s s e s s the r e l a t i v e e f f e c t i v e -

ness of two teaching methods in t-aecbing h i g h s choo l

geomet ry . One method was s i m i l a r to what ia u s u a l l y c a l l e d

a t r a d i t i o n a l method and the other method was a type of d i s -

cove ry method . -In o rde r to a s s e s s the r e l a t i v e e f f e c t i v e -

ness of t h e s e methods an experiment was conduc ted in which

f o r t y - t w o f reshmen a t the U n i v e r s i t y of I l l i n o i s High School

served as s u b j e c t s . The c r i t e r i o n t e s t was g iven as a p r e -

t e s t two months be fore the s t u d y b e g a n . On -the b a s i s of

t he s c o r e s on t h i s t e s t and an i n t e l l i g e n c e t o s t the

Eugene Douglas N i c h o l s , "Comparison of Two Approaches to the Teaching of S e l e c t e d Topics in Plana Geometry," unpub l i shed d o c t o r a l d i s s e r t a t i o n , U n i v e r s i t y of I l l i n o i s , Urbana, I l l i n o i s , 195-6.

6?

s u b j e c t s were d iv ided i n t o two matched groups of equal s i z e .

I n s t r u c t i o n was given dur ing seventeen c l a s s s e s s i o n s of

s i x t y minutes each . The two groups were t augh t a l t e r n a t e l y

by t h r e e t e a c h e r s In order to randomise the t eacher e f f e c t .

At the end of the t e ach ing p e r i o d the c r i t e r i o n t e s t was

aga in adminls t e r e d . 'Phis tea t was des igned to measure know-

ledge of vocabu la ry , c r i t i c a l t h i n k i n g a b i l i t y , a b i l i t y to

so lve pr "ibleras, and a c q u i s i t i o n of fundamental s k i l l s . No

a i g n i f i c a r j t d i f f e r ence was found between the group t a u g h t

by the t r a d i t i o n a l method and the group t augh t by the' d i s -

covery method with r e s p e c t to the a b i l i t i e s measured by the

c r i t e r i o n t e s t .

A s e r i e s of s t u d i e s in which s u b j e c t s were t augh t t e c h -

n i c a l s k i l l s have been conducted . Although such s t u d i e s

may not be d i r e c t l y r e l a t e d to the l e a r n i n g of mathematics

they do provide some evidence concerning the e f f e c t i v e n e s s

of d i s c o v e r y methods of t e a c h i n g . S ines the e x t e n t to which

the r e s u l t s 01 th<38e s t u d i e s g e n e r a l i z e to the I s s r n i n g of

mathematics i s q u e s t i o n a b l e only two of these s t u d i e s a re

reviewed h e r e .

In order to provide a d d i t i o n a l exper imenta l and a p p l i e d

r e s e a r c h evidence as to the r e l a t i v e e f f e c t of us ing a

guided d i scove ry method of teaching in s i t u a t i o n s p r o v i d i n g

nunberoua problem so lv ing o p p o r t u n i t i e s , Ray^2 conducted a

hî,1 "Willis ti. Ray, " P u p i l Discovery vs • D i r e c t I n s t r u c t i o n , " J o u r n a l of Exper imental Educa t ion , XXIX (March, 1961} ,~"57Z=25o"7 " " L

68

s tudy in which the l e a r n i n g t a sk c o n s i s t e d of l e a r n i n g

micrometer measurement p r i n c i p l e s , , JL guided d i s c o v e r y

ne thod of t each ing these p r i n c i p l e s was compared wi th a

t r a d i t i o n a l d i r e c t and d e t a i l e d method of i n s t r u c t i o n .

The s u b j e c t s used in t h i s s tudy were 1.35 n i n t h - g r a d e b o y s .

F i f t y - f o u r s u b j e c t s were t augh t by a d i r e c t e d d i s c o v e r y

method, f i f t y - f o u r were t augh t by a t r a d i t i o n a l d i r e c t and

d e t a i l e d method, and a t h i r d group of twenty-seven s u b j e c t s ,

c a l l e d the c o n t r o l g roup , r ece ived no i n s t r u c t i o n . The

s u b j e c t s in each group ve ra c l a s s i f i e d accord ing to t h r ee

l e v e l s of mental a b i l i t y . The two exper imenta l groups

r ece ived f o r t y — m i n u t e s of i n s t r u c t i o n in one

i n a t r u c bioaa 1 a a a a l o » .

Three c r i t e r i o n t e s t a were g i v e n . One t e a t was given

imraedistoly a f t e r the i n s t r u c t i o n a l s e s s i o n . The second

t e s t was g iven one week l a t e r and the t h i r d t e s t was given

s ix weeks a f t e r i n s t r u c t i o n . The f i r s t t e s t vsa des igned

to measure i n i t i a l l e a r n i n g . The second and t h i r d t e s t a

were des igned to measure r e t e n t i o n and t r a n s f e r . The

r e s u l t s were analyzed using an a n a l y s i s of v a r i a n c e wi th

t r e a tmen t s by l e v e l a t e c h n i q u e .

Ana lys i s of da ta r epea l ed DO s i g n i f i c a n t d i f f e r e n c e in

i n i t i a l l e a r n i n g due to the exper imenta l t e a c h i n g methods .

There was no s i g n i f i c a n t d i f f e r e n c e in r e t e n t i o n a f t e r one

week, b u t s i x weeks - a f t e r i n s t r u c t i o n t h e r e was a s i g n i f i -

can t d i f f e r e n c e In r e t e n t i o n in f avo r of the group t augh t

69

by the d i r e c t e d d i s c o v e r y method , I t wag found t h a t t he

group t a u g h t by the d i r e c t e d d i s c o v e r y method was s i g n i f i -

c a n t l y s u p e r i o r to t he group t a u g h t by the t r a d i t i o n a l

method when the c r i t e r i o n was pe r fo rmance on t e a t a d e s i g n e d

to measure a b i l i t y t o t r a n s f e r l e a r n i n g to new s i t u a t i o n s .

This was t r u e f o r t h e t r a n s f e r t e s t g iven one week a f t e r

i n s t r u c t i o n and f o r t he t e s t g iven s i x weeks a f t e r

ins t r u s t i o n . A h y p o t h e s i s of no i n t e r a c t i o n between the

two t e a c h i n g methods and l e v e l of m e n t a l a b i l i t y was t e s t e d

and a c c e p t e d in a l l c a s e s . I t was conc luded t h a t a d i r e c t e d

d i s c o v e r y approach to t e a c h i n g i s s u p e r i o r t o d i r e c t and

d e t a i l e d i n s t r u c t i o n w i t h r e a p e c t to r e t e n t i o n of m a t e r i a l

i n i t i a l l y l o c n s a d end w i t h r e s p e c t t o e n r o l i n g p u p i l s to

make wide a p p l i c a t i o n s of m a t e r i a l l e a r n e d to new and

r e l a t e d s i t u a t i o n s .

Moss^3 a t t e m p t e d to f o r m u l a t e a t e a c h i n g method b a s e d

on t h e t e a c h i n g s of g e a t a l t p s y c h o l o g y . The t e a c h i n g me th -

od deve loped v;aa e s s e n t i a l l y a gu ided .d iscovery method of

t e a c h i n g . In o rde r to de t e rmine the e f f e c t i v e n e s s of t h i s

t e a c h i n g method i t was compared w i t h a more t r a d i t i o n a l

method of t e a c h i n g . The s u b j e c t s used in t h e compar ison

s t u d y were 131 j u n i o r end s e n i o r s t u d e n t s in v o c a t i o n a l -

i n d u s t r i a l c u r r i c u l a a t a t e c h n i c a l h igh s c h o o l . -The

j. O Jerome Moaa, J r . , "An E x p e r i m e n t a l Study of t he R e l a t i v e E f f e c t i v e n e s s of the D i r e c t - D e t a i l e d and t h e Dl J?OCtedL D "f & f* F? v & v rr ?/ .<=» *M-\ ^ ~ M ~ ~ - n. J» ._

70

learning task involved tbe learning of a highly organized

d0D-raanipulative learning task, letterpress imposition.

The scope and complexity of this material does not differ

greatly from many typical classroom lessons in which tech-

nical material is presented to a group of students.

The subjects were randomly assigned to three groups.

One group was taught by the guided discovery method, one

group wag taught by the traditional method, and the third

group received no instruction. Instruction was given to

two of tbe groups during one instructional session. All

instruction was presented with the use of a tape recorder

in order to insure constancy of presentation. A test of

initial learning was administered iranediately after the

learning period. lie tent ion and transfer tests were given

one and six weeks after instruction.

The results showed that the performance of the two

groups receiving instruction was superior to the perfor-

mance of tbe group receiving no instruction. The results

showed further that there was no significant difference

between the group receiving instruction by a guided dis-

covery method and the group receiving instruction by a

traditional method with respect to initial learning, re-

tention, and transfer of training.

In this section six studies have been reviewed in

which discovery methods of teaching junior or senior high-

school ma thematic a were compared with traditional methods

71

of t each ing . Wolfe ar,d Nicho ls found t h a t s t u d e n t s l e a r n e d

e q u a l l y w e l l under d i s c o v e r y .methods of t e a c h i n g and ex-

p o s i t i o n methods of t e a c h i n g . Eowitz and Price reported

s t u d i e s in which s t u d e n t s taught by d i s c o v e r y methods vrere

s u p e r i o r to s t u d e n t s taught by t r a d i t i o n a l methods , a t

l e a s t in c e r t a i n f a c e t s of l e a r n i n g . That t h i s s u p e r i o r i t y

was a c t u a l l y a r e s u l t of the d i f f e r e n t t e a c h i n g methods i s

d i f f i c u l t t o determine because the s t u d e n t s t a u g h t by a

d i s c o v e r y method a c t u a l l y l e a r n e d a d i f f e r e n t s e t of f ac ta

than the s t u d e n t s t a u g h t by an e x p o s i t i o n method . Aa a

r e s u l t the f i n d i n g s of t h e s e s t u d i e s should be viewed wi th

c a u t i o n . oobe 1 found th3 t , wi th s t u d e n t s of above ave rage

a b i l i t y , teaohlug by a d i s c o v e r y method produced b e t t e r

r e s u l t s than those ob ta ined when t e a c h i n g by a t r a d i t i o n a l

method. In c o n t r a s t , Michae l found t h a t an e x p o s i t i o n

method of t e a c h i n g produced s u p e r i o r r e s u l t s .

Two s t u d i e s which were conducted w i t h t e c h n i c a l edu-

c a t i o n s t u d e n t s were r e v i e w e d . Ray found t h a t a d i s c o v e r y •

its 8 wQ cd of t e ach ing was b e t t e r tb n n a d i r e c t and d e t a i l e d

iflethod of t e a c h i n g , waereaa Mosa found t h s t s t u d e n t s

learned equa l ly w e l l when e i t h e r teaching method wag used.

S t u d i e s Conducted in t he Col lege

Ma tb sks t i c a C la a s r o era

Two s t u d i e s have been conducted in which d i s c o v e r y

and e x p o s i t i o n methods of t e a c h i n g mathemat ics to c o l l e g e

72

s t u d e n t s have been compared. Both et-udiaa ware conduc ted

w i t h c o l l e g e f reshmen ea s u b j e c t s .

W e i n e r ^ i nvs s t i g s t ed the r e l a t i v e e f f e c t i v e n e a a of

two t e a c h i n g t e c h n i q u e s in d e v e l o p i n g the f u n c t i o n a l

competence of c o l l e g e s t u d e n t a in a f i r s t series t a r c our a a

in ma t h e m a t i c a . The s u b j e c t s were s t u d e n t s s c r o l l e d i n a

conation i t y c o l l e g e and who were i s t he same c u r r i c u l u m ,

e l e c t r i c a l t e c h n o l o g y . Only male s t u d e n t s were used as

s u b j e c t s . Two c l a s s e s wore used as a c o n t r o l group and

two c l a s s e s were used aa en e x p e r i m e n t a l g r o u p . Each

c l a s s c o n t a i n e d approxiroa fcely twenty- two s t u d e n t s . The

t e ach ing -me thod used w i t h the c o n t r o l group c o n s i s t e d ,

e s s e n t i a l l y - , of s t a r t i i jg wi th an e x p l a n a t i o n of t he

g e n e r a l c o n c e p t s and then p r o c a d i n g t o i l l u s t r a t e sad

a p p l y t h e s e concep t s to v a r i o u s e x e r c i s e s . In the t e a c h -

ing method uaed w i th the experin-ionts1 group a t t e m p t s were

made to l e a d up to the concep t s t o bs l e a r n e d th rough

p r o b l e m a t i c s i t u a t i o n s . The i n s t r u c t i o n a l p e r i o d c o n s i s t e d

of f o u r i i f t y - r i s i a u t e c l a s s s e s s i o n s per week f o r approx**

i rna te ly s i x t e e n weeks . The same sequence and tiraa s c h e d u l e

was used by a l l c l a s s e s . There was no random a s s i g n m e n t

of s u b j e c t s to c l a s s e s , b u t an examina t ion of s c o r e s

e l v i n Weiner , MA Comparison of the E f f e c t of Two Teaching Tecbniquea in Developing the F u n c t i o n a l Competence of C o l l e g e S t u d e n t s in a F i r s t S'eraeater Course in Matheraatica , w unpub l i shed d o c t o r a l d i s s e r t a t i o n , New York TTv* 1 *!? S 4 "f -rr *kY /-• *.* \T -«r * • X 1 . x . TIT -rr . v ^ f •*

73

achieved by the s u b j e c t s on s e v e r a l t e s t a given b e f o r e the

exper iment was begun i n d i c a t e d no s i g n i f i c a n t d i f f e r e n c e

between the t r ea tmen t g roups .

At the end of the i n s t r u c t i o n a l pe r i od the t e a t , Davis

Tea t of F u n c t i o n a l Competence in Mathemat ics , vraa admin-

i s t e r e d aa a c r i t e r i o n t e a t . No s i g n i f i c a n t d i f f e r e n c e was

found between s u b j e c t s of high a b i l i t y in the two t r e a t m e n t

g r o u p s . I t was found t h a t the performance of the s t u d e n t s

of average a b i l i t y in the exper imenta l group was s u p e r i o r

to the performance of the s t u d e n t s of average a b i l i t y in

the c o n t r o l g roup . I t was concluded t h a t f u n c t i o n a l com-

pe tence in mathematics can be improved by a t each ing method

based upon a p rob lem-can te red technique and s t u d e n t s of

weak t o average mathemat ica l background show the g r e a t e s t

improvement in f u n c t i o n a l competence when t augh t by such a

t e c h n i q u e .

C u m m i n d e v e l o p e d a s e r i e s of s tudy-gu ide shee t s

wnich could be used in a beg inn ing course in polynomial

c a l c u l u s . These m a t e r i a l s ware designed to develop under-

s t and ing in the use of a ome of the fundamenta l ideas b e f o r e

these concepts were s u b j e c t e d to c r i t i c a l d i s c u s s i o n by the

c l a s s or b e f o r e . t h e r e s u l t s sugges ted as hypotheses were

f i n a l l y shown to be t rue by deduc t ive methods . Thesi ie

Kenne uh Cummi r, s , "A Student Hxper i ence-Discovery Approach to the Teaching of Calculus ,« The Mathematics

L I I I (March, I960) , 162-170.

- • 71+

m a t e r i a l s "were d e s i g n e d to c r e a t e an a tmosphere in which

the s t u d e n t s could d i s c o v e r ger,ei'£ .ilze t i o n s which c o u l d be

v e r i f i e d d e d u c t i v e l y a f t e r the s t u d e n t s had become f a m i l a r

w i t h them.

Dur ing each of two q u a r t e r s one c l a s s was t a u g h t u s i n g

t h e s e m a t e r i a l s in a d d i t i o n to the s t a n d a r d t e x t b o o k and

a n o t h e r c l a s s was t a u g h t by the t r a d i t i o n a l method us ing

only the t e x t b o o k . Two t e a c h e r made t e s t s were uaed as

p r e t e a t s and poo t i e s t a . A f t e r a s t a t i s t i c a l a n a l y s i s was

made of the t e a t s c o r c s i t was conc luded t h a t the p e r f o r -

mance of the experimental group waa s i g n i f i c a n t l y s u p e r i o r

to the per f o r r a n e e of the group t a u g h t by the t r a d i t i o n a l

me thod . Accord ing to Cummins, "The r e a u l l s i n d i c a t e t h a t

the method of t e a c h i n g under examina t ion was e s p e c i a l l y

e f f e c t i v e in p romot ing a deeper u n d e r s t a n d i n g of the c a l -

cu lus and t h a t t h i s ga in waa n o t a t the s a c r i f i c e of

p r o f i c i e n c y in m a n i p u l a t i o n s and a p p l i c a t i o n s

In t h i s s e c t i o n two s t u d i e s conduc bed-with c o l l e g e

ma themat i c s s t u d e n t s have been rev iewed. V/oiner found t h a t

a p r o b l e m - c e n t e r e d t e c h n i q u e of beaching was e f f e c t i v e in

i n c r e a s i n g the f u n c t i o n a l competence of s t u d e n t s w i t h a v e r -

age and below ave rage a b i l i t y . , Cummins found t h a t a

d i s c o v e r y method of t e a c h i n g polynomial c a l c u l u s waa e f f e c -

t i v e in promoting u n d e r s t a n d i n g . The r e s u l t s of the s t u d y

^ I b i d . , p. 168.

75

by Cummins do not n e c e s s a r i l y p r o v i d e ev idence support ing

d i s c o v e r y methods of tcaching b e e a a s s the s tudents in the

exper iinenta 1 group a c t u a l l y learned a d i f f e r e n t s a t of

f a c t a than the s u b j e c t s in the c o n t r o l group.

Summary

The pu rpose of- t h i s c h a p t e r was to p r e s e n t a compre-

h e n s i v e summary of r e s e a r c h l i t e r a t u r e r e l a t e d to d i s c o v e r y

methods of t e a c h i n g . The s t u d i e s have been p r e s e n t e d in

groups w i t h the o r g a n i z a t i o n a l baa i s f o r the groups b e i n g

the type of l e a r n i n g m a t e r i a l us ad i n t he s t u d y .

In the f i r s t s e c t i o n the s t u d i e s of George KG ton a and

t h r e e r e l a t e d s t u d i e s wars rev iewed. The s t u d i e s conduc ted

by Katons and the s t u d y r e p o r t e d b y H i l g a r d , I r v i n e , and

Whipple p r e s e n t e d ev idence ind ie s t ing t h a t t e a c h i n g methods

emphasizing unde.ratanding and p u p i l d i s c o v e r y a r e s u p e r i o r

t o t e a c h i n g methods emphasising r o t s m e m o r i s a t i o n . Gorman

found t h a t a p p r o p r i a t e gu idance i s b a t t e r than no gu idance

and l e a v i n g the s t u d e n t to d i s c o v e r f o r h i m s e l f t he s o l u t i o n

to a problem w i l l n o t p r e v e n t undera tending, b u t p r o b a b l y

w i l l d e l a y i t . Scanaura found t h a t e x p o s i t o r y methods of

t e a c h i n g are j u s t as e f f e c t i v e aa and in some c s saa mora

a f f e c t i v e than d i s c o v e r y methods of t e a c h i n g .

In the second s e c t i o n three s t u d i e s wore reviewed in

which the l e a r n i n g t a s k s involved l e a r n i n g methods of d e c i -

p h e r i n g c o d e s . Two of the s t u d i e s i n d i c a t e d t h a t l e a r n i n g

76

through discovery is no more e f f e c t i v e than lea rn ing by

memorization.

A group of four s tud ies was reviewed in which s u b j e c t s

were requ i red to learn word r e l a t i o n s h i p s . Two of tha

s tud ies ind ica ted t ha t l ea rn ing is moat e f f i c i e n t when a

maximum amount of d i r e c t i o n is given during i n s t r u c t i o n .

In the other two s tud ies i t was concluded t h a t soma form

of l ea rn ing by discovery ia more e f f e c t i v e than l ea rn ing

by a method in which a l l necessary information ia i d e n t i -

f i e d by the i n s t r u c t o r .

In the four th group of s tud ie s f i v e s tud ies were

reviewed in which a ur;s of s e r i e s were l ea rned . In four

of these s tud ies i t was concluded tha t a discovery method

of teaching is mors e f f e c t i v e than a d i r e c t method of

t each ing .

Five s tud ies were reviewed which repor ted the r e s u l t s

of experiments conducted in elementary school is a thematics

c lassrooms. In four of the f i ve s tud ies aorae evidence was

found which supported the hypothesis that; l ea rn ing by d i s -

covery is more e f f e c t i v e than l ea rn ing by a d r i l l method.

In the s i x th sec t ion e igh t s tud ies conducted in junior

and senior high school classrooms were reviewed. In three

s tud ies i t was .found t h a t s tudents learned equal ly wel l

under d iscovery methods of teaching and expos i t ion method

of t each ing . Pour s tud ies repor ted conclusions in favor

of d iscovery methods of t each ing . A s ix th study found tha t

77

an expos i t i o n method of t each ing produces r e s u l t s s u p e r i o r

to those ob ta ined when 8 d i s cove ry M e t h o d i s u sed .

In a f i n a l s e c t i o n two s t u d i e s were reviewed which

were conducted a t the c o l l e g e l e v e l . Tn each s tudy i t waa

found t h a t a form of d i s cove ry method of t each ing i s more

e f f e c t i v e than a t r a d i t i o n a l method.

I t cao ba a can t h a t , a l though agreement ia n o t

unanimous, the m a j o r i t y of the s t u d i e s p u r p o r t to p rov ide

evidence suppor t i ng the c l i ira t h a t l e a r n i n g by d i s c o v e r y

i s snore e f f e c t i v e than l e a r n i n g by memor i sa t ion , d r i l l , or

a u t h o r i t a t i v e i d e n t i f i c a t i o n . F u r t h e r examinat ion r e v e a l s

t h a t the s u p e r i o r i t y of l e a r n i n g by d i s c o v e r y i s rcore

l i k e l y to ba revaaltsci viisn Ilia c * i i e i i o a la r e t e n t i o n ,

t r a n a f o r of t r a i n i n g , or mo t iva t ion than whan the c r i t e r i o n

ia i n i t i a l l e a r n i n g . Caution should be used when i n t e r -

p r e t i n g the r e s u l t s of t hese s t u d i e s because the t e ach ing

methods used in the s t u d i e s vary w i d e l y . What ia c a l l e d

the d i s cove ry method of t each ing in one s tudy ia no t nec -

c e s a a r i l y tha a sice t each ing rcatbod as the t e a c h i n g method

l a b e l l e d wi th the saras nsma in ano ther s t u d y . In some i n -

s t a n c e s what ia c a l l e d a d i s c o v e r y method of t each ing in

one s t u d y is c a l l e d an e x p o s i t i o n method of t e a c h i n g in

s o o t h e r s t u d y . A f t e r rev iewing r a a c a r c b l i t e r a t u r e r e l a t e d

to d i acovery methods of t e a c h i n g , Kersh concluded t h a t

. . . as i s t r u e in o ther a r ea s of r e s e a r c h , tha evidence ia aoraawhat e q u i v o c a l , p a r t l y because i t i s d i f f i c u l t to equate a t od i e s la terras of tha

78

amount and kind cf d i r e c t i o n t h a t i s p r o v i d e d . The exper imen t s 1. s u b j e c t s r a r e l y i f ever are r e q u i r e d to l e a r n c o m p l e t e l y w i t h o u t h e l p , and the k inds of h e l p p r o v i d e d commonly d i f f e r s .4-'

S i m i l a r l y Wor then has s t a t e d t h a t

To d a t e t he se s t u d i e s have f a i l e d to c l a r i f y many of the q u e s t i o n s p a r t s i n i n g to d i s c o v e r y and e x -p o s i t o r y i n s t r u c t i o n ; r a t h e r , the f i n d i n g s of t he v a r i o u s s t u d i e s , when taken a t f a c e v a l u e , o f t e n seem t o be c o n t r a d i c t o r y . Perhaps the g r e a t e s t f a c t o r which c o n t r i b u t e s to such e q u i v o c a l r e s e a r c h ev idence i s the d i f f e r i n g s p e c i f i c a t i o n among r e -s e a r c h e r s as to what they mean by such terms a a " d i s c o v e r y , " Mgu ided d i s c o v e r y , " and " e x p o s i t i o n . ^ S ince t he se t e r n s have n o t y e t been reduced to g e n e r a l l y a c c e p t e d o p e r a t i o n a l d e f i n i t i o n s , i t i s h i g h l y p r o b a b l e t h a t r e s e a r c h e r s working i a what i s n o m i n a l l y the seme domain a r e n o t a c t u a l l y i n v e s -t i g a t i n g the same phenomena a t a l l . 4 $

Prom such s t a t e m e n t s i t can be seen t h a t when a t t e m p t i n g

to i n t e r p r e t the r e s u l t s of s t u d i e s on d i s c o v e r y methods

of t e a c h i n g i t ia Impe ra t i ve t h a t the c h a r a c t e r i s t i c s of

each t e a c h i n g method used, be i d e n t i f i e d . Soma of the

c h a r a c t e r i a t i e s of s p e c i f i c c l a s s p r o c e d u r e s t h a t s h o u l d

be c o n s i d e r e d a r e amount of i n f o r m a t i o n g iven as g u i d a n c e ,

type of i n f o r m a t i o n g iven as g u i d a n c e , " t i d i n g " of i n f o r -

ma t ion g iven or sequence of p r e s e n t s t i o n of i n f o r m a t i o n ,

and mean i s g f u l a e s a of the i n f o r m a t i o n g i v e n . More d e t a i l e d

a n a l y s e s of some of the s t u d i e s r ev iewed in t h i s c h a p t e r

^ K e r s h , "The M o t i v a t i n g E f f e c t of Lea rn ing by D i r e c t e d D i s c o v e r y , " p . 6 5 .

^"Bla ine R. Wor t h e n , "A Corapar i s on of D i s c o v e r y and E x p o s i t o r y Sequencing in E l e m e n t a r y Mathemat ics I n s t r u c t i o n , " Research in Mathemat ics E d u c a t i o n , e d i t e d

79

h a v e b e e n w r i t t e n b y VJ i t t r o c k , ^ G r o r b a c b , ^ ® a n d B e c k e r

a n d McLeod.^-'-

. G * Wi t t rock , "The Learning by Discovery H y p o t h e s i s , " L e a r n i n g b j D i s c o v e r y ; A C r i t i c a l Appra iaaJL, e d i t e d b y L e e 8 . SbulmsH a n d T l v a n ~R. Ke'is lar" fChTc a g o , 1966), p p . 33-75.

''^Lee -J. Cronbecb, " The Logic of Harperimenta on Discovery ," Learning by Discovery , pp . 76-92.

^"'•Jerry P. Becker and Gordon K. McLeod, "Teaching, Discovery , and the Problems of Transfer of Tra in ing in Ma tbsiaa t i c s Research in Mathematics Education, pp. 93-107. — — — — ~—

CHAPTER I I I

EXPERMENTAL DESIGN AND EXPERIMENTAL

PROCEDURES


Fr am. the p r e c e d i n g c h a p t e r i t can b e seen t h a t t h e

r e s u l t s of expev i m e n t a l s t u d i e s r e l a t e d t o d i s c o v e r y m e t h -

ods of t e a c h i n g a r e n o t w h o l l y c o n c l u s i v e . As a r e s u l t ,

g e n e r a l i z a t i o n s c o n c e r n i n g d i s c o v e r y methods of t e a c h i n g

roust be a c c e p t e d w i t h c a u t i o n . In o r d e r t o p r o v i d e f u r t h e r

e v i d e n c e c o n c e r n i n g t h e e f f e c t i v e n e s s of d i s c o v e r y methods

of t e a c h i n g in an a r e a where few i n v e a t i g a t i c n s have b e e n

r . s d e . sn e x p e r i m e n t a l s t u d y vras c o n d u c t e d . The p u r p o s e of

t h i s s t u d y v;as t o a s c e r t a i n t h e v a l u e , as d e t e r m i n e d by

s t u d e n t a c h i e v e m e n t , of u s i n g a g u i d e d d i s c o v e r y me thod of

t e a c h i n g , in a c o l l e g e f r e s h m a n iaa thesis t i e s c o u r s e f o r n o n -

ma therca t i e c and n o n - s c i e n c e ma j or a , The p u r p o s e of t h i s

c h a p t e r i s t o d e s c r i b e t h e s t u d y t h a t was c o n d u c t e d . The

s e t t i n g of t h e s tudy . , t h e e x p e r i m e n t a l d e s i g n used in t h e

s t u d y , t he t e a c h i n g ms-tbods c o m p a r e d , ' t h e t a s t i n g p rog ram

c o n d u c t e d in o r d e r t o e v a l u a t e t he r e l a t i v e e f f e c t i v e n e s s '

of t h e t e a c h i n g m e t h o d s , and t h e p r o c e d u r e s u sed in a n a -

l y z i n g the t e s t d a t a a r e d e s c r i b e d , in t h i s c h a p t e r .

An

81

The S e t t i n g of t h e S t u d y

The purpose of th ia s e c t i o n i s to d e s c r i b e the s c h o o l

a t which the experiment was c o n d u c t e d , the s u b j e c t s used in

the e x p e r i m e n t , and the course of s t u d y in which t h e s e sub-

j e c t s were e n r o l l e d . The exper iment wss conducted dur ing

the s p r i n g semes ter of the 1967-1968 s c h o o l year a t Sou t h -

wea t e r n S t a t e C o l l e g e , Weather ford , Oklahoma . Weatber ford ,

Oklahoma has a p o p u l a t i o n of a p p r o x i m a t e l y 6 , 0 0 0 and ia

l o c a t e d in we a t ern Oklahoma . V7en t h e r f o r d i s a r u r a l com-

m u n i t y , w i t h moat of i t a r e s i d e n t a hav ing an a g r i c u l t u r a l

backgrond.

S o u t h w e s t e r n S t a t e C o l l e g e ia a s t a t e - s u p p o r t e d and

f o l l y a c c r e d i t e d four y<*ar c o l l e g e . The e n r o l l m e n t a t

Southwes t e r n S t a t e C o l l e g e d u r i n g t h e s p r i n g te rm of t h e

1967-1968 s c h o o l y e a r was a p p r o x i m a t e l y 14,300. Comple te

p rograms of s t u d y a r e a v a i l a b l e i n many m a j o r a r e a s . The

c o l l e g e i n c l u d e s a r e p u t a b l e s c h o o l of pharmacy and a g r a d -

u a t e s c h o o l which o f f e r s the Master of T e a c h i n g d e g r e e .

The c o l l e g e \ms o r i g i n a l l y c o n c e i v e d aa a t e a c h e r s c o l l e g e ;

a l t h o u g h l i b e r a l a r t s d e g r e e s a r e a v a i l a b l e in s e v e r a l

f i e l d s of s t u d y , the m a j o r i t y of the s t u d e n t s are s t i l l

e n r o l l e d in t h e s c h o o l of e d u c a t i o n . Many-of t h e s t u d e n t s

who a t t e n d Southwestern S t a t e C o l l e g e l i v e in r u r a l com-

m u n i t i e s in w e s t e r n Oklahoma and commute to s c h o o l d a i l y .

The s u b j e c t s f o r t h e s t u d y were s t u d e n t s u n r o l l e d in

f o u r of t h e f o u r t e e n s e c t i o n s of C o l l a g e Ka tfcema t i e s t a u g h t

o,o U £L

a t S o u t h w e s t e r n S t a t e C o l l e g e d u r i n g t h e s p r i n g semes t a r of

the 1967-1968 s c h o o l y s a r • C o l l a g e 1-1 a tbexna t i c s i s a r e -

q u i r e d c o u r s e f o r a l l s t u d e n t s who do n o t c o m p l e t e C o l l e g e

Algebra and T r i g o n o m e t r y a s a p a r t of t h e i r program of

s t u d y . S t u d e n t s who do n o t n o r m a l l y e n r o l l i n C o l l e g e

Ma thema t i c s a r e t h o s e r . a j o r i n g in mafcbematics , p h y s i c s ,

c h e m i s t r y , b i o l o g i c a l s c i e n c e , and s t u d e n t s e n r o l l e d in the

s c h o o l of pbar raacy . A l l o t h e r s t u d e n t s a r e r e q u i r e d t o

comple t e C o l l e g e Ka tbezsa t i c s . C o l l e g e M a t h e m a t i c s h a s a

t w o f o l d p u r p o s e . The c o u r s e i s d e s i g n e d to s e r v e ©a an

adcroducfcion t o ma thsrns t i c s f o r 1 ibe r a 1 a r t s s t u d e n t s s n d

i t i s a l s o d e s i g n e d to s e r v e as one of too r.a the ma t i c s

c o u r s e s r e q u i r e d of a l l p r o s p e c t i v e e l e v e n t a r y - g e b o o l

t e a c h e r s , A d e t a i l e d o u t l i n e of t h e n s t e r i a l t a u g h t In t h e

f o u r s e c t i o n s of C o l l e g e ' l a t h e r,;a t i c s u t i l i s e d in the s t u d y

i s p r e s e n t e d in Appendix A.

The E x p e r i m e n t a l Des ign

i.n vhi» a*3c;..icn toe n a s u r e of tli3 e x p e r I m e n t s 1 s t u d y

conducted, i s p r 3 sen t ed and t h e p r o c s d u r e s f o r ex s e a t i n g

•che e x p e r i m e n t a r e d e s c r i b e d . The e ^ p a r i n a c t c o n d u c t e d

was d e s i g n e d t o conform t o t h e modal f o r a two by t h r e e

l a c t o r i a l e x p e r i m e n t . The two i n d e p e n d e n t ? a r i a b l e s were

method of t e a c h i n g and l e v e l of a b i l i t y . The d e p e n d e n t

v a r i a b l e va3 a c h i e v e m e n t .

83

In the des ign of the exper iment snd in the a n a l y s i s of

da ta t h r e e l e v e l s of a b i l i t y if ere c o n s i d e r e d . In o rder to

de te rmine the l e v e l of a b i l i t y of each s u b j e c t the s u b j e c t ' s

composi te score on the American Col lege Tea t i n g Fr ogrsra va s

u t i l i z e d . The l e v e l s of a b i l i t y were de te rmined on the

b a s i s of t h e - c o m p o s i t e s co re s on the- American Co l l ege Tesjt-

in^ Prograra of a l l s t u d e n t s e n r o l l e d in Col lege Mathemat ics

d u r i n g t e a s p r i n g semes te r of the 1966-196? school y e a r .

Scores were a v a i l a b l e f o r ? 8 . 9 p e r c e n t of these s t u d e n t s .

On the b a s i s of t hese scores i t was de t e rmined t h a t s-tudenta

wi th a composi te s c o r e of twenty or above ranked above the

s i x t y - s i x t h p e r c e n t i l e whi l e s t u d e n t s wi th a s co re of f i f -

teen or-below ranked below the t h i r t y - t h i r d p e r c e n t i l e , As

a r e s u l t , in the exper imen t Leve l I c o n s i s t e d of a l l sub-

j e c t s w i t b a compos i te s c o r e on the Avr.erlean Co l l age T e s t -

ing Program of twenty or a b o v e . Leve l I I c o n s i s t e d of a l l

s u b j e c t s w i th a composi te s c o r e h i g h e r tbao f i f t e e n b u t

lower than twenty and Level I I I c o n s i s t e d of a l l s u b j e c t s

w i th a composi te s c o r e of f i f t e e n or l o w e r . In the a n a l y s i s

of data only the c r i t e r i o n s c o r e s of t h o s e s u b j e c t s en-

r o l l e d in the f o u r s e c t i o n s of Co l l ege -Mathematics u t i l i z e d

m the s t u d y and f o r whom s c o r e s on the Arnerican Co l l ege

!££££££ " 6 r Q a v a i l a b l e - a r c u . ' cd . Composite, s c o r e s

were a v a i l a b l e f o r 79 .3 p e r c e n t of the s t u d e n t s e n r o l l e d

in the fou r s e c t i o n s of C o l l e g e Ka t h e m a t i c a .

, . • 8i+

For t he metfood of t e a s b l e g v a r i a b l e t h e r e were two

l e v e l s . Ona aafchod of t e a c h i n g was c a l l a d the Guided D i s -

covery Method and the o the r method of t e a c h i n g waa c a l l e d

the E x p o s i t i o n Method. The n a t u r e of each t e a c h i n g method

i s d e s c r i b e d in a s e p a r s t e s e c t i o n . Of the f o u r s e c t i o n s

of Co l l ege Mathemat ics u t i l i z e d in the a tody two s e c t i o n s

were t a u g h t by the Guided D i s c o v e r y Method and two were

t a u g h t by the E x p o s i t i o n Method. Two of the f o u r flections

of Co l l ege Mathemat ics mat from 10:00 a . m . to 10:50 a . m .

on Tuesday , Thur sday , and F r i d a y d u r i n g each we.ek of the

a femes t e r . One of t h e s e s e c t i o n s waa t a u g h t by the Guided

D i scove ry Method and the o t h e r s e c t i o n waa t a u g h t by the

E x p o s i t i o n Method. The r s n s i n i n g two s e c t i o n s of Co l l ege

Mathemat ics raet from 2 : 0 0 p „ir. t o 2*.$0 p .m. on Monday,

Tuesday, and Thursday of each waek of the s e m e s t e r . One of

t h e s e a a c t i o n a waa t a u g h t by the Guided D i s c o v e r y Method

and the o tha r s e c t i o n was t a u g h t by rn 0 "Ex p o a :i t i o n M e th o d .

For b o t h the morning c l a s s e s and tha a f t e r n o o n c l a s s e s the

method to he used in a p a r t i c u l a r c l a s s was d e t e r m i n e d b y

the f l i p of a c o i n . The ten o ' c l o c k and two o ' c l o c k time

p e r i o d a were chosen in order to minimize the e f f e c t s of

t ime of day on l o s r n i n g . These t ima p e r i o d a ware chosen

in o rde r to a v o i d the d e p r e s s i n g e f f e c t s on l e a r n i n g of

e a r l y time p e r i o d a , l a t e t ime p e r i o d s , end time p e r i o d a

nea r noon . A time p e r i o d f o r b o t h worn ing and a f t e r n o o n

85

was chosen in o rde r t o a c h i e v e a b a l a n c e of e f f e c t s of

morn ing and a f t e r n o o n t ime p e r i o d s .

A a a c r i t e r i o n f o r d e t e r m i n i n g t h e r e l a t i v e e f f e c t i v e -

n e s s of t h e two t e a c h i n g methods a s i n g l e d e p e n d e n t v a r i a b l e

was c o n s i d e r e d . This d e p e n d e n t v a r i a b l e was s t u d e n t

2ena.eveirsen u• S t u d e n t a c h i e v e m e n t was measured, by f o u r

t e s t s . Three of t h e s e t e s t s v e r a s t a n d a r d i z e d t e s t s and

the f o u r t h t e s t was t e a c h e r made t e s t and wss a d m i n i s t e r e d

in s e v e r a l p a r t s . A d e t a i l e d d e s c r i p t i o n of the t e s t s used

and t h e methods and fcirce of a d m i n i s t r a t i o n of t h e s e t e s t s

i s g i v e n in a s e p a r a t e s e c t i o n .

Two i n s t r u c t o r s p a r t i c i p a t e d in t h e s t u d y . B o t h i n -

3 t r u e cor a were r e g u l a r ôoiobera of the s t a f f of t h e

m a t h e m a t i c s d e p a r t e o n t a t S o u t h w e s t e r n S t a t e C o l l e g e . Both

i n s t r u c t o r s bad s e v e r a l y e a r s e x p e r i e n c e as c o l l e g e t e a c h e r s

and b o t h i n s t r u c t o r s were fern 1 l i a r w i t h d i s c o v e r y methods

of t e a c h i n g . One of t he i n s t r u c t o r s has p a r t i c i p a t e d in sn

_ns t i tu t e . f o r .< ec onua r y rsis the re a t i c s t e a c h e r s c o n d u c t e d a t

l l S e d i v e r s i t y of I l l i n o i s . Max Beberraan, one of t ha f o r e -

Most a d v o c a t e s of d i s c o v e r y methods of t e a c h i n g m a t h e r a a t l c s ,

v:as t he d i r e c t o r of t h i s i n s t i t u t e . I n o r d e r t o b a l a n c e

t he e f f e c t of the i n s t r u c t o r upon s t u d e n t a c h i e v e m e n t , each

i n s t r u c t o r t a u g h t one s e c t i o n of C o l l e g e M a t h e m a t i c s by t h e -

Guided D i s c o v e r y Method and one s e c t i o n by the E x p o s i t i o n

Me thod . I n o rde r t o b a l a n c e t he e f f e c t of t ime of day each

i n s t r u c t o r t a u g h t by one method in the morn ing and b y t h e

86

o t h e r method in the a f t e r n o o n , Tha method of t e a c h i n g t o

be used by each i n s t r u c t o r a t a g iven t i n e p e r i o d was

d e t e r m i n e d by a random p r o c e s s . The r e s t r i c t i o n a on r a n -

d o m i z a t i o n were such t h a t each t e a c h i n g method was used

d u r i n g each time p b r i o d and each i n s t r u c t o r t a u g h t by each

t e a c h i n g me thod . I n s t r u c t o r A t a u g h t u s i n g the Guided

D i s c o v e r y Method in tho morning and the E x p o s i t i o n Method

in the a f t e r n o o n . I n s t r u c t o r B t a u g h t us ing the E x p o s i t i o n

Kethcd in the morning and the D i s c o v e r y Method in t he

a f t e r n o o n * I

In o rder t o p r o v i d e s means f o r a s t a t i s t i c a l co ra l -

par i s on of the r c la t i v e effectiveness of t he two t e a c h i n g

methods used in the s t u d y , i t was n e c e s s a r y to o-ssign t h s

s u b j e c t s to the- t e a c h i n g methods th rough a randorn p r o c e s s .

S p e c i a l arrangements were made so t h a t t h i s would be p o s -

s i b l e . In the s p r i n g s c h e d u l e of c l a s s e s a t S o u t h w e s t e r n

S t a t e Collage- t h r e e s e c t i o n s of C o l l e g e Mathemat ics were

l i s t e d under t he 10 a ,m. t i r . e p e r i o d . Thesa c l a s s e s were

l i s t e d as s e c t i o n s two, f o u r , and f i v e . A l l of t h e s e

e l 8 3 s o a were l i s t e d as b e i n g t a u g h t a t the sama time and on

the same d a y s . Mo i n s t r u c t o r s were l i s t e d f o r s e c t i o n s

four and f i v e . During e n r o i l m e n t a l l s t u d e n t s e n r o l l i n g in

e i t h e r s e c t i o n f o u r or 5 a c t i o n f i v e v e r s g i v e n a c l a s s c a r d .

f o r s e c t i o n f i v e and were i n s t r u c t e d t o mee t t h e i r f i r s t •

c l a s s in t he room l i s t e d f o r flection f i v e . The names of

a l l of the s t u d e n t s e n r o l l i n g in t h e s e c l a s s e s were o b t a i n e d

87

d u r i n g e n r o l l m e n t . Three roc Si ens of Col lege Mathemat ics

were a l s o l i s t e d under the 2 p .m. t ime p e r i o d . They were

s e c t i o n s s e v e n , e i g h t , and n i n e . No i n s t r u c t o r was l i s t e d

f o r s e c t i o n s e i g h t and n i n e . Dur ing e n r o l l m e n t a l l s t u d e n t s

e n r o l l i n g in e i t h e r s e c t i o n e i g h t or s e c t i o n n ine were g iven

a c l a s s ca rd f o r s e c t i o n e i g h t and were i n s t r u c t e d t o meet

t h e i r f i r s t c l a s s in the room l i s t e d f o r s e c t i o n e i g h t . The

nfiines of a l l s t u d e n t s e n r o l l i n g in s e c t i o n s e i g h t and n i n e

were o b t a i n e d d u r i n g e n r o l l m e n t .

• B e f o r e the f i r s t day of r e g u l a r c l a s s e s a t the b e g i n -

n ing of the s e m e s t e r , ' the r e c o r d s of a l l s t u d e n t s e n r o l l i n g

in s e c t i o n s f o u r , f i v e , e i g h t , and n ine of Col lege Mathe-

m a t i c s were checked . Of the s t u d e n t s e n r o l l e d in these

a e c t i o n a , s co re s on the Amer i ccn Col lege Tea i i ng P r j ^ r a m ,

(A»G.T. ) , were a v a i l s o l e f o r 115 s t u d e n t s . A l l t he s t u d e n t s

were d i v i d e d i n t o f o u r c a t e g o r i e s a c c o r d i n g to compos i te

s c o r e s on the American Col lege T e s t i n g Program. Group one

c o n s i s t e d of a l l s t u d e n t s w i th a compos i t s s co re of twenty

or a b o v e . Group two c o n s i s t e d of a l l s t u d e n t s w i t h a com-

p o s i t e s c o r e l e a s than twenty b u t more tfc&n f i f t e e n . Group

t h r e e c o n s i s t e d of a l l s t u d e n t s w i t h a composi te s c o r e f i f -

t een and l e s s and group f o u r c o n s i s t e d of a l l s t u d e n t s fop

whom no s c o r e s were - a v a l i a b l e . Each of t h e s e fou r c a t e g o r i e s

were then d i v i d e d i n t o two c a t e g o r i e s , one c a t e g o r y f o r t hose

e n r o l l e d in a morning c l a s s and one c a t e g o r y f o r t hose

e n r o l l e d in an a f t e r n o o n c l a s s . 3a ch of the r e s u l t i n g e i g h t

88

ca tegor i e s was then dividsd in to tv?o c a t e g o r i e s , one c a t e -

g o r y fo r mala s tudents and ens category fo r female

s t u d e n t s . Through t h i s process each s tudent was placed in

one of s ix teen c a t e g o r i e s . The number of s tudents in each

ca tegory is given in Table I .

'PA5LB I

CL-4 5 ?.TP J OA ? I OH OF SUBJECTS AT THE 3E0IS1ITKG OF THE EXPERIMENT

A .G.T. Composite Score

20 and Above Mala Feres la

16 through 19 Ma la Fer.a la

15 aad Below Ma la Feir.a le

R'o score

Time of Day

Male Feiaale

Tot a 1

Tota l Hi---bar of Sab.focta_

10 a«m.

Q J

IS) 9

16 7

10 8

8"

2 x> »Xti.

8 cf

10 12

10 JL

8 JL

62

1U5

A f t e r a l l tbs s t'.:dcr<ts v: ;.r s' c Is« 2 i f i s i r ib»> .s tudents

in each category were assigned to one of the. two c l a s se s

to be taught, during ths time .period fo r which they had

89

p r o c e s s and t h e s t u d e n t s in e a c h c a t e g o r y w e r e a s s i g n e d

S e p a r a t e l y . The randore a s s i g n m e n t of s u b j e c t s t o c l a s s e s

was a c c o m p l i s h e d b y a s s i g n i n g a number t o e a c h s t u d e n t .

Theaa numbers w e r e p l a c e d on s l i p s of p a p e r and t h e s e s l i p s

of p a p e r w e r e p l a c e d i n ^ o x c a l a b e l e d w i t h t h e A p p r o p r i a t e

ca ueôr i e s . Then , t h e two i n s t r u e tors? t o o k t u r n s d r a w i n g

s l i p j o u t of tioe v a r i o u s ca t e g or is s . The c a t e g o r i e s w e r e

nuir/oered a n d i t was d e t e r m i n e d b y t h e f l i p of a c o i n w h i c h

i n s t r u c t o r w o u l d ciraw a number f r o m t h e f i r s t c a t e g o r y .

A f t e r t h e f i r s t d r a w , the i n s t r u c t o r s d r ew number s a l t e r -

n a t e l y f r o m t h e f i r s t c a t e g o r y u n t i l a l l numbers h a d b e e n

d r a w n . The i n s t r u c t o r v h a bad. n o t drawn t h e l a s t number

i-i oiii uue f i r s c c a t e g o r y urew the f i r s t number f r o m the

s e c o n d c a t e g o r y , , I n g e n e r a l , number s w e r e drawn f r o m a

c a t e g o r y u n t i l a l l number s h a d b e e n d r a w n . The i n s t r u c t o r

who h a d n o t drawn t h e l a s t number f r o m t h e c a t e g o r y d rew '

t he f i r s t number f r o m t h e n e x t c a t e g o r y . A f t e r a l l numbers

h a d b s e n d r a w n , c l a s s r o l l s w e r e c o n s t r u c t e d f o r the- two"

i?nn,jxx;p c l a s s e s and t o e two a f t e r n o o n c l a s s e s . T h e n , b y

i* Ob a i xp fx. s c o i n i t was d e t e r m i n e d w h i c h i n s t r u c t o r

s h o u l d v e a c n u s i n g t h e G u i d e d D i s c o v e r y Me thod i n t h e m o r n -

i n g and w h i c h i n s t r u c t o r s h o u l d t e a c h u s i n g t h e E x p o s i t i o n

Method in th e m oyn in T) ff

Two v e r y s i m i l a r c l a s s r o o m s were u s e d f o r t h e m o r n i n g

c l a s s e s and two v e r y s i m i l a r c l a s s r o o m s , w e r e u s e d f o r t h e

90

e n r o l l e d f o r one of the e l a s s e s t h a t met a t 10 a . m . were

a l l i n s t r u c t e d to meet In the same c l a s s r o o m . The s t u d e n t s

who e n r o l l e d in one of the 2 p .m. c l a s s e s were g iven s i m i -

l a r i n s t r u c t i o n s . During t h e f i r s t day of c l a s s the c l a s s

r o l l s f o r each c l a s s were r e a d and the s t u d e n t s were t o l d

in which room t h e i r c l a s s ws-a to m e e t . A f t e r tha c l a s s

. ro l l s v;ere r e a d , the c l a s s e s d i v i d e d snd from then on the

c l a s s e s met in s e p a r a t e rooms.

During the semes te r s e v e r a l s t u d e n t s were e x c e s s i v e l y

a b s e n t and i t became n e c e s s a r y t o drop them from the c l a s s

in which they had been e n r o l l e d . F u r t h e r m o r e , compos i te

s c o r e s on the American Co l l ege T e s t i n g Program were n o t

a v a i l a b l e , f o r a l l of the s t uden t s who e n r o l l e d in the f o u r

s e c t i o n s of Co l l ege Mathemat ics used in the e x p e r i m e n t . As

a r e s u l t , the number of s t u d e n t s whose s c o r e s were used in

the a n a l y s i s of da ta was a c t u a l l y s m a l l e r than i n d i c a t e d by

Table I . The d i s t r i b u t i o n of the s t u d e n t s whose s c o r e s were

used in t he f i n a l a n a l y s i s of da ta i s g iven in Table I I .

At t he b e g i n n i n g of the e x p e r i m e n t , a l l s u b j e c t s were

a s s i g n e d t o c l a s s e s through a random p r o c e s s . T h e r e f o r e ,

excep t f o r chance d i f f e r e n c e s , the c l a s s e s shou ld n o t have

been s i g n i x i c a n t l y d i f f e r e n t from esch o ther in terms of

f a c t o r s which cou ld a f f e c t sobJsvc.s icot . This s i m i l a r i t y

of c l a s s e s could have been a f f e c t e d by the f a c t t h a t e l even

s t u d e n t s f o r whom composi te s c o r e s were a v a i l a b l e dropped

from the c l a s s e s d u r i n g the s e m e s t e r . Of those t h a t dropped

'TABLE II

CLASSIFICATION- OP SUBJECTS WHOSE SCORES WERE USED IN THE ANALYSIS OP DATA

91

A.C.T. Class C onp o s i t a Score

10 a,m. Iks true tor A

Discovery Method

10 a »m. Instructor B

Exp os i t i on K ethDd

20 and Above Mala Fern a le

k 3

k 2

16 through 1.9 Mali Female

7 k

8 5

15 and BqIow Mala Ferns la -

6 1+

8 3

Total 28 ... .. 30

2 p .m. Instructor A

Exp ds i ti on. Me thod

2 p.m. Instructor B

_Discovers Method

20 and Above Hale Feraale

3 3

5 2

16 through 19 Male Female I

15 and Below Ma la F err; a 1«

5 3-

k 1

• Total 2)x L_ - - 22

Total Numb or of 3nb Jects 10[i

92

nine vrere male3 and two w6'ro feiasieS • Proia t h i s i t can be

seen t h a t these s t u d e n t s did no t drop from c l a s s through a

random process » In order to determine whether t he re is any

reason to b e l i e v e t h a t the group c o n s i s t i n g of a l l s t uden t s

fo r whom composite scores were a v a i l a b l e , who did no t drop

from c l a s s , and who wars t augh t by the Guided Discovery

Method was s i m i l a r to the group c o n s i a t i n g of a l l s t u d e n t s

f o r when composite scores ware a v a i l a b l e , who d id no t drop

from c l a s s , and who were t augh t by the Expos i t ion Method,

a t tbe beg inn ing of the exper iment , an a n a l y s i s of va r i ance

wis performed on the composite scores of these s u b j e c t s .

These scores ware ob ta ined b e f o r e the experiment began, so

they should give ?ome i n d i c a t i o n of how the two groups corn-

pared b e f o r e the beg inn ing of the exper iment . The a n a l y e i s

used to compare the two groups t augh t by the two t each ing -

methods was an ana l y s i s of va r i ance in which t h r e e l e v e l s

of a b i l i t y were cons ide red as we l l as the two groups as

determined by tbe two t each ing methods. Froia Table I I I i t

can be seen t h a t the ? - r a t i o f o r the two groups determined

by the two t each ing methods did no t reach s i g n i f i c a n c e and

the P - r a t i o fo r i n t e r a c t i o n d id not r each s i g n i f i c a n c e .

The P - r a t i o fo r the two groups determined by the two t e a c h -

ing kcfchoda i n d i e s t a s t h a t the neons f c r these two groups

a r e not s i g n i f i c a n t l y d i f f e r e n t . The P - r a t i o f o r i n t e r -

a c t i o n i n d i c a t e s f u r t h e r t h a t a t each l e v e l of a b i l i t y the

two groups determined by the two t each ing methods ware

'ABLE I I I

ANALYSIS OF VARIANCE TABLE FOR A.C.T. COMPOSITE SCORES AT THE BEGINNING OP THE EXPERIMENT

(N = 101+)

93

Source __ j [ » "i f:. y p <J

• ~j Mean Square F .. P

Me tood a 0 x j. 1 ty In t e rae felon

.013 1)491.936

17.108

1 2 2

t

.013 7US.968

8.505

.003 166.797

1.902

p > . 2 5 p < . 0 0 1

* 1 ^ p ^ #25

Err or U36.290 98 I i 4 7 2

To ta l 19a7.3U6 103

I i 4 7 2

s i m i l a r . These s t a t i s t i c a l t e a t s i n d i c a t e t h a t the two

groups determined by t each ing methods were not s i g n i f i -

c a n t l y d i f f e r e n t a t the beginning of the experiment in

terms of those a b i l i t i e s measured by the composite score

of "the Aro-er lean Col lege Tea t i n g Pr ogârru -

In summary, an experiment was conducted which was d a -

s igned to csafer® to the model f o r "a two by t h r e e f a c t o r i a l

exper imen t . The two independent v a r i a b l e a were method of

t each ing and l e v e l of a b i l i t y . The dependent v a r i a b l e was

ach ievement . The e f f e c t s of two methods of t e a c h i n g , the

Guided Discovery Method and the Expos i t ion Method, were

cOi.ipartu x'.i tulias ox vnexi: 6 i i e c u upon fcuoent schieveinent .

The s u b j e c t s f o r - t h e experiment were IOJ4. s t u d e n t s e n r o l l e d

in fou r s e c t i o n s of College Mathematics 01 Southwestern

S t a t e College dur ing the sp r i ng semester of the 1967-1968

91+

s c h o o l y e a r . These s t u d e n t s v e t o d i v i d e d into- s i x t e e n •

c a t e g o r i e s a c c o r d i n g t o l e v e l of a b i l i t y , t h e t ime of day •

d u r i n g which t h e c l a a s in which the;/ had e n r o l l e d m e t , and

a c c o r d i n g t o s e x . Through a random p r o c e s s each s t u d e n t

was a s s i g n e d t o a c l a a s t a u g h t b y t h e Guided D i s c o v e r y

Method or t o a c l a s s t a u g h t by t h e E x p o s i t i o n M e t h o d . An

a n a l y s i s of v a r i a n c e r e v e a l e d - t h a t , a t t he b e g i n n i n g of t h e

e x p e r i m e n t , t h e g roup c o n s i s t i n g of a l l s t u d e n t s t a u g h t by •

t h e Guided D i s c o v e r y Method v:as s i m i l a r t o t h e g roup c o n - '

ss i s t i n g of a l l s u b j e c t s t a u g h t b y t h e E x p o s i t i o n M e t h o d .

Pour s e c t i o n s of. C o l l e g e Matbsrc.at iea were u t i l i z e d f o r

t h e e x p e r i m e n t . Two c l a s s e s n e t a t 10 a .m. and two c l a s s e s

iriet a t . 2 p .in. - Two i d s Li1 uc t oro. p a r t i c i p a t e d io t h e s t u d y .

I n s t r u c t o r A t a u g h t u s i n g t h e Guided D i s c o v e r y Method in

t h e n o r r . i n g and t a u g h t u s i n g the E x p o s i t i o n ' M e t h o d in t he

a f t e r n o o n . I n s t r u c t o r B. t a u g h t u s i n g t h e E x p o s i t i o n Method

in t he morn ing and t a u g h t u s i n g the Guided D i s c o v e r y Method

in t h e a f t e r n o o n . ' ' •

I n o rde r t o d e t e r m i n e t h e r e l a t i v e s f f G C ^ i v e j i e s a of

t h e two t e a c h i n g methods t h s two t e a c h i n g methods were com-

p a r e d in t e rms of t h e i r e f f e c t upon s t u d e n t a c h i e v e m e n t .

S t u d e n t a c h i e v e m e n t was m e a s u r e d by f o u r d i f f e r e n t t e s t a .

On each of t h e . f o u r t e s t s t he s c o r e s of a l l t h e r fcudanta '

t a u g a t by t h e Guided D i s c o v e r y Method -were c o n s i d e r e d as

one gi oup ox s c o r e s ^nd t h e s c o r e s of a l l t h e s t u d e n t s

95

group of s c o r e s . By c o p i n i n g scores ir; th i s way, c o n t r o l s

were provided f o r the d i f f e r e n c e s between the two i n -

s t r u c t o r s and for the e f f e c t s of t ine of day upon l e a r n i n g .

The Teaching Metfaods

In fcg od uc t ion • -

The purpose of t h i s s e c t i o n i s t o d e s c r i b e the two

teaci:or;g rcetnods -compared - In the exper i m e n t t o ahow in

which v sys the two t e a c h i n g methods d i f f e r e d , and to show

xn which ways the two t e a c h i n g methods were s i m i l a r . The

two t e a c h i n g methods d i f f e r e d in t h r o e b a s i c aape.cts . In

the Guided D i scove ry Method p r i n c i p l e s and concepts were

p r e s e n t e d indue t i v a i y whereas in the .Exposit ion M e t h o d '

concepts and p r i n c i p l e s were p r e s e n t e d d s d u c t i v e l y . In

the Guided D i scovery Method concep t s and p r i n c i p l e s were

i n t r o d u c e d &nd t a u g h t to rough examples whereas in the Ex-

p o s i t i o n h e t n o a concep t s and p r i n c i p l e s were i n t r o d u c e d by

the i n s t r u c t o r g i v i n g an a u t h o r i t a t i v e s t a t e m e n t of the . '

p r i n c i p l e or c o n c e p t and then a t t e m p t s "were made to j u s t i f y

the s t a t e m e n t through d e d u c t i v e r e a s o n i n g . The t h i r d a s -

p e c t in which the two t e a c h i n g methods d i f f e r e d was the

order of p r e s e n t a t i o n of the m a t e r i a l s used to a i d l e a r n i n g .

I n - t h e Guided D i s c o v e r y Method s t e n t s were presented w i th

examples of the concep t or p r i n c i p l e to be l e a r n e d and -then

a s tatement of the concep t or p r i n c i p l e was g iven e i t h e r by

96

i n s t r u c t o r . In t h e E x p o s i t i o n Method t h e s t u d e n t s w e r e

f i r s t p r e s e n t e d w i t h a s ts. t sin e a t of Sbe p r i n c i p l e and

e x a m p l e s of t h e p r i n c i p l e were g i v e n l a t a r .

T h e r e ware s e v e r a l a s p e c t s w i t h r e s p e c t t o which t h e

two t e a c h i n g me thods w e r e s i m i l a r , The g r o u p s t a u g h t b y

t h e two m e t h o d s s t u d i e d t h e same, c o n c e p t s and w i t h o n l y a . .

few e x c e p t i o n s t h e s e c o n c e p t s were s t u d i e d in t h e same

o r d e r . I n a few c a s e s t h e o r d e r of p r e s e n t a t i o n of c o n -

c e p t s was v a r i e d i n o r d e r t o b a s t u t i l i z e ' t h e d i f f e r e n t

c h a r a c t e r i s t i c s of t h e two t e a c h i n g m e t h o d s . In g e n e r a l , •

t h e s a n e c o n c e p t s w e r e p r e s e n t e d in each l e s s o n t o a l l

c l a a a e s . A l l c l a s s e s met t h e same number of t i m e s a n d

were g i v e n t h e 32:53 number of homework a c s i g u n e n ba . A t -

t e m p t s w e r e made t o make a i l homework a s s i g n m e n t s of

app rox ima t e l y t h e some l e n g t h . :

In o r d e r t o a s s u r e - t h e e f f e c t i v e n e s s of t h e Gu ided

D i s c o v e r y M e t h o d , s p e c i a l i n s t r u c t i o n a l m a t e r i a l s were p r e -

p a r e d . These m a t e r i a l s were in t h e form cf' d u p l i c a t e d

l e s s o n s wh ich were h a n d e d c u t t o nbe s t u d e n t s d u r i n g e a c h

c l a s s p e r i o d . In o r d e r t o a v o i d a n y s p e c i a l e f f e c t t h a t

s u c h m a t e r i a l s m i g h t h a v e , s i m i l a r m a t e r i a l s w e r e p r e p a r e d

f o r t h e s t u d e n t s who were t a u g h t by t h e E x p o s i t i o n M e t h o d .

Care was taken- t o a s s u r e t h a t t h e cs/uc c o n c e p t s were p r e - ' "

sen t e a in t h e two a .eta of ma t e r i a l s and t h a t t h e two s e t s

of l e s s o n s c o n t a i n e d a p p r o x i m a t e l y t h e same amount of w r i t -

t e n m a t e r i a l .

97

The two teach ing methods ware s i m i l a r in t h a t bo th

methods s t r e s s e d unders tanding on the p a r t of the s t u d e n t .

In the Guided Discovery Method of t e a c h i n g , a t t emp t s were

made to he lp s tuden t s unders tand what they l ea rned by r e -

q u i r i n g the s tuden t s to l e a r n through examples and through

a c t i v e p a r t i c i p a t i o n in the i n v e s t i g a t i o n of these examples.

In the Expos i t ion Method of t each ing a t t empts were made to

he lp s t u d e n t s understand what they l ea rned by j u s t i f y i n g

each concept by a deduct ive prococs and by i l l u s t r a t i n g the

concept through examples a f t e r the concept hod bter> p r e -

s e n t e d . Thus tha two t each ing methods were s i m i l a r in

terras of the kinds and amounts of m a t e r i a l p r e s e n t e d b u t

were d i f f e r e n t in tex;.;a of the order and the way in which

these m a t e r i a l s were p resen ted f o r the i n d i v i d u s l concep t s .

A more d e t a i l e d d e s c r i p t i o n of each of the two t each ing

methods i s p r e sen t ed n e x t .

Tire Guided Discovery Method. of Teaching

The Guided Discovery Method of Teaching can be cb&r- •

a c t e r i z e d sa a method of t each ing in whvcb s t u d e n t s l e a r n

through i n d u c t i v e p r o c e s s e s , l e a rn through sxsrcples, and a

sfnethuti of reaching in which formal s t a t ements of concepts

or p r i n c i p l e s a re not given to the s t u d e n t s u n t i l the s t u -

dents have had some exper ience wi th the concept or p r i n c i p l e

«nd u n t i l tlj8 s tuden t s a re conv.vncsd t h a t the concept or

pr iBcipj.8 i s ??3lid . Cn par t ilc u l a r , osoh concept or

98

p r i n c i p l e is t augh t through an In s tr uc t i o n a l sequence

c h a r a c t e r i z e d by a sequence of s t eps a.° desc r ibed balow.

In the Guided Discovery Method of t each ing each i n -

s t r u c t i o n a l sequence is begun by a a king the s t uden t s t o

work e x e r c i s e s which are examples of the concept or p r i n -

c i p l e to be l e a r n e d . As the e x e r c i s e s proceed they become

more e l c a e l y r e l a t e d to the concept to be l e a r n e d . These

e x e r c i s e s a re s e l e c t e d and p r e s e n t e d in such a way aa to

guide the s t u d e n t to d i scove ry of the concept to be l ea rned ,

During the second s t ep of the i n s t r u c t i o n a l sequence

a t t empts a re ir.oda to determine whether d i scovery has oc-

c u r r e d . A f t e r the s tuden t s have completed the i n i t i a l

e x e r c i s e s , the s tuden t s s ro risked ques t ions which a r e based

on the concept to bs l e a r n e d , These ques t ions a r e s e l e c t e d

in such a way t h a t they can be answered e a s i l y and qu ick ly

i f the s t uden t s understand, the concept to be l e a r n e d , b u t

i f the s t u d e n t s have not d i scovered the c o n c e p t , t h e y a re

very d i f f i c u l t to answer. I f s t u d e n t responses i n d i c a t e

t h a t the concept is not f u l l y unders tood , the t h i r d s t ep of

tne i n s t r u c t i o n a l sequence is in the form of more gu idance .

Phis guidance may be in en e form of lea d ins ques t i ons or in

the . fo rm of a d d i t i o n a l e x e r c i s e s . As it ore guidance i s

g iven , t h e guidance bee ones »nore s p s e i f io , A d d i t i o n a l

-guidance is given u n t i l s t u d e n t responses i n d i c a t e t h a t

d i s cove ry has occur red . I f the concept to be l ea rned i s

99

espec i s l l y d i f f i c u l t y the gu idance may e v e n t u a l l y become an

a c t u a l s t a t e m e n t of -the p r i n c i p l e to be l e a r n e d .

U s u a l l y the s t u d e n t ' s f i r s t i n t r o d u c t i o n to a concep t

i s in the form of e x e r c i s e s g iven a t the end. of the preced-

ing l e s s o n . Thus, the s t u d e n t f i r s t works w i t h examples of

the c o n c e p t on h i s ovm. Then, d u r i n g the n e x t c l a s s s e s s i o n

tbe s tuc .en ts a r e asked to work more e x e r c i s e s d e a l i n g w i t h

the c o n c e p t to be l e a r n e d . Then, the i n s t r u c t o r t e s t a for

d i s c o v e r y by a s k i n g a a e r i e s of q u e s t i o n s . Then, i f n e c e s -

s a r y , the ins true tor g i v e s a d d i t i o n a l gu idance in the form

of l e a d i n g q u e s t i o n s . These ques t ions a r e d i s c u s s e d in

c l a s s and answered "by th3 c l a s s . Thus, the s t u d e n t ia

f i r s t g iven an o p p o r t u n i t y co d i s s over the p r i n c i p l e on h i s

own. I f t h i s i s t ?o d i f f i c u l t , the c l c s s i s then g iven an

o p p o r t u n i t y to d i s c o v e r the concep t a a a g r o u p .

'ihe f i n a l s t e p in the i n s t r u c t i o n a l , sequence c o n s i s t s

of naming tbe concep t or p r i n c i p l e d i s c o v e r e d and in some

caae& t h i s a tep -aia o includes, the development of a formal

s t a t e m e n t of the concep t or p r i n c i p l e d i s c o v e r e d . . I f th.e

s t u d e n t s p o s s e s s the v e r b a l c a p a c i t y to do s o , the s t a t e -

ment of the c o n c e p t or p r i n c i p l e i s f o r m u l a t e d by the

s t u d e n t s , i f the s t u d e n t s do n o t p o s s e s s s u f f i c i e n t v e r b a l

a b i l i t y -to x.'orrsi.'lste a p r e c i s e s t s tTZ ' r r . t of the p r i n c i p l e ,

such a s t a t e m e n t ia g iven to them e i t h e r by the i n s t r u c t o r

or in the i n s t r u c t i o n a l m a t e r i a l s .

100

The steps of the instructional seqc-ence need not be

•presented as a continuous unit. Id a or, a esses it ia desir-

able to introduce a concept in one sot of exercises, present

more exercises dealing with the concept in the next exercise

set, and then present a final exercise set in which the stu-

dent ia brought to a full understanding of the concept.

Then* the naming of the principle and the presentation of a

formal statement of the principle may be delayed for one or

more lessons. A delay in the presentation of a formal

- statement of the principle ia often desirable because- this

provides those students who have not yet discovered the

principle wi i-h more t ivoe to reach a better unders Landing of

the principle.

Since the Guided Discovery Method of teaching is based

on the process of inductive reasoning end on students learn-

ing through examplea the students are often asked to atata

gsneralizations on the bsaia of an investigation of several

exercises. Since these exercises have been carefully se-

lected, the student can become overconfident-and develop a

tendency to accept generalisations after investigating only

a minimum of cases of the generalization. In order to avoid

this tendency, several special exercise sets were developed

and presented during the goiter.' In these exercise a«ta,

exercises were presented which seemed to indicate that a

certain generalization was valid. After the e x e r c i s e ast

bad been completed the students were questioned concerning "

101

th i s g e n e r a l i z a t i o n . I f the s t u d e n t s seemed to tbin& t h a t

the g e n e r a l i z a t i o n was v a l i d , a d d i t i o n a l e x e r c i s e s were

g iven which showed t h a t the g e n e r a l i z a t i o n was no t v a l i d .

This p rocedure was used to d i s c o u r a g e s t u d e n t s from a c c e p t -

ing g e n e r a l i z a t i o n s a f t e r an inadequa te amount of i n v e s -

t i g a t i o n .

One of the f a c t o r s which has made the r e s u l t s of o the r

s t u d i e s d e a l i n g wi th d i s c o v e r y methods of t e ach ing ques t ion-

ab l e i s the f a c t t h a t in msny s t u d i e s more i n s t r u c t i o n a l

time was devoted to the s t u d e n t s who l e a r n e d by the d i s -

covery Method than, t o the s t u d e n t s who l e a r n e d by the

e x p o s i t i o n method. In order to equate i n s t r u c t 3 onal time

f o r the two ta ' -ching raothods, i t became n e c e s s a r y to p r o -

vide the s t u d e n t s t a u g h t by "oha Guided Discovery Method

wi th f a i r l y e x p l i c i t g u i d a n c e . P rov id ing d i r e c t forms of

guidance f o r the s t u d e n t s t a u g h t by the Guided Discovery

Method was a l s o c o n s i d e r e d a p p r o p r i a t e because of the

r e s u l t s r e p o r t e d In the s t u d i e s conducted by Cra ig^ and'

Underwood and .Richards o n . 2 By p r o v i d i n g s x p l i c l t forria of

g u i d a n c e , i t was p o s s i b l e f o r the s t u d e n t a taught by the

Guided Discove ry Method and t h e s t u d e n t s t a u g h t by the

^Robert C. C r a i g , " D i r e c t e d Versus Independen t Discovery of E s t a b l i s h e d R e l a t i o n s The J o u r n a l of E d u c a t i o n a l Psycho logy , XLYII ( A p r i l T 9 5 5 T 7 " 2 2 3 ~ 2 1 ' i | .

2.. , ~~ yen ton J . Underwood and Jactf R i c h a r d s o n , "Verba l

Concept Learning as a Func t ion of I n s t r u c t i o n and Dominance L e v e l , " J o u r n a l of Expe r imen ta l P s v c b o l o e y . VI ( A p r i l , 1956) , ' — *—

10 i

Expos i t ion Method to study tbs sa»T.s concepts and p r i n c i p l e s

dur ing equal amounts of i n s t r u c t i o n a l t ime .

Sines no co l l ege textbooks f o r isathejnaties have been

w r i t t e n in which d i s c o v e r y methods of t each ing a re empha-

s i z e d , i t "-."aa nece s sa ry to p repare s p e c i a l i n s t r u c t i o n a l

m a t e r i a l s f o r the s t u d e n t s who -were t augh t by the Guided

Discovery Method, Before the beg inn ing of the series t e r , a

complete o u t l i n e of t op i c s to be p r e s e n t e d dur ing the s e -

mester was p r e p a r e d . Than, t h i r t y - t h r e e l e s sons were

w r i t t e n in which these top ics were p r e s e n t e d . These -lea-

sons were d u p l i c a t e d and one l e s s on was given to the s t u -

dents dur ing each c l a s s s e s s i o n . Severa l of these l e s sons

a re p r e sen t ed in Appendix B. j?or the sake of comparison

cor responding sample l essons f o r the Expos i t ion Method of '

t each ing a r e p r e sen t ed in Appendix C.

The Expos i t ion Mjrthod of Teaching

The Expos i t ion Method of t each ing can be c h a r a c t e r i s e d

as a method of t each ing in which s t u d e n t s l ea rn through

deduc t ive p r o c e s s , l e a rn through s tudy ing a u t h o r i t a t i v e

s t a t emen t s of p r i n c i p l e s and c o n c e p t s , ana a method of

t each ing in which p r i n c i p l e s and concepts a re s t a t e d in

a fo rmal manner b e f o r e the s t u d e n t s have had exper ience

witn the concepts and p r i n c i p l e s . In p a r t i c u l a r , each con-

cept: or p r i n c i p l e i s t augh t through an i n s t r u c t i o n a l

sequence c h a r a c t e r i z e d by a sequence of s t eps as d e s c r i b e d

be low.

103

In t h e E x p o s i t i o n Method of • fceschS.Bg each i n s t r u c t i o n a l

Sequence i s begun b y p r e s e n t i n g a f o r m a l s t a t e m e n t of t h e

c o n c e p t or p r i n c i p l e t o be l e a r n e d t o the s t u d e n t s . Th i s

s t a t e m e n t i s p r e s e n t e d e i t h e r by t h e i n s t r u c t o r or in t he

i n s t r dc t i on a 1 ma t e r ia I s .

A f t e r Vha s t a t e m e n t of t h e p r i n c i p l e has been p r e -

s a n t e d , m a t e r i a l s d e s i g n e d t o h e l p t h e a t u d a n t u n d e r s t a n d

t h e p r i n c i p l e and d e s i g n e d t o c o n v i n c e t h e s t u d e n t t h a t t h e

p r i n c i p l e i s v a l i d a r e p r e s e n t e d . These m a t e r i a l s may i n -

c l u d e a d e d u c t i v e p r o o f of t h e p r i n c i p l e a s w e l l a s s e v e r a l

.examples of a p p l i c a t i o n s of t he p r i n c i p l e or- s e v e r a l exam-

p l e a of i n s t a n c e s of t he p r i n c i p l e . The s t a t e m e n t of t h e

p r i n c i p l e and the d e d u c t i v e p r o o f way be p r e s e n t e d e i t h e r

s s o p a r t of t h e ' v r i t t s n ma t s r i s l s or S3 a l e c t u r e g i v s n

b y the i n s t r u c t o r .

A f t e r t h e f o r m a l s t a t e m e n t of t h s p r i n c i p l e , t h e d c - •

due t i v e p r o of or t h e p r i n c i p l e . , and examples of t h e p r i n c i p l e

have been p r e s e n t e d t h e s t u d e n t ia p r o v i d e d w i t h o p p o r t u - '

n i t i e s t o use t h e c o n c e p t or p r i n c i p l e p r e s e n t e d t o s o l v e

s p e c i f i c p r o b l e m s . Th is ia u s u a l l y in t h e form of an e x e r -

c i s e s e t g i v e n t o t h e s t u d e n t and t o be c o m p l e t e d b e f o r e

tha n e x t c l a s s s e s s i o n . In aoroe e a s e s s e v e r a l p r i n c i p l e s

m a y b e p r e s e n t e d and . d i s c u s s :=d d u r i n g a . s i ng l e c l fess p e r i o d .

Then, t n e s t u d e n t s a r e p r e s e n t e d w i t h an e x e r c i s e s e t c o n -

s i s t i n g of p rob l ems r e q u i r i n g the use s a d a p p l i c a t i o n of

- a l l t h e p r i n c i p l e s p r e s e n t e d - c u r i n g the c l a s s s e s s i o n .

I01|.

I a c o n t r a s t to the Guided Discovery Method of teaching,

a l l the s t eps of the i n s t r u c t i o n a l sequence for the Expo-

s i t i o n Method of t eaching a re u s u a l l y p r e sen t ed in the sane

l e s s o n . A l l the ins t rue tioii r e l a t e d to a given concept or

p r i n c i p l e is given as a complete u n i t ; a l t hough , i n t e r -

re la t ionsh ips among the concepts and p r i n c i p l e s a r e r ecog-

nized. sod in many e s ses p resen ted to the s t u d e n t s .

Since the Exposi t ion Method of teaching is a deduc t ive

method of teaching, i f a student can not f ind the' so lut ion

to an e x e r c i s e , he is re ferred to the s p e c i f i c formula or

pr inc ip le that can be. used to solve the problem. In con-

t r a s t , i f a s t uden t in e c l a s s taught by the Guided

D* - * ... ^ * A . r-. a IP, r .n * v . , , » „ • . . _ -* -* t t „

i-^vUirary xn n iAj u O. ii w uiix -l u a X u v w U ii xu g k& p X' D u X olfi $ (18 1B

presan-ted wi th an s a m p l e problem t h a t i l l u s t r a t e s how the

p r o b l 6 rn e a o b a w o r k c d .

Spec ia l in s t ruc t iona l lessona were prepared f o r the

-classes t aught by the Expos i t ion Method. These l essons

were p r spa r sd ir, o r d e r ' t o a s su re t h a t tha «,t«daats t augh t

by the Exposi t ion Method and the students t aught by the

Guided Discovery wsfcaod wou1i receiva cowpsrsble i n s t r u c t i o n

and study the same cono«spts in very n e a r l y the same sequence«

In th i s way t he re was do d i f ferenoa in the two groups taught

by the two d i f f e r e n t i n s t r u c t i o n ! methods in tezzas of t h e

type of i n s t r u c t i o n a l materials used. neither" group used

une taxIbootc u s u a l l y used m Col!6556 Matheins t i c s * Thirty—

tbr ea. lessons war a p repared fo r the c l a s s e s t augh t by the

105

Exposit ion Method. One l e s son was given to the s t u d e n t s

dur ing each c l a s s s e s s ion * Severa l of these l e s sons a r e

p r e s e n t e d in Appendix C. For the sake of comparison,

cor responding sample l essons f o r the Guided Discovery

Method of t each ing a re p r e s e n t e d in Appendix B.

The Tes t ing Program

In order to p rov ide a means fo r comparing the r e l a t i v e

e f f e c t i v e n e s s of the Guided Discovery Method of t e ach ing

and the 'Exposit ion Method of t e a c h i n g , a t e s t i n g program

was conducted in which t h ree s t a n d a r d i z e d t e3 t a and a

s e r i e s of t eacher raad-i t r-sts were administered. Each of

the s t a n d a r d i z e d t e a t s was chosen because i t was des igned

to measure an a s p e c t cf achievement s imi lar to the type of

achievement emphasized in some p o r t i o n of the course of

s tudy used in the exper iment . The th ree s t a n d a r d i s e d t e a t s ,

combined, measured knew ledge of c o n t e n t a n d ' a b i l i t y to use

f a c t s l ea rned fo r near ly a l l a s p e c t s of the course cf s tudy

used in the experiment. The teacher made t e s t s were i n -

c luded because tbey were des igned to s p e c i f i c a l l y measure

the achievement of the studenta in the s u b j e c t ma t t e r

t augh t dur ing the semester in the course of s t u d y .

In the f o l l o w i n g s e c t i o n s a d e t a i l e d d e s c r i p t i o n i s

given of the t e s t s used in the tasfclag program. The' com-

p o s i t e score of the American College Tes t i ng Prograra waa

used SG S measure of a b i l i t y in order to e s t a b l i s h l e v e l s

106

of a b i l i t y . These l e v e l s of a b i l i t y were used in order t o

de t e rmine whether the r e l a t i v e e f f e c t i v e n e s s of t he Guided

D i scove ry Method of t e a c h i n g and the E x p o s i t i o n Method of

t e a c h i n g a r e dependen t upon s t u d e n t a b i l i t y . S ince the

s c o r e s o b t a i n e d from the Am.er i can Col lege Te s t i n g ?r pgr am

were an i m p o r t a n t p a r t of too e x p e r i m e n t a l p rogram, t b i a

t e s t i n g program i s a 13o d e s c r i b e d .

•rhfi Araerican Col lege Program Examinat ion •

Tfea Araer i can Co l l ege T e s t i n g Program3 examina t ion ia

a t h r e e - h o u r t e s t b a t t e r y d e s i g n e d to t e a t a b road a r e a of

e d u c a t i o n a l s k i l l s . I t ia d e s i g n e d t o t e s t knowledge of

f a c t a as w e l l as a b i l i t y t o use knowledge in the ? o l u t i o n

of complex p r o b l e m s . The examina t ion was deve loped in order

to p r o v i d e a p r e d i c t o r f o r the s u c c e s s of c o l l e g e - b o u n d

h i g h s c h o o l s e n i o r s and. j u n i o r c o l l e g e s t u d e n t s who i n t e n d

t o t r a n s f e r to a f o u r - y e a r c o l l e g e . During each year over

300 ,00 a t u d e n t a complete the t e s t and the r e s u l t s a r e s e n t

to over 700 c o l l e g e s .

The t e s t c o n s i s t s of f o u r s u b t e s t s . The i tems in each

s u b t e . i t a r e m u l t i p l e - c h o i c e i tenia • T e s t I i s an e i g h t y -

i t e m , f i f t y - m i n u t e t e s t of E n g l i s h u s a g e . T e s t I I i s a

f o r t y ~ i t e x n , f i f t y - m i n u t e t e a t of mathemat ies u s a g e . T e s t

I I I i s a f i f t y - t w o-iten, f o r ty -minu te s o c i a l s t u d i e s ' r e a d i n g

Ôacar K r i s e n E u r o s , e d i t o r , The S i x t h Menta l Measurements Yearbook •(Highland' P a r k 7 1 ^ 5 T > ~ P P •' 1 - 1 3 .

10?

t e a t a n d T e s t IV i s a f I f t y - t w o - i t e r n » f o r t y - m i n u t a n a t u r a l

s c i e n c e r e a d i n g t e a t . Each e d i t i o n c f the e x a m i n a t i o n i s

p u b l i s h e d a s a s i n g l e t h i r t y ~ t w o - p a g e b o o k l e t c o n t a i n i n g

t h e f o u r s u b t e s t s .

I n d e v e l o p i n g new forma c f t h e t e s t , s p e c i f i c a t i o n s

f o r t e s t 5.terns a r e d e v e l o p e d . Than w r i t e r s a r e emp loyed

t o £3e lec t and w r i t e t e s t Items w h i c h m e e t t h e s e s p e c i f i -

c a t i o n s . " T r y o u t " u n i t s a r e t h e n a d m i n i s t e r e d t o l a r g e

r e p r e s e n t s t ' v e s a m p l e s of s t u d e n t s , Then i t e m a n a l y s i s 5.3

c o n d u c t e d and t h e r e s u l t s on t h e new u n i t s a r s coxnpar.ed

w i t h t o e s c o r e s t h s s t u d e n t s i n t h e a a mp 1 e h a v e a c h i e v e d

on t h e lotca Teg t s of E c. c a 11 on a 1 p e v a 1 opm e n t . C-n t h e b a s i s

of t h i s pj?ograw of a n a l y s i s oa b i o ; j a l per-cau b i l e u c«rus' o r e

d e v e l o p e d . In a d d i t i o n l o c a l norms and o t h e r d a t a a r e

p r o v i d e d f o r c o l l e g e s t h a t p a r t i c i p a t e i n t h e p r o g r a m .

The o d d - e v e n r e l i a b i l i t y c o e f f i c i e n t s of t h e f o u r s u b -

t e s t s s r e . 9 0 , , 3 9 , . 8 6 , and . 83 f o r E n g l i s h , m a t h e m a t i c s ,

s o c i a l s t u d i e s , a n d n a t u r a l s c i e n c e r e s p e c t i v e l y . The

r e l i a b i l i t y of t h e c o m p o s i t e s c o r e i a , 9 5 . Tha i n t a r e o r -

r - e l a t i c n s b e t w e e n s u b t e s t s a r e g e n e r a l l y g r e a t e r t h a n . 5 0 .

Such h i g h i n t e r c o r r e l a t i o n a c a u s e t h e e f f e c t i v e n e s s of t h e

t e s t i n d i f f e r e n t i a t i n g arc on g t h e v a r i o u s t y p e s of a b i l i t y

s u p p o s e d l y t e s t e d t o b e s u s p e c t . As a r e s u l t , i n t h e

e x p e r i m e n t a l s t u d y r e p o r t e d h e r e . , o n l y t h e c o m p o s i t e s c o r e

was c o n s i d e r e d .

108

Teat I : The S t r u c t u r e of tbe Berber System

The Coopera t ive V^tht-r.n t5?a Te h i s a s e r i e s of

t e s t s des igned to measure achievement in tbe major c o n t e n t

a r ea s of matberas t i c s from jun io r h igh school a r i t h m e t i c

through c o l l e g e ca lcu lus , . In cons t rue t i n g the t e s t s ,

f o r t y - s i x mathematics tea chars» j un io r high through c o l l e g e ,

were engaged to w r i t e items fo r tbe t e a t s . These items

wGje reviewed and e d i t e d and p r e t e s t forms were assembled .

These p r e t e s t forma were adra in i s te red to a n a t i o n a l sample

of s t u d e n t s in May, I960 . The t e s t a were then r e v i s e d and

r e ~pr e fces t ed in a n a t i o n a l pi' eg ram. The r e s u l t s i n d i c a t e d

t h a t the t e s t a are now a p p r o p r i a t e for the in tended popu-

l a t i o n s . Content v a l i d i t y is c la imed on the b a s i s bbst

persons w e l l - q v s l i f i s d to judge the r e l a t i o n s h i p of t e s t

c o n t e n t to t each ing o b j e c t i v e s wrote the t e s t i t ems .

Toe t e s t , tbe S t r u c t o r s of the ffutiiber Sya tern, is a

f o r t y - i t e m , for ty-minuts t e a t . A l l Items a re m u l t i p l e -

choice items 'with f i v e a l t e r n a t i v e s . The t e s t i s des igned

to measure knowledge of number syatems inc lud ing bases

o tb or than t a n , p r op e r t i e £ o f opera t i o n a w i th n umb e r a

through f r a c t i o n s , a r i t h m e t i c judgment, modular a r i t h m e t i c ,

and number l i n e a .

The c h a r a c t e r i s t i c s of the t e s t ".-/ere determined by

a d m i n i s t e r i n g the t e a t to a n a t i o n a l sample of over 1,000

^"Kandb ook: Cooperat ive l-la themat ic a Testa , Educa t i ona l Tes t ing Service (Prliticeta£*7l?%'|3

109

seventh and e igh th grade 2 t u d e n t s . On tbs b a s i s of the

sco res ob ta ined from t h i s sample r e l i a b i l i t i e s were c a l -

c u l a t e d using the Ruder-Richards on Formula 20. The

r e l i a b i l i t y of form A, the form used in t h i s s t u d y , was

found to be .86 • Although the t e s t was des igned fo r

seventh and. e i g h t h grade s t u d e n t s , an examinetion of the

t e s t items lead to the conc lus ion t h a t the t e s t was a p p r o -

p r i a t e f o r measurement of the c o n t e n t of the course used

In the exper imenta l s t u d y . The t e s t was a drain 5 a t a r e d to

two s e c t i o n s of College Mathematics dur ing the f a l l semes-

t e r of the 1967-1968 school year and an examination of

those t e s t scores i n d i c a t e d t h a t the t e s t was of a p p r o -

p r i a t e d i f f i c u l t y and provided a s a t i s f a c t o r y d i s t r i b u t i o n

of s c o r e s .

The t e a t was admin is t a r e d dur ing the l a s t week of -

r e g u l a r c l a s s e s dur ing the s p r i n g semester and was admin-

i s t e r e d as the f i r s t of a s e r i e s of t h r ee s t a n d a r d i z e d

t e s t s . A l l t e s t s were scored by hand. A l l t e s t s were

scored twice in order to i n su re a c c u r a c y .

Tegt I I : Algebra X

The second t e s t admin i s t e r ed as a s e r i e s of t h r ee

s t a n d a r d i z e d t e a t s was the t e a t Algebra I . Thi3 t e s t i s

one of the t e a t s in the Coopera t ive Mathematics Testa

s e r i e s . The t e s t ia a f o r t y - J bem f o r t y - m i n u t e t e s t . A l l

i tems a re m u l t i p l e - c h o i c e items wi th f i v e a l t e r n a t i v e s .

110

The t o s t i s conta ined as s s i n c i e a n i t in a seven page t e a t

b o o k l e t . The t e a t was w r i t t e n and r e v i s e d using the same

procedure a a used fo r S t r u c t u r e of the I-Iumber System. The

c h a r a c t e r is t i c s of the t e s t were determined by adminis -

t e r i n g the t e s t to a sample of 1,200 e igh th and n i n t h grade

s t u d e n t s . The r e l i a b i l i t y of form A, the form used in t h i s

s t u d y , was found to be .35 , as determined by the uae of the

Kuder-Richardson Formula 20.

The t e s t is des igned to measure a b i l i t y to manipula te

algebrts ic e.xpresaions , a b i l i t y to so lve a l g e b r a i c e q u a t i o n s ,

and a b i l i t y to solve l i t e r a l problems. In order to d e t e r -

mine the a p p r o p r i a t e n e s s of using t h i s t e a t in College

Ma fcheicatica, tb<? t e s t i t e;i;s wore 3axr.ined '• r d corcpared vrifch

the top ic s d i scus sed in the course College Mathemat ics . In

a d d i t i o n the t e s t wss adrainis feared to two s e c t i o n s of

College Mathematics dur ing the semester p receding the ex-

pe r imen t . On- the b a s i s of these inves.tigra t iona i t was

determined t h a t the t e s t is an a p p r o p r i a t e ^ a s s u r i n g in- '

a t rument f o r c c r t s i n a spec t s of the course College

Ma theraa t i c s .

The t e s t was admin i s t e red dur ing the l a s t week of

r e g u l a r c l a s s e s during the exper iment . I t ws s adminis -

t e red during1 the e l s as pe r iod fo l lowing the c l a s s pe r iod '

-during which the S t r u c t u r e of the Huiuber Syatexn ,waa adiain-

i s t e r c d . Standard answer shee t s were used and a l l answer

I l l

s h e e t s were s c o r e d by b a n d . A l l scr-r ir .g was checked t w i c e

in o r d e r t o i n s u r e a c c u r a c y .

T e s t I I I : The V/a t son-GIase r G r i t i c a l

Th in k in g ' A p or a i s a 1

^ - e o n - G l a s e r C r i t i c s I T h i n k i n g Appra i s a 1- i a a

t e s t d e s i g n e d t o measu re c r i t i c a l t h i n k i n g a b i l i t y . The

t e s t e x e r c i s e s i n c l u d e p r o b l e m s , s t a t e m e n t s , a r g u m e n t s ,

and i n t e r p r e t a t i o n s of d a t a s i m i l a r t o t h o s e which an i n -

d i v i d u a l m i g h t e n c o u n t e r in d a i l y l i f e . The t e s t c o n s i s t s

oT f i v e s u b t e s t s . Each s u b t e s t i s d e s i g n e d t o measu re a

s p e c i f i c a s p e c t of c r i t i c a l t h i n k i n g . S u b t e s t I i s a

t w e n t y - i t e m , m u l t i p l e - c h o i c e t e a t . Kach i t em h a s f i v e a l -

t e r n a t i v e s . Th is s u b t e s t i s d e s i g n e d t o measure a b i l i t y to

d i s c r i m i n a t e among d e g r e e s of t r u t h or f a l s i t y of i n f e r e n c e s

drawn f rom g i v e n d a t a . S u b t e s t I I i s a s i x t e e n - i t e m ,

m u l t i p l e - c h o i c e t e s t in which each i t em p r e s e n t s two c h o i c e s ,

Th i s s u b t e s t i s d e s i g n e d to m e a s u r e a b i l i t y t o r e c o g n i z e

u n s t a t e d as3ui , ip t iona or p r e s u p p o s i t i o n s which a r e t a k e n f o r

g r a n t e d in g i v e n s t a t e m e n t s • ' S u b t e s t I ' l l i s a t w e n t y - f i v e -

i t e m , m u l t i p l e - c h o i c e t e s t in :<rhicb each i t em p r e s e n t s two

a l t e r n a t i v e s . Th is t e s t i s d e s i g n e d t o m e a s u r e a b i l i t y t o

- t h i n k d e d u c t i v e l y f rom g i v e n s t a t e m e n t s or p r e m i s e s . Sub -

t e s t IV l a a t r f e n t y - f o u r - i t e m , m u l t i p l e - c h o i c e t e s t in which

t h e r e a r e two c h o i c e s f o r each i t e m . Th i s t e s t s amples

"G oodwin ""a t s on and Edward M „ C l a s e r , Manua l : Watson-Gla a e r Or i t i c a ? . Th i n k i n g Aj?»ra_.iaal (KYorTF7"T9SI j . ) .

112

a b i l i t y to weigh e v i d e n c e and to d i s t i n g u i s h be tween g e n -

e r a l i z a t i o n s w h i c h , a l t h o u g h n o t a b s o l u t e l y c e r t a i n , do

seem to be w a r r a n t e d . The f i n a l s u b t e s t i s a f i f t e e n - i t e m

t e s t d e s i g n e d to measure a b i l i t y to d i s t i n g u i s h between

arguments which a r e s t r o n g and r e l e v a n t and t h o s e which a r e

weak or i r r e l e v a n t to a p a r t i c u l a r q u e s t i o n . The a d m i n i s -

t r a t i o n t imes f o r each of the sob t e s t a a r e t h i r t e e n m i n u t e s ,

s i x rclnutca, e l even m i n u t e s , twelve m i n u t e s , and e i g h t

minute a. r e s p e c t i v e l y . For t h i s t e s t t he t o t a l raw s c o r e

was used as tha c r i t e r i o n sco re*

The y s t s p n - G l a 5 s r Cr i t i c a 1 Th ink ing A p p r a i s a l i s a

w e l l known t e s t end i t s c o n s t r u c t i o n and s u b s e q u e n t r e v i s i o n

were b a a e d on over tw cnoty-f ive y e a r s of s tudy* r e s e a r c h and

e x p e r i m e n t a t i o n of the rceosurement of c r i t i c a l t h i n k i n g

a b i l i t i e s . Two forma of t he t e s t a r e a v a i l a b l e * For t h i s

form c o n s i s t e d of $.,237 f r e s h n e n a t f i f t e e n f o u r - y e a r

l i b e r a l a r t s c o l l e g e s l o c a t e d in e l e v e n d i f f e r e n t s t a t e s .

Prom t h i s saiople tha s p l i t - h a l f r e l i a b i l i t y c o e f f i c i e n t was

c a l c u l a t e d sna found to be . 6 5 .

The t e a t , Wataon-Glaaer G r i t i c a 1 Tbirsk 1 ng; Appraisa 1 ,

was a d m i n i s t e r e d s s t h e t h i r d of a a e r i e s o f - s t a n d a r d i z e d

t e a t s . The t e s t was s d m i n i s t e r e d d u r i n g the l a s t week of

r e g u l a r c l a s s e s d u r i n g the s e m e s t e r and was a d m i n i s t e r e d

d u r i n g the c l a s s p e r i o d f o l l o w i n g the c l a s s p e r i o d d u r i n g

which the Algebra I t e s t was ac te in i s t e r e d . A l l "answer

, • • i i 3

sheets were scored by bend. A l l scores were checked twice

in order to insure a c c u r a c y .

Tea t IV; Teacber-»Made Tejsta

In a d d i t i o n to the t h r e e s t a n d a r d i z e d t e s t a , a s e r i e s

of f o u r - t e a c h e r made t e a t s were admin i s t e r ed during the

course of the semes t a r . These t e s t s were des igned to meas-

ure the s p e c i f i c course c o n t e n t covered dur ing the time

pe r iod between the a d m i n i s t r a t i o n of the p rev ious t e a t and

the given - tea t . N one of the t e s t a were cornprehsns ivo in

52Sture• J?one of the t e s t s were designed to measure cor.;tent

l ea rned f o r the e n t i r e s e i sea t e r . In addit ion a s e r i e s of

homework ass ignis en's? was c o l l e c t e d snd graded. The scores

on a. 11 the hossewo^K « ssignmeafca for each s t u d e n t were

t o t a l e d , m u l t i p l i e d by one hundred and d iv ided by the t o t a l

ma a irfiuci p o s s i b l e s c o r e . The raw scores on the four tea ts

and the homework score were then t o t a l e d . This t o t a l score

waa ussd as the c r i t e r i o n score f o r Teat IV.

' Sb0 f i r s t teschsr-reade t e s t was acciinlster-ed du r ing

che l i l t o visek oI school snd wss des igned to t o s t knowledge

of the c o n t e n t of l e s sons one through twe lve . The second

t e a t was a d m i n i s t e r e d dur ing the e igh th week of school and

was des igned to measure knowledge of the content of l essons

t o i r t e e n through s e v e n t e e n . The t h i r d "taacher-KCde t e a t ' '

was administered du r ing the t h i r t e e n t h week of school and

covered l e ssons e igh teen through twenty - s ix . The fourth

~ ' 111+

t e a t was a d m i n i s t e r e d d u r i n g the r e g u l a r t ime fo r . f i n a l

e x a m i n a t i o n s . This t e s t was a d r o l n i s t o r e d d u r i n g the week

a f t e r the t h r e e s t a n d a r d i z e d t e s t s had been a drain i s t a r e d .

The f o u r t h t eache r -mads t e a t vjas n o t a comprehensive t e s t

and was d e s i g n e d to t e s t -knowledge of the c o n t e n t of l e s s o n s

twenty-seven th rough t h i r t y - t h r e e . Each of the t e a c h e r

made t e s t s wan g raded on the b a s i s of one hundred p o s s i b l e

p o i n t s . By the way the borne work 3 cor a was c a l c u l a t e d , the

h i g h e s t p o s s i b l e homework s c o r e was a l s o one hundred p o i n t s .

Thus , each of the t eacher -made t e s t s and the homework s c o r e

ccn t r ibu ted e q u a l l y to the f i n a l s co re used f o r Tea t IV.

The te&cher-nada i e 2 t s were c o n s t r u c t e d in the way

jdd c3 ,ior Co,Lj.Uj5.c i'U'J t,rioii.u i , f j ajjo u sua lxy coni' oi u o t e d . {To

s t a t i s t i c s in t a r a a cf r e l i a b i l i t y , v a l i d i t y , and i tem

d i s c r i m i n a t i o n .'.-era c a l c u l a t e d . The t e s t a were c o n s t r u c t e d ,

in c o o p e r a t i o n , b y the two i n s t r u c t o r s who p a r t i c i p a t e d in

the s t u d y . The same t e s t s v/ore a cm in i a t e r e d t o a l l f o u r

s e c t i o n s ' of Co l l ege Ka thema t i c s and each i n s t r u c t o r s c o r e d

the bea t s aamin is t e r e d in the c l a s s he

the f o u r t e s t s a r e p r e s e n t e d in Appendix I>.

• ogtvfc. Copies of

<?,,o v,

In o rder t o p r o v i d e a means f o r comparing the r e l a t i v e

e f f e c t i v e n e s s of t h e .Juided .v iscovery I ' e thod of t e a c h i n g

and one e x p o s i t i o n >Iev.or>a of t e sca i i ' j g , f o u r t e a c h e r- re a d e

t e s t a and t h r e e a t a n d a r d i z e d t e s t s were a d m i n i s t e r e d . In

115

addition* hosmv?rk papers r-iers collected and graded during

the semester. Three of the "teacher-Made testa were admin-

istered during the a arrester. Than, during the last week

of classes three standardised testa were administered. The

Structure of the Number System, A Igebra I, and Wa taon**Gla.aer

Critical ThJ.nkirj_g Appr»>is:\l. Than, during the next week,

which wss the week cf final examinations, the fourth teacher

mads test W8 3 sdaiinis tiered.

Tlia raw scores for each of the standardized testa were

usr;d as criterion scores, Each of the teach or-made testa

w&3 baaed on one hundred possible points. In addition, the

homework score was calculated in-such a way at to be baa ad

on a score of one hundred points. The four test

scores and vbe homework score were totaled and the total

was ti3od as vie or iter ion score for Test IV. •

Methods For Tr.eeting the Data

For purposes cf a ta fcxa tiea 1 ana lysis tha scores on

esch of the four criterion testa of the .10'4 subject-a who

participated in the study ware divided in to six categories.

The criteria for determining the categoric a wa a teaching

metnoa and J.ovel of ability. There were two teaching metto —

oda, the Guided Discovery Method and the Exposition Method,

and three levels of ability. The ability level of each

student was determined by his composite score on the

116

American Col lege Tes ting: Program. Level I c o n s i s t e d of a l l

s t u d e n t s w i t h a composi te s co re of twenty or more . Level

I I c o n s i s t e d of a l l s t u d e n t s w i t h a composi te s c o r e g r e a t e r

than f i f t e e n and l e s s than twenty . Level I I I c o n s i s t e d of

a l l s t u d e n t s w i th a composi te score of f i f t e e n or l e s s . The

a ix c a t e g o r i e s then were Level I and-Guided Discove ry Meth-

od, Leve l I and E x p o s i t i o n Method, Level I I and Guided

Discovery Method, Leve l I I and E x p o s i t i o n Method, Level I I I

and Guided Di scove ry Method,, and Leve l I I I and E x p o s i t i o n

>i u t.h od.

Each c a t e g o r y c o n t a i n e d s t u d e n t a from two d i f f e r e n t

c las ; jed . In t h i s way s t u d e n t a e n r o l l e d in each time of day

^nd t « J * . • oy eacn of cce two xna t rue t o r s were r e p r e s e n t e d

in esch c - i t z p o r y . In t h i s way c o n t r o l s ware p r o v i d e d f o r

the of time of day and f o r t eacher e f f e c t s .

The "analysis of d n a used was the s t a n d a r d a n a l y s i s of

v a r i a n c e . The c o m p u t a t i o n a l fo rmulas used a r e those g iven

by L i n d q u i a T h e s e fuxmulaq do n o t depend on an equa l

numoer ox enu r i ea xn each c a t e g o r y . Ail, cs 1c ula t i o n s were

dent on a de#h. es Xcuin t o r and were checked twice in o rder

t o i n s u r e a c c u r a c y . The c r i t e r i o n used f o r a c c e p t i n g or

r e j e c t i n g hypo theses wad the .0$ l e v e l of s i g n i f i c a n c e .

6 T-i ^ „ © s* & L.!.x?ciqu i. s G y 1)6 s ig"n 8 iid A D& l y s i.s of Exp a r xto6 D "T S

I E ilSSMkSMl £iH £ducaTroH"T3'o3~robTHT^TT

11?

Seminary

An e x p e r i m e n t a l s t u u y K$S aon&ucv6d in which two

t e a c h i n g methods we're c o m p a r e d . "The t e a c h i n g methods ware

the- Guided D i s c o v e r y Method of t e a c h i n g and t h e E x p o s i t i o n

Method of t e a c h i n g . In a d d i t i o n t h e r e l a t i v e e f f e c t s of

t he two t e a c h i n g method? aa r e l a t e d t o s t u d e n t a b i l i t y ware

c o n s i d e r e d . The s u b j e c t s f o r the e x p e r i m e n t were 10lj s t u -

d e n t s e n r o l l e d in J4 s e c t i o n s of C o l l e g e M a t h e m a t i c s a t

3 0 u t h w e s t e r n 31 a t e C o l l e g e , \-f e a tb e r f or d , Ok l a h oraa , d u r i n g

the s p r i n g s e m e s t e r of t h e 1967-1968 s c h o o l y e a r . Two of

the C o l l e g e M a t b e m a t i c s G l a s s e s wore t a u g h t a t 10 a . m . and

two of t he c l a s s e s were t a u g h t a t 2 p . m . Two in a t r u e t o r s

p a r t i c i p a i-ed in t he s h u d y . Bach in s t r u e cor t a u g h t one

c l a s s us in:; t h e Guided D i s c o v e r y Method and one c l a s s u s i n g

the E x p o s i t i o n M e t h o d . At each t ime p e r i o d t h e s t u d e n t s

were a s s i g n e d a t random t o a c l a s s t a u g h t by t h e Guided

D i s c o v e r y Method or t o a c l a s s t a u g h t by t h e i m p o s i t i o n

M e th cd .

Three s t a n d a r d i s e d b e s t s and a s e r i e s of t e a c h e r - m a d e

t e s t s were used as c r i t e r i o n t e s t a . The s t a n d a r d i z e d t e s t s

were t h e b t ruc tu r_a of t h e Number S[,£s tern, A l g e b r a I , and the

v/ats o n - G l s g c r C r i t i c a l Th i n k i n g A p p r a i s a l . These t e s t s were

a d m i n i s t e r e d d u r i n g the l e s t week of r e g u l a r c l a s s e s d u r i n g

the s e m e s t e r . D u r i n g the s e m e s t e r t h r e e t e a c h e r - m a d e t e s t s

wexe duiii in i s l e r e d . In a d d i t i o n , a t e a c h e r - m a d e f i n a l exam-

i n a t i o n was a d m i n i s t e r e d . For each s t u d e n t t h e s c o r e s on

118

the four t«ochcr~rnads te.utn wars to ' ; e l ed ' a long with a horse*

work score • These t o t a l acoroa c o n s t i t u t e d the sco res f o r

a f o u r t h c r i t e r i o n t e a t in ddci It. ion to the thru a s t a n d -

a r d i s e d t e s t a .

Tha raw scores f o r asch of the four c r i t e r i o n t e s t s

were d iv ided i n t o s ix c a t e g o r i e s . The f a c t o r s used i a

doi«rmiiji3)g the c a t e g o r i e s ware tha two methods of t e a c h -

ing snd t h r e e la vela of a b i l i t y . These sco res ware then

a n a l y z e d , us ing a two by th ree f a c t o r a n a l y s i s of va r i ance ,

A l l hypotheaea were accep ted or r e j e c t e d a t the .05 l e v e l

of a i g n i f i c a n c e •

CHAPTER IV

THE RESULTS OP THE EXPERIMENTAL STUDY

In troduction

An oxperircenfcal s t u d y waa c o n d u c t e d i n which the

re lative effectiveness of a g u i d e d d i s c o v e r y method of

teaching end sr, position method of teaching wera com-

par qd. Tbe subjects for tha a tudy were a feudents snr o 11 ed

i n f o u r sections of College V.*thc&atics a t Southwcstern

State C o l l e g e during the spring scses-sr nf t h e 1967-1968

school year. A total of 10!j subjae bs p a r t i c i p a t e d in the

s t u d y . Two of t h e s s o t i o n a o f C o l l e g e l is t hema tics were

taught at 10 a .ra. On a of t h o s e sections was taught by t h e

0-uidsd Disc ovary Mefcti.d s;-d bhe e t h e r ssotion wa a taught

by t h e E x p o s i t i o n K - s t h c d . Tba r wâ i n i n g lwo s s c t i o n a of

G o 1 l e g e Ms t!:•«ms t i c a w ? r e fcs ;::v h t a t 2 p >n Cn e o f th e s e

section* k s s taught by t h e E x p o s i t i o n Xetncd and t h e o t h e r

section was t a a p h t by the G u i d e d v i s e o v a r y I l e t h o d . A t

o a o h time p e r i o d , the s t u d e n t a T-iere assigned t o t h e classes

t h r o u g h a random p r o c e s s .

In order to determine t b a r e l a t i v e offsotiveness of

the G u i d e d D i s c o v e r y T l c t h o d of t e a c h i n g e n d t h e E x p o s i t i o n

'Method o f teaching, f o u r c r i t e r i o n t e s t a w a r e a d m i n i s t e r e d .

I n o r d e r t o b s l s n e e t h e effects o f t i m e o f d a y a n d the

. . . ' 1 2 0

d i f f e r e n c e s be teaen the two Kho p a r t i c i p a t e d in

the s t u d y , on each c-rit-ex- ion t e a t , tbe acoraa of the two

c laaaos taught by the Guided Discovery Method were con-

s i d e r ad as oiaa group of scores and ths scores of the two

c l a s s e s tsu'gbt by fcba Expos i t ion Method were cons idered aa

one group of a c o r e s . l a a d d i t i o n , t h ree l e v e l s of a b i l i t y

were cons ide red . On each c r i t e r i o n tea t s the a corea wore

d iv ided in to s i s c a t e g o r i e s • The f a c t o r s de terrain tog t h s

c a t e g o r i e s were the tw*o t each ing methods and the throe,

lovula of a b i l i t y .

The data were analysed us ing the s t andard th ree by two

cnslyala of vor i anca• The compute t i o n a l forma If s used ase

those given by Libcquia t • A.II c s l c u l a t i o n a vie 1*8 performed

on a Monroe desk tor • A11. c a l c u l a t i o n s were chocked

twice in order- to In.iare accuracy , 'the s t a t i s t i c s of i n -

t e r e s t were the P - r a t i o for the two groups d>?terrained by

the two teaching raethoda f-:ncl the f - v a t i c fo r the inUxrac t ion

between the two groups defcarasined by the two teach ing ineth-

oda and fcba th ree groups determined by ths tbr«<3 l-avsla of

a b i l i t y . In urder f o r the F ~ r a t i o to l o d i c s t * that a given

"factor or- a e t of f o c t o r a have a s i g n i f i c a n t e f f e c t in" ten-is

of the c r i t e r i o n measure , the F - r a t i o must be l a r g e r than

o?:-a, ;:-ad in most cases even l a r g e r . ^Therefore, en F - r a t l o

of .roe or leaa would i n d i c a t e t h a t the combina t ion 'o f

' * * • i'inc.tjw.) a t , Des ign find & rial?3ia of - "Exoeriroenta lP_g_*Zgfcp-o.&f.. »aA saucatEoif T^'ton7T9S3TrP j ~ ~

' • • • 1 2 1

f a c t o r s f o r which the r a t i o was c a l c u l a t e d d id not s i g -

n i f i c a n t l y a f f e c t the scores achieved by the s u b j e c t s on

tba c r i t e r i o n t e a t . This would be t rue r e g a r d l e s s of what

c r i t e r i a f o r 3 i g n i f i c o n c o is u sed . In order f o r the P - r a t i o

to i n d i c a t e t h a t a given combination of f a c t o r s haa a s i g -

n i f i c a n t e f f e c t , the F - r a t i o nu* t be l a r g e r than one and

the exact s i s a which the s a t i o inuat be in order to I n d i c a t e

a s i g n i f i c a n t e f f e c t depends on the c r i t e r i a fo r p i g -

n i f i c a n c e uaed and the degrees of freedom fo r tba f a c t o r s .

For each of the four c r i t e r i o n t e s t a , t h r ee F - r a t i o a

were c a l c u l a t e d . ? - r a t i o s ware c a l c u l a t e d fo r the t each ing

la a th o d f a c t o r , th e a b i l i t y l e v e l fa e t o r , a n d - th a i n t e r a c t i o n

betv*e-«n- the teaching-roe thod f a c t o r and the a b i l i t y l e v e l

fac tor - . A s i g n i f i c a n t F - r a t i o fo r tba a b i l i t y f a c t o r would

i n d i c a t e t h a t tba ativ.lonts in the t h r ee l e v e l s of a b i l i t y

achieve d i f f e r e n t l y . That achievement ia dependent upon

a b i l i t y ia a u e l l e s t a b l i s h e d f a c t , so the ? - r a t i o fo r the

a b i l i t y f a c t o ? ia n o t - o f pr imary i n t e r e s t . A s i g n i f i c a n t

F-ra t i o f o r the i n t e r a c t i o n between tba t each ing method

f a c t o r and the a b i l i t y f a c t o r would i n d i c a t e t h a t tha o f -

f a c t a tba two t each ing methods on achievement a r e dependent

.upon . a b i l i t y and t h a t tb.3 e f f e c t s of tha two t e a c h i n g meth-

ods h i s y b a d i f f e r e n t a t each l a v a l of a b i l i t y . An i n a l g -

n i f i c a n t F - ra t i o fo r tha i n t a i - s c t i on a f f e c t would i n d i c a t e

t h a t the tuo t e ach ing methods have s i m i l a r e f f e c t on

achievement a t e.ac'b l e v e l of a b i l i t y . I f the F - ra t i o f o r

122

the interaction effect is insi&nlficar.it, then a significant

F-ratio fo-r the teaching method fee tor would indicate that

one teaohing inethod ia. more effective than the other® An

insignificant F-ratio for the teaching method factor would

indicate that both teaching methods are equally effective.

In .all .cas.es . the • ...05 _l?>vel of significance was used to

'de terrains tha sign if ice. uca of the effects of the various

"factors." Tha raw data used in the statistical analysis are

given in Appendix S.

The Findinga for Test I

The criterion scores for Teat I were tha raw scores

achieved by tha subjects on the test Structure of the

Kur.foog System. This tost is a foi'ty-aiinufce, forty'-item

rau 111pla-cboice tsst. '.Era teat is designed to measure

knowledge 'of the proper bias of operations with 'numbers,

number, ays tews including bases other than ten, modular

arithmetic, arid arithmetic judgment.

- Prora • IVo 1« IV it can ha 3a.on that the F for tha

intcrectlon effect'-of tha ability factor and - the fa.ctor • •

de j.siined- by the teaching-i-iothods is not largo enough to

Indicate a- significant effect at any Uvnl of aigkrificanca.

On this basis, the following hypothesis is accepted: the

relative effects of a guided discovery method of teaching

snd an expos i t ion »ie "boa of teaching on achleveiaeat as mea fl-

u-red by the tost, C&ppogatlva Ha thereat log ssts, structure

123

TAB'S IV

ANALYSIS OP ~7J) RIANOE TABLK FOR TEST I

(N --= IOJ4)

-d puree

K e til od A V, 4? 1 * A- *rr

c L i t. <.y

I n t e r a c t i o n

5rt' or

T o t a l

Sam of Sm5Bj-0.J

5.009 913 >010 33.71?

22143,

_df_

1 2 2

98.

321 Lu 76O j 103

Mean _3qaa.gea

5-009 if.66 «5i]9

16.858

22.888

.219 20.381; • 737

P

p < . 001

of fae IJumb^r System, w i l l not be dependent upon ntudont

a b i l i t y .

From Table IV i t can be 3Jen t h a t the P f o r ' the f a c t o r

gg t-31'reined by the teaching methods i s no t large . enough to

i n d i c a t e a s i g n i f i c a n t e f f e c t a t any l e v e l of s i g n i f i c a n c e .

This ind ica tes tha t the Guldad Diac0vory X atbod of teaching

snd t»pe i t ion Ma vhud of t a i c h i n g org ocvuslly e f f e c t i v e

•when the c r i t e r i o n i s t h a t c speo t of «tude-j b a o M a v m m t

tmuxco ia .inec«0^c-;d by the Last , J?true t a r * of the j.Tui:Vber

* Therefore, tne f o l lowing hypo thes i s i s r-a-jcctedi

s t u d e n t s t augh t by a guided discovery method w i l l score •

a .i,gi.si 1 i c s r s t ly h igher en the t e s t , Co ope r a t i v e Mathematics

ZSJjLJ.* .£,5 ™?„£ Nmnber 2 , t han s t u d e n t s

taugn-fc by f?r; e x p o s i t i o n method.

!21\

The Findings f o r Tes t I I

The scores used in the -analysis of data f o r Tes t I I

were the raw scores achieved on the t e a t , AIgebra I . This

t e a t i s a s t a n d a r d i z e d t e a t and i s one of the t e s t a in the

Coopera-tive Mo theing t i c s Tesjta s e r i e s . Algebra I ia a

f o r t y - m i n u t e , forty-iI'c-m m u l t i p l e - c h o i c e t e a t . Tbo t e a t

i s des igned to measure a b i l i t y to manipula te a l g e b r a i c

e x p r e s s i o n s , a b i l i t y to perforin ope ra t i ons wi th a l g e b r a i c

e x p r e s s i o n s , a b i l i t y to so lve a l g e b r a i c e q u a t i o n s , and

a b i l i t y to aolve l i t e r a l p roblems.

Prom Table V i t can be seen t h a t the F f o r the

i n t e r a c t i o n e f f e c t of the a b i l i t y f o o t e r and the f a c t o r

determined by tbo t each ing methods does not i n d i c a t e a

s i g n i f i c a n t o f f e e t a t any l e v e l of s i g n i f i c a n c e , Aa a

r e s u l t , the fo l lowing hypo thes i s ia a c c e p t e d : the r e l a t i v e

TABLE V

ANALYSIS OF VARIANCE TABLE FOR TEST I I

(N - IOI4)

Source Sura of f'eva r 3 elf

Method ' 10.961 1 A b i l i t y 936.797 2 I n t e r a c t i o n 6.q. , 21|6 2

Err or 98

Tot a 1 ill 10 7 /oSh 103

Mean Scuares

10.981 Jj6o 3 93

32,123

—31:262

13 •2dli p ^ . 0 0 1

12Jp

e f f e c t s of a guided d iscovery method of teaching and an

expos i t ion method of teaching or; schieveroent aa measured

by the test . , Coopers t i v e Mr bh era a t i c a Te s t a , Algebra I ,

w i l l not be dependent upon student a b i l i t y .

Prom Table V i t can be seen t h a t vhe F f o r the f a c t o r

determined by the teaching raathod.a is Dot largo enough to

i nd i ca te a s i g n i f i c a n t e f f e c t a t any l e v e l of s i g n i f i c a n c e .

From t h i s i t la concluded, tha t the Guided Discovery Method

of teaching and. the Expos i t ion Method of teaching are

squa l ly a f f e c t i v e •when the c r i t e r i o n is tha t aspect of

a iudant achievement measured by the t e s t , Aleebra I . On

the basis of tb ia i n fo rms t i on , the f o l l o w i n g hypothesis la

r e j ec ted : a tudeuts laught by -3 guided d iscovery method

w i l l score s i g n i f i c a n t l y h igher on tba t e s t , Cooperative

Hathfcicatlca, Alggbxa 1 , than a tudents taught by an 02 op-

s i t ion method. *

Toe Findings f o r Teat I I I

'file ci5 i t e r ion 3 c ore s for- Test I I I wars the raw so ore a

achieved by the subjects on the t e n t , Wajtsc-n-Gla.ier C r i t i c a l

Thinking App ra i sa l . This t e s t i s designed to rsesnuta

a b i l i t y to d i sc r im ina te among.degrees of t r u t h or f a l s i t y

of inferences drawn from givon da ta , a b i l i t y to reaaoo

deduc t i ve ly tro:n given statements or premises, a b i l i t y to '

recognise una vate»i assumptions which ^ rs taken f o r granted

i n g iven statements or a s s e r t i o n , a b i l i t y to weigh

1Z6

TABLE VI

AHA LYSIS OF VARXA3CB I'ABL'S PGR TEST I I I

(N - 101.)

Sum of _ Source ^Squares _ d f

Me tbod 1|7.2O9 1 A b i l i t y 1599.687 2 Irrteiu'seiion . 62,592 Ok

Err or - 5358.39U _$8.

!„ a 1 Jt. i.i |

v nAi oA o 103

gf'uarea

. U7.289 799.8 Mi

31.296

P

,865 111 .628

.572

...... Jl„

p < .001

evidence and to d i s t i n g u i s h bsivaen genera l isa t ions frora

given data that are not warranted and genera l i sa t ions which

.isora to be warranted, and a b i l i t y to d i s t i n g u i s h between

a rgiwienta which ore strong and re levant snd those which ara

He ak or i r r e l e v a n t to a p a r t i c u l a r quest ion a t issue *

As revealed in Table VT, tha F for i n t e r a c t i o n i s loss

than one. This i n d i c a t e s tha t tbo Guided Di^c-ovory Ho tbod

of t.-c-oMng and the E x p o s i t i o n Method of teaching nave tbo

sauie r s l s t i v o e f f s e t on i h « t aspect of acbievemanfc measured

by- tha- fcson- Qlap,or S r - i t i e a l Appra i sa 1 a t edcb

l e v e l 'of a b i l i t y . Accord ing ly , tbo f o l l o w i n g hypothesis

i s a c c a p t s d : t ha ' r e l a t i v e a f f a c t a of a gu ided d i scovery "

ractbod of t e a c h i n g and an e x p o s i t i o n me thod of t e a c h i n g on

achievement aa maosruted by tha ison-G- lcer C r i t i c a l Think-

i r ^ A p p r a i s a l - w i l l no t be dependent op on a t u d a n t a b i l i t y , ,

12?

Aa revealed in Table VI, the F for the effects of the

two teaching methods is leas than on 3. Since the inter-

action effect is not significant ttola indicates that the

difference in the effects of tha two teaching methods is

not significant when the criterion .la achievement aa

measured by tha ataor;-Gls szp Critical Thinking Apora trial..

Cm the basis of this inforiaation, tha following hypothesis

ia r.ejnoted: students taught by a guided discovery method

will a cor a significantly higher on tha V&.t a on~ (Kisser

Critical. Thinking Appraiaal than students taught by an

exposition method.

The Findings'for Teat IV

During the semester of the experimental a tudy, four

teacher-made to at a wore administered. Those teats war a

designed to measure knowledges of the material presented

during a given period of the smentsr. Each teat tea tad

knowledge of c particular unit of r,iafcsrial. None of the

testa wore d.?eignad to Unit kuwledga of tha content of

ttio onvire course of study,, -ffh6 fourth certcher—mads beat

was & din in is tared curing the last week of the semester, •

but the teat was not n co;.opr'shenaive test. Each of the

four" tea char-made taata way gradad" on the -basia of on a

bandied possiole points. During the serasatsr a seriea

ol nomwork assigna nfcs ware collected end graded. At

the end oi tha aaznea tier tho poxnto ocbieved by each student

128

TABLE VII

AHA LYSIS' OP VABIAKCB TABLE PGR TEST IV

(N = lOLj.)

• Sum of Mean Source • Squares £f 3 011 area F

Method 6565 „ 171 1 6565,171 1.857 Ability 5S'60iu6i[0 2 27o02. ".20 7.86)4 Idtsraction >658 .¥2 2 2829.231 .800

Error 3)4oU8l.lll _ 2 l |

3535.522

?otsil__ Itiy091l8£ -XPjJ 1

P

,l<p<.2 p <,001

on tils homework aaaigcaiants were totaled. This total was

then divided by the total number of points possibles and

then multiplied by one hundred. Tbua each student bad a

possible bCiVjt:'work score o£ one hundred. At the end of

the seme3 tor the scopes achieved by each student oa the

four teacher made testa sod tba student's homework score

hfjr3 totaled. Tba resulting scores v?cre u.'isd as the

eritcilou '/.coraa for Teat IV.

A3 i'fidicatod in Table VII, the P for ir: tcsi-ao ti 03 on

Teat IV la leas tliG a one. Therefore, the followisjg

hypothesis is accepted: tbo relative effecba of a guided

discovery method of teaching and an exposition net-hod of

teaching on• achievement as raes ur-?d by the a fcudsnt'a' grade

in toe course will not be dependant upon 9tudcKt sbility•

129'

A a indicated in Table VII s the P for the effects of

t h e two teaching methods ia greater tbaio one but not great

enough to indicate a significant effect at the ..G level

of s i g n i f i c a n c e . This indicates that there w a s no signifi-

cant d i f f e r e n c e between the G u i d e d Discovery Method o f

t e a c h i n g and the Exposition Method of t e a c h i n g y f e e n the

c r i t e r i o n ia a t u d s n t a c h i e v e m e n t a a m e a s u r e d b y four

t e o c h « r m a d a t s a ta a n d h ora e v o r k .1 c 0 r qs . T h e r e f o r e , t b e

f o l l o w i n g h y p o t h e c i a ia rejected: atudenta taught by a

guided d i s c o v e r y method w i l l make significantly higher

grades in tha course than s t u d e n t s t a u g h t by an exposition

Method.

S u m m a r y

our cr i'corion tc313 \ip.re used l;c determine the

rela c..n/a eiiec oiveneas of the Guided D i s c o v a r y Method of

t e < a c n i n g and une li-xpos ix'ion ri€tood of tfu'iobing. The s 0 0 r 0 3

on e a c h t e s t were a n a l y z e d uîng t h e atsndiu-d three b y t w o

f a 0 t o r a n a l y s i s o f v a r i a n c e . The J'-rsfclo v.-og calculated

urio i.n ier.iC'cion e f f e c t of i'he two rjefchods of -tsDching

•and torea levala of ability. In sddXtioo, the P-ra tio for

the ef 1 ec t>i of tbe tit? 0 tea ch i n g r<ie th oda w a s ca 1 c u l a t o d .

On each of tha four criterion feoata the P-ratio for

tha e i f e c t a of tbo two teaching r c e f c h o d a ima not large

"Hough t o indicate a ,1 ignificanb difference between the

two reaching mc vbeda. A a a romjlft, of the hyoptbeaea

130

identified in Chapter I , the following vera rejected:

1. Students taught by 3 guided disc ovary m e t h o d will

3 core significantly higher on the tea t, Co opera tlva

Mathematics Testa, Structure of the Rubber S^stora, than

students taught b y an exposition method.

2. Students taught b y a guided discovery method

.will -icora significantly 'higher on the teats Cooperative

Mathematics Te £ta , Algebra I , than s i;uds«ts taught h y an

fizpoiiition method.

3. Students taught b y a guidad discovery m e t h o d

will score significantly higher on the Viatsop-Glason

Critical Thinking Apyrîtjal than atadsnfcs taught h y an

expoiition method.

h(. Students taught b y a guided diacovary method w i l l

make significantly higher gr-idaa in the course tbsn stu-

dents taught b y fin expos lfcic-8 Method,

Co each of the four criterion tests the F~ratio for

the interaction of tho ability factor and the teaching

method fsctor '-/as not large enough to indicate a signifi-

cant interaction of these factors. Ibis implies that the

comparative sffectivenssa of the Guidad Discovery M e t h o d

of teaching aod the Exposition M e t h o d of teaching was

similar at each lev si of ability. A <3 a result, of the

hypotheses identified in Chapter I, the following were

accepted:

1.31

3'. The r e l a t i v e e f f e c t s of s g u i d e d d i a c o v e r y

m e t h o d of t e a c h i n g a n d an ' i m p o s i t i o n m e t h o d of t e a c h i n g

on, a c h i 6 v e r o e n t a s measured b y t h e t e s t , C o o p e r a t i v e .

Mathawa t i c a T e a t s , S t r i ae t u r e o f t h e Huraher Syg tern, w i l l

n o t be d e p e n d e n t upon s t u d e n t a b i l i t y .

6 . The r e l a t i v e e f f e c t s o f a g u i d e d d i s c o v e r y

m e t h o d of t e a c h i n g a n d an e x p o s i t i o n me thod of t e a c h i n g

on a c h i e v e m e n t aa m e a s u r e d b y t h e t e s t , C o o p e r a t i v e

i-la t h e is 3 t i c s Tea t a , AIgjebra I , w i l l n o t be d e p e n d e n t upon

.1 iu d e n t a b i 1 ;t t y .

7 . The r e l a t i v e e f f e c t s of a g u i d e d d i s c o v e r y

m e t h o d of t e a c h i n g a n d an e x p o s i t i o n n e t h o d of t e a c h i n g

on a c h i e v e m e n t a s m e a s u r e d b y t h e t a o n - t r l e a e r C r i t i c a l

T b l n k l i ^ w i l l n o t b e d e p e n d e n t upon s t u d e n t

a b i l i t y .

8 . The r e l a t i v e e f f e c t s of a g u i d e d d i s c o v e r y

m e t h o d of t e a c h i n g a n d an e x p o s i t i o n m e t h o d of t e a c h i n g

on a c h i e v e m e n t a.- rcoaaured b y t h e s t u d e n t ' s g r a d a i n t h e

c o u r s e n o t be d e p e n d e n t upon s t u d e n t a b i l i t y .

I n 3uaia:3ry, i n tarrna of a l l of t h e f a c t o r s c o n s i d e r e d ,

t h e r e waa no s i g n i f i c a n t d i f f e r e n c e b e t w e e n t h e G u i d e d

D i s c D v e r y Mefchod o f - t e a c h i n g a n d t h e E x p o s i t i o n M e t h o d of

t e a c h i n g ' w h e n tb,= ci<i I c r i o n i s s t u d e n t ~ eh i e vera e n t .

CHAPTER V

SUMMARY, FIHDIKGS AND CONCLUSIONS,

AND RE'COI-IMEHDATIONS

£urr,raa ry

The purpose of tbl'S study vras to provide a reference

for research related to tba discovery method of teaching

K2 tbeiaatics aad to ascertain the vs lu e, o a d? terra in ad by

student achievement, of U3i»g a discovery method of teach-

ing mathematics in a college freshman ma thoaa ties c our a a

for non-matherastics and non-science majors. la order to

provide a reference for rea ;srcb related to tba 'discovery

method of tsscbing iaa thorns tics ,• a atMprobenaivo sunmiary of

research literature related to this method of teaching

rjstbematlca was presented. In order to ascertain the value

of using a discovery aa thod of teaching inothsiaatics in a

college : C > * n th'jso tic,i c our a 3 for con-ks tbm®tics

a ad ijctn-jJciencc iia joi'S ©n experimental study xma conducted.

Ars excei i ssatal atudy was c endue ted in which the

effects of too rsethoda of teaching on student auhieveroeat

were compared. The ras \« ll oda of teaching were the Guided

Discovery Method of teaching and the Exposition Method of

teaching. The subjects for the e-periment were IOI4 stu-

dents enrolled in four -ections of College Mathematics at

112

133

Southwestern S ta t s College f "rfc-tfcerfcrd5 Oklahoma. The

experiment was conducted dur ing tbn spr ing semester of the

1967-1968 school y e a r . College Ma their..** t i e s ia a course

designed f r o l i b e r a l a r t a s tudents and ia a r equ i red course

fo r a l l s tu dents v:ho do not complete co l lege algebra and

tr igonometry aa s p a r t of t h e i r degree program. Two of the

four sec t ions of Collage Mathematics wore taught a t 10 a.m.

snd the remaining two sec t ions wara taught a t 2 p.m. Two

i n s t r u c t o r s p a r t i c i p a t e d in the s tudy . The four c l a s sc s

ware arranged In such a way t h a t a t each of the two time

per iods one c lass was taught by the Guided Discovery Method

and one c l a s s was taught by the Exposi t ion Method. Each

i n s t r u c t o r taught one c la a a during each time per iod and

each i n s t r u c t o r taught one c l a s s using the Guided Discovery

Method of teaching and one c l a s s using tha Exposi t ion

Method of t each ing .

ConpoJite scores on the American College Test ing

Program were used to e s t a b l i s h three l eve l s of a b i l i t y .

Level I cons i s t ed of a l l s tudents with a composite score

of twenty or h i g h e r . Level I I cons i s t ed of a l l a fcud-'inta

with a composite score higher than f i f t e e n » bu t lower than

twenty, and Level I I I cons i s t ed of a l l s tudents with a com-

p o s i t e score of f i f t e e n or lower. The s tudents who had

en ro l l ed fo r the 10 a.m. c lasses and the s tudents who bad

enro l l ed for the 2 p,in. c lossss were divided in to c a t e -

gor ies according to sex snd l e v e l of a b i l i t y . The s tudents

131+

in each ca t ego ry were then c iv id a a Jn to groups through

a random p r o c e s s . In i b i s n a n u s r , of the s t u d e n t s e n r o l l e d

a t 10 a . m . , two groups were formed; snd , of the s t u d e n t s

e n r o l l e d a t 2 p , m . , two groups ware formed. During each

time p e r i o d , one of the two groups was t augh t by the

E x p o s i t i o n Method and one of the groups was t augh t by tha

Guided Discovery Method. Tha t each ing method to be used

wi th each group vaa detsrrninad by a random p r o c e s s .

The purpose of tha exper imen ta l s tudy was to determine

the r e l a t i v e e f f e c t i v e n e s s of the Guided Discovery Method

of t each ing and the Expos i t ion Method of t e a c h i n g . The

Guided Discovery Method of t each ing i s a method of t e a c h i n g

in which s t u d e n t s l ea rn through Induc t ive p r o c e s s e s , l o o m

through examples, and a method of t each ing in which formal

s t a t emen t s of concepts or p r i n c i p l e s a r e no t given to the

s t u d e n t s u n t i l tha s t u d e n t s have had exper ience wi th tha

coneopt or p r i n c i p l e snd u n t i l the s t u d e n t s a r e convinced

t h a t tha conospt or p r i n c i p l e i s v a l i d . The Expos i t ion

Method of t each ing i s a mafchod of t each ing in which s t u -

den ts I ca ro through deduct ive p r o c e s s e s , l e a r n through

s tudy ing a u t h o r i t a t i v e s t a t emen t s of p r i n c i p l e s and con-

c e p t s , and a method of teaching- in which p r i n c i p l e s and

concepts a r e s t a t e d in a formal mennev a t the beg inn ing

of the i n s t r u c t i o n a l sequence in which the p r i n c i p l e or

concept is i n t r o d u c e d . S p e c i a l J u s t r u e t i o n a 1 m a t e r i a l s ware

p repared , f o r .both i n s t r u c t i o n a l Methods, in order to a s s u r e

•• ' 135

t h a t tha inst ruct iona l , m a t e r i a l s "vonlc be. presented in a

manner approprlate f o r e teb i n s t r a c t i o n a l method and to

i n su re t h a t the s t u d e n t s t augh t by the two t each ing methods

a c t u a l l y l ea rned the same c o n c e p t s .

In order to determine the r e l a t i v e e f f e c t i v e n e s s of

the two t each ing methods, t h r ee s t a n d a r d i z e d t e s t s were

admin i s t e r ed and a a e r i e s of tea char-made t e a t s wore a d -

m i n i s t e r e d . The s t a n d a r d i z e d t e s t s were Coopera t ive

Ma the ma t i e s T e_s t s , Structure of th<a Number System,

Coopers t i v e Ma thoma t i c s Teats , AJj^zebra and the Watson-

Glaaer C r i t i c a l Thinking A p p r a i s a l , Thaae t e s t a were

adminis t a r ed dur ing the la a t week of the somas ta r* 'Three

teacher-made t e a t s were admin i s t e r ed dur ing the semester

and a f o u r t h teacher-ma do t e s t va a a d m i n i s t e r e d a t the

end of tha s emes t e r , Tha raw scores on each of tha s t a n d -

a r d i z e d t e a t s and the t o t a l scores ob ta ined by summing

each s t u d e n t ' s scores cn the fou r tea chsr-made t e s t s and

a homework score ware used as four c r i t e r i o n measures .

For each of the fou r c r i t e r i o n measures a s t a n d a r d t h r e e

by two f a c t o r a n a l y s i s of v a r i a n c e v;ss pe r fo rmed . This

method of a n a l y s i s was used to determine the r e l a t i v e

e f f e c t i v e n e s s of the Guided, Discovery Method of t each ing

and the Expos i t i on Method of t e a c h i n g .

136

Find ings &nd CpnclusioKS

For each of the fear c r I t s r i o n yieasuroa the F - r a t i o

for the e f f e c t s of the two t each ing methods was c a l c u l a t e d .

Of the four F - r a tios clsculs ted, none wo 3 l a r g e enough to

i n d i c a t e a s i g n i f i c a n t d i f f e r e n c e between the tvro t e a c h i n g

methods at tbe .05 l e v e l of sigrsiflessee. As a r e s u l t , of

the e i g h t hypo theses f o r m u l a t e d a t the beg inn ing of the

s t u d y , the following "were r e j e c t e d :

1 . S tuden t s t a u g h t by a guided d i s c o v e r y method will

score s i g n i f i c a n t l y h i g h e r on the t e a t , Coopera t ive

Ma thsraa tic a T e a t 3 , S t r u c t u r e of the Number System, than

s t u d e n t s t a u g h t by an e x p o s i t i o n mot-bed.

2 . Students t augh t by a guided d i s c o v e r y method w i l l

score significantly h i g h e r on t b e t e s t , Cooperative

Ma them a t i c a Test 8 , Algebra I , than s t u d e n t s t a u g h t by an

e x p o s i t i o n method.

3 . S tuden t s caughb by a guided d i s c o v e r y method will

score igr.ificently h i g h e r on the I fa t . ion-Glsaer C r i t i c a l

Thinking A p p r a i s a l than s t u d e n t s t augh t by an exposition

me thod .

k ' S tuden t s t augh t by a gu ided disc ovary method will

make significantly h i g h e r gradea in the course than atu-

den t s t a u g h t by an e x p o s i t i o n method.

For each of the f o u r criterion ro^-sres the F - r a t l o

f o r the i n t e r a c t i o n a f f e c t s of the two beaching methods

and the t h r e e levels of ability vrsa c a l c u l a t e d . Of the

137

fou r F - r a t i o s c a l c u l a t e d none was l a r g e enough to i n d i c a t e

a s i g n i f i c a n t i n t e r a c t i o n between the t each ing method

f a c t o r end the a b i l i t y f a c t o r . As a r e s u l t , of the e i g h t

hypotheses fo rmula ted a t the beg inn ing of the s t u d y , the

fo l lowing were accep t ed :

The r e l a t i v e e f f e c t s of a guided d i s c o v e r y method


achievesient a.i measured by the t e a t , Coopera t ive Mathematics

T e s t a , S t r u c t u r e of the Nuraber System, w i l l no t be dependent

upon s t u d e n t a b i l i t y .



achievement as measured by the t e a t , CoojDera t i v e Ma th eroa t i c s

To a t a , Algebra I , w i l l no t be dependent upon s t u d e n t

ab i l i t y .

7 . The r e l s t i v e e f f e c t s of a guided d i s c o v e r y method

of t each ing and an expos i t i on method of t-se?hing on

achievement as Measured by the Watigop-Gttccev C r i t i c a l

Thinkipg A p p r a i s a l w i l l no t be d-apancient upon s t u d e n t

a b i l i t y .


of t each ing end an expos i t i on r.iethod of t each ing on

achievement -aa measured by the ? t u d e n t ' a grade in the

course w i l l no t be dependent upon s t u d e n t a b i l i t y .

On the b a s i s of the t e s t i n g pj?i>£3?axa and the s t a t i s t i c a l

a n a l y s i s , i t iu concluded t h a t , in a c o l l e g e frasbmon

133

ma thema t i c a c o u r s e f o r non-xna theraa t i c a and n o n - s c i e n c e

M a j o r s , the Guided D i s c o v e r y Method of t e a c h i n g and the

E x p o s i t i o n Method of t e a c h i n g a r e e q u a l l y e i f e c t i v e wtien

the c r i t e r i o n i s s t u d e n t a c h i e v e m e n t .

H s c oiniTs e n da t i o n a

The r a a u l t a of t he expe r imen t r e p o r t e d in t h i s s t u d y

i n d i c a t e - tha t in a c o l l e g e ma tb ems t i c 3 c o u r s e d e s i g n e d f o r

l i b e r a l a r t s s t u d e n t s t he Guided D i s c o v e r y Method of t e a c h -

ing and the E x p o s i t i o n Method of t e a c h i n g a r e e q u a l l y

e f f e c t i v e i f the c r i t e r i a a r e r e t e n t i o n and a b i l i t y t o

a p p l y l e a r n i n g t o s i t u a t i o n s s i m i l a r t o , b u t d i f f e r e n t

f rom, the s i t u a t i o n s in which the l e a r n i n g o c c u r r e d . I f

the o b j e c t i v e s of i n s t r u c t i o n fire t o f o s t e r r e t e n t i o n of

l e a r n e d f a c t s and t o f o s t e r the a b i l i t y to use l e a r n i n g in

new s i t u a t i o n a , then the E x p o s i t i o n Method of t e a c h i n g ia

as e f f e c t i v e aa the Guided D i s c o v e r y Katbod of t e a c h i n g

f o r o b t a i n i n g -the d e s i r e d o b j e c t i v e s ,

S t u d e n t s rought by the Guided Di scove ry Method of

t e a c h i n g were r e q u i r e d to d i s c o v e r coco opta and p r i n c i p l e s .

They were a la o r e q u i r e d to formula t e g e n e r a l i z e t i o n a

through i n d u c t i v e p r o c e s s e s of r e a s o n i n g and to d i s c o v e r

methods ,of s o l u t i o n s to p r o b l e m s . I t would appear t h a t i f

the o b j e c t i v e s of i n s t r u c t i o n o r e t o deve lop the a b i l i t y

to d i s c o v e r c o n c e p t a , p r i n c i p l e s , and methods of s o l u t i o n s

t o p r o b l e n s and the a b i l i t y to f o r m u l a t e g e n e r a l i ^ a t i o a a

139

through i n d u c t i v e p r o c e s s e s , therj tbo Guided Discovery

Method of t e a c h i n g .'should be mora e f f e c t i v e than the

E x p o s i t i o n Method, of t e a c h i n g . The t e s t i n g program con-

duc ted in r e l a t i o n to the exper iment r e p o r t e d in t h i s

s t u d y was n o t des igned to measure t he se a s p e c t s of

a ch i evemen t . A rev iew of r e s e a r c h l i t e r a t u r e r e v e a l s t h a t

va ry few e x p e r i m e n t a l s t u d i e s have been conducted in which

t he se a s p e c t s of achievement have been c o n s i d e r e d . There-

f o r e , f u r t h e r r e s e a r c h i s needed to de te rmine whether

gu ided d i s c o v e r y methods of t e a c h i n g a r e mora e f f e c t i v e

chan e x p o s i t i o n methocis of t e a c h i n g v*hen tb o.? <3 a s p o c fcs of

achievement a re used as the c r i t e r i a f o r d e t e r m i n i n g the

r e l a t i v e e f f ec t iveness ..'j of t b s e crop a red t e a c h i n g ne thod i j .

As has fceea i n d i c a t e d , iv-:r>y a s p e c t s of the r e l a t i v e

e f f e c t i v e n e s s of d i s c ovary methods of t e e c b i n g and expo™

a i t ion methods of t e a c h i n g have no t be or- a d e q u a t e l y s t u d i e d .

In p a r t i c u l a r , r-a s ea r ch i s needed which u i l l h e l p aoswe*

tho f o 11 cvring qu QS t i c-n-i:

1 . Does l e a r n i n g by d i s c o v e r y a f f e c t the r e l a t i o n

the l e a r n e r p e r c e i v e s between bl icael f end the f i e l d of

knowledge in v;o 1 eh the dJ. scovery l e a r n i n g occur red? In

p a r t i c u l a rj does l e a r n i n g by d i s c o v e r y f o s t e r a c o n f i d e n c e

in cue l e a r n e r ous'c be has vhe s b l l i c y to ope ra t e i n d e -

p e n d e n t l y , l e a r n f o r b i n s e l f , and cope wi th the p r o b l e m

he f a c e s through the use of h i s own i n t e l l e c t u a l a b i l i t i e s ?

L l { 0

2. Doea learning by discovery affect the attitude of

the learner toward.a the field -f ksovrledga in which the

discovery learning occurred? Doea learning by discovery

produce an enduring interest in that field of knowledge

and a das ire to continue to learn from that field of know-

ledge?

3. Doea learning by discovery create in the learner

an fspprecia tion of the value of the knowledge in the field

in which the discovery learning occurred'?

4. Does learning by discovery affect the way a learner

behaves when he encounters an unprecedented problera? Doea

learning by discovery lead the learner to be lesa rigid, in

the way he attacks problems?

5* Doea learning by discovery affect the? way a learner

behaves when he forgets how to solve a problem?

APPENDIX

1 1 , 1

APPENDIX A

A COURSE OUTLINE FOB

COLLEGE MATRTSMATICS


The pu rpose of Appendix A i s to p r e s e n t en o u t l i n e

of the t o p i c s c o n s i d e r e d l a the course Co l l ege Mathemat ics

d u r i n g the spr ing s e m e s t e r of the 1967-1968 s c h o o l y e a r .

The s u b d i v i a i o n a of the o u t l i n e w i l l be according t o the

lesaona aa t h e y were g iven t o the s t u d e n t s .

Th3 Course O u t l i n e

Lasacn 1 A. . D e f i n i t i o n of term a

1 . S e t 2 , Slemen.t

• 3 * N a t u r a l number l | . B i n a r y o p e r a t i o n 5 • Ordered p a i r

B. E x e r c i s e s • . 1 . The b i n a r y o p e r a t i o n B d e f i n e d b y

( a , b) B (c> d) - (a x b , c) 2 . Toe b i n a r y o p e r a t i o n d e f i n e d by

{a , b) 3 ( c , d) = (a x d , b) 3• Tho b i n a r y o p e r a t i o n B d e f i n e d by

( a , b) B ( c , d) - (a x c , b + d)

Le s a on 2 A. The corarnutative p r i n c i p l e

1 . + i s commu ta t i v e 2 . x l a comrautative 3 . The b i n a r y o p o r a t i o n B da f i n a d by

( a , b) B ( o , d) ™ (a x c , b + d) ia coraiau ta t i v e .

1*. I f ( a , b) B ( c , d) = {2a + 2 c , bd) , then B i s comnutativ-. j .

l/;2

1!|3

B. The saaoeia t ive p r inc ip le 1. + ia assoc ia t ive 2. x i s assoc ia t iva

C. Exercises

Loss on 3 A» I d e n t i t y elements

1, 0 i3 an i den t i t y for + 2. 1 is an i d e n t i t y fo r x 3• (0, 1) ia on i d e n t i t y for B i f

{a, b) B (c , d) - (a + e , bd) B. Exe.rcia oa

Lesson A. The inverse of an element with respect to a

binary operation and an i d e n t i t y 1. -a ia the inverse of a with respect to i-2 . If (3, b) if (c , d) = (a + c , b + d) , then

(0, 0) ia aa i den t i t y for $ and ( - a , ~b) ia an inverse of (a , b)

B• Tlia solut ion of cer ta in l inear equations 0 . Kxerciaaa

Le 53 3 01] ^ A. If 9 Ob •- x and a = b B x have the aarae

so lu t ion , then 0 ia said to be the inverse of B 1. - ia the inversa of -r 2. If ( a , b) § (c , d) = (a + c , b + d) and

(ai b) & (c , d) = (a - c , b ~ d ) , then § ia the inverse of &.

B. Exercises

Lesson 6 A. Division la the inverse of mul t ip l i ca t ion B. Eserciaea

Lesson 7 A. The d i a t r ibut ive p r inc ip le

1 . Mul t ip l ica t ion ia d i s t r i b u t i v e over addi t ion 2 . Addition i s not d i s t r i b u t i v e over m u l t i p l i -

cat ion 3» I f (s > o) *« ( c j d) ™ (sc) bd) s nd

(a» b) § (e , d) - (a + e , b + d ) , then * is d i s t r i b u t i v e over # .

k- I f (a , b) 0 ( c . d) = (a + c , bd) and (a , b) & {c, d) ~ {a i- d , b e ) , then 0 is not d i a t r i b u t i v e over &.

3 . Exercises

lljlj.

Lesson 8 A. Equivalence relations

1. = is an equivalence relation on the sat of integers.

2. If (a, b) = (c, d) means a -- c and b = d, then = is an equivalance relation on the set of a 11 ordered pairs of Integers.

3. If tha relation B is defined by (a, b) R (c, d) if and only if a + d -b - c, then R is not an equivalence relation.

i].. The relation R- defined by {a, b) R^ (c, d)

if and only if a + d ~ b * c is an equivalence relation.

B. Exerc is es

Leas on 9 A. Equivalence clasao3

1. Tha equivalence classes of modular arithmetic 2. Binary operations defined on equivalence

classes 3. Exercises

Lesson 10 A. Clock arithmetic B. Exercises

Lesson 11 A. The rel'ationship between arithmetic on a clock

and arithmetic with equivalence classes of natural numbers

B. Tha properties of clock arithmetic C. Exercises

LeS3 on 12 A. The relation = defined by a •= b if and only

if the remainder obtained by dividing a by n is the same as the remainder obtained by dividing b by n* 1. ==£ is an equivalence relation

2. The relationship bot:-reen ™ and arithmetic on the n-'nour clock ~i

3. Exercisea

Lesson 13 A. The binary operafcion © defined by

(a, b) 0 (c, d) - (ad 4- be, bd)

i ks

B. 'The binary o p a i a t i o a Q d a fined by (a, b) (x) (c, a) ~ (ac, bd)

C. Exorcises

Lesson 11}. A. Some properties of(0

1. (5) ia cororau ta tive 2. (0, 1) la an identity element for © 3 • Not <3very ordered pair (as b) baa an

inverse vitta respect to B. Some properties of x

1. (x) is comrnufcativa 2. (1, 1) is an identity element for (>c) 3. Hot every ordered pair (a, b) baa an

invars e with respect t o © C. Sxerciaes

La a a on If? A. 'Fme equivalence relation = defined by

( a , b) = (c, d) if and onTy if ad - be B. R is the" set of all equivalence classes of

ordered pairs (a, b) where a is an integer and b ia a non-aero integer with respect to the relation =

C. Exercises ~~

L'3 3 :3 on 16 A. Properties of elements of R B. The binary operation + defined on R as follows:

If X and Y are elements of R, (a, b) ia an element of X and (c, d) is an element of Y, then X i* Y ia the element of R containing (a, b) 0 (c, d).

G. Th3 binary opera fci-on x defined on R aa follows: If X and Y are elements of R, (a, b) is an element of X and (c, d) is an element of Y, then X x Y is the element of R containing (a , b) 0 ( c , d).

D. x and 4- are well defined E. Exercises

Lega on 17 A. Properties of + and x

1. The element of R containing (0, 1) ia an identity element for +

2. The element o:C R containing (1, 1) ia an identity element for x

3. The element of R containing {-a, b) ia the .inverse of the element of R containing

j o} w5.'co respect "co

11*6

i|» i f a la n e t z e r o , then the e lement of R conta In i n? i"o s a) la the i n v a r s a of tbe element of H c o n t a i n i n g ( a , b) w i t h r e s p e c t to x

B. E x e r e i s e a

Lesson 18 A. S o l u t i o n of l i n e a r equa t i ons w i th e lements of R B. S u b t r a c t i o n w i th e lements of R C. D i v i s i o n wi th e lements of R D. 3 .xercisea

Laa a on 19 , . A. The e lement of R c o n t a i n i n g ( a , b) may be

considered, to be e q u i v a l e n t to the f r a c t i o n a / b

B . Opera t ions wi th f r a c t i o n s 1 . A d d i t i o n 2 . M u l t i p l i c a t i o n 3 . S u b t r a c t i o n !{.. D i v i s i o n 5 . E q u i v a l e n t f r a c t i o n s

C. E x e r c i s e s

Le3 2 on <c!G A. D e f i n i t i o n of a m a t h e m a t i c a l system B. The r i g i d t r a n s l a t i o n s of an e q u i l a t e r a l

t r i a n g l e G. Ilia b i n a r y o p e r a t i o n -"<• d e f i n e d on the a s t of

r i g i d t r a n a l a t i o n a of a t r i a n g l e D. Exere i s e3

Leas on 21 A. P r o p e r t i e s of the b i n a r y opera felon -> B. The i n v e r s e of the b i n a r y o p e r a t i o n •& C. Subse t s of the s a t of t r a n s l a t i o n s of a t r i a n g l e

f o r which ia a complete b i n a r y o p e r a t i o n D. E x e r c i s e s

Lesson 22 A . B ina ry o p e r a t i o n s d e f i n e d by Caley t a b l e s B. Procedures f o r i n v e a t i g s t i n g the p r o p e r t i e s

of b i n a r y o p e r a t i o n s d e f i n e d by Calay t a b l e s . C. E x e r c i s e s

Lesson 23 A. P r o p o s i t i o n s

1 . The n e g a t i o n of a p r o p o s i t i o n 2 , P r o p o s i t i o n s of the form p and q

B. E x e r c i s e s

4 ?

Leaaon 2l\. A-, More on propos i t i o n a

1 . P r o p o s i t i o n s of the form p or q 2 . The nega t ion of a p r o p o s i t i o n of the

form p and q 3 . Tlia nega t i o n of a p r o p o s i t i o n of the

form p or q B. E x e r c i s e s

Leaaon 25 A. P r o p o s i t i o n s of the form i f p , then q B. E x e r c i s e s

Lasson 26 A. Mora on p r o p o s i t i o n : ! of the form i f p , than q

1 . The converse of an i m p l i c a t i o n 2 . The c o n t r a p o s i t i o n of an i m p l i c a t i o n

B. E x e r c i s e s

Less on 27 A. Methods of d e d u c t i v e r e a s o n i n g

1 . The n a t u r e of axiom ays terns 2 . The law of de tachment

B. S.xerc i s es

Lesson 28 A. Typea of arguments t h a t l e a d to v a l i d

c o n c l u s i o n s B. Types of arguments t h a t l e a d to i n v a l i d

ooBclus ions C. E x e r c i s e s

Lesson 29 A. Drawing c o n c l u s i o n s u.iing Venn Diagrams B. E x e r c i s e s

Leaaon 30 A. An example of a s imple axiom sya tam 3 . Methods of p roof C. E x e r c i s e s

Lesson 31 A. Another a b s t r a c t axiom system B. E x e r c i s e s

Leaaon 32 A. The d e f i n i t i o n of a group B. Sorae s imple group theorems G. E x e r c i s e s

1)4.8

Lea a on 33 A. Example 3 of s j s te rss t h a t a r e groups B. Examples of systems t h a t a r e no t groups C. Exercises

APPENDIX 3

SAMPLE LESSONS FOR TBS GUIDED DISCOVERY

METHOD OF TEACHING

i 5o

Math 163

Col lege Mathemat ics

Throughout th i s course the term aat w i l l ba uaed to

r e f e r to any w a l l - d e f i n e d c o l l e c t i o n , g r o u p , or c l a s s of

o b j e c t s or i d e a s . The phrase "wal l -dof ined" meana t h a t

the s e t ia d e s c r i b e d p r e c i s e l y enough so t h a t one can

t e l l whether or no t any g iven o b j e c t b e l o n g s to the s e t .

The o b j e c t s c o n t a i n e d in a s e t a r e c a l l e d e1ereent a .

In a s e t of d i s h e s , esch d i s h ia an e lement of the s e t

of d i s h e s . In the case of a co in c o l l e c t i o n , each co in

ia an e lement of the s e t of c o i n s .

Ona way of d e n o t i n g a s e t i s by l i s t i n g a l l of i t a

e l e m e n t s . I f the elements of a s e t a r e l i s t e d , the

l i s t i3 placed, w i t h i n b r s c e a . Thus 1 , 2 , 3, i | , 5 , 6

d e n o t e s the set c o n s i s t i n g of a l l n a t u r a l numbers from

1 through 6 i n c l u s i v e . In soma case s n o t a l l the

e lements of a s e t can ba l i s t e d . One such s e t ia t he

s e t of tia t u r a i nunbera . This f is t may ba deno ted a3

f o l low3:

1, 2 , 3 , h> 5 , 6 , 7 , 8 , 9 , 10, 11 , 12, . . . .

Another concep t which w i l l be used e x t e n s i v e l y

obrpughouiu chis course i s t he concep t of b i n s r y oper01ion,

A b inary operat ion ia a way of a s s o c i a t i n g w i th two

e lements of a s e t a t h i r d e lement of t h a t s e t . A d d i t i o n

i s a b i n a r y o p e r a t i o n on t h e s e t of n a t u r a l number* .

151

For example, the o p e r a t i o n of a d d i t i o n a s s o c i a t e s w i th

the n a t u r a l numbers 3 snd 5 the n a t u r a l number 8 . I t

a s s o c i a t e s wi th the numbers 598 and 683 the number 1281.

M u l t i p l i c a t i o n ia a l s o a b i n a r y o p e r a t i o n on the s e t of

n a t u r a l numbers. This ope ra t i on a s s o c i a t e s wi th the

numbers 3 snd 5 the-number 15» I t a s s o c i a t e s wi th tha

number a 598 and 683 the number 4 08,1|3^ •

I t ia p o s s i b l e to d e f i n e many b i n a r y o p e r a t i o n s on

the a e t of n a t u r a l numbers . Suppose t h a t wi th two

n a t u r a l numbers we a s s o c i a t e the f i r s t of the two

numbers. Thia ia an example of a b i n a r y o p e r a t i o n on

the a e t of n a t u r a l numbers . Thia ope ra t i on a s s o c i a t e s

wi th 3 and 5 the number 3- I t a s s o c i a t e s wi th 1? and

9 the number 17«

Tha concept of b i n a r y ope ra t i on leoda ua to c o n s i d e r

the concept of o rdered £3 JLr. A b i n a r y o p e r a t i o n

a s s o c i a t e s wi th a p a i r of e lements of a a e t an element

of the a e t . The b i n a r y o p e r a t i o n d i s c u s s e d in the

p receed ing paragraph a s s o c i a t e s wi th the p a i r of numbers

3 and 5 the number 3* I t aaaoc ia tea wi th the p a i r of

numbers 5 and 3 the number 5• In each caae t he same

p a i r of numbers ia ua ed, b u t the r e s u l t s a r e d i f f e r e n t .

Thia la t r u e because the order in which the elements of

the p a i r a r e l i s t e d i s d i f f e r e n t . AM o rdered p a i r

c o n s i s t s of a p a i r of elements l i s t e d in a s p e c i f i c

i Co

o r d e r . An o r d e r e d p a i r w i l l be deno ted by H a t i n g tba

e lements of tbe p a i r w i t h a comma between thera and

p l a c i n g tba l i s t in p a r e n t h e s e s . Thus (3> 5) d e n o t e s

tbe o r d e r e d p a i r c o n s i s t i n g of t he numbers 3 and 5 w i t h

3 coming b e f o r e 5* (5> 3) d e n o t e s tbe same p a i r , b u t

with 5 corning b e f o r e 3• This shows t b a t (3> 5) ia not

the same o r d e r e d p a i r as ( 5 , 3 ) • In t be o r d e r e d p a i r

( 5 , 3), 5 i s c a l l e d tbe f i r a t component of tba o r d e r e d

p a i r and 3 i s c a l l e d tba second conpoaen t of tba

o r d e r e d p a i r » Two o r d e r e d p a i r s a r e s a i d t o be equa l

i f t hey have the same f i r s t component and the same

second component .

L e t us d e n o t e t b e s e t of a l l o r d e r e d p a i r s o f

n a t u r a l numbers by tbe symbol ? . Some of tbe e lements

of ? are ( 3 , 9 ) , ( 5 , D , ( 5 , 5) , ( 2 8 7 1 , 1 ) , and

( 2 , 3800972} ' Many i n t e r e s t i n g and unusua l b inary

o p e r a t i o n a can be d e f i n e d on t b e s a t P . Remember

tbat a b i n a r y o p e r a t i o n i s a way of a s s o c i a t i n g w i th

two e l e m e n t s of a a e t a n o t h e r e l e m e n t o f t b e s e t . Thus

a b i n a r y o p e r a t i o n on P a s s o c i a t e s w i t h a p a i r of

o r d e r e d p a i r s an o r d e r e d pa i r . An e x a m p l e of a b i n a r y

o p e r a t i o n on P i s tbe o p e r a t i o n wbicb a s s o c i a t e s w i t h

a p a i r of o r d e r e d p a i r s the second o r d e r e d p a i r . This

b i n a r y o p e r a t i o n a s s o c i a t e s with (k» 1) and ( 9 , $)

the o r d e r e d p a i r ( 9 , 5 ) . I t a s s o c i a t e s w i th ( 1 , 17)

lfJ3

and ( 3 9 , 111) t h e o r d e r e d p a i r ( 3 9 , l ! | ) .

I f we wish to c o n s i d e r t h e number which the b i n a r y

o p e r a t i o n of a d d i t i o n on the s e t of n a t u r a l numbers

a s s o c i a t e s w i t h 7 and Lj. we v r i t s 7 + S i n c e t h e

o p e r a t i o n of a d d i t i o n s s s o e i a ta.i w i t h 7 a nd t h e number

1 1 , we w r i t s 7 + i| ™ 11 • Th i s i s a v e r y compac t and

c o n v e n i e n t way of s a y i n g t h a t t he b i n a r y o p e r a t i o n

a d d i t i o n a s s o c i a t e s w i t h 7 snd i|. t he n a t u r a l r>ur.;ber

1 1 . When c o n s i d e r i n g b i n a r y o p e r a t i o n s on P wa s h a l l

u s e t h e n o t a t i o n ( a , b ) B ( c , d) = ( e , f ) t o d e n o t e

t h a t t he b i n a r y o p e r a t i o n B a s s o c i a t e s w i t h t h e p a i r

of o r d s r e d p a i r s ( a , b ) and ( c , d) the o r d e r e d p a i r

( e , f ) . Thud i f 3 d s a o t a a t h e b i n a r y o p e r a t i o n on ?

d i s c u s s e d i n t he preCOGding p a r a g r a p h wa hava t h a t

d j , 1) B ( 9 , 5) ( 9 , 5 ) • A l s o

( 1 , 17) 3 ( 3 9 , l h ) =* ( 3 9 , 110 .

A b i n a r y o p e r a t i o n on ? can o f t e n b y d e f i n e d u s i n g

a g e n e r a l f o r m u l a , ( a , b ) B ( c , d) - ( c , d) i a a f o r m u l a

which can be ua gd t o d e f i n e t h e b i n a r y o p e r a t i o n on P

d i s c u s s e d in tha p r o c e e d i n g p a r a g r a p h . I f a i a r e p l a c e d

b y a n a t u r a l number and i f b ia r e p l a c e d b y a n a t u r a l

numbor , t h e n ( a , b) becomes an e l e m e n t of p . S i m i l a r l y

i f c i s r e p l a c e d b y e n a t u r a l number and i f d i a r e p l a c e d

b y a n a t u r a l number , t h e n ( c , d) becomes an e l e s ion t of

P . Then ( a , b) 3 ( c , d) •- ( c , d) i a a way of s a y i n g

1514

t h a t the b inary o p e r a t i o n B a s s o c i a t e s w i t h ( a , b) and

( c , d) the ordered pa ir £c» d ) .

Cons ide r the b i n a r y o p e r a t i o n on P d e f i n e d by tha

formula

( a , b) B ( c , d) - (a + o , b x d) .

Thia formula a t a t e a t h a t t o f i n d the f i r s t component of

the ordered p a l r a s s o c i a t e d wi th ( a , b) and { c , d)

add the f i r s t components of ( a , b) and ( c , d ) . To

f i n d the second component m u l t i p l y the second components

of ( a , b ) and ( c , d ) . According t o the formula the

ordered p a i r a s s o c i a t e d wi th ( , 8) and ( 9 , 7) i s

(If + 9 , 8 x 7) or (13 , 5 6 ) . S i m i l a r l y

( 1 2 , 3) B (U, 17) « (12 + he, 3 x 17) or

(12 , 3) B U , 17) ™ (16 , 51) .

E x e r c i s e 1 .

In each of the f o l l o w i n g e x e r c i s e s a b i n a r y o p e r a t i o n

i s d e f i n e d . TJae the d e f i n i t i o n to f i n d the o rde red

p a i r a s s o c i a t e d w i t h the g iven p a i r of o rde red p a i r a ,

3.x ample: {a , b) B ( c , d) •- (a x b , c ) .

Complete the f o l l o w i n g .

( 3 , 2) B {!+, 7) =

( 5 , 1) B ( 9 , 18) =

Solu t ion:

( 3 , 2) B ( l | , 7) ~ (3 x 2 , k) - ( 6 , k ) .

( 5 , 1) 3 ( 9 , 13) = (5 X 1 , 9) - ( 5 , 9 ) .

i55

1. (a j b) B (c, d) = (a x d, b).

Example: {1, 9) B (7, 1|) = (1 * k, 9) -= (!|» 9)

C oarp lata tbo following .

(1|, 3) B (18, 5) :

(15, 2 ) B (1, 1) =

(1, 1} B (15, 2 ) *

(5, 7) B (9, 3) =

(9, 3) B (5, 7) =

2. (a, b) B (c, d) = (a x c, b + d).

Example: ( 3 , 2) B ( 9 , 1) = (3 * 9 , 2 + 1) = ( 2 7 , 3)

Coiaplate the following.

(7, l) 3 (9, 5) =

(9, 5) B (7, l) =

(8, 2) B (13, 11) =

(13, I D B (8j 2) =

(1, 1) B (8, 3) s

(8, 3) B (1, 1) =

(6, 9) B (11|, 7)

(111, 7) B (6, 9)

3. Make up (invent) a formula of your own which definea

a binary operation on P. Us a your formula to complete

the following,

156

(3, 1) B (2, 2) = _

(8, 2) B (7, 5) 83 _

(3, 11) B (8, 6) «

(8, 6) B (3, 11) «

(6, 8) B (11, 3) =

(6, 8) B (3, 11) =

(ll, 7) B (1, 15) «

(15, 1) B (]4, 7) •-•=

(7, h) 8 (1, 15) =

13'7

L-3 3son 2

Consider tbe binary ops ra tion on P defined by the

following formula:

(a, b) B ( c , d) - (a x c, b + d).

U3ing this formula, complete the following aa quickly aa

poaslble. If you discover a abort-cut, do not hesitate

to use it. Be sure tbe short-cut works.

(1, 3) B (8, J|) =

(8, h) B (1, 3) =

(2, 55 B (3, 7) -

(3, 7) B (2, $) =

(8, 2) B (1, 6) =

(1, 6) B (8, 2) *

(12, 1) B O4, 9) =

{1|, 9) B (12, 1) ~

(5, 11) B (2, 2) «

(2, 2) B (5, ID «

Note: If wa write 2a we shall me on 2 x a. If we write

ab, wg shall me a a a x b.

Consider tbe binary operation of P defined by the

following:

(a, b) B (c, d) — (2a + 2c, bd)

Examples:

(3, k) B (1, 2) ™ (2 x 3 + 2 x 1, J4 x 2) « (8, 8)

(2, 5) B (1, 9) « £2 x 2 + 2 x 1, $ x 9) - (6, kS)

158

TJalng the above formula complete the f u l l owing a a q u i c k l y

a a p o 3 3 i b i s .

( 1 , 3) B ( 2 , 8) = *

( 2 , 8) B ( 1 , 3) =

(ll, 6) B (1 , 1} -

(1 , 1) B (U* 6) =

(5» 10 B ( 2 , 11)

( 2 , 11) B ( 5 , H)

(12, 15) B (21, lj.2)

(21 , 1-2) B (12 , 15) -

Waa i t neces sa ry to c a r r y out a l l tha impl ied c a l c u l a t i o n s

or could, you f i n d the answers to some of tbe e x e r c i s e s

and from these conclude what the enaver3 to tha remaining

e x e r c i s e s whould be?

A b i n a r y ope ra t ion on a s e t i s a way of a s s o c i a t i n g

wi th two elements of a s e t an element of t h a t s e t ,

Addi t ion ia a b i n a r y ope ra t ion on the s e t of n a t u r a l

numbers. This b i n a r y ope ra t ion a s s o c i a t e s wi th 9 and 12

the n a t u r a l number 21 . 21 ia c a l l e d the sum of V and 12.

Suppose t h a t I t la n e c e s s a r y to f i n d the sura of a l i n t

of n a t u r a l numbers. Consider the l i s t 7? 3 * 9 , k, and 6 .

The sum can be found by f i n d i n g the aura of 7 and 3 . This

sum. ia 10. Then f i n d the sum of 10 and 9 . This sum ia

19. Then f i n d 19 + k . 19 + ^ ~ 23. Then f i n d 23 + 6 .

2-3 + 6 = 29 . This ia tha sum of 7 , 3, 9 , 1}, and 6 .

159

U s i n g t h i s p r o c e d u r e two numbers were added i n eacb

s t e p of t h e p r o c e d u r e . Th is t y p e of p r o c e d u r e "wsa

n e c e s s a r y b e c a u s e a d d i t i o n i a a bir i&ry o p e r a t i o n .

Only two numbers can be added a t a t i m e . C o n s i d e r t h e

t a s k of f i n d i n g t h e aura of t he l i s t k , 15» "and 2 .

[j. + 15 = 19 and "19 + 2 -- 2 1 . Th i s p r o c e d u r e can be

r e p r e s e n t e d a a f o i l o*/ a :

(J| + 1$) + 2 - 21.

The p s r on t h e s e s a r o u n d 1|. -!• 1$ i n d i c a t e s t h a t t h e aura

s h o u l d be f o u n d f i r a t and t h a t t l i i a r e s u l t s h o u l d b e '

added to 2 . P a r e n t h e s e s and b r a c k e t s , £ ^ , a r e

o f t e n u a e d t o i n d i c a t e which operafc iona s h o u l d be

p e r f o r m e d f i r s t . . .

E x m c p l e s :

1 . [1+ .-5 - 2 ^ + 6 =

The b r a c k e t s i n d i c a t e t h a t l | x 2 s h o u l d be f o u n d

f i r s t . The r e s u l t s h o u l d then be added t o 6 .

[ l | x 2 ] + 6 ~ 8 + 6 » 11|.

2 . 3 + (2 + 7) 58

The p a r a n t h e a e s i n d i c a t e t h a t 2 + 7 s h o u l d be

f o u n d f i r s t . Then 3 s h o u l d be added t o t h e r e s u l t .

3 + (2 + 7) « 3 + 9 - 6.

3• I f the b i n a r y o p e r a t i o n B i s d e f i n e d b y

( a , b ) B ( c , d) « ( a d , c ) , t hen f i n d

1(1 , 3) 3 ( 9 , 2 ) 3 3 ( 8 , 6) .

160

Solution: Tha brackets indicate that (1, 3) B (1, 2)

should be found first.

(1, 3) B (9, 2) « {1 x 2, 9) « (2, 9).

Than (2, 9) B {8, 6) nbould be found.

(2, 9) B (8, 6) = (2 x 6, 8) = (12, 8).

Therefore:

i d . 3) B {9, 2)] B (8, 6) = (12, 8)

Exercise 2.

Complete the following:

[k + 3 J + 7 = 18 + (3 9) -

{18 + 3) + 9 :

(1|7 + 3) + 18

l|7 + (3 .+ 18)

[lit + 6 ] + 27

11+ -4- £6 ... 27J

When finding the sum of a list of natural numbers,

doe3 the way in which the numbor3 are grouped affect

the final 3urn?

Using the binary operation on P defined by

( a > ft) B (e, d) = (a + c, bd),

complete the following aa quickly aa possible. If you

discover a short-cut, uae it.

[(1, 2) B (3, l4}] B (9, 1)

(1> 2) B [(3, k) B (9, 1)]

161

\ ( 5 , 2 ) B ( 9 , 9) } B ( 6 , 5 )

( 5 , 2 ) 6 £ ( 9 , 9} B ( 6 , S)] =

( U , 7 ) B [ ( 3 , 8 ) B ( 1 , 1 ) ] «

^ ( l t , 7 ) B ( 3 , 8 ) 3 B ( 1 , 1) a

( 1 , 6 ) B [ ( 2 , 1 1 ) B ( 1 1 , 2 ) ] =

[ ( 1 , 6 ) B ( 2 , 11)" ] B C H , 2 ) «

- - Cooa the p o s i t i o n of the b r a c k e t s a f f e c t the f i n a l

r e s u l t ?

Using the same b inary operat ion aa used above,

comple te the f o l l o w i n g :

( 2 , 1) B ( 3 , i | ) =

( 8 , i+) B ( 2 , 1 ) =

( 1 , 6 ) B ( 2 , 11 )

( 2 , 11 ) B ( 1 , 6 )

(h> 7) 3 ( 3 , 8 ) =

( 3 , 8 ) B {J | , 7 ) = __

In c o n s i d e r i n g the f i n a l r e s u l t does i t m a t t e r

which o r d e r e d p a i r cornea f i r s t ?

Can you w r i t e ( d i s c o v e r , i n v e n t ) a formula f o r a

b i n a r y o p e r a t i o n on P so t h a t fcba r e s u l t w i l l not

depend on which ordered pair cornea f i r s t ? Try to f i n d

such a f o r m u l a .

Try to w r i t e a formula f o r a b i n a r y operat ion on P

such t h a t when the b i n a r y o p e r a t i o n 5.1 used w i t h t h r e e

o r d e r e d p a i r s , -the r e s u l t s w i l l n o t depend on the

162

placement of th». b r a c k e t s . That l a , «r I fca a formula so

t h a t [ ( a , b) B ( c , d ) ] 3 ( e , f ) - ( a , b) 3 [ ( c , d) B ( e , f ) J

16'

L63 3OR 1

Complete tha following.

2 + 3

3 + 2

19 + 2?

27 + 19

3 x k

8 * 7 =

7 x 8 ™

9 x 6 =

6 x 9 -

12 x 28

28 x 12

When finding the sum of two numbers, does the

order in which tha nuiv-bers occur affect the sum?

When finding the product of two numbers doea the' order

in which tha. number3 occur affect the product?

Using the binary operation defined by

(a, b) B (c, d) - {ab, c)

complete tha following.

(1, 3) B (2, bt) :=

(2, 1+) B (1, 3) =

U , 3) B (9, 2) *

(9, 2) B (l4, 3) =

Does tha order Id which the ordered pairs occur affect

tha final result?

16!+

Using the b i n a r y opera l i o n d e f i n e d 'by

( a , b) B ( c , d) = (fi + c , bd)

comple te the f o l l o w i n g .

(1, 3) B ( 2 , k) =

<2, k) 3 ( 1 , 3) =

( 8 , ?) B ( 0 , 1) =

( 0 , 1} B ( 8 , ?) -

Does tha o rder in which the o rde red p a i r a occur

a f f e c t the f i n a l r e s u l t ' ?

A b i n a r y o p e r a t i o n a s s o c i a t e s wi th two elements of

a s a t an e lement of tba s e t . I f the o rde r in which tha

f i r s t two elements appear doas n o t a f f e c t tha r o s u l t ,

tha b i n a r y o p e r a t i o n Is s a i d to be comraufcativa. la

a d d i t i o n on the s e t of n a t u r a l numbers a eoiimiuta t i v a

b i n a r y o p e r a t i o n ? Is m u l t i p l i e s t i o n on the aat of

n a t u r a l numbers a coiamutatlva b i n a r y operat ion?

I s the b i n a r y o p e r a t i o n on P d e f i n e d by

( a , b ) 3 ( c , d) -- ( a b , c)

a coiiimutative b i n a r y o p e r a t i o n ? l a tha b i n a r y o p e r a t i o n

on P d e f i n e d by

( a , b) B ( c , d) = (a + c , bd)

a commutative b i n a r y o p e r a t i o n ?

Cons ider the e x e r c i s e s on the use of g roup ing

symbols ( s ee page IfeO ) • When f i n d i n g tha si-m of t h r e e

n a t u r a l numbers, doea tha way the numbers a r e g rouped

a f f e e t tha aur.i?

16£

Complete the fo l lowing .

(2 X l+) x 7 =

2 x U x 7) =

(3 x 9) x I4 =

3 x (9 x if) =

(i | x 1) x 18

l | X ( 1 X 1 8 )

Does the way the numbers are grouped affect t h e

product of t h r ee na t u r o l numbers?

Using the b i n a r y ope ra t ion on ? d e f i n e d by

( a , b) B ( c , d) ~ ( ab , c)

complete the f o l l o w i n g .

[ (1 , 2) B (k> 7)] B (9 , 2) =

(1 , 2) B [(! ; , 7) B (9 , 2)}

( 3 , i|) B 1 (8 , 1) 3 (2 , 7)] =

t(3, U) D (8, 1)] 3 (2, 7) =

Does the way the ordered p a i r a a re grouped a f f e c t the

f i n s l r e s u l t ?

I f 3 ia d e f i n e d by

( a , b) B ( c , d) = (a + c , bd)

complete tbe f o l l o w i n g .

[ ( 1 , 2) B (!+» 7)] B (9 , 2)

( 1 , 2} B [(k, 7) B (9 , 2)J

(0 , 1) B [ ( 8 , 1) B ( 2 , 7)3

[ ( 0 , 1) B (8 , 1)J B (2 , 7)

166

Do a a the way the ordered pairs ore grouped affect the

final result?

Suppose that a-binary operation ia to be used on a

list of elements of a set. If the way the elements of

the list are grouped does not affect the final result,

the binary operation ia aaid to be aaaocia tlve. Which

of the following are asaocia tive binary operations?

1. Addition on the set of natural numbers .

2. Multiplication on the set of natural numbers.

3. The binary operation on P defined by

(a, b) B (c, d) — (ab, c).

!|. The binary operation on P defined by

(a, b) B (c, d) - (a + c, bd) .

Ex 6 r c i 3 •

Using the binary operation defined by

(a, b) B (c, d) - (a + c, bd)

complete tha following.

(8, 3) B (0, 1) « _ _ _ _

(l4, ?) B \{Z, 9) B (0, 1)] «

D s . 9) B (0, B (2, 6) =

(5, 9) B 1(0, 1) 3 {?., 6)J =

{0, 1) B (8, k) «

Pill in the blanks,

U 5 , 3) B - (15, 3)

(9, 5) B _ (9, 5)

167

( 7 , U) B = ( 7 , 1-0

B ( 8 , 111) - ( 8 , II.)

B ( 9 , 17) = ( 9 , 17)

Using the b i n a r y o p e r a t i o n d e f i n e d by

(a* b) B ( c , d) -» ( a c , bd)

comple te tho f o l l o w i n g .

( 1 5 , 3) B ( 1 , 1) :

( 1 , 1) B ( 9 , 8)

( 2 7 , i |2) B ( 1 , 1)

[ ( 2 9 3 , 3k) B ( 1 , 1 ) ] 3 ( 0 , 1)

( 2 9 3 , 3h) B £ ( 1 , 1) B ( 0 , 1 ) ]

F i l l in t he b l a n k s .

( 2 , h) B = ( 2 , 1+)

( 3 , 7) B =* ( 3 , 7)

B ( 5 , 1) - ( 5 , 1)

B ( 8 , 16) = ( 8 , 16)

Con-aider the b i n a r y opera t5on d e f i n e d by

(a y b) B ( c , d) - (a + c , b + d ) .

Can you f i n d an o rde red p a i r ( x , y) auch t b a t

( a , b ) B ( x , y) « ( a , b)

no m a t t e r which n a t u r a l number ia used in p l a c e of a and

no 10a t t e r which n a t u r a l nutrber ia used in p l a c e of b?

I f s o , "which number shou ld be uaed f o r x? Which number

shou ld be used f o r y?

Lesson ?

U s i n g the b i n a r y o p s r a t i o n § d e f i n e d b y

( a , b ) § ( c , d ) » (a + c , b + d)

and t h e b i n a r y o p e r a t i o n d e f i n e d b y

( a , b ) * ( c , d ) = ( s o ,

f i l l in the blanks

(1), -1) * t i l , 6) f ( - 8 , 2)]

\ll-s, -1) » (1. 6)3 i { l l j , -1) «• ( - 8 , 2 ) ] »

(3, 9) « ft-2, 7) * ( - 3 , - D ] «

ft 3, 9) * ( - 2 , 7)^ # r i 3 , 9) 6 ( -3 , - I f ) =

( -6 , 3) » 1.(14, 9) # ( - 5 , 8) ] =

t ( - 6 , 3) * (U, 9)1 i"r H-b, 3) * ( -S, S)] =

£ ( - i , -3) * d(, 7)3 t I ( - l . -3) * (8, 9 ) ] -

( - 1 , -3) » I d s , 7) # (8, 9)] =

( 8 , - 7 ) » £ ( 3 , - 5 ) # ( f t , 9)" ] -

[ ( 8 , -7) * (3, - 5 j ] # [ (8 , -7) # ]

(1, 1) * £(3, -IS) t (2, l i t ) ] »

£ * (3, - i5:Q a I d , i ) » (2, l i t

[(-»(, 3) * (8, 2)"j # 0 - ! t , 3) « ( -3 , 7 ) ] -

^ £<6, ?.) # ( - 3 , 7) 3

Using the binary operation 0 defined

(•9) b) 0 ( c , d) ~ {a + a, bd)

and tha binary operation <k defined by

( s f b ) 6c ( c , d) (s •!* d , b e ) ,

f i l l in tha b1a nks.

169

(3, 5) 0 \ s ^ 1) & (8, 2}]

\j 3> 5) 0 (9, 1)'] & £(3, 5) o (8, 2)3

(8, -1) 0 L«3. 2) & (2, -3)"]

[(8, -1) 0 (3, 2)"] & \{8, -1) 0 (2, -3)']

(-1+, 7) 0 \j-l, 1) & (8, -6)3 =

Jj-.'l, 7) 0 (-1, I)} & fc-li, 7) 0 (8, -6)3

(-2, -5) 0^(3, 6) & (-J+, 7)] »

-2, -5) 0 (3, 6)*3&£(~2, »5) 0 (-!{., -7)3

Fill In the blanks.

7 x (8 + 9) =

(7 x 8) * (7 x 9)

li4 x[k+ (-9)] =

fru * 43 + 1 1 ! + x (-9).]

37 x £(-12) + 93 -

[37 /(-12)] 4-^37 x 9 ] ~

t ( " l 8 ) * 7 J + L(~l8> x 1°3 a

(-18) x ^7 + 10] = _ _ _ _ _ ___

^( -12) x l|l] + ^(-12) x (-18)3 " x 1^1 + (-18) J

£16 3 533 + \l6 x 92]= 16 x £ + 92

^-k-7 x 983 -5- £(-i|) x 106] = -l. x\jQ + ]

A binary operation associates with two elements

of a 3qt an element of that safe. Suppose the elements

of a set are themselves seta . Gouaider the- sat 8

'which has as eleraents the seta

1 7 0

A = ^ 1 ) Ij.$ i f 1 0 $ 1 3 » » • • • 3 •>

3 = \ 2 , 5 » 8 , 1 1 , 111i 1 ? , . . 3 , e n d

C = %3, 6 , 9 , 1 2 , 1 5 , 1 8 , . . . 3 .

We s l i f l l l d e f i n e a b i n a r y o p e r a t i o a 0 o d t h e s a t S .

L e t X and . Y d e n o t e e l e m e n t s o f 5 . T h e n X (+) Y d e n o t e s

t o e e l e m e n t o f S . w h i c b © 3 S S o c 1 g t e a w i t h X a n d Y .

"I'D f i n d X 0 Y , p i c k a n y e l e m e n t o f X , p i c k a n y e l e m e n t

o f Y a n d f i n d t h a i r s u m . T h e n X 0 Y l a t h e s a t

c o n t a i n i n g t h i a s u r a .

E z s m p l s s :

F i n d A 0 B . 1 0 i a e n e l e m e n t o f A . 8 i a a n

e l e m e n t o f B . 1 0 + 8 ™ 1 8 . 1 8 i a a a e l e m e n t o f 0 .

T h e t s Co v a A (? ) 3 » 0 .

F i n d G © A . 6 i a a n e l e m e n t o f 0 a n d l o i a a n

o l e i f j a n f e o f A . 6 4- 1 6 — 2 2 a r .d 22 i a a a e l e m e n t o f A .

T h e r e f o r e G 0 A ~ A .

E ^ a g o i a a 7 »

P i l l i a fcbe b l a n k s ,

- 1 2 x f j | l 4- ( - 1 8 ) ] = \ . ( - i 2 ) x ] t- ft-12) X ( - 1 8 ) ]

1 6 X ^ 5 8 4- 9 2 J - X 5 8 1 + t . 1 6 x 9 2 ]

- > i X ^ 9 8 + 1 0 6 " ] « ^ X 9 8 ] + £ _ x 1 0 6 ]

3 7 x ft-12) + 9 * ] - [ 3 7 x 1 3 7 x _ _ ]

L 6 x 1 2 ] + £ 6 x 8 2 | ] = _ _ _ _ _ x £ 1 2 + 81j. ]

£ 7 * u V £ 7 * 6 > 7 * £ + 1

171

i ® x ^ " 1 + t 1 9 x = \ *s* _ j| ^ 6

Tlh X 9 3 + t . 2 1 X 9"]™ L X H + 3 *

2 + T_a x 7 ] ~

^ 2 + 8 l x £ 2 + 7 ] -

~8 * \.k * 9 ] - _ _

L~ 8 + -*A X \1-Q + 9 ] ~

7 + ^ - 6 x 1 2 ] =

\> + (-6)1 X £ 7 + 12] - _ _

R s f e r i n g to the b i n a r y o p e r a t i o n (^ d e r i n e d j o o t

before exorcise 7> fill in tbe blanks.

3 © 3 =

Q A »

c © o -

A S O =

x>

3 O .... - A

la + a cornrflutotlve b i n a r y o p e r a t i o n ?

A CD (c © B ) « _ _ _ _ _

(A 0 0 O B =

B © ( B © A ) ®

( B 0 B ) 0 A -

Is 0 a n a s s o c i a t i v a b i n a r y o p e r a t i o n ?

Is there >in i d e n t i t y e lement for \£) ?

172

Lea a on 3

Le t S be a s e t and l e t 0 .-and. 0 be b i n a r y o p e r a t i o n s

d e f i n e d on 8. I f f o r any e lements A, B, and C of S

A 0 (B 9 C) = (AOS) 0 (AOG),

than 0 i s s a i d to be d i a t r i b u t i v a over 0 .

1 . I s m u l t i p l i c a t i o n d i s t r i b u t i v e over a d d i t i o n ? (Sea

Loa3on ? • )

2 . I s a d d i t i o n d i s t r i b u t i v e over m u l t i p l i e s t i o n ?

(3as Leason ?• ) _

3 . 13 d i a t r i b u t i v e over #? (Sea Leason 7 . )

1|. I s 0 d i s t r i b u t i v e over &? (See Lea a on 7 . )

In Lea a on .1 i t i s afcated t h a t two o r d e r e d p a i r s a r e

equa l i f t h e y have the same f i r s t c Disponent end tha asm a

second component . Me i n d i c a t e the f a c t t h a t two o r d e r e d

p a i r a a r e equa l by p l a c i n g the symbol between tbsm.

I f two o r d e r e d p a i r s a r e n o t e q u a l , t b i a can be i n d i c a t e d

by p l a c i n g the symbol V * between there. P l a c e the a p p r o -

p r i a t e .symbol m or /• between tha fo l lovxiag p a i r s of o r d e r e d

p a i r s .

(3, 7) _ (8, 7)

(~h , 2) (->4, 2)

(7 - 6 , 9) ( 1 , 9)

(-8 + 3", 6 - (-2) ) ( 5 / 8 )

(6 - 9 , )+ + ( - 6 ) ) _ ( - 3 , -2)

173

I f a + d » b - c , tb«3B wq t ' h s l l p l a c e t h e symbol

T'R" between tba p a i r s ( s , b) end ( c , d) and wa s h a l l s a y

t b a t ( a , b ) 13 r o l a t a d to ( c , d ) . Cons ider t he p a i r s

(3s I'-l) a n d (7f k) • 3 + k ~ 7 snd lij. - 7 — 7 • Tiier of or a

( 3 i 11|) R ( 7 , U) and we say t h a t (3# l i t) ia r e l a t e d t o

(7> l|) • I s ( - 6 , 3) r e l a t e d to ( 8 , 1 )?

Cons ide r tba p a i r s ( 6 , 4) and ( 9 , 3 ) . 6 + 3 = 9 and

| | - 9 = . 5 . S ince 6 + 3 snd I4. - 9 a r e n o t e q u a l , wa gay

t h a t ( 6 , I4.) and (9., 3) a r e n o t r e l a t e d . We i n d i c a t e t h i s

by p l a c i n g tba symbol " jri'5' b 3 tv; 3 e n t b e o r d e r e d p a i r s .

Thus ( 6 , I;) ^ {9, 3 ) - P l a c e tba a p p r o p r i a t e symbol "R"

or between tba f o l l o w i n g p a i r a of o r d e r e d p a i r s .

( 6 , li+) (3» 5)

( 3 , 5) ( 6 , Ik )

(7* 2) ( 9 , 3)

( 4 , -2 ) ( k , 6)

U4, 6) (U, - 2 )

( 3 , 7) (2, 2)

( 2 , 2) ( - 5 , 5)

(-Pt ( 3 , 7)

Many d i f f e r e n t r e l a t i o n s can be d e f i n e d on s e t s of

o r d a r a d p a i r a . Cons ide r tba r e l a t i o n , R 1 ? d e f i n e d aa

f o l l o w s : ( a , b) ia r e l a t e d to ( e , d) by R^ i f

a + d - b - f c . I f a + d « b + c than we w r i t e

( a , b) R- (Qs <j) a I f a + d p b •!* c , tbe?) we H„?tta

17U

(a , b) ^ ( c , d) and kg say t h a t ( a , b) is not r e l a t e d to

{c, d) by the r e l a t i o n . Consider the pa i r s (1|, 8) and

(2 , 6 ) . + 6 - 10 and 8 + 2 = 10. Therefore (l»., 8) ia

r e l a t e d to {2, 6} by the re la cion and ve i nd i ca t e th ia

by w r i t i n g (I4., 8) (2 , 6) *

Consider the pa i r s (9 , 2) and (8 , 3) • 9 + 3 - 12 and

2 + 8 ™ 10, Sicca 9 + 3 i f l not equal to 2 + 8 we say t b a t

(9# 2) ia not r e l a t e d to (8 , 3) by the r e l a t i o n R and we

ind ica t e thia by wr i t i ng (9 , 2) (8 , 3) .

Ex egolag 8 .

Datannine which p a i r s of ordered p a i r s are r e l a t e d by

R]_ and. which p a i r s of ordered p a i r s are not r e l a t e d by R^.

Tbat i a , f i l l in the blanka using the appropr ia t e symbol,

Rx , or jf1#

(3, 7) (3, 7)

U , 8) (1 , 5)

( l , 5) U , o)

(9 , 6) (5s l )

(5 , l ) (9 , 6)

(8, 16) (-3> •$)

( - 3 , 5) (8 , 16}

( 7 , 2) (8 , 3)

(8, 3) (7 , 2)

(li, 11) (7, 1U)

175'

( - 6 , 9) ... ( - 6 , 9)

( 1 5 , 3) (15 , 3)

(18, 21) (18, 21)

(7, 5) __ (10, 8)

( 1 0» 8) (5, 3)

<7. 5) (5,3)

U , 3) _ _ _ ( 8 , 7)

( 8 , 7) ( 9 , 8)

(!+» 3) ( 9 , 8)

Pill In the blanks. (First see question 3 a t the

baginning of Lssaon 8.)

(8 , ..7) » 0 3 , -S) jt (It, 9 ) ] =

t • ( 3 . -5)3 # ! _ _ * (4, 9)3

(-1* 1) ic £(3^ 12) $ (12, *"3]T) ~

I * \ * i * 3

(5, -2) » 1.(6. It) # (6, 3)3 ~

£ — - • » •}

[ ( f ) , 6) # ( - 7 , 2)1 * ( - 1 2 , 3) »

* 3 # E * _ 1

l l - k , 3) * (8 , 2 ) 3 # 3) * ( -3 , 7)-] =

* t(8. 2) # (-3, 7)3

£(3, 9) » (-2, 7 ) 3 # 1(3, 9) * (-3, -1)3 =

» £ _ # (-3, -1)3

[ ( - 6 , 3) * <!j, 9)3 # D - 6 , 3) * ( - S , 6)3 =

* 1 ' 3

176

[ ( - 1 , -3 ) * 04, 7)1 # [ { - 1 , - 3 ) *• ( e , 9 )3

» £ _ . * 1

O i l , 9) -* (-6, 1)1 -tr 1 ( 9 , 3) * (-6, 1 ) } =

[ ( H , 9) # ( 9 , 3 ) ^ *

( 8 , 7) * ( 9 , 9)1 § t ( 2 , - I ) * ( 9 , 9 ) 3 =

i — * — > —

Le t S ba the s e t con?;aining the e lements

A = 5 , 9 , 13 , 17 , 2 1 , 25 , 2 9 , . . . } ,

B ~ ^ 2 , 6 , 10, 1!|, 18 , 2 2 , 26 , 30, . . . J ,

c = £ 3 , 7 , 11, 15, 19 , 23, 27 , 3 1 , . . . 3 , and

D « 8 , 12, 16, 2 0 , 21., 28 , 32 , . . . j .

x ) i a the b i n a r y o p e r a t i o n of 8 d e f i n e d a a follows?: I f X

and Y a r e e lements of 3 , a ia an e l emen t of X ami b i s an

e l emen t of Y, thon x Q y ia the e leraant ( s e t ) of 8 con-

t a i n i n g a x b . For exes .p ie , f i a d A 0 C. 9 i s on e l e m e n t

of A and -3 i s an ql6i'nois t of G. 9 x 15 ™ 135• 135 i s sn

e l emen t -of C. T h e r e f o r e A (£) C « C.

F i l l i n the b l a n k s .

A

B ©

G 0 G

A ( k ) D « BQD

D 0 A « " D(BB

177

Los. a on 9

Let S be a s e t . A r e l a t i o n on 8 a s s o c i a t e s witta two

elements of S e i t h e r the v o r d , ! ! yes ," or tho word, "DO."

I f R ia a r e l a t i o n on S, A and B a re elements of 3 , and R

a s s o c i a t e s wi th A and 3 the word " y e a t h e n we say t h a t

A ia r e l a t ed to B and we i n d i c a t e t h i s by w r i t i n g ARB.

I f R a s s o c i a t e s wi th A -?nd B tbe word " n o , " than we aay

t h a t A ia no t r e l a t e d to 3 and we i n d i c a t e t h i s by w r i t i n g

AJt3. I f R ia a r e l a t i o n on S, then e x a c t l y one of the

fo l l owing ia a t r ue s ta tements

1. A ia r e l a t e d to B.

2 . A ia nob r e l a t e d to B.

A 1" r e l a t e d to B i f and only i f R a s s o c i a t e s wi th A and

B the word ' 'yea ." A ia not r e l a t e d to 3 i f and only i f

R a s s o c i a t e s wi th A. and 3 the word "no . "

1 . I f o and b a re i n t e g e r s , ia e x a c t l y one of the

fo l lowing t r u e ?

a . a -- b .

b . a •/- b .

I f the answer i s yea , then " —' i s a r e l a t i o n on the

s e t of i n t e g e r a .

2 . . I f ( a , b) and ( c , d) a re o rdered p a i r a of i n t e g e r s .

ia e x a c t l y one of the - fo l lowing t r u e ?

a . {a, b) = ( c , d ) .

b , ( a , b) •£ ( c , d) . .

1?8

13 »=« a r e 1 s t I on oa t b s s a t of o r d e r ad -pa i r a of

i n t e g e r s ? _

3 . I s R a r e l a t i o n on t h e s e t of o r d e r e d p a i r s of

i n t e g e r s ? (See Lesson 8 . )

If.. I s R-j_ a r e l a t i o n on t h e s e t of o r d e r e d p a i r s of

i n t e g e r s ? (See Lesson 8 . )

D e f i n i t i o n : Lo t S be a a e t and l a b R be a r e l a t i o n on S .

Then R i s an e q u i v a l e n c e r e l a t i o n i f f o r any e l e m e n t s A,

B, and G of S ,

1 . ARA ( R e f l e x i v e P r i n c i p l e )

2 . I f ARB, then BRA. {-Symmetric P r i n c i p l e ) -

3* I f ARB and BRG, then ARC. { T r a n s i t i v e P r i n c i p l e )

E x a m p l e s .

1 . I s an e q u i v a l e n c a r e l a t i o n on t h e a e t of i n t e g e r s ?

For t o be on e q u i v a l e n c e r e l a t i o n , t h r e e t h i n g 3

rauat be t r u e .

a . I f a i s an i n t e g e r , t h e n "a - a" m u s t be t r u e .

I s any i n t e g e r e q u a l t o i t s e l f ?

b . I f a = b» then b ~ 3 . l a t h i s t r u e ?

c . I f o •- b and b » o t t h e n a = c . I f a - b and

b s= e , t h e n a and c a r e b o t h e q u a l t o b . Does

t h i s imply t h a t a and c a r e e q u a l t o each other-?

2 . P l a c e t h e a p p r o p r i a t e symbol , »~!f o r b e t w e e n t h e

f o l l o w i n g p a i r a of o r d e r e d pa I r a .

179

a . ( 3 , 7) (3s 7)

( 9 , 2) (9S 2)

(-1*, 6) _ <-!i, 6)

( - 1 8 , - 3 D ( - 1 8 , - 3 D

b . ( 6 , 9) (3 + 3 , 5 + 1].)

(3 + 3 , 5 + U) ( 6 , 9)

( - 8 , 5) _ _ _ (2 - 10, 1£ ~ 10)

(2 - 10, 15 - 10) ( - 8 , 5)

(11 , 6) (5 6)

(5 + 6 , 6) (11 , 6)

c . (3 , 12) (1 + 2 , 6 + 6)

(1 + 2 , 6 + 6) (1+ - 1, 13 - 1)

( 3 , 12) „ _ • . • (k - 1» 13 » 1)

(-1+, 7) (5 - 9, 9 - 2)

.. (5 - 9 , 9 - 2) _ (8 - 12, 12 - 5)

7) (8 - 12, 12 - 5)

I s " an equiva lence r e l a t i o n on the s e t of ordered

p a i r s of i n t e g e r s ?

3• I s R an equivalence - r e l a t i o n oil tbo s a t of ordored

p a i r a of integers? (See Lessen 8 . )

i|» l a Rx an equiva lence r e l a t i o n on tbe s e t of o rdered

pa i r a of i n t e g e r a ? (See Lea aon 8„) _

La t S be tli 9 3et containing the *1. em ant a

A « 5 , 9J 13, 17, 21, 2$# 29, 33, 37, I j l , . . . } ,

B - £2, 6 , 10, 4 , 18, 2 2 , - 2 6 , 30, 3!-b 38, 1*2, . . ,

180

G = \ 3 > 7 , 1 1 , 15* 19> 2 3 , 2 7 , 3 1 , 3 5 , 3 9 , 4 3 , . . - j , a n d

D = 8 , 12, 16, 20, 2)-, 28, 32, 36, 1|0, i|J4, . . , } .

Le t <£) be a b i n a r y ope ra t i on on S da f i n e d a 3 f o l l o w s : I f

X and Y a re elements of S, a ia an element of X, and b ia

an element of Y, then X ® Y ia the element ( s e t ) of S

conta in ing a + b . For example, f i n d A 0 C . 9 i s AN e l e -

ment of A and 15 ia cn <3lament of G. 9 *!- 15 ~ 21|. 2)4. ia

an element of D. T h e r e f o r e , A 0 G 3 D.

Exegois3 9.

F i l l in the b l a n k s .

A Q B =

B 0 A -

33 © D =

D(S>A -

0

A © ( C © D )

(B 0 A ) 0 B

B © ( A © B )

0 (?) (D

181

l a the b i n a r y o p e r a t i o n (*>csramu fcative?

I s the b i n a r y o p e r a t i o n ( ^ a s s o c i a t i v e ? :

I s t h e r a an i d e n t i t y e lement f o r the b i n a r y o p e r a t i o n a l ?

Us ing the b i n a r y o p e r a t i o n 0 a n d the b l e a r y o p e r a t i o n

Cxâs d e f i n e d i n Les son 8 , f i l l i n tho b l a n k a .

B Q ( G © A ) «

( B © C ) 0 ( B @ A ) =

D 0 ( B 0 C )

( D © B ) 0 ( D © G ) a.

(A Q B ) © ( A §>A) =

Does ( x j d i a t r i b u t e ever .

13 0 3 commutat ive b i n a r y o p e r a t i o n ?

I s (g!)aa aaaooiafc ive b i n a r y o p e r a t i o n ?

I s theirs an i d e n t i t y e lement f o r © ?

P i l l i n the b l a n k s .

A Qb. ^ D

B ^ ~ D

C « D

D - D

A % « A

B © " A

0 © = A

D<S> = A

APPEHDIX C

SAMPLE LESSOIS PGR THS EXPOSITION

METHOD OP 'mOHING

183

1VT ^ H 1 A O i ' i OS I; k j .L O J

Coll 6 g e K 31 h s i';i g t i e s

Throughout thia course the terra a e t w i l l be used to

r e f e r to any wel l—def ined c o l l e c t i o n , g r o u p , or c l a s s of

o b j e c t s or i d e a s . The p h r a s e "we l l -de f ined" means t h a t

the a at ia d e s c r i b e d p r e c i s e l y enough so t h a t one can t e l l

whether or n o t any g iven o b j e c t be longs to the s a t . The

o b j e c t s con t a ined in a s a t e r a c a l l e d ejleinenta. In a s e t

of d i s h e s , each d i s h ia an e lement of the s e t of diahea .

In the caae of a co in c o l l e c t i o n , ec-cb co in la an e l emen t

of the s e t of c o i n s .

One way of d e n o t i n g a s e t la by H a t i n g a l l of Ita

e l e m e n t s . I f the elements of a s e t are l i s t e d , the l i s t

i s p l a c e d w i t h i n b r a c e s . Thus ^ 1 , ?., 3 , i | , 5> 6 ^ , de -

no te s the s e t c o n s i s t i n g of a l l n a t u r a l numbers from 1

through 6 i n c l u s i v e . In some cases no t a l l the elements

of a a at can bo l i s t e d . One such s e t i s tha s o t of

na tura l numbers. Th1£ £ 0 i aay be denoted a a ' f o l l o w a j '

£ l , 2, 3, H, 5, 6, 7, 8, 9, 10, 11, 12, . . . J .

Another concept which w i l l be used e x t e n s i v e l y

t h r o u g h o u t t h i s cour se la tha concept of b inary o p e r a t i o n .

A b i n a r y o p e r a t i o n ia a way of a s s o c i a t i n g with two e l a - .

menta of a s e t a t h i r d e lement of t h e t s e t . A d d i t i o n ia

a b i n a r y o p e r a t i o n on the s e t of n a t u r a l numbera . For

example , the operat ion of a d d i t i o n a s s o c i a t e s wi tb the '

].8)|

natural nurabera 3 and - 5 the aa tural nuxaber 8. It associates

with the numbers 593 and 683 tbe nurcbsr 1281. Multi-

plication is also a binary operation on to0 set of natural

numbers. This operation associates with. tbe nurabora 3 and

5 tbe number 1.5. It associates with the numb era 598 and

683 tbe number ' C3 ,)|3H•

It ia possible to define many binary operations on

tbe set of natural numbers. Suppose that with two natural

numbers we associate tbe- first of tba two mucberfl. This

ia an example of a binary operation on tbe set of natural

numbers . This operation- associates with 3 and 5 the num-

ber 3. It associates with 17 and 9 tbe number 17.

The concept of binary operation leads ua to consider

toe concept ol .ordered pa ir. A binary operation associates

sfxtb a pair ol elements of 3 set :;n element of the set.

Ibe binary operation dlccussed ia the preceding paragraph

s3ijocj.3 fce3 ivir.h tba pair of nunbsr'J 3 &nd 5 the number 3.

It associates with the pair of numbers 5 and 3 the number

5. In each case the 3 arcs pair of nun/onra is uaed, but the

i-.e.-iulta are different. Tbi.i ia iruo because tbe order in

v.hich in e e 1 em en 1:3 of tbe pair are listed is different,

A n P *<***«.<* Hl£. cons ia ts of a pair of - eleven ts listed in

^ specific order, A n ordered pu ir uili bo demoted by

lis ting., the elegants of ibe pair with a corcrna between them

and placing., tbe list in parentheses. -fbus (3, $) denotes

185 p

the ordered p a i r c o n a i s i i r g of tha autsbers 3 and 5 wi tb 3

coming be fo re $» (5 , 3) denotes the same p a i r , bu t wi tb

5 coming b e f o r e 3. This abowa t h a t (3> 5) i s not tha 3 sine

ordered p a i r as (5 , 3 ) • In the ordered p a i r (5 , 3 ) , $ ia

c a l l e d the f i r s t c proponent of the ordered p a i r and 3 ia

c a l l e d the second component of tha ordered p a i r . Two

ordered p a i r s a re s a i d to be equal i f they have tha soma

f i r art component and the same second component.

Let us danota tha s a t of a l l ordered p a i r s of n a t u r a l

numbers by tba symbol P . Sdioa of tha elements of P a r e

(3 , 9 ) , ( 5 , 1 ) , {<> 5 ) , (2871, 1 ) , and (?., 3800972). Many

i n t e r e s t i n g and unusual 'binary opera t ions can be d e f i n e d

011 the s e t P . Remember t h a t a b i n a r y opera t ion ia a way

of a s s o c i a t i n g with two elements of a s a t another element

of tha a e t . Thus a b i n a r y opera t ion on P a s s o c i a t e s wi th

a p a i r of ordered p a i r a an ordered p a i r . An example.of a

b i n a r y opera t ion on P ia the ope ra t ion which a s s o c i a t e s

with a p a i r of ordered pa i r a tha second ordered p a i r . This

b i n a r y opera t ion aascciata. ' i with (! | , 1) and (9 , i?) the

o'rderad p a i r ( 9 , - 5 ) . I t a s s o c i a t e s with. .(1, 17) -and

(39 , 1?|) the ordered p a i r (39, II4).

I f we- wish to cons ider tha number which the b i n a r y " " "

opera t ion of a d d i t i o n on tha safe of n a t u r a l numbers a s s o -

c i a t e s wi th 7 and J4 we w r i t e 7 + Since the ope ra t ion

of a d d i t i o n a s a o c t a t e a wi th 7 acd l.j. the cumber 11, we

186

w r i t e 7 + 2| = I I . This ia a- very compact and conven ien t

way of saying t h a t tha b i n a r y opera t ion n d d i t i o n a s s o c i a t e s

wi th 7 and the aa t u r a l -number 11 . Whan- cons ider lug

b i n a r y opera t i ons on P v/a alia3.1 usa the n o t a t i o n

( a , b) B ( c , d) « ( e , f ) to denote t h a t tha b i n a r y

ope ra t ion 3 a s s o c i a t e s v/ith the p a i r of ordered p a i r a

( a , b) ana (c f d) tha ordarcd p a i r ( e , f ) . Thtia i f B d e -

no tes tha b i n a r y ope ra t Ida on .? d i s c u s s e d In tha p reced ing

paragraph we haira t h a t (Ij., 1) 3 (9 * $) ~ (9# 5 ) » A l a o

( 1 , 1?) B (39j, 1)4) - (39 , H i ) .

A b i n a r y ope ra t i on on P can r f t a n be du f ined us ing

a g ^ n a r a l forsHila. ( a , o) B ( a , d) - ( c , d) ia a formula

which aan be used to d a f i n a the b i n a r y operation- -on P

di"cu3.rad in t-ho proeodlng p&fagraph. I f a i s r e p l a c e d

by a n a t u r a l tiusthc.r and i f b i s rop lucad by a n a t u r a l num-

b e r , than ( a , b) b a c o n s an olexnont of P . S i m i l a r l y i f c

.la r a p i a e a d by a n a t u r a l nurah or and i f d in r a p l a e a d by a

n a t u r a l Bupjbor, than ( c , d) beoorgs i;,n <a Ism-ant of P, Then

(a j, u) B ( g j cu *" ('3# o) A3 a 'way ox -• • j ••-> •'ij t n a t x'hu b i n a r y

ope ra t i on 3 as i oo;la wi th ( a , b) ana ( c , d) tha ordered

pa i r {c, d) .

Coaaidar t h a . b i n a r y opera t ion on ? d e f i n e d by tha

forEUla ( a , b) B ( e , d) (a * s , b x d ) .

This foru;ula n U U a tha fe to f5,?;d th;; f i r a t component of

tha or dared p a i r a s s o c i a t e d Xi% Rj ( a , b) , a d ( c , cl) add tha

18?

f i r s t components of ( a , b) and ( c , d ) . To f i n d the s econd

component m u l t i p l y ths s ee r e d *-Cv.:pc-r.-viit.? of ( a , b) and

( c , d) . A c c o r d i n g to the fo rmula t h e o r d e r e d p a i r a s s o -

c i a t e d w i t h (Li, 8} ana ( 9 , 7) ia (Lj. + 9 , 8 x 7) or

( 1 3 , £ 6 ) . S i m i l a r l y (12, 3) B O4, 17) » (12 + !*, 3 x 17)

or ( 1 2 , 3) B U | , 17) *-= ( 1 6 , 5 D .

EXERCISE

ID each of t h e f o l l o w i n g e x e r c i s e s a b l e a r y o p e r a t i o n

i a d e f i n e d . Use t h e d e f i n i t i o n to f i n d t h e o r d e r e d p a i r

a s s o c i a t e d w i t h the g i v e n p a i r of o r d e r e d p a i r s .

Example: ( a , b) B ( c , d) = (a x b , c ) .

Complete the f o l l o w i n g .

(3» 2) B U;, 7) ^

i<> I) B ( 9 , 18) «

S o l u t i o n :

(3» 2) B (J+, 7) - (3 x 2 , 10 « ( 6 , k)

( 5 , l ) B ( 9 , 18) - (5 X 1 , 9) - ( 5 , 9)

1 . ( 3 , b) B ( c , d) - (a x d , b)

sxarapla: (1, 9) B (7, If) « (1 x !j, 9} « (![, 9).

C carpi a fee t he f o l l o w i n g .

O4, 3) B ( 1 8 , £) a

( 1 , 1) B ( 1 5 , 2 )

( 5 , 7) B ( 9 , 3) -

( 1 5 , 21) B ( 1 , 1)

( 9 , 3) B ( 5 , 7) '-=

188

2. (a , b) B (c, d) f "7" f t !'•- 4 - * ^ <('» /S V .Jt* «<" V» ft

Example: (3» 2) B (9* 1) ~ (3 x 9, 2 + 1) - (27, 3) •

Complete the following.

(7, 1) B (9, 5') =

(1, 1) B (8, 3) =

(6, 9) B (ll|, 7) 3

(9, 5) B (7, 1) =

(1)4, 7) B {6, 9) =

(13, 11) B (8, 2)

(8, 3) B (1, 1)

(8, 'd) 3 (13, 11)

3. Make up (invent) a formula of your own which defines

a binary operation on P. Uss your fori.iula to ccisplata

the following,,

(3, 1) B (2, 2) =

(6, 8) B (3, ID

( 1 5 , 1 ) B ( I 4 , 7 )

(8, 2) 3 (8, 6) -

(8, 6) B (3, 11)

(7s it) B (1, 15)

(8, 2) B (7, 5) =

(6, 8) B (11, 3)

U, 7) B (1, 15)

189

XJ e c s o 4 J 2

A b i n a r y operation a « a o c i c t a a t / i t b two e l ements of e

s o t an e lement of tbo set. I f the order in which the f i r s t

two e lements appear does no t a f f e c t the f i n a l r e s u l t , the

binary o p e r a t i o n Is said to be commutative.

Examplesj

Addition on the set of natural, numbers ia a corarautative

binary operation. If a list of numbers is to be added,

the result d o e s not depend on the order in v;bich the num-

ber a ere listed.

7 + 3 ~ 1 0

3 -i 7 - 10

Lj. Jr 11 — 3-5

11 + l< - 15'

M u l t i p l i e s t i o n on the set of natural numbers i s a com-

mutative binary operation. For example:

7 x 12 « 813.

12 x 7 - 81*

14 X 26 a loll,

26 x Ij. s IO/4.

The binary operation on P defined by

(s, b} B (e, ci) ss (q x c, b d)

i s a coBffiiutative b i n a r y o p e r a t i o n . Verify this by com-

p l e t i n g the following,

(1, 3) B (8, U) =

190

(8 , k) B (1 , 3) » „

(2 , 5) B (3 , 7)

(3? 7} B (2 , S) a s _ T1J .*

(8 , 2) 3 (1 , 6 ) a _

(1 , 6) B (8 , 2) s

(12 » 1) B (k . 9) - _

()4, 9} B (12, 1) =

Note: I f ve w l t a 2a we 2Ma 11 inean 2 x a . I f wa wr i t e

ab us aha 11 moan a x b .

Consider the b inary cperdfcloa ya ? def ined by the

f o l i o ; log:

( a , b) S ( e , d) - (2s * 2c, bd)

This ia a cmtiakat iva b inary o p e r a t i o n .

Eâfw-ples;

(3 , k) B (1 , 2} « (2x3 2x1, 1^2} •« (8, 8}

{1, 2.) B (3 , k) ~ (2x1 f 2x3» 2x4) « (8 , 8)

V e r i f y fcbst t'nia b ina ry operat ion ia comaiufcafcivQ by eons-

p 1611 a g ' tb »3 £ o 11 ov i n g*

(1 , 3) B (2 j 8)

(2 , 8) B (1 , 3) » ........

0|» 6) B (1 , 1) » _

(1 , 1) B 6) « _

(5 , h) B' (2 , 11) k

(2 , i d B (£ , ;«.) «

(12, lf>) B (21, J42)

191

{21, l<2) B (12 , 15) »

The b i n a r y operation of a d d i t i o n on the s a t of n a t u r a l

numbers a s s o c i a t e s with 9 and 12 the n a t u r a l number 21.

21 ia c a l l e d the sura of 9 and 12. Suppose t h a t i t ia nec-

e s s a r y to f i n d the aura of a . H a t of n a t u r a l authors .

Consider the l i s t 7> 3* 9> !|> and 6 . The sum can be found

by f i n d i n g the surn of 7 and 3 . Thia aura ia 10. men f i n d

the a urn of 10 and 9 . This sura ia 19. Then f i n d 19 + !|.

19' + k ~ 23. • To en f i n d 23 > 6.- 23 * 6 a 29. This ia the

sum of 7 , 3, 9, lj» a ad 6 .

Us log t h i a procedure two numbers were added in each

atep of the p r o c e d u r e . Thia type of procedure was necea-

Sary oecou3o a d d i t i o n i-3 3 binorjr opsr<2tion. Only two

numbers can be added- a t a t ime . Consider the t a sk of

f i n d i n g one audi of uho l i t i t 1}, 15, and 2 . Ij. + 15 ~~ 19.

19 + 2 21 . There fo re the d e s i r e d sura ia 21. Thia pro-

c e d u r e can be represented as follow,*:. . . .

(U + 15) + 2 « 21.

- Ihe poi entnea 03 around ]j + 15 i n d i c a t e s t h a t t h i a aura

. aaould be iound . f i r s t , and t h a t ihe r e s u l t should be" added

to 2 . Pa ren theses and brackets , £ jj, -are of tan used to

indioa ca which ope ra t i ons should be perforated f i r s t .

Examples: .

-1. £lt -x 2 J-5- 6 ss

192

' The brackets i iKl laa ta t h s t I| .s 2 shou ld be found

f i r s t . The r e s u l t should thea he added, to 6 .

x 2 *} + 6 — 8 + 6 ll\.

2 . 3 + . (2 *»• 7) «

The p a r e n t h e s e s i n d i c a t e t h a t 2 + 7 shou ld be found

f i r s t . Then 3 shou ld be added to t he r e s u l t .

3 + (2 + 7) 55 3 9 s3 12»

3 . I f t he b i n a r y o p e r a t i o n B i«i d e f i n e d by

( a , b) B ( c , d) - ( a d , c ) ,

than f i n d t d . 3) B ( 9 , 2)'} B ( 8 , 6} .

S o l u t i o n j The b r a c k e t s i n d i c a t e t h a t ( 1 , 3) B ( 9 , 2)

shou ld be found f i r s t .

( 1 , 3) B (9, 2) - (1x2 , 9) « (2, 9)

Then ( 2 , 9) B ( 8 , 6) should be f o u n d .

( 2 , 9) B ( 8 , 6) « (2x6 , 8) « (12 , 8 ) .

T h e r e f o r e ,

1 ( 1 , 3) B (9s 2 ) 1 B ( 8 , 6) « (12 , 8) .

Suppose t h a t a b i n a r y o p e r a t i o n i s t o be used on a

l i s t of e lements of a s a t . I f the way fcba e lawenta of the

l i s t a r e grouped does n o t e f f e c t t h e f i n a l r e s u l t , t h e

b i n a r y o p e r a t i o n ia a a i d to be a a s p c i a t i v e .

E x e r e l a e 2 .

. A d d i t i o n on the s e t of n a t u r a l numbers i s aa

a s s o c i a t i v a b i n a r y o p e r a t i o n . V e r i f y t h i s by cample t i n g

the f o l l o w i n g .

193

18 -«• (3 + 9) -

(18 + 3) + 9 -

(!|7 *• 3) + 1 6 =

kl + (3 + 18) =

till + 61 + 27 =

l!| + 1,6 4- 27 3 ~

Consider the binary operation on F defined bj

(a, b) B (c, d) - (a + c3 bd).

This binary operation la an associative binary operation.

Verify thia by completing the following.

\ u , 2) b (3, b (9, 1) «

(1, 2) B t(3, U) B (9, 1)1 «

1(5, 2} 3 (9, 9)1 3 (6, 5) ~

(5* 2} B [(9, 9) B (6, 5)] = .

(1|, 7) B 1(3, 8) B (1, 1)] «

[ U , 7) B (3, 8)1 3 (1, 1} -•=

(1, 6) B £(2, 11) B (11, 2)]

[(1, 6) 3 (2, 11)3 B (11, 2)

The binary operation used 3bovo Is a comniuta fcive binary

operation. Verify this by completing the followlag.

(2, 1) B (8, l4) .

(8, k) B (2, 1) =

(1, 6) B {2, 11)

(2, II) B (1, 6)

(I4, 7) 3 (3, 3) =

( 3 » 8 ) 3 ( 1 | , 7 ) ••

19I.J-

Lesson 3

Review.

Which of tba fo l lowing b i n a r y ope ra t i ons a re

cors,muta t i v e ?

1 . Addi t ion on the safe of n a t u r a l numbera .

2 . M u l t i p l i c a t i o n on the s e t of n a t u r a l number3 .

3 . The b i n a r y ope ra t i on on £ d e f i n e d by

( a , b) B ( c , d) ~ ( ab , c ) .

i | . The b i n a r y opera fcion on P d e f i n e d by

( a , b) B ( c j d) - (a 4- c , bd) .

Which of tba fo l lowing b i n a r y opera t iona a r e

a g a o o i a t i v e ?

1 . Addi t ion on the s o t of n a t u r a l numbers.

2 . M u l t i p l i c a t i o n on the sot of n a t u r a l numbers.

3 . ITi e b i n s r y o p e r a t i o n on P d e f i n e d by

( a , b) B ( c , d) ™ ( a b , c ) .

I | . The b i n a r y ope ra t ion on ? d e f i n e d by

( a , b) B ( c , d) - (a + c , b d ) .

A b i n a r y opc ra t i on a s s o c i a t e s with two elements of a

a e t an element of the s e t . Let 3 denote a a e t and l a t "a"

and Mb" denote elements of the s e t . Lat "0" denote a b i -

nary ope ra t ion on 3 . Then we s h a l l denote tbe element of

3 which 0 ' a s s o c i a t e s with a and b by a Ob. Suppose t h e r e

ia an element of S, denoted b y the symbol "e" , aucb t h a t

f o r each element a of S, sOa - a a a < j aOa •- a . I f such an

element, a, s x i o t a it ia c a l l e d g neutral e l e m e n t or en

element.

Examplea.

1, Given the set

\ 0 , 1 , 2 , 3 , k > 5 , 6 , ? , 8 , 9 , 1 0 , 1 1 , 3 2 , . . . }

0 is an identity element for addition on this set.

For example:

7 + 0 -= 7

0 + 7 - 7

18 + 0 -•= 18

0 + 18 = 18

921 + 0 - 921

0 + 9 2 1 - 9 2 1

2. 1 ia an identity element x'or Multiplication on ibe set

of natural numbers. For example:

3 x 1 = 3

1 x 3 = 3

1 7 x 1 •« 1 7

1 x 1 7 = 1 7

bt2 x 1 i}2

1 x i(2 = 1|2

3 8 5 .X 1 a 3 8 5

1 X 385-« 385

3• (0, 1) ia an identity element for the biu&x'y operation

on P defined by {a, b) B {c, d) ~ (a + c, bd) .

1 / I h

Ver i fy tbia by completing the following.

( 3 , 3) 3 ( 0 , 1]

(0, 1) B {8, 3

(2, 9) B (0, 1

{0, 1) B (2, 9

(0, 1) B (5, 9

(5, 9) B (0, 1

F i l l in the blanks .

( 1 5 , 3) B = ( 1 5 , 3)

(9, 5) B « (9, 5)

( 7 , U) B - ( 7 , If.)

B ( 8 , 111) « ( 8 , 11,}

B (9, 17) - (9, 17)

Id each o f the l a s t f1va exe rc iaaa the b l a n k shou ld have

been f i l l e d w i t h (0 , 1 ) . The. aniswoi* shou ld be (0, 1) i n

each eaaa because (0, 1) is aa i d e n t i t y element for the

b i n a r y o p e r a t i o n B .

Exarclag 3 •

(I.^ 1) is so identi ty o 1 gib3n u for tbc biBsry opoyy t i d.o oo

P d a f i n a d by

(a, b) B (c, d) = (nc, bd).

Verify thia by cornplotiog the following.

(15, 3) B'(1, 1) s

(1, 1) B (9, 8)

(9, 8) B (1, 1)

197

(27, 31-1-5 B (1, 1) =

(1, 1) B (2?, 3h) =

(293, bl) B (1, 1)

(1, 1) B (293, kl)

Fill in tha blanks.

(2, ij.) B = (2, 1 )

(3, 7) B = (3, 7)

B (5, 1) « (5, 1)

B (8, 16) = (8, 16)

(0, 0) ia an identity element for tha binary operation

defined by

(a3 b) 3 (c, d) ~ (a + c, b d).

Verify this by completing the following.

(2, )|) B (0, 0) «

(0, 0) B (2, k) =

(3, 19) B (0, 0) =

(0, 0) B (8, 19) ~

[Uk» 3) B (9, 2)3 3 (0, 0)

(0, 0) B 1(8, 7) B ()., 13)3

Pill in tha blanka.

* 7 ~ 7

4 + -- ii-

18 X _ _____ s= 18

92i| « 92!|

138 + -3 138

193

296 x ~ 296

x 148321 » 1|&321

+ 1-8321 ™ Lj.8321

199

Lesson ?

Le t S be and aat and l e t 0 cud 0 ba b i n a r y o p a r a t i o n a

d e f i n e d on S . I f f o r any e lement3 A, B, and G of S

A 0 (B 9 C) ~ (A03) 0 (A 0 0 ) ,

then 0 i s s a i d to ba t_ r ibu t ive over 0 .

Examplea.

1 . M u l t i p l i c a t i o n i s d i s t r i b u t i v a over a d d i t i o n .

'That i s , a x {b + c) = (a x b) + (a >: c) .

For example,

7 x (8 + 9) « 7 x 17 ~ 119 and

( 7 ^ 8 ) + ( 7 x 9 ) ~ £6 63 - 119

T h a r e f o r e

7 x (8 9) ™ (7 :< 8) + ( 7 x 9 )

D - 1 3 ) * 7 £ ( - 1 8 ) * 1 0 ] = -126 + -180 « -306

- 1 3 x £7 + 10*3 = >18 x 17 "-= -306

T h e r e f o r e

I ( - l 8 ) x 7 ] + £{~l8) x 1 0 J ~ -18 x £7 •> 1 0 ^ .

2 . A d d i t i o n ia' n o t d i s t r i b u t i v a over m u l t i p l i c a t i o n .

For example ,

2 + (8 x 7) ~ 2 + 56 ~ 58

(2 + 8) x (2 1- 7) ~ 10 x 9 » 90

T h e r e f o r e

2 + (8 x 7) {2 -b 8) "x (P. + 7) »

3 . Tba b i n a r y o p e r a t i o n # d e f i n e d by

( a , b) * ( c , D ) •- ( G O , bd)

200

ia d i s t r i b u t i v e over the b i m i j ops.rafcion § d e f i n e d by

{a, b) if ( c , d) ~ {a + c , b + d) .

For example,

( ! ; . , - 1 ) » £ ( 1 , 6 ) # ( - 8 , 2 ) ] « ( I t , - 1 ) « ( - 7 , 8 )

= (-28, -8)

£(l+, ' - l ) « ( 1 , 6 ) ] # £ U , - 1 ) « ( - 8 , 2 ) 3 " Ct»-6) # ( - 3 2 , - 2 ) '

= (-28, -8)

There fo re

( l » , - l ) » t ( l , 6 ) # ( - 8 , 2 ) 3 - £ ( l t , - l ) - » ( l , 6 ) 3 f I ( U , - X ! » < - 3 , 2 ) 3

J|. The b i n a r y o p a r a t i c a 0 d a f i n e d by

( a , b) 0 ( c , d) ~ (a 4 c , bd)

i3 no t d i s t r i b u t i v e over tba b i n a r y o p e r a t i o n &

d e f i n e d by

( a , b) & ( c , d) ~ (a t- d , b e ) .

This can ba shown aa f o l l o w a :

( 3 , 5 ) O f t 9 , l ) & ( 8 , 2 ) > ' (3 ,5 ) 0 (11,8) ( l l | , i | 0 )

l( 3 ,5) 0( 9, i Q & O 3 , 5 ) 0 ( 3 , 2 ) 1 = {12»5)&( 11,10) ----- (22 ,55)

Th ere fo r a

( 3 , 5 ) 0 1 ( 9 , 1 ) 4 ( 8 , 2 ) 3 f. C ( 3 , S ) 0 ( 9 , l ) 3 & t ( 3 , S ) 0 ( 8 , 2 ) ]

One countsvaxamp 1 ft ia s u f f i c i e n t to sbow tba t 0 ia not

d i s t r i b u t i v e over

Use the f a c t tba t m u l t i p l i c a t i o n ia d i s t r i b u t i v e ovar

a d d i t i o n to f i l l i n - t b a b l a n k s .

(16 x 58) + (16 x 92) - 16 x { %• 92)

12) * I t ] + [ ( - 1 2 ) z ( - 1 3 ) ] = 2 -[it ( - 1 8 ) ]

B - 4 ) * 983 + | ( - H ) X 106 } » -)$ X £98 + "J

- 1 2 * C W * ( ~ i 3 ) 3 = * 3 + t ( - i 2 ) * ( - 1 0 )

16 x (53 + 92) ™ ( x 58) + (16 z 92)

-!| x (S'B f 106) ~ ( _ x 98)' ( _ x 106)

37 x [ ( - .12 ) -s- 9 . ] - [ 3? * ] [*37 x „

(7 x 1;.) + (7 .x 3) 3 ? 31 ( + )

(8 x 6) »- (19 x 6) = ( + ) ~i 6

{ l i | x 9) + (21 x 9) = (__ * ) x

Es eye lag 7 .

'CJa3.Bg tbe f a c t t h a t l a d i a t r i b u t i v a over § $ f i l l

i n the b l a oka .

( i w - u s ' O i , 6 j # ( - 8 , 2 ) 3 » r ; I . . , - 1 ) . - ( 1 , 6 0 # 0 i t , - 1 ) « ]

( 3 , 9 ) 4 ( - 2 , 7 ) # ( - 3 , - l ) j - £ < 3 , 9 ) » „ _ 1 # C < 3 . 9 ) » 3

{ - 6 , 3 ) « l ( l » , 9 ) « - 5 , 3 ) 3 = 1 » C M > ] # C * < - 5 , 8 ) ] .

i 1 , 1 ) « { ( 3 , - IS) # (?-, U»)] = £( 1 ,1 ) # _ J # g l , X ) » 'J

( 8 , ~ 7 ) » £ ( 3 , - 5 ) # ( M ) 3 - t _ • C 3 . - 5 ) J ? £ _ »<U»9)3

U "^>3)-*('•- if £ ( - - ! j , 3 ) * ( "3>7)3 ™ *£ (6 ,2 ) j? ( -3»7)^J

D 3 , 9 ) » ( - 2 , 7 ) j # I ( 3 ,9 ) * { - 3 , •-1 )3 - . * 1 / ( - 3 , - 1

8 - 6 . 3 ) » < l » , 9 3 # B - 6 , 3 > * C - $ , S i } » „ . < • [ J L

E » - 3 j ^ £ n , 7 j J r / y ~l,-3) :--~tS ,9}J j //

f < V 9 M - 6 , I ) ] # g 9 , 3 > * < - 6 , i a - f (lt,9M9,3f]e

B l . 8 ) * ( 3 , l S ) l # 8 - 9 , J 0 ) » ( 3 , a 2 ) 3 « £ ( 1 , 2 ) #

0 ' • s ?) *( 9 , 9 i j u 2 , 9 , 9 ) J £

H® ^~ i f " " " ~ |

202

\iU,6)#(-7,2)3«-(-12,3) « T _»

(-1,1)»\( 3,12)#( 12, -3)} * f * ] # { * ]

Review Exarciaea.

Solve the following squa felons.

1+ = 7 • x

12 ~ x - 7

-3 + x = . - 1 0

x - ~ -6

(1+, -6) - (9, k) # (*, y)

(1, -1) « U , y) // {?, -3)

(-2, ?) // (x, y) « (£*, 5)

(5, -6) - (3, 7) # (x, y)

(x, y) # (6, -93 (-ij, 12)

2x •" 16

32 = l6x

17x - 68

LB 3 3 o n 8

la Lea son 1 it la stated that two ordered pairs are

e q u a l if they h a v e the aaiae first c o m p o n e n t a n d tba same

second c o m p o n e n t . Via indicate the fact that two ordsred

p a i r s are equal b y placing the s y m b o l w « " b e t w e e n there.

If two ordered pairs sr a n o t e q u a l , this can be i n d i c a t e d

b y p l a c i n g the synbol V " between tbem. For example

(-k, 2) = (-4, 2) and (3, 7) A (8, 7).

F o r integers the f o l l o w i n g ore t r u e .

1. If a is an i n t e g e r , then a s= a ,

2 . If a •- b , tfcen b ~ a .

3- If a = b and b - g, then a -- c.

F o r pairs- of Integers, it can ba v e r i f i e d thnt the

f o l l o w i n g are true:

1. (a, b ) S3 {a , b ) .

2* I f {a, b ) ™ { c * d ) , than ( c s d ) ~ (o > b) .

j« A1 (s j 0 } — {C , d) rtijd (Cj d) - • { 3 # f ) > tb dT>

(a, b ) ~ (e, f).

Let iJ be a set, A tlon on 8 a s s o c i a t e s vrlth two

elements of 3 either bbe word, "yea," or tba w o r d , "no."

If R la a r e l a t i o n on 3, A and B are e^sraeufcs of £?, and R

a s a o c i a t e a with A *nd B the word nje&» then we say tbat A

is related, to B and we indicate tlx.la by -urlting A B B . If

R asaociatea with a and B tbo v o - d "no" than wo say tbat

j% i.i noc re la ved to B and we indicate this b y writJng AjiB •

2C

I f R Is a r e l a t i o n on S sad A sod B a r e elements of S, then |

e x a c t l y one of the f o i l e d log i s a t r e e s t a tertian ts

1 . A i s r e l a t e d to B. |

2 . A ia no t r e l a t e d to B. !

A i s r e l a t e d to B i f and only i f R a s s o c i a t e s wi th A and B

the uord "yea . " A ia not r e l a t e d to B i f and only i f H I

a s s o c i a t e s wi th A and B the word "no . "

Exaraplaa of E o l a t i o n s . j •

1 . "ss" ia a r e l a t i o n on the s e t of i n t e g e r s . I f a ©nd b

a re i n t e g e r s , then e x a c t l y one of the fo l l owing la a

t r u e 3 ta'teraer»t;

a . a -- b .

b . a / b , .

2 . ia & r a l a t ion on th.o s a t o f . ordered p a i r a of

i n t e g e r s . I f ( a , b) and {c, d) a r e ordered p a i r a of

i n t e g e r s , then e x a c t l y one of the fo l l owing ia a t r u e

8 ta t era en k:

a . (a , b) J S (c , d) » b . ( a , b) p { o y d) .

3. The s t a t e m e n t s ,

3 • (a) b ) R ( c , d ) , { (9 , b) i i r e l a t e d to ( a , d}) i f

a + d •-= b - c , and

b . ( a , b ) ^ ( c , d ) , {{a , b) ia n o t r e l a t e d to ( e , d ) )»

i f a + d -f b ~ c»

d e f i n e a r e l a t i o n on the s s t cf ordered p.aira of i n t e -

g e r s .

o 2 05

For example, (3, 9}R(U» 2) beesuae 3 + 2 = 5> and

9 - '4 ~ 5- (7, 6)k(2.» 8} because 7 + 8 88 l£.

7 - 2 = Ij. and 15 r '4*

)4« Another exsrnpla of a relation on the sat of ordered

pairs of Integers la the relation R^ defined by,

a. (a, bjR-^c, d) if a + d » b + c.

b. (a , b) ( c, d) if .3 -5- d 7-- b + c.

Another way of defining R- is by the statement,

(a, b)R^(c, d) if and only if a + d = b + c .

(h> 8)Rl(2, 6) be 0 3 use ,'4 + 6 « 10 and 8 + 2 *- 10.

(9, 2)^(6, 3) bo causa 9 + 3 r~ -2, 2 + 8 10 and

12 /• 10.

Definition: Let 3 be a sr<it and let R be a 5;elation on 8.

Then'"R ia an ' equivalence relation if for any elements A,

B, and 0 of 3,

1. ARA. (Reflexive Principle)

•2. If ARB, • then BRA . {Syr®etrie Pr Idcip 16)

3. If ARB and 3RC,. then ARC.- (Transitive Principla)

Example a of Equivalence Relations.

1. "a" is an equivalence relation on the sat of integara.

2. ia an 0quivalence relation on the sat of ordered

pairs of integers. Thia ia true because,

a. Any order ed pair is equal to its elf. That ia,

(a, d) ™ \a, b).

b. If (a, b) « (c, <i) , then a « c and b = d.

206

I f a - c , tbers c - a . I f b ~ d , t h e n d - b .

T h a r e f o r a ( c , a) ( a , b ) .

c . Suppose ( a , b ) •= { c , d) and ( c , d) -- ( e , f } .

S i n c e ( a , b) ••= ( o , d) , a - c and b ™ d .

S i n c e Co, d) - ( e , f ) , o ~ e and & » f .

S i n c e a ™ c «nd c = a , t h a n a = e .

S i n e a b = d and d -- f , t h e n b -- f .

T h e r e f o r a , ( a , b ) -- ( e , f ) .

3 . The r e l a t i o n R d e f i / i e d b y ( a , b ) R ( c , d) i f and o n l y

i f a + d ss b. - c i s p o t an e q u i v a l e n c e r e l a t i o n . '

- a . R d o e s n o t s a t i s f y t h a r e f l e x i v e p r i n c i p l e .

Fo r exam p i e , (3» 7 ) / f ( 3 , 7) b e c a u s e 3 * 7 r 7 - 3»

b . R do0a n o t sa fcisfy t h o r y / ^ e t r i e p r i n c i p l e

b e c a u s e ( 3 , 6 ) R ( 1 , 2 ) , b u t ( 1 , 2 ) f ( { 3 , 6 ) .

c . R d o e s eofc s a t i s f y t h e t r a n s i t i v e p r i n c i p l e

b e c a u s e ( 3 , 6 ) R ( 1 , 2) a n d ( 1 , 2 ) 8 ( 1 , 0 ) , b u t

( 3 , 6 ) t f ( l , 0 ) ,

l b s • r e l a t i o n R x d e f i n e d b y ( a , b ) f l 1 ( c , d ) i f a o d o n l y

i f a + d b + c i s 3n o q u i v a l r e l a t i o s i .

P l a c e t h e a p p r o p r i a t e symbol , or be to; e en t b a

f o l l o w i n g p a i r a of o r d e r e d p a i r s ,

( 3 , 7) _ ( 8 , 7)

(~k> 2) 2)

(7 - 6 , 9) ( 1 , 9)

207

( -8 + 3 , 6 - ( - 2 ) ) ( £ , 8)

(6 - 9 , k + ( " 6 ) ) (-• 3J -2)

(12, -3) _ (6 + 6 , 9 - 6)

Place the appropr ia te symbol, S1R" or 81 between the

f o l l o w i n g pa i rs of ordered p a i r s .

( 6 , l l4) _ (3 , 5)

(7, 2) (9, 3)

(1[, 6 ) (1| , - 2 )

(3s 7) „ (2, 2)

(3» 5) (&> ii}-)

(1|, -2) (l\, 6)

(3» 7) ( -5 , 5)

Place the appropr ia te symbol, "R j * or between

the f o l l o w i n g pai iM of ordered pa i r a.

(3i 7) (3, 7)

( i | , 8) (1 , 5)

1) (9 , 6)

( - 6 , 9} ( - 6 , 9)

(It , 11) (7 , 13)

(7 , 5) ( 9 , 8)

( 8 , 3) ( 7 , 2)

F i l l i n t h a blaaka i n sueb a way tha t so oh - o f tba

f o l l o w i n g ia t r u a .

(3# ) « (3, -17)

( * ) = ( - 9 , /*47)

208

(6, 9)R( , 1}

3) H (2,

( 7)R(5, l)

U , 3)#( , 3)

(6, 9 ) R 1 ( _ _ , 6)

( /, J ^ R ^ , 7)

( / 1 5 ) R 1 ( 3 , 21)

( 7)R 1(20, 9)

(-3$ 8}r,( , 5)

(-If., >-9)jpf-L< , !{.)

In each of the following state the noma of tfaa

principle illuatea ted.

(3, 2).Rr{3, 2). . _ : • •

(3> 2}H 1(6, 55 and (6, $)R-L(i<, 3). Therefore

(3, 2)r1(1|, 3). • ,

(k»- 7)8^(3*- 6) . therefore, (3> 6)R- (iLj., 7).

(9, 8)R 1(6, 5) and (6, 5)R 3(9, 6). Therefore,

{9, 8}R 1(9 j 8).

209

Lesson 9

A binary operat ion asaoc io t sa with two elements of a

act en element of t ha t s e t . -Suppose tba elements of a ae t

are themselves s e t s . Consider the s e t 3 which baa aa e l e -

ments the 3ata

A = t l , 5 , 9, 13, 17, 21, 25, 19, . . ,

3 - £2, 6 , 10, 1)|, 18, 22, 26, 30, . . , j ,

G = $3, 7, 11, 15, 19, 23, 27., 31, . . . J , and

D = |]|.» 8 , 12, 16, 20, 2)|_, 28, 32, . . . J .

We s h a l l de f ine a b inary operafclon <J) on tbe s e t S. Lot

X find Y denote elements of 8 . 'Then X Q Y denotes the olo-

iiiont of 8 which © a a 3oeia tea with X and Y. To f i n d X£f) Y,

pick any ol^m-eat of X, pick any element of Y and f i n d t h e i r

sum* Then X © Y is the s a t conta in ing t h i s aum.

Examples.

Find A (T) B. 9 ia an element of A. 18 i s qd eleuiont

of B. 9 + 18 ~ 27. 2-7 is an eleraent of C. I 'barefora

A (s)B ~ 0 . "Find 0 © A. 7 i s "n elojssnt of C and 17 la

an element of A, 7 17 « 2)4 arid 2lj. ia art e lament. of D.

Therefore 0 OA — D.

1 . Tlie b ina ry opera t ion 0 ' i a a coorautdtlve b ina ry

- ope ra t i on . This can be abovm as fo l lows : Let X and

Y be elements of S• Td -find X (%)Y choose an -element

a of X and aw oleiaent b of Y* Than XC;P)Y ia the e l o -

raant ( s a t ) of 3 conta in ing a + b . To f i n d Y$)X wo

23.0

we mus t choose an e l e m e n t of Y and an e l e rusn t of X."

Then t h e aura of t h o i e i h o .oâbers irro^t be f o u n d . Y (*) X

i s the s e t c o n t a i n i n g t h i 3 sura, b ia an e l e m e n t of Y,

so c h o o s e b . a i a an e l e m e n t of X, so choose a . Than

Y 0 X ia t h e s a t c o n t a i n i n g b + a . b -s- a ^ a + b and

X . © Y con-tains a- + b . '1^1 a a bow a t h a t Y 0 X ~ X 0 Y»

There fore 0 i s a corornutative b i n a r y o p e r a t i o n .

•an a s s o c i a t i v e b i n a r y o p e r a t i o n , Thia can be

shown u s i n g a pr ocedure a i m i l a r to t h a t u sed a b o v e .

3 . D ia an i d e n t i t y e l e m e n t f o r the b i n a r y o p e r a t i o n © •

i | . Each e l ement h a s an i n v e r s e w i t h r e s p e c t ' to tho

i d e n t i t y D. Tha i n v e r s e of A i s 0 b e c a u s e A 0 0 *= D.

rrha i n v e r s e of B *.,'3 B b a c a u s a B O b = D. The i n v e r s e

of 0 ia A b e c a u s e C (£)A " D. The . inverse of D i a D

because D 0 D ~ D.

We now d e f i n e another b i n a r y o p e r a t i o n on t h e s a t S ,

the b i n a r y o p e r a t i o n ® . I f X and Y a r e e l ements j 0 f t h e

a e t S , a l a an e lement of X ana b ia an e l e m e n t tj»£ Y, t h e n i

X @ Y ia t h e eIcmient ( s e t ) of 3 c o n t a i n i n g 3 x b | Fon

example , f i n d A (£) C. 9 i a an a l e / sent of A and IfjJ ia an !

e l e m e n t o f C. 9 x 15 = 135 . - 135 i a an e l ement of C- "

T h e r e f o r e A (S) C - = 0 . .-

1 . © l a -a c Drain u ta t i v o b i n a r y o p e r a t i o n .

2 , © ia an a s a o c l a t i v e b i n a r y o p e r a t i o n , Tbia can be

shown sa f o l l o w a : L e t X, } Kind >3 be eleraenta of 3 .

211

Let a be an element r>t X, t be s-x? I'leu'ent of Y, and c

-be an element of Z. Then (X Q Y ) Q Z ia the "set con-

t a i n i n g {a x b) x c . X Q (Y O Z) ia the s a t con ta in ing

a x (b x c} . Since (a x b) x c ~ a x (b x c) ,

( X 0 Y ) 0 Z and X Q U © 2 ) rssuat be the same s a t .

Therefore ( X Q Y ) 0 Z « X <3> (Y © Z) .

3 . A ia an i d e n t i t y element for the b i n a r y opera t ion (£) .

Tois can be v e r i f i e d by showing t h a t each of the

fo l lowing ia true,"

A © A a A •' C ® A « C

B © <\ ~ 3 ' A 0 C -•= 0

A © B -- B D 0 A « D

A © D = D

1|. Each of the e l era an': a A Bad G has an inverse wi th r e s p e c t

to the b i n a r y opera t ion 0 and the i d e n t i t y element A.

The inverse of each element can be determined by f i l l i n g

the b l a n k s .

A © ~ A

0 Q a A

Since c e r t a i n olemonta have i nve r se s under the b i n a r y

opera t iona @ a n d •(£), c e r t a i n types of equat ions a r e s o l v a b l e .

1 . Solve the equat ion B © X *-= G f o r X.

iSolution j

The inverse of B with r e s p e c t to © ia B s ince B @ B = D.

' Since B © X = 0 , * B 0 C . ' Since © ia

2 IP

associative , (3 ® B) Q X « B 0} 0 . I) 0 X = B

Since D la cn Identity for 0, X = B(

X « A.

2. Solve the aquation C 0 X = B.

Solution:

C ia the inverse of G since C © G ~ A. Then

0 <D(C © X ) = C

( C © C ) © X = B

• A (D X = B -

X = B since A is nn identity element for 0 .

Exercise 9.

2)c.

1 G and

Fill in the blank3.

A ® 3 - .

B © D -

0 © D =

B © 0 -

0 © 0 =

D © B =

8 © C =

c © c •.

3 0 0 -

4 B ™

Solve the followiug equations for X.

A O = D

X €>G = B

D © x - A

G © X D

A 0 x D

0 ' (K\X « B

13

D 0 X = D

B Q X = A

X 0 G ~ A

Consider the set S which baa as elanenta the seta

\ st j, Lj. j f s 101 13 # l1# • » • j J

B ™ ^2, 5, 8, 11, 1!;, 17, . . .J , and

C ~ |3, 6, 9, 12, 15, 18, . . .3 .

Let + be the binary operation on 3 defined aa follows:

If X and T 311 a oleraenta of S, a ia an element of X end b

ia aa element of Y, fchan X- + Y la ths element (sot) con-

ta in log a + Id . Pill in the blanks.

C + 3 ™

B * A a

G + G =

A + 0 «

6 + G «

G "8" A ~

APFxSEDIX D

« C H E R MADE M3-3TS

215

MATE 163

Teat I

I." Using the binsry operation B defined, by

(a, b) B (e, d). = . (a + e, b - d) complete

the following. . '

1. (3, U) 3 {2; 6) » , .

2. (7, 3) 3 (1, ~k)

3» (-8, 9) B (i|, -3) «

k- (t>, 7) B <11, 2} »

5. (-3, 5) B (-7, h)

6. (ht -6} B {3? "9)

7* (2, - 1) B (-6, -IX) -

II. Using fcb<3 binary operation 0 da fined by

(a, b) 0 (c„ d) •- (a 4- 2b, 0 x d) complete,

the following.

1. (3, i|) 0 (2, 6) =

2. .(1, 3) 0 (9, 2) «

3, (9, 2) 0 (1, 3) -

]|. U4, 5) 0 (-1, 6}

5. (-1, 6) 0 (14, 5) «

.6. (3, -7) 0 (-2, -9) "

-7. (-2, -95 0 (3, -?) * '

8. la tbo -biaary op so. fcion '0 cororauta tive?

216

I I I . P l ace the o p p r o p r i a t s ayxafcol, 02 ^ in each of

the fo l l owing blanks, ,

1. 3 _____ y

P.. 1+ 10

3 * 1 0 .

1|. 7 18

5 . 19 2?

6* 8 __ 20

7. 20 _ hh

8 . 8 )+U

IV. 1 . Doe a fcba r e l a t i o n ~ ^ ;jofclafy the r e f l e x i v e

p r i n c i p l e ?

2 . Does the r e l a t i o n » , s a t i s f y the aymitiotric w" O

p r i n c i p l e ? _ _

3 . Does the r e l a t i o n « (0 s a t i s f y the t r a n s i t i v e

p r i n c i p l e ?

k* Is3 the r e l a t i o n » ^ an equivalence r e l a t i o n ?

V. The b i n a r y ope ra t i on i s d o f i n a d as fo l lows*

a x& b i 3 the who la Dumber lea.-a then 6 r e l a t e d to

a x b by the r e l a t i o n ~ ^ . Using the b i n a r y

o p e r a t i o n complete tha fo l l owing t i b l e .

21?

nmTT

VI.

Is the binary operation eoinnrntotiva?

la thQ'fQ an identity element for the binary

operation x^? ^

T.-rnicb eX&nsenfca have lovarsea with respect to the

binary operation .

If* there ara solutions 3 find a XX possible a olu tiona

to eacb of the following equations.

1. 5 *6 X < - 3 x -

2. x x6 X a If X

3» 5 *6 X 1 X '•••

!|» X x6 X 3 X »

h * 2 s6 X ss 5 X "

6. 0 •7iS X _ as 0 "W #»-» . % ftwj*

7. 2 *6 X ss k X ~

8. 3 ;x KS 0 X =5

VII. la the binary operation ooiativs?

Give examplea to support youv cnower.

o i A

ii

' " MATH 163

Teat XI

Perform the indicated operations

1. (3,5) + (7,8) =

2. (1,7) + (8,3) « • " " - _

3. (13,10) + ('21,7) =

'4 • (8,13) X (1,1)

5. (11,9) x (2,3) »

6. (5','i) x (16,8) =

7. (12,57) x (2314,531) «

8. r~27TT s- TO777 -

9. X57r7J + T-TT3T as _

10. "T-I2787 + TT,JJ - _

11. "(^7-87 +1127^187 =

12. T277L07 + r~'o 7-17 ---

13* (sBty'Tbl,89~X*>6f -i- (Cs,53&SI5) "

i;4. T o T ^ F ^ j m T o T + T-9003S17TJ512&97.« _

15. TF7r?7 x T - r p n -

16. T^37T17 * • c&; ~vj -

17. T u ; ~ m x T-'BTISy ^ - :

18. T - T 3 7 P X T = 7 ^ 7 - .

19. TSffSVWPmSTSl x T6T3"95TrT7r^SS7

20. {-^6975*3 ,1772067 ^ T3T?o¥9i^ÎWfeWroJ e

Place the appropriate symbol, , «•««, 0r »j0*

in each of the following blaokn.

3"i 1 9

— ( 7 , 9 ) ( 3 , k )

2. (2,1) . (-6 ,-3)

3 . ( 6 , - 2 } ( - 3 , - 1 )

l { . ( 1 8 , 2 ? ) 1 2 * 3 )

$ ' ( i J " 8 ) ( ~ S V " 1 0 )

6 « ( 3 , 7 ) ( 6 , 1 ^ )

? • ( 3 , 6 ) ( - 9 , 2 )

9 . T 5 F 5 3 l i T 7 3 ^ 3 0 T ) "

i o . T o r r m w f _ _ ( ' o 7 : T I 1 9 3 5 S T

I I I . S o l v e the i*ollcwir;3 equations •

1. T6'79T + X

£

[sO

j"-—^

11 X a

2. T T > - B T "* - > c « T 3 T - 8 T X S

mJ ^ ( T 3 T + X --- I ' ViT X • - ____ _

,*|j A TS7FT -i - } i ' t ' o T i ' T X ~

5 . T ' ^ n n , x « ( o ; i j X _

6 . r ^ y r + X « I ' 0 7 , 1 9 7 X : 3

7. 17 rm x X - i i . ? t r X « _

8. (-5,11) X x * u ; i i X a

9. T 6 7 3 7 .x X « ( a , 3 ) X . , _

10. T T 7 2 T x X » n : o 7 5 T }f £2

B a l o w a r e l i .8 1 6 d S 0'/?i Q 0 r t b 8 p . ? l n c I p l e s w s h a v e s t u d i e d .

GL Hgjp •H _ i-U £3 k DlTill] -i,• CJ i< L, 'o o' i.' X1 Xii 0 J. i 1 '.J •X01< 'T »

2 . T h S C DBITil ;-31 :a t i v e p x» I b c Ip 1 e £ o r x „

3 - T b a a a s o Xjt «V a 1 1 v e p j? 1 ; j c i p i a f o r + .

220

ij. The associativa principle £oi1 x. .

$. x la clist? ibutiv-a ever +*

6. Identity element for +.

7. Identity element for x .

8. Inverse of an element with respect to +.

9. Inverse of an element with respect to x.

10. Reflexive principle of n ^ .

11. Syrurne trio principle of M =" .

12. Tranaitive pr inc. ipla of

Below ore insigncpa of soma of the above principles.

Place the number of the principle in the blank4

A. VoiTTTx n r 3 T » V(?J) x T87I7T

b. D T 9 T x ("97JT « riTif

c. 3*1) + (67tf)'Jz 12,57 ™ £(371'f*H?7+tST^J)xr£7$r3

d. rBTTiT + JjTTIF X IT,12)"] « [1773T x ITT^fJ * T^HTF

E. Tir;rr • 1 1 7 9 7 « 1 3 7 9 7 + i o f

F. (18 j12) =j (18,12)

s. W7BT-\W73T x ni7r}^-|jfa7bT x 78737}:* T2*7iT

H. IljriTf + T o 7 U = 177171*

I. If <3,3) 2 then (2,b) ^ (3,3)

j. nr.Tr * o r s r - n n z r

221

MATH 1 6 3

T e s t I I I

I I ,

Ps rforro the indicated t

1. (2/5) + < s — 2. (1/2) + (3/2) * _

3> (l/2) + (1/5)

1| • (3/5) + ( -1/6)

(-2/7) * O A S ) ;= _

6» (1/3) - Ul/3) »

7. (7/5) - ( -3/5)

8. (3/5) - (2/3) ~

9. (-i/5) -• (2/3)

10. (1/3) - (2/5) =

11. (-1/5) * (2/-3)

12, i'i/k) x (~l/5)

13. (3/-7) -x (5/3) = _

I'l. (5/6) - (1/3)

If? • (1/12) - (-7/6) a

Solve the following equations.

1 . (3/3) + X - (1/2) X ™

2. X + (3/7) - (-7/5) X ™

3. Y x ( l/li) - ( 2 / 3 ) Y "

i*.. (-2/7) ™ (1/5) x Y Y 3

5. (2/3) « (1/2) + (3/5) x Y Y »

GOilSiOSI' ift/9 0xrj3i.'J' OpOjFQ '0"i. O'D 0 '.I Q'f' "ii ft (1 lb,y «h0

222

following tPfcls

L L U J L L L . Q pi

F

P

Q | H

H

H

"O T P

Q

Q

M

.iL

J L ft

n i p

1 . l a there an identity e l e m e n t for the binary

operation o ? _

2. Is t h e binary op a rat ion o c o m m u t a t i v e ?

3. Give an example to support your aaawar to

question 2.

i|. W h i c h e l e m e n t s of the s e t M, H, ?, Q, have invoices

with r e s p e c t to o ? _

5 . I 3 t h e binary o p e r a t i o n o a s s o c i a t i v e ? _

6, Give a n e x a m p l e to support your s n s w e r to

IV.

question 5 «

Cons Ider the "o Idsrj ' Op 61' atioca t" and

the following tables.

t«t 2 J J

5 x"

2 2 3 h 5 2

3 3 k 7 ] 1 3 I I I

J? u j. tktC~ si

5 2 3 h j

5 ' 1 5 j 2 3 li 1 |

2 I 3 .7-23 ; SMSTt WX • i

2

3

! { ,

5

u

2

5 •ig&iTESSWfl

2

5

1

3

223

X. la tba bir-ary operat ion t* commutative? .

2 . C-ive so example to support your answer to

quest ion 1 .

3 . Ia there en i d e n t i t y elaraont for x J ?

I f so, what la i t ?

1^,5,6, and ? , Write the i nve r se , i f any, of each of

the fol lowing uifch recpoct to t" .

3 ^ „

6 ,9 ,10 ,and 11. V/rito the i n v e r s e , i f a ay , of each of

the fol lowing with r e s p e c t to x,! •

k 2

r ]>

12» .13 .X'1 ciis ferx'>utiv 3 oViSi* t*'? ^

example to support your answer,

Giva an

13- la f d i s t r i b u t i v e over x*?

example to support your ensue*

Give an

V. Write the negat ion of each of the fol lowing,

1 . I t ia c l ea r ,

2 • X'c i 3 f 0 ],0 6 t'."3'3 u i 0 5.3 wiiiuy»

2Zk

3. He i s h u r t or be Is p re t^nu i i ig .

!|. I t ia cloudy and v.indy.

VI. Given the p r o p o s i t i o n , " I f a toorn ia a boora, than a

zoom is a moorn,"

1 . Write the converse of the p r o p o s i t i o n .

2 . Write tli3 c o n t r s p o s i t t v a of the p r o p o s i t i o n .

V I I . Answer the fo l lowing true or f a l s e

1 . 3 + k ~ 7 find 2 + 9 = 11

2 . i f 3 + 7 ~ 10 - ten 9 + 5 =* 12.

3 . H + 8 - 9 or 7 « 6 + 1

!|. 5 + 7 13 12 and 12 - I; « 6

5 . I f 3 ™ 9 - 7» tbeu 6 + 2 = 8 .

<1 rf ef..C$

MA 'M 163

X* c o t» X v

I . In eacb of the fo l lowing s t a t e whether the arguraant

i s v a l i d or i n v a l i d .

1 . Given: I f a person la a c o l l e g e s t u d e n t , than

he i3 I n t e l l i g e n t .

Given: John i s a c o l l s g a s t u d e n t .

Conclus ion: John 13 i n t ' S l l i g e n t .

2.. Given: I f a person la b l i n d , then he ia no t an

a i rp1ane p i l o t .

Given: V'eslsy i s no t b l i n d .

Conclus ion: Waslay i s an a i r p l a n e p i lo t®

3 . Given: I f a person I<i a a ingGr, then he ia

teirjpGj«:iaontol.

Given: S a l l y ia not temperamenta l .

Conclusion: S s l l y i s no t a a i n g a r .

J4. Given: I f you a re a Democrat, then you a r e no t

3 Republ ican .

Given: You a r e not a Democrat.

Conclus ion : You a r e a Republ ican ,

5 . Given: I f a polygon i s a s q u i r e , than i t s

226

d I s g o a a 1 s St re e qu s» 1.

Given: :ifhe dispone; Is of this polygon 0 equal.

Conclusion: Thia polygon 'is a square.

6. Given: If a natural number ends in % or 0, then

it ia divisible by 5>.

Given: If a 'natural number n ia divisible by 5*

than there oxisfca a natural number ra

guch that a - 5 x a.

Conclusion: If a natural nuxabar n enda in 5 or 0,

than there exists a natural number in

such that n •- 5 x m.

7. Given: If a qusdrilatarsi ia a parallelogram

then ifca opposite aides are pnrallel.

Giver.': The opposite sides of chi3 quadrilateral

are not parallel.

'Co-*'Cl(.i5 3.od: This qufldilcfceral i s 'not a

paralielogr am.

y. t* a,

8. Given: If two anglea ora right angles, then they

are equal.

Given:. Auglas A and 3 ere right angles.

Goncluaion: Angle A ia equal to angle B,

22?

9. Given: If a triangle Is equilateral, then it ia

equi:mgul&r.

Given: If a polygon ia equiangular, then it ia

a regular polygon.

Concilia ion: If a triangle ia equilateral, then

it i3 a regular polygon.

10. Given: If two angles cro right angles, than they

ara equal.

Given: These two angles are equal..

Conclusion: These two angle a are right angle a.

II. In caeh of tho following draw a Venn diagram and

stata whether the argument ia valid or invalid.

1. Given: Sor/io politicians ara dishone.it paopla.

Given: Ho hanker ia a dishonest para on.

Conclusion: No banker i.i a politician*

2. Given: No horses ara doga.

• Given; Mo dogi ara cats.

Cooclusion: Ho boraea are cats

Given: 3.ally is t/ndep-JKctehle parson.

Given: Kally fess bi'oirr* ckiin.

Conclusion: Soros people *;itb brown skin ere

an dependable.

It- Given: All'wine people drive carefully on

icy streets.

Given: Mr. Brown drives carefully on icy streets

Conclusion: Mr. Br own is wise.

Given: All squares are rectangles.

Given; All rectangles are parallelograms.

Conclusion: All squares are parallelograms.

XIX• In 63cti of ijliq .following cwo propositions are

given. If these are accepted aa true, "what con-

clusion, if any, .xVllova from thorn. If no conclusion

follow a , write f,no conclusion" .

1. Given; All college students are clever.

229

Given: -John 5.3 a college sl-udont.

Conclusion:

3. Given: Soma coeds are beautiful.

Given: Some blondes ara beautiful.

Conclusion:

I4. Given: Ho traffic policeman baa a ••sense of

humor.

Given: A H fat people ba ve a sense of huvuor.

ConciliaIon:

$• Given: All rectangles are psrallelograiti.a.

Given: All parallelograms are quadrilatarala .

Concluaion:

IV. Consider the following axiom system.

Undefined terras .

1. Divifjiblo by 3

2. Odd

3. Not divisible by Ij.

Multiple of 3

Axioma • . .

PI. 8 ia divisible by )x.

P2. If a number ia divisible by 3S then it ia odd.

230

P3. If a number It? odd, than it. ia not divisible by I4.

Pi|, A nursber is divisible* by 3 if and only if it ia

a multiple of 3*

"Prove" tha following theorems.

Tl. If a Dumber ia divisible! by 3? then it ia not

dlviaibla by )|.

T2. 8 ia not divisible by 3

T3 • If 3 number ia a multiple of 3 > than it is

divisible by 3•

T).|. 8 ia not a nulttpls of 3

T£. If a n-uraber ia diyiaible by br3 then it is not odd,

V. Complete tli3 following truth tables.

not p

231

p q

T T

P T

T P

P P

p and q ! n

T T

P T

T F

P P

If p, then q

T

P

q t

T p

F

p or q P

T F

T

P

If p, then not q

T

F

T

T

F

P

T

P

T

T

F

T

F

T

1?

T

T

T

T

If p, then q or r

T

A P P E N D I X S

T H E RAW DATA

P j

233

Tfca Cod 5 rig Procedure

In order to f a c i l i t a t e the hand l ing of the raw data

each s t u d e n t waa a s s igned a s t u d e n t number. The number

c o n s i s t e d of a c a p i t a l l e t t e r fo l lowed by a four d i g i t

number. This "number i d e n t i f i e d the s t u d e n t in terms of .

a ex , a b i l i t y l e v e l , .and c la as . Prom the c lans numb or i t

could be de termined a t v h a t time the c laaa met , who the

i n s t r u c t o r f o r the c laaa was, and what t each ing method

wa s u a d In th e c la a 3 .

The f i r s t l e t t e r in the s t u d e n t number i n d i c a t e s the

3ex of the 3ti.rl.3nt. J\a '1M:l i n d i c a t e s male and an "F"

i n d e 'J tea f era a l e .

'Each s t u d e n t miraber c o n s i s t e d of a l e t t e r fo l lowed by

a four d i g i t number. The f i r s t d i g i t in the number I n d i -

es tea the c l a s a in ~faieh tha a tuden t was e n r o l l e d . t h e

d i g i t uln danot-3 3 the c l aaa taught by I n s t r u c t o r A a t 10

a.:n. and t sugh t by the Guided Mseotrery Method. The d i g i t

n2 f t denotes the c laaa t augh t by I n s t r u c t o r B a t 10 a .m.

and t augh t by the Expos i t ion Method, lb a d i g i t i,3s' dsnotea

the c l a s a t augh t by I n a t r u e t o r A a t 2 p . a . and t augh t by

the Expos i t ion Method. The d i g i t denotes the c l aaa

t augh t by - I n s t r u c t o r B a t 2 p.rc. and t augh t by the Guided

D i s c o v « r.y M a th o d .

'Foe second d i g i t of the fou r d i g i t a timber i n d i c a t e s

the s t u d e n t ' s a b i l i t y l e v e l as determined by h i s cctapoaite

sco re on the .hner.lcan Col lege .âJîjog; Program. The numeral

9 " ? h

« 1 « i n d i c a t e s a c o m p o s i t e s c o r a o f t w e n t y o r h l g b o r . T h e

n u m e r a l u 2 t t i n d i c a t e s a c o m p o s ' I t o s c o r e o f m o r e t h a n f i f -

t e e n b u t l a s s t h a n t w e n t y . T h e n u m e r a l s , 3 ' ? l i n d i c a t e s t h a t

t h e s t u d e n t ' s c o m p o s i t e s c o r e i s f i f t e e n o r l e s s .

T h e l a s t t w o d i g i t s o f t h e f o u r d i g i t n u m b e r w e r e u s e d

t o d i f f o r e n t i a b e a m o n g t b a - s t u d e n t s o f t h e a s m s s e x a n d i n

t h e 3 a o ) 6 c l a s s . T h e I s a t t w o d i g i t . i r a n g e f r o m ! 1 Q 1 S ' a n d

u p .

I n bc i ' p r e t i n g t h e D a t a

I n t h e c o l u m n l & o e l e d " A C T S c o r e " t h e i n i t i a l c o m -

p o s i t a . s c o r e s o n t h e A w c r i c a a C o i l o g e T e s t i n g P r o . ^ r G m o f

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BIBLIOGRAPHY

Books

A us libel, David P., "In Dafansa of Verbal Learning," Readings in the Psychology of Cognition, editad by RliljharS 07 Xndar'aon* and*"'Da''vfd P. Ausuftel, New York, Holt, Rinohart and Wi/j 3 ton, Inc., 1965, pp • 87-102.

Euros, Oscar Krlsen, edj tor, Tha Sixth Mental Meaauromenta Yearbook, Highland Fa r k/'"l€e~"G'i^hoi'r'Pra a'IT, "".Vyb'ST" ppTi^rj.

Handbook: Coopera ti-</a Ms thematic;? Taata, Princeton, """~~~¥duca tToh¥rTfoTt^

Katona, Gaorga, Organising and Honor is ir>g, New York, Colu rib ia IJn'ivVri.Cty "Proi'a / l9l|.6V~

Lindquist, B. P., Djsaign ogd Analgia of Expariraonfea In Psychology anH"l£&u'c3*t £onBoston, Houghton .Mli'I'lin 0 orepany7'"X9B'3"7

Shulasn, toe S., and Evan R. Keialar, editors, by Discovery: A Critical Appraisal, Chicago, Sana t'iCNW"IXy'lTna ~0oinpaDy",'' I.ySS-,""

Sifi;n.ior?, l?i{fchar J., G» Looter Anderson, ;vod Chalraara L» Si;scayf learning 'Theory in School Situation a, KinnccpoTi¥,'njKi'vHraTiy 6if 14ermeio~ta " P r ' a l a •

This la, C. L., Tea Cont.:'ibiJ fcion of Genera Hy,a t ion to jtha Learnin/? oi*"The"/:"cuiTtion FQCta',"**1 eVr York'CoiVwibTa Uoli'iFa i'ty "PreiTaT"1936"7~

Us ta on, Goodwin, and Edward M. Glass?, Manual: ¥a ta on -Glaser Critical 'Thinking A ppr s i s aiTHTaw""YorKV Harc0urTJ"*Brici" 5nS"¥oFX'd'7"cTV~T96ij..

Wertbairaar, Max, Productive Thinking, enlarged edition, How' Yorks jiaYjTeT "fc'17

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Gorman, Barnard R., s,i;be Effect of Varying Amounts and Kinds of Information as Guidance in Problem Solving," Psycbolpgical Monographs,. LXXII (19£>7)» Whole Hoi.'ibor Ifjiiu """

Crsig, Robert C., "Directed. Versus Independent Discovery of Established Hols felonsTha Journal of Educational Paycboloffg-, XLVII (April, -

Cuwralna, Kenneth, "A Student Experience - Discovery Approach to the Teaching of C a l c u l u s T h e Ma the-resfclca Teacher, LIXI (March, I960) , 162-1707"""""

Gabsi, flyman, "An Experimental Twelfth-Grade Ma theraa tic a Course," The Mathematics Tea char, LX (April, 196*7), 375-380. • ~~

Gogne', Robert M., and Larry T. Browrn, "Some Factors In ths Programing of Conceptual Learning,n Journal of Experiraoi ' Experimental Psychology, LXII (October, 196l'f,"""

Haalorud, G. M., and Shirley Meyers, "The Transfer Value of Given and Individually Derived Principlea," The «Tou,'rr.:31 of !?ducc» tiou$1 Psychology, LIX (D9ceinbeF7~ i-v5o")'ra3-

Hor.drix, Gertrude, "A llew Clua to Transfer of Training," The Rlaaontsi'y School Journal, XLVI1I (December,^19hl) »

Ililgard, Eroest R., Robert ?, Irvine, and Jaraaa E• Whipple, "Rote Mernor Ikg tlon, Una***1rei:4i3jt, -nd prasefers An Extension of Ka ton* *s C^rd-rvlckÊxpori&en hs journal i;;A ExpwJf^crjtal P3£uhoXo£7, X£VI (Octobar, 1953)7'""" lid8 - 292«

Kerah, Sort Y., "The Adequacy of ^Meaning' aa an Explanation for the Superiority of Learning by Independent Discovery," Journal of Education-si PkycholoirY* XT,IX (October, 195ST7""2H2-"a9'2' 2 """* "*•

> ,f ueuining by Discovery; Instructions! ft'ero t Q g x e s T h a Arithmetic 'Peceher, XII (October, 1 / 6 ) , 4" J4-4I ( 0

— "'x'bQ Motivating Effact of Learning by Di* <,c \,ed UJ. u c s.. v 6i' y J o u r n a l of EducBfcionsl P 3 voh 010 » LI1 (April, 1962), 65™?"l7 ~ ~

2 I . f . O

Kitfcell, Jack E,. , "An FxperiT3i-.ritsl S'ta?dy of the Effect of External Direction During Learning on Transfer and Retention of Principles ,!f rr1:*s .Tc-uynal of l?duca Wjrmal Psycho logy, XLV11T {Kovo.xaT)sr:7 19&5T7 *"3"^14j!0'S'V""* "

McDonnell, T. R., "Discovery V3. Authoritative Identification in tils Learning of Children," Studies in Education, IX (193^)» 11-62 . -

Michael, R• C., ''The Relative Kffectiveneaa of Two Methods of Teaching Certain Topics in Ninth Grade Algebra," 'Eiw Mathematics Teacher, XLII (February, 1949), tij-s3J«

Ray, Willis E., ?,f-upil Discovery Va . Direct Instruction," Journal of Experir-iantal Education, XXi'X (March, 1961), ^'72-231*. — * —

Scandura, Joaaph M., "An Analyaia of Expoaition and Dis-covery Mods a of Problem Solving Instruction," The Journal of Experimental Education, XXXIII (W.!n"t"eF, i ; } j o h r r v ^ ~ m 7 ~ - - —

SigurdaoQ, .3. E., and ilalia Boycbuk, "A Fifth-Grade Student Discovery Zero," Tbo Arithmetic Tcacbor, XIV (April, 196?), 278-279 7™ - —

Oobel, Max ft ., ^Concept Learning in Algebra ,H The Ma the -mablca Teacher, XLXX (October, 1VI>6), ! j25" ) j307 "

Standiah, Henry, "Seventh Gredars Voluntas.? for After-School Classes in Algebra t

n The Ha tic a Teacher, LI I (Dec crab as?, I960) , 6ij.0~6l.f37 — •* -

'j-aylor, Roaa , -'First Com* 2* in A Xgsbva - IT. 1*0 ,M . and S.M.S.G, - A Goifipn'e 1-ton'rbe Katbo^n i-.ica Tcacbsa?, L-Y (October, 1962), — -

Underwood, Benton J., owl Jac'c Rioba^d.jon, "Verbal Concept Learning aa 3 Fuucti.ua 02* Cos ti-uciion Dominance-Lavol," Journal of Sxps2?J;ueatr»l Psychology, VI (Apr 11, T955TT^^>3'.* ' kJL

Wittroek, M, 0,, "Verbal Stimuli in Concept Formation: Lear n i;;g by D13 c 0v a £ y," Jour 0a 1 of nd u 001.10.aa 1 Payeholoj-rff, LIV (July, l ^ y f T Q l - l W .

2]j.l

P u b l i c a t i o n s of Learned O r g a n i s a t i o n s

D a l l a - P l a n a , G a b r i e l M G a r t h M. ISldr edge , and Bla in a R. Worthen, Sequence CP..-u/acjia v i < t l •:a of Tsx t Ma t e r i j i l s and T^£.n£V¥r""l5F"jje£rrn £ng'j 15**'™ D'fa cQ?ary"'Lsar n ing","~£a I t Lalfa' "oTty ,"""Krr5i"u*"bT"*Edu -csT^fofT âa"eaFcH7~Univ3rs i by of Utah , 196$.

Scandura , Joseph M. , e d i t o r , Kasearob in Ma ttojjmafclca Edooa t l o n , Washington, Toe" l:Ia"cToaST ^"ouncTT'lTf^Tsjachara of MatTaematics, 1967.

P u b l i c Documents

G a g s e ' , R. M. , " I m p l i e s t i o n s of Soma Doc t r i nea of Ma t h e -sr-a t i c s Teaching f o r Rasa s r cb in Human L e a r n i n g , " ReKoarob ProblGipa in Ma fchanafcica E d u c a t i o n , Wi%iivjt%QnJ G o v e r n l i i b T l ^ I n H B ^ p p . i t9-56.

Unpubl ished Ma to r i a l a

Fu l l e r t on , Cra ig K e r r , "A Comparison of th* E f f e c t i v e n e s s of Two Proscribed_ Matheds of Teaching M u l t i p l i e s U.on of Vfoole Number a unpub l i shed d o c t o r a l d i s a or ti: l i o n , S t a t a Univcra i t y of lovra, Iowa C i t y , l o v a , 1955.

I lowi tz , Thomas A l l e n , "The Di scove ry £pjj*orfob: A Shady of i t a Re l a t t v e E f f e c t i v e s ® s a in Ma fcheraa t i o a ,w

unpub l i shed d o c t o r a l d i a a ^ ' k i t i o n , U n i v e r s i t y of Minneso t a , M i n n e a p o l i s , MSosasote , 19653®

Xreba , S;buphen Orne , "An Xiw.utig-a t i on of -Jhrauafa? B*ffeefc3 of Given _ and Der ived C-d irjg ? r in a i p l c s a t Three Lava I s oi Mental A b i l i t y , ' ' o v f ^ t i i i s h a d d o c t o r a l d.1330r to 'c i0n Michigan S t a t e Ua iva r -a i iy , Saafc L a n s i n g , Mich igan , X / U €*m «

M o s s , J e r o m e , J r . , "An Expe r imen ta l S tudy of the R e l a t i v e E f f e c t i v e n e s s of tho Di rec t -Da t a i l e d and D i r e c t e d Discovery He t&ods of Tea oiling L e t t e r p r e s s Impos i t i o n , u n p u b l i s h e d d o c t o r a l d i s s e r t a t i o n , U n i v e r s i t y of I l l i n o i s , Urban* , IXlir joi-a , I960,

3

n

N i c h o l s , Eugene Douglas , "Comparison of Two Approaches to uua j?eaoo.u}g 01 Seleciicsd Topic.? J.t'j Plains CsoiTiatry," u np a b 1 i s h ed do c t o r a l d i 3 a q r fca t i m> ITn i v s r a i t y o f I l l i n o i s , Urbana, I l l i n o i a , 1956.

2)[Z

Fries , J a c k Stanley, ecovery: Its Sffoct on the A chieve m ? n t a n d C r i c i c a 1 T h I r, k 5. n g A b i 1111 a a of Tenth Grada Gecera 1 M a f h ? t i e s £11;denta ," «npubliahad doctoral diasor tat'on, Wayn* State University, Detroit; M icb ius a, 1.965 •

Walner, Kelvin, "A Gomparisoo of the Effect of Two Teaching Techniques in Developing tba Functional Corafafccuca of College Students in a First Seaoster Couraa in Ma tberr-a tin a ,M unpublished doctoral dissertation, New York Univaraity, Now York City, New York, 1961.

Wolfe, Martin Sylvester, '"Effects of Expo a 1 tor y Instruction in Mathcmatics an Students Accustomed to Discovery Hodos ,ff urroobliahed doctoral diaaorta tion, Univarsity of Illinois, Usbnna, Illinois, 1963.

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